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Biometrics & Biostatistics International Journal

Research Article Volume 5 Issue 6

Zero- truncated discrete shanker distribution and its applications

Munindra Borah, Krishna Ram Saikia

Department of Mathematical Sciences, Tezpur University, India

Correspondence: Krishna Ram Saikia, Department of Mathematical Sciences, Tezpur University, Napaam, Tezpur, Assam, India

Received: March 10, 2017 | Published: May 16, 2017

Citation: Borah M, Saikia KR. Zero- truncated discrete shanker distribution and its applications. Biom Biostat Int J. 2017;5(6):232-237. DOI: 10.15406/bbij.2017.05.00152

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Abstract

Discrete analogue of the continuous Shanker distribution, which may be called a discrete Shanker distribution, has been introduced. The probability mass function and probability generating function of the distribution have been obtained. Zero truncated form of the distribution has been investigated. Certain recurrence relations for probabilities and moments have been also derived. The parameters of Zero- truncated discrete Shanker distribution have been estimated by using Newton- Raphson method. The distributions have been fitted to eight numbers of well- known data sets, which are used by other authors. A comparative study has been made among ZTP, ZTPL and ZTDS distributions, using the same data set based on the goodness of fit test. It has been observed that in most cases ZTPL gives much closer fit than ZTP distribution. While ZTDS gives very closer fit to ZTPL and in some cases ZTDS gives better fit than ZTPL distribution.

Keywords: discrete shanker distribution, zero-truncated discrete shanker distribution, zero- truncated Poisson- lindley distribution, recurrence relations, survival function

Abbreviations

DS, discrete shanker; ZTP, zero–truncated poisson; ZTPL, zero–truncated poisson lindley; ZTDS, zero–truncated discrete shanker; PDF, probability density function; pmf, probability mass function; S( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaabofajuaGdaqadaGcpaqaaKqzGeWdbiaabIhaaOGa ayjkaiaawMcaaaaa@3B66@ , survival function; r( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaabkhajuaGdaqadaGcpaqaaKqzGeWdbiaabIhaaOGa ayjkaiaawMcaaaaa@3B85@ , failure hazard rate; r ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaabkhajuaGdaqadaGcpaqaaKqzGeWdbiaabIhaaOGa ayjkaiaawMcaaaaa@3B85@, reversed failure rate; f z ( x ; θ ) : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaabAgal8aadaWgaaqaaKqzadWdbiaabQhaaSWdaeqa aKqba+qadaqadaGcpaqaaKqzGeWdbiaabIhacaGG7aGaaeiUdaGcca GLOaGaayzkaaqcLbsacaGG6aaaaa@4163@, pmf of DS distribution; f D ( x ; θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadAgajuaGpaWaaSbaaSqaaKqzadWdbiaadseaaSWd aeqaaKqba+qadaqadaGcpaqaaKqzGeWdbiaadIhacaGG7aGaeqiUde hakiaawIcacaGLPaaaaaa@40EC@ , pmf of ZTDS distribution; η [ r ] ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaabE7al8aadaqhaaqaa8qadaWadaWdaeaajugWa8qa caqGYbaaliaawUfacaGLDbaaa8aabaqcLbmapeGaae4jaaaaaaa@3E7B@ : r t h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaca WGYbqcfa4aaWbaaeqabaqcLbmacaWG0bGaamiAaaaaaaa@3B57@, factorial moment of ZTDS distribution; μ [ r ] ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaabY7al8aadaqhaaqaa8qadaWadaWdaeaajugWa8qa caqGYbaaliaawUfacaGLDbaaa8aabaqcLbmapeGaae4jaaaaaaa@3E80@ : r th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaca qGYbWcdaahaaadbeqaaiaabshacaqGObaaaaaa@39AC@, raw moment of DS distribution; P r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaabcfal8aadaWgaaqaaKqzadWdbiaabkhaaSWdaeqa aaaa@3A16@, r th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaca qGYbqcfa4aaWbaaeqajyaGbaqcLbmacaqG0bGaaeiAaaaaaaa@3BDF@ Probability of DS distribution; P z r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaabcfal8aadaahaaqabeaajugWa8qacaqG6baaaSWd amaaBaaabaqcLbmapeGaaeOCaaWcpaqabaaaaa@3C8E@, r t h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadkhajuaGdaahaaqabeaajugWaiaadshacaWGObaa aaaa@3B77@ Probability of ZTDS distribution

Introducton

It is sometimes inconvenient to measure the life length of a device, on a continuous scale. In practice, we come across situation, where lifetime of a device is considered to be a discrete random variable. For example, in the case of an on off switching device, the lifetime of the switch is a discrete random variable. If the lifetimes of individuals in some populations are grouped or when lifetime refers to an integral numbers of cycles of some sort, it may be desirable to treat it as a discrete random variable. When a discrete model is used with lifetime data, it is usually a multinomial distribution. This arises because effectively the continuous data have been grouped. Such situations may demand another discrete distribution, usually over the non negative integers. Such situations are best treated individually, but generally one tries to adopt one of the standard discrete distribution. Some of those works are by Nakagawa and Osaki,1 where the discrete Weibull distribution is obtained; Roy2 studied discrete Rayleigh distribution; Kemp3 derived discrete Half normal distribution. Krishna and Pundir4 investigated the discrete Burr and the discrete Pareto distribution. Gomez-Deniz5 derived a new generalization of the geometric distribution obtained from the generalized exponential distribution of Marshall and Olkin.6 Borah et al.7,8 studied on two parameter discrete quasi- Lindley and discrete Janardan distributions respectively. Borah and Saikia9 introduced discrete Sushila distribution. Dutta and Borah10 studied zero- modified Poisson- Lindley distribution.

Derivation of the proposed distribution

One parameter continuous Shanker distribution introduced by Shanker11 with parameter  is defined by its probability density function (pdf)

f( x:θ )=  θ 2 θ 2 +1  ( θ+x )  e θx .x>0. θ>0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGMbqcfa4aaeWaaOWdaeaajugib8qacaWG4bGaaiOoaiab eI7aXbGccaGLOaGaayzkaaqcLbsacqGH9aqpcaGGGcqcfa4aaSaaaO Wdaeaajugib8qacqaH4oqCl8aadaahaaqcgayabeaajugWa8qacaaI YaaaaaGcpaqaaKqzGeWdbiabeI7aXLqba+aadaahaaWcbeqaaKqzad WdbiaaikdaaaqcLbsacqGHRaWkcaaIXaaaaiaacckajuaGdaqadaGc paqaaKqzGeWdbiabeI7aXjabgUcaRiaadIhaaOGaayjkaiaawMcaaK qzGeGaaiiOaiaadwgal8aadaahaaGcbeqaaKqzadWdbiabgkHiTiab eI7aXjaadIhaaaqcLbsacaGGUaGaamiEaiabg6da+iaaicdacaGGUa GaaiiOaiabeI7aXjabg6da+iaaicdacaGGUaaaaa@670B@                                         (2.1)

Discretization of continuous distribution can be done using different methodologies. In this paper we deal with the derivation of a new discrete distribution which may be called discrete Shanker (DS) distribution. It takes values in {0, 1, 2, . . .,}. This distribution is generated by discretizing the survival function of the continuous Shanker distribution

S( x )= x f( x:θ )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGtbqcfa4aaeWaaOWdaeaajugib8qacaWG4baakiaawIca caGLPaaajugibiabg2da9KqbaoaaxadabaqcLbsacqGHRiI8aKqbag aajugWaiaadIhaaKqbagaajugWaiabg6HiLcaajugibiaadAgajuaG daqadaGcpaqaaKqzGeWdbiaadIhacaGG6aGaeqiUdehakiaawIcaca GLPaaajugibiaadsgacaWG4baaaa@5031@

=  θ 2 +1+ θx θ 2 +1   e θx ,x>0.θ>0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqGH9aqpcaGGGcqcfa4aaSaaaOWdaeaajugib8qacqaH4oqC l8aadaahaaqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaaGymai abgUcaRiaacckacqaH4oqCcaWG4baak8aabaqcLbsapeGaeqiUde3c paWaaWbaaeqabaqcLbmapeGaaGOmaaaajugibiabgUcaRiaaigdaaa GaaiiOaiaadwgajuaGpaWaaWbaaSqabeaajugib8qacqGHsislcqaH 4oqCcaWG4baaaiaacYcacaWG4bGaeyOpa4JaaGimaiaac6cacqaH4o qCcqGH+aGpcaaIWaGaaiOlaaaa@5B1E@                                                 (2.2)

S( x+1 )=  θ 2 +1+ θ( x+1 ) θ 2 +1   e θ( x+1 ) ,x>0.θ>0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGtbqcfa4aaeWaaOWdaeaajugib8qacaWG4bGaey4kaSIa aGymaaGccaGLOaGaayzkaaqcLbsacqGH9aqpcaGGGcqcfa4aaSaaaO Wdaeaajugib8qacqaH4oqCl8aadaahaaqabeaajugWa8qacaaIYaaa aKqzGeGaey4kaSIaaGymaiabgUcaRiaacckacqaH4oqCjuaGdaqada GcpaqaaKqzGeWdbiaadIhacqGHRaWkcaaIXaaakiaawIcacaGLPaaa a8aabaqcLbsapeGaeqiUde3cpaWaaWbaaeqabaqcLbmapeGaaGOmaa aajugibiabgUcaRiaaigdaaaGaaiiOaiaadwgal8aadaahaaqabeaa jugWa8qacqGHsislcqaH4oqClmaabmaapaqaaKqzadWdbiaadIhacq GHRaWkcaaIXaaaliaawIcacaGLPaaaaaqcLbsacaGGSaGaamiEaiab g6da+iaaicdacaGGUaGaeqiUdeNaeyOpa4JaaGimaiaac6caaaa@6B8D@                       (2.3)

Where f( x;θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGMbqcfa4aaeWaaOWdaeaajugib8qacaWG4bGaai4oaiab eI7aXbGccaGLOaGaayzkaaaaaa@3DD1@ denotes the pdf of Shanker distribution.

The pmf of discrete Shanker distribution f D ( x;θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGMbqcfa4damaaBaaaleaajugWa8qacaWGebaal8aabeaa juaGpeWaaeWaaOWdaeaajugib8qacaWG4bGaai4oaiabeI7aXbGcca GLOaGaayzkaaaaaa@40CB@  may be obtained as

f D ( x;θ )=S( x )S( x+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGMbWcpaWaaSbaaeaajugWa8qacaWGebaal8aabeaajuaG peWaaeWaaOWdaeaajugib8qacaWG4bGaai4oaiabeI7aXbGccaGLOa GaayzkaaqcLbsacqGH9aqpcaWGtbqcfa4aaeWaaOWdaeaajugib8qa caWG4baakiaawIcacaGLPaaajugibiabgkHiTiaadofajuaGdaqada GcpaqaaKqzGeWdbiaadIhacqGHRaWkcaaIXaaakiaawIcacaGLPaaa aaa@4E47@

= ( θ 2 +1+ θx)( 1  e θ )θ  e θ θ 2 +1   e θx ,x=0,1,2,3.... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaaOWdaeaajugib8qacaGGOaGaeqiUde3cpaWaaWbaaeqa baqcLbmapeGaaGOmaaaajugibiabgUcaRiaaigdacqGHRaWkcaGGGc GaeqiUdeNaamiEaiaacMcajuaGdaqadaGcpaqaaKqzGeWdbiaaigda cqGHsislcaGGGcGaamyzaSWdamaaCaaabeqaaKqzadWdbiabgkHiTi abeI7aXbaaaOGaayjkaiaawMcaaKqzGeGaeyOeI0IaeqiUdeNaaiiO aiaadwgal8aadaahaaGcbeqaaKqzadWdbiabgkHiTiabeI7aXbaaaO Wdaeaajugib8qacqaH4oqCl8aadaahaaqabeaajugWa8qacaaIYaaa aKqzGeGaey4kaSIaaGymaaaacaGGGcGaamyzaSWdamaaCaaabeqaaK qzadWdbiabgkHiTiabeI7aXjaadIhaaaWcpaGaaiilaKqzGeGaamiE aiabg2da9iaaicdacaGGSaGaaGymaiaacYcacaaIYaGaaiilaiaaio dacaGGUaGaaiOlaiaac6cacaGGUaaaaa@70B8@                      (2.4)

Proposition 1: The probability generating function (pgf) of DS distribution is given by

G D ( t )= ( θ 2 +1 )( 1  e θ )θ ( θ 2 +1 )( 1  e θ t ) + θ( 1  e θ ) ( θ 2 +1 ) ( 1 e θ t ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGhbWcpaWaaSbaaOqaaKqzadWdbiaadseaaOWdaeqaaKqb a+qadaqadaGcpaqaaKqzGeWdbiaadshaaOGaayjkaiaawMcaaKqzGe Gaeyypa0tcfa4aaSaaaOWdaeaajuaGpeWaaeWaaOWdaeaajugib8qa cqaH4oqCl8aadaahaaGcbeqaaKqzadWdbiaaikdaaaqcLbsacqGHRa WkcaaIXaaakiaawIcacaGLPaaajuaGdaqadaGcpaqaaKqzGeWdbiaa igdacqGHsislcaGGGcGaamyzaSWdamaaCaaakeqabaqcLbmapeGaey OeI0IaeqiUdehaaaGccaGLOaGaayzkaaqcLbsacqGHsislcqaH4oqC aOWdaeaajuaGpeWaaeWaaOWdaeaajugib8qacqaH4oqCjuaGpaWaaW baaOqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaaGymaaGccaGL OaGaayzkaaqcfa4aaeWaaOWdaeaajugib8qacaaIXaGaeyOeI0Iaai iOaiaadwgal8aadaahaaGcbeqaaKqzadWdbiabgkHiTiabeI7aXbaa jugibiaadshaaOGaayjkaiaawMcaaaaajugibiabgUcaRKqbaoaala aak8aabaqcLbsapeGaeqiUdexcfa4aaeWaaOWdaeaajugib8qacaaI XaGaeyOeI0IaaiiOaiaadwgajuaGpaWaaWbaaOqabeaajugWa8qacq GHsislcqaH4oqCaaaakiaawIcacaGLPaaaa8aabaqcfa4dbmaabmaa k8aabaqcLbsapeGaeqiUdexcfa4damaaCaaakeqabaqcLbmapeGaaG OmaaaajugibiabgUcaRiaaigdaaOGaayjkaiaawMcaaKqbaoaabmaa k8aabaqcLbsapeGaaGymaiabgkHiTiaadwgal8aadaahaaGcbeqaaK qzadWdbiabgkHiTiabeI7aXbaajugibiaadshaaOGaayjkaiaawMca aKqba+aadaahaaGcbeqaaKqzadWdbiaaikdaaaaaaaaa@9402@

Proposition 2: The cumulative distribution of DS distribution is given by

F( x )= ( θ 2 +1 )( θ 2 +1+θ ( x+1 ) ) e θ( x+1 ) ( θ 2 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGgbqcfa4aaeWaaOWdaeaajugib8qacaWG4baakiaawIca caGLPaaajugibiabg2da9Kqbaoaalaaak8aabaqcfa4dbmaabmaak8 aabaqcLbsapeGaeqiUdexcfa4damaaCaaaleqabaqcLbmapeGaaGOm aaaajugibiabgUcaRiaaigdaaOGaayjkaiaawMcaaKqzGeGaeyOeI0 scfa4aaeWaaOWdaeaajugib8qacqaH4oqCl8aadaahaaqabeaajugW a8qacaaIYaaaaKqzGeGaey4kaSIaaGymaiabgUcaRiabeI7aXjaacc kajuaGdaqadaGcpaqaaKqzGeWdbiaadIhacqGHRaWkcaaIXaaakiaa wIcacaGLPaaaaiaawIcacaGLPaaajugibiaadwgal8aadaahaaqabe aajugWa8qacqGHsislcqaH4oqClmaabmaapaqaaKqzadWdbiaadIha cqGHRaWkcaaIXaaaliaawIcacaGLPaaaaaaak8aabaqcfa4dbmaabm aak8aabaqcLbsapeGaeqiUde3cpaWaaWbaaeqabaqcLbmapeGaaGOm aaaajugibiabgUcaRiaaigdaaOGaayjkaiaawMcaaaaaaaa@6EDF@

The survival function of DS distribution has obtained as

S D ( x )= ( θ 2 +1+θ ( x+1 ) ) e θ( x+1 ) ( θ 2 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGtbWcpaWaaSbaaKGbagaajugWa8qacaWGebaajyaGpaqa baqcfa4dbmaabmaak8aabaqcLbsapeGaamiEaaGccaGLOaGaayzkaa qcLbsacqGH9aqpjuaGdaWcaaGcpaqaaKqba+qadaqadaGcpaqaaKqz GeWdbiabeI7aXLqba+aadaahaaGcbeqaaKqzadWdbiaaikdaaaqcLb sacqGHRaWkcaaIXaGaey4kaSIaeqiUdeNaaiiOaKqbaoaabmaak8aa baqcLbsapeGaamiEaiabgUcaRiaaigdaaOGaayjkaiaawMcaaaGaay jkaiaawMcaaKqzGeGaamyzaKqba+aadaahaaGcbeqaaKqzadWdbiab gkHiTiabeI7aXTWaaeWaaOWdaeaajugWa8qacaWG4bGaey4kaSIaaG ymaaGccaGLOaGaayzkaaaaaaWdaeaajuaGpeWaaeWaaOWdaeaajugi b8qacqaH4oqCjuaGpaWaaWbaaOqabeaajugWa8qacaaIYaaaaKqzGe Gaey4kaSIaaGymaaGccaGLOaGaayzkaaaaaaaa@6916@

The failure hazard rate may be obtained as

r D ( x )= ( θ 2 +1+ θx)( 1  e θ )θ  e θ ( θ 2 +1+θ x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGYbWcpaWaaSbaaeaajugWa8qacaWGebaal8aabeaajuaG peWaaeWaaOWdaeaajugib8qacaWG4baakiaawIcacaGLPaaajugibi abg2da9Kqbaoaalaaak8aabaqcLbsapeGaaiikaiabeI7aXLqba+aa daahaaWcbeqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaaIXaGaey 4kaSIaaiiOaiabeI7aXjaadIhacaGGPaqcfa4aaeWaaOWdaeaajugi b8qacaaIXaGaeyOeI0IaaiiOaiaadwgal8aadaahaaqabeaajugWa8 qacqGHsislcqaH4oqCaaaakiaawIcacaGLPaaajugibiabgkHiTiab eI7aXjaacckacaWGLbWcpaWaaWbaaeqabaqcLbmapeGaeyOeI0Iaeq iUdehaaaGcpaqaaKqba+qadaqadaGcpaqaaKqzGeWdbiabeI7aXLqb a+aadaahaaWcbeqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaaIXa Gaey4kaSIaeqiUdeNaaiiOaiaadIhaaOGaayjkaiaawMcaaaaaaaa@6FE1@

The reversed failure rate

r * ( x )= [ ( θ 2 +1+ θx)( 1  e θ )θ  e θ ] e θx ( θ 2 +1 )( θ 2 +1+θ ( x+1 ) ) e θ( x+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGYbWcpaWaaWbaaeqabaqcLbmapeGaaiOkaaaajuaGdaqa daGcpaqaaKqzGeWdbiaadIhaaOGaayjkaiaawMcaaKqzGeGaeyypa0 tcfa4aaSaaaOWdaeaajuaGpeWaamWaaOWdaeaajugib8qacaGGOaGa eqiUde3cpaWaaWbaaeqabaqcLbmapeGaaGOmaaaajugibiabgUcaRi aaigdacqGHRaWkcaGGGcGaeqiUdeNaamiEaiaacMcajuaGdaqadaGc paqaaKqzGeWdbiaaigdacqGHsislcaGGGcGaamyzaKqba+aadaahaa WcbeqaaKqzadWdbiabgkHiTiabeI7aXbaaaOGaayjkaiaawMcaaKqz GeGaeyOeI0IaeqiUdeNaaiiOaiaadwgajuaGpaWaaWbaaSqabeaaju gWa8qacqGHsislcqaH4oqCaaaakiaawUfacaGLDbaajugibiaadwga juaGpaWaaWbaaSqabeaajugWa8qacqGHsislcqaH4oqCcaWG4baaaa GcpaqaaKqba+qadaqadaGcpaqaaKqzGeWdbiabeI7aXTWdamaaCaaa beqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaaIXaaakiaawIcaca GLPaaajugibiabgkHiTKqbaoaabmaak8aabaqcLbsapeGaeqiUde3c paWaaWbaaeqabaqcLbmapeGaaGOmaaaajugibiabgUcaRiaaigdacq GHRaWkcqaH4oqCcaGGGcqcfa4aaeWaaOWdaeaajugib8qacaWG4bGa ey4kaSIaaGymaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaqcLbsaca WGLbqcfa4damaaCaaaleqabaqcLbmapeGaeyOeI0IaeqiUde3cdaqa daWdaeaajugWa8qacaWG4bGaey4kaSIaaGymaaWccaGLOaGaayzkaa aaaaaaaaa@93F4@

The second rate of failure is obtained as

r * ( x )=log[ s( x ) s( x+1 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGYbWcpaWaaWbaaeqabaqcLbmapeGaaiOkaaaajuaGdaqa daGcpaqaaKqzGeWdbiaadIhaaOGaayjkaiaawMcaaKqzGeGaeyypa0 JaciiBaiaac+gacaGGNbqcfa4aamWaaOWdaeaajuaGpeWaaSaaaOWd aeaajugib8qacaWGZbqcfa4aaeWaaOWdaeaajugib8qacaWG4baaki aawIcacaGLPaaaa8aabaqcLbsapeGaam4CaKqbaoaabmaak8aabaqc LbsapeGaamiEaiabgUcaRiaaigdaaOGaayjkaiaawMcaaaaaaiaawU facaGLDbaaaaa@51DB@

=log[ ( θ 2 +1+θ ( x+1 ) ) e θ ( θ 2 +1+θ ( x+2 ) ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqGH9aqpciGGSbGaai4BaiaacEgajuaGdaWadaGcpaqaaKqb a+qadaWcaaGcpaqaaKqba+qadaqadaGcpaqaaKqzGeWdbiabeI7aXL qba+aadaahaaWcbeqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaaI XaGaey4kaSIaeqiUdeNaaiiOaKqbaoaabmaak8aabaqcLbsapeGaam iEaiabgUcaRiaaigdaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaaWd aeaajugib8qacaWGLbqcfa4damaaCaaaleqabaqcLbsapeGaeyOeI0 IaeqiUdehaaKqbaoaabmaak8aabaqcLbsapeGaeqiUde3cpaWaaWba aeqabaqcLbmapeGaaGOmaaaajugibiabgUcaRiaaigdacqGHRaWkcq aH4oqCcaGGGcqcfa4aaeWaaOWdaeaajugib8qacaWG4bGaey4kaSIa aGOmaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaaaaaGaay5waiaaw2 faaaaa@6840@

The Proportions of probabilities is given by

f D ( x+1;θ ) f D ( x;θ ) = e θ [ 1+ θ( 1  e θ ) ( θ 2 +1+ θx)( 1  e θ )θ  e θ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaaOWdaeaajugib8qacaWGMbqcfa4damaaBaaaleaajugW a8qacaWGebaal8aabeaajuaGpeWaaeWaaOWdaeaajugib8qacaWG4b Gaey4kaSIaaGymaiaacUdacqaH4oqCaOGaayjkaiaawMcaaaWdaeaa jugib8qacaWGMbWcpaWaaSbaaeaajugWa8qacaWGebaal8aabeaaju aGpeWaaeWaaOWdaeaajugib8qacaWG4bGaai4oaiabeI7aXbGccaGL OaGaayzkaaaaaKqzGeGaeyypa0JaamyzaKqba+aadaahaaWcbeqaaK qzadWdbiabgkHiTiabeI7aXbaajuaGdaWadaGcpaqaaKqzGeWdbiaa igdacqGHRaWkjuaGdaWcaaGcpaqaaKqzGeWdbiabeI7aXLqbaoaabm aak8aabaqcLbsapeGaaGymaiabgkHiTiaacckacaWGLbqcfa4damaa CaaaleqajyaGbaqcLbmapeGaeyOeI0IaeqiUdehaaaGccaGLOaGaay zkaaaapaqaaKqzGeWdbiaacIcacqaH4oqCl8aadaahaaGcbeqaaKqz adWdbiaaikdaaaqcLbsacqGHRaWkcaaIXaGaey4kaSIaaiiOaiabeI 7aXjaadIhacaGGPaqcfa4aaeWaaOWdaeaajugib8qacaaIXaGaeyOe I0IaaiiOaiaadwgajuaGpaWaaWbaaSqabeaajugWa8qacqGHsislcq aH4oqCaaaakiaawIcacaGLPaaajugibiabgkHiTiabeI7aXjaaccka caWGLbWcpaWaaWbaaKqaGfqabaqcLbmapeGaeyOeI0IaeqiUdehaaa aaaOGaay5waiaaw2faaaaa@8A3F@

Probability recurrence relation:

 Probability recurrence relation of DS distribution may be obtained as

P r+2 =  e θ ( 2 P r+1   e θ P r )r=1,2,3, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGqbqcfa4damaaBaaaleaajugWa8qacaWGYbGaey4kaSIa aGOmaaWcpaqabaqcLbsapeGaeyypa0JaaiiOaiaadwgal8aadaahaa qabeaajugWa8qacqGHsislcqaH4oqCaaqcfa4aaeWaaOWdaeaajugi b8qacaaIYaGaamiuaSWdamaaBaaabaqcLbmapeGaamOCaiabgUcaRi aaigdaaSWdaeqaaKqzGeWdbiabgkHiTiaacckacaWGLbqcfa4damaa CaaaleqabaqcLbmapeGaeyOeI0IaeqiUdehaaKqzGeGaamiuaKqba+ aadaWgaaGcbaqcLbmapeGaamOCaaWcpaqabaaak8qacaGLOaGaayzk aaqcLbsacaqGYbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaaio dacaGGSaGaeyOjGWlaaa@61B3@                                                                                (2.5)

Where P 0 = ( θ 2 +1)( 1  e θ )θ  e θ θ 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGqbqcfa4damaaBaaaleaajugWa8qacaaIWaaal8aabeaa jugib8qacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiaacIcacqaH4o qCjuaGpaWaaWbaaSqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIa aGymaiaacMcajuaGdaqadaGcpaqaaKqzGeWdbiaaigdacqGHsislca GGGcGaamyzaKqba+aadaahaaWcbeqaaKqzadWdbiabgkHiTiabeI7a XbaaaOGaayjkaiaawMcaaKqzGeGaeyOeI0IaeqiUdeNaaiiOaiaadw gajuaGpaWaaWbaaSqabeaajugWa8qacqGHsislcqaH4oqCaaaak8aa baqcLbsapeGaeqiUdexcfa4damaaCaaaleqabaqcLbmapeGaaGOmaa aajugibiabgUcaRiaaigdaaaaaaa@61C8@ , and

P 1 = ( θ 2 +1+ θ)( 1  e θ )θ  e θ θ 2 +1   e θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGqbWcpaWaaSbaaOqaaKqzadWdbiaaigdaaOWdaeqaaKqz GeWdbiabg2da9Kqbaoaalaaak8aabaqcLbsapeGaaiikaiabeI7aXL qba+aadaahaaGcbeqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaaI XaGaey4kaSIaaiiOaiabeI7aXjaacMcajuaGdaqadaGcpaqaaKqzGe WdbiaaigdacqGHsislcaGGGcGaamyzaSWdamaaCaaakeqabaqcLbma peGaeyOeI0IaeqiUdehaaaGccaGLOaGaayzkaaqcLbsacqGHsislcq aH4oqCcaGGGcGaamyzaKqba+aadaahaaGcbeqaaKqzadWdbiabgkHi TiabeI7aXbaaaOWdaeaajugib8qacqaH4oqCl8aadaahaaGcbeqaaK qzadWdbiaaikdaaaqcLbsacqGHRaWkcaaIXaaaaiaacckacaWGLbWc paWaaWbaaOqabeaajugWa8qacqGHsislcqaH4oqCaaaaaa@6A2B@                                                                   (2.6)

 Here P r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGqbWdamaaBaaabaqcLbmapeGaamOCaaqcfa4daeqaaaaa @3A7B@  denotes Pr(X= r).

Factorial moment recurrence relation

Factorial moment generating function (fmgf) may be obtained as

MD( t )= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytaK qzadGaamiraKqbaoaabmaakeaajugibiaadshaaOGaayjkaiaawMca aKqzGeGaeyypa0daaa@3E8B@ ( θ 2 +1 )( 1  e θ )θ ( θ 2 +1 )( 1 e θ   e θ t ) + θ( 1  e θ ) ( θ 2 +1 ) ( 1 e θ   e θ t ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaaOWdaeaajuaGpeWaaeWaaOWdaeaajugib8qacqaH4oqC juaGpaWaaWbaaSqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaaG ymaaGccaGLOaGaayzkaaqcfa4aaeWaaOWdaeaajugib8qacaaIXaGa eyOeI0IaaiiOaiaadwgal8aadaahaaqabeaajugWa8qacqGHsislcq aH4oqCaaaakiaawIcacaGLPaaajugibiabgkHiTiabeI7aXbGcpaqa aKqba+qadaqadaGcpaqaaKqzGeWdbiabeI7aXLqba+aadaahaaWcbe qaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaaIXaaakiaawIcacaGL PaaajuaGdaqadaGcpaqaaKqzGeWdbiaaigdacqGHsislcaWGLbWcpa WaaWbaaeqabaqcLbmapeGaeyOeI0IaeqiUdehaaKqzGeGaeyOeI0Ia aiiOaiaadwgal8aadaahaaqabeaajugWa8qacqGHsislcqaH4oqCaa qcLbsacaWG0baakiaawIcacaGLPaaaaaqcLbsacqGHRaWkjuaGdaWc aaGcpaqaaKqzGeWdbiabeI7aXLqbaoaabmaak8aabaqcLbsapeGaaG ymaiabgkHiTiaacckacaWGLbWcpaWaaWbaaeqabaqcLbmapeGaeyOe I0IaeqiUdehaaaGccaGLOaGaayzkaaaapaqaaKqba+qadaqadaGcpa qaaKqzGeWdbiabeI7aXXWdamaaCaaaleqabaqcLbeapeGaaGOmaaaa jugibiabgUcaRiaaigdaaOGaayjkaiaawMcaaKqbaoaabmaak8aaba qcLbsapeGaaGymaiabgkHiTiaadwgajuaGpaWaaWbaaSqabeaajugi b8qacqGHsislcqaH4oqCaaGaeyOeI0IaaiiOaiaadwgal8aadaahaa qabeaajugWa8qacqGHsislcqaH4oqCaaGaamiDaaGccaGLOaGaayzk aaqcfa4damaaCaaaleqabaqcLbmapeGaaGOmaaaaaaaaaa@969F@ .                                                     (2.7)

First four factorial moments may be obtained as

μ [ 1 ] ' =   e θ [ ( θ 2 +1 )( 1 e θ )+θ ] ( θ 2 +1 ) ( 1 e θ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH8oqBl8aadaqhaaqaa8qadaWadaWdaeaajugWa8qacaaI XaaaliaawUfacaGLDbaaa8aabaqcLbmapeGaai4jaaaajugibiabg2 da9Kqbaoaalaaak8aabaqcLbsapeGaaiiOaiaadwgal8aadaahaaqa beaajugWa8qacqGHsislcqaH4oqCaaqcfa4aamWaaOWdaeaajuaGpe WaaeWaaOWdaeaajugib8qacqaH4oqCjuaGpaWaaWbaaSqabeaajugW a8qacaaIYaaaaKqzGeGaey4kaSIaaGymaaGccaGLOaGaayzkaaqcfa 4aaeWaaOWdaeaajugib8qacaaIXaGaeyOeI0IaamyzaSWdamaaCaaa beqaaKqzadWdbiabgkHiTiabeI7aXbaaaOGaayjkaiaawMcaaKqzGe Gaey4kaSIaeqiUdehakiaawUfacaGLDbaaa8aabaqcfa4dbmaabmaa k8aabaqcLbsapeGaeqiUdexcfa4damaaCaaaleqajyaGbaqcLbmape GaaGOmaaaajugibiabgUcaRiaaigdaaOGaayjkaiaawMcaaKqbaoaa bmaak8aabaqcLbsapeGaaGymaiabgkHiTiaadwgal8aadaahaaqabe aajugWa8qacqGHsislcqaH4oqCaaaakiaawIcacaGLPaaal8aadaah aaqabeaajugWa8qacaaIYaaaaaaaaaa@767C@

μ [ 2 ] ' = 2  e 2θ [ ( θ 2 +1 )( 1 e θ )+2θ ] ( θ 2 +1 ) ( 1 e θ ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH8oqBl8aadaqhaaqaa8qadaWadaWdaeaajugWa8qacaaI YaaaliaawUfacaGLDbaaa8aabaqcLbmapeGaai4jaaaajugibiabg2 da9Kqbaoaalaaak8aabaqcLbsapeGaaGOmaiaacckacaWGLbWcpaWa aWbaaeqabaqcLbmapeGaeyOeI0IaaGOmaiabeI7aXbaajuaGdaWada GcpaqaaKqba+qadaqadaGcpaqaaKqzGeWdbiabeI7aXTWdamaaCaaa keqabaqcLbmapeGaaGOmaaaajugibiabgUcaRiaaigdaaOGaayjkai aawMcaaKqbaoaabmaak8aabaqcLbsapeGaaGymaiabgkHiTiaadwga juaGpaWaaWbaaSqabeaajugWa8qacqGHsislcqaH4oqCaaaakiaawI cacaGLPaaajugibiabgUcaRiaaikdacqaH4oqCaOGaay5waiaaw2fa aaWdaeaajuaGpeWaaeWaaOWdaeaajugib8qacqaH4oqCjuaGpaWaaW baaSqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaaGymaaGccaGL OaGaayzkaaqcfa4aaeWaaOWdaeaajugib8qacaaIXaGaeyOeI0Iaam yzaSWdamaaCaaabeqaaKqzadWdbiabgkHiTiabeI7aXbaaaOGaayjk aiaawMcaaKqba+aadaahaaWcbeqaaKqzadWdbiaaiodaaaaaaaaa@78BB@

μ [ 3 ] ' = 6  e 3θ [ ( θ 2 +1 )( 1 e θ )+3θ ] ( θ 2 +1 ) ( 1 e θ ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH8oqBl8aadaqhaaqaa8qadaWadaWdaeaajugWa8qacaaI ZaaaliaawUfacaGLDbaaa8aabaqcLbmapeGaai4jaaaajugibiabg2 da9Kqbaoaalaaak8aabaqcLbsapeGaaGOnaiaacckacaWGLbWcpaWa aWbaaeqabaqcLbmapeGaeyOeI0IaaG4maiabeI7aXbaajuaGdaWada GcpaqaaKqba+qadaqadaGcpaqaaKqzGeWdbiabeI7aXTWdamaaCaaa beqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaaIXaaakiaawIcaca GLPaaajuaGdaqadaGcpaqaaKqzGeWdbiaaigdacqGHsislcaWGLbqc fa4damaaCaaaleqabaqcLbmapeGaeyOeI0IaeqiUdehaaaGccaGLOa GaayzkaaqcLbsacqGHRaWkcaaIZaGaeqiUdehakiaawUfacaGLDbaa a8aabaqcfa4dbmaabmaak8aabaqcLbsapeGaeqiUde3cpaWaaWbaae qabaqcLbmapeGaaGOmaaaajugibiabgUcaRiaaigdaaOGaayjkaiaa wMcaaKqbaoaabmaak8aabaqcLbsapeGaaGymaiabgkHiTiaadwgaju aGpaWaaWbaaSqabeaajugWa8qacqGHsislcqaH4oqCaaaakiaawIca caGLPaaal8aadaahaaqcgayabeaajugWa8qacaaI0aaaaaaaaaa@78BA@

μ [ 4 ] ' = 24  e 4θ [ ( θ 2 +1 )( 1 e θ )+4θ ] ( θ 2 +1 ) ( 1 e θ ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH8oqBl8aadaqhaaqaa8qadaWadaWdaeaajugWa8qacaaI 0aaaliaawUfacaGLDbaaa8aabaqcLbmapeGaai4jaaaajugibiabg2 da9Kqbaoaalaaak8aabaqcLbsapeGaaGOmaiaaisdacaGGGcGaamyz aSWdamaaCaaajyaGbeqaaKqzadWdbiabgkHiTiaaisdacqaH4oqCaa qcfa4aamWaaOWdaeaajuaGpeWaaeWaaOWdaeaajugib8qacqaH4oqC l8aadaahaaqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaaGymaa GccaGLOaGaayzkaaqcfa4aaeWaaOWdaeaajugib8qacaaIXaGaeyOe I0IaamyzaKqba+aadaahaaWcbeqaaKqzadWdbiabgkHiTiabeI7aXb aaaOGaayjkaiaawMcaaKqzGeGaey4kaSIaaGinaiabeI7aXbGccaGL BbGaayzxaaaapaqaaKqba+qadaqadaGcpaqaaKqzGeWdbiabeI7aXT WdamaaCaaabeqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaaIXaaa kiaawIcacaGLPaaajuaGdaqadaGcpaqaaKqzGeWdbiaaigdacqGHsi slcaWGLbqcfa4damaaCaaaleqabaqcLbmapeGaeyOeI0IaeqiUdeha aaGccaGLOaGaayzkaaqcfa4damaaCaaaleqabaqcLbmapeGaaGinaa aaaaaaaa@7A05@

Proposition 3: The general form of factorial moment may also be written as

μ [ r ] ' = r!  e θr [ ( θ 2 +1 )( 1 e θ )+θr ] ( θ 2 +1 ) ( 1 e θ ) r+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH8oqBl8aadaqhaaqaa8qadaWadaWdaeaajugWa8qacaWG YbaaliaawUfacaGLDbaaa8aabaqcLbmapeGaai4jaaaajugibiabg2 da9Kqbaoaalaaak8aabaqcLbsapeGaamOCaiaacgcacaGGGcGaamyz aSWdamaaCaaabeqaaKqzadWdbiabgkHiTiabeI7aXjaadkhaaaqcfa 4aamWaaOWdaeaajuaGpeWaaeWaaOWdaeaajugib8qacqaH4oqCjuaG paWaaWbaaSqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaaGymaa GccaGLOaGaayzkaaqcfa4aaeWaaOWdaeaajugib8qacaaIXaGaeyOe I0IaamyzaSWdamaaCaaabeqaaKqzadWdbiabgkHiTiabeI7aXbaaaO GaayjkaiaawMcaaKqzGeGaey4kaSIaeqiUdeNaamOCaaGccaGLBbGa ayzxaaaapaqaaKqba+qadaqadaGcpaqaaKqzGeWdbiabeI7aXTWdam aaCaaabeqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaaIXaaakiaa wIcacaGLPaaajuaGdaqadaGcpaqaaKqzGeWdbiaaigdacqGHsislca WGLbWcpaWaaWbaaeqabaqcLbmapeGaeyOeI0IaeqiUdehaaaGccaGL OaGaayzkaaqcfa4damaaCaaaleqabaqcLbmapeGaamOCaiabgUcaRi aaigdaaaaaaaaa@7B8B@                                                      (2.8)

Hence, mean and variance may be obtained as

Mean =   e θ [ ( θ 2 +1 )( 1 e θ )+θ ] ( θ 2 +1 ) ( 1 e θ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabg2da9Kqbaoaalaaak8aabaqcLbsapeGaaiiOaiaa dwgal8aadaahaaGcbeqaaKqzadWdbiabgkHiTiabeI7aXbaajuaGda WadaGcpaqaaKqba+qadaqadaGcpaqaaKqzGeWdbiabeI7aXTWdamaa CaaabeqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaaIXaaakiaawI cacaGLPaaajuaGdaqadaGcpaqaaKqzGeWdbiaaigdacqGHsislcaWG Lbqcfa4damaaCaaaleqabaqcLbmapeGaeyOeI0IaeqiUdehaaaGcca GLOaGaayzkaaqcLbsacqGHRaWkcqaH4oqCaOGaay5waiaaw2faaaWd aeaajuaGpeWaaeWaaOWdaeaajugib8qacqaH4oqCjuaGpaWaaWbaaS qabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaaGymaaGccaGLOaGa ayzkaaqcfa4aaeWaaOWdaeaajugib8qacaaIXaGaeyOeI0IaamyzaK qba+aadaahaaWcbeqaaKqzadWdbiabgkHiTiabeI7aXbaaaOGaayjk aiaawMcaaKqba+aadaahaaWcbeqaaKqzadWdbiaaikdaaaaaaaaa@6EA5@ , and

Variance=   e θ [ ( θ 2 +1 ) 2 ( 1 e θ ) 2 +( θ 2 +1 )( 1 e θ )θ e θ θ 2 ] ( θ 2 +1 ) 2 ( 1 e θ ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGwbGaamyyaiaadkhacaWGPbGaamyyaiaad6gacaWGJbGa amyzaiabg2da9Kqbaoaalaaak8aabaqcLbsapeGaaiiOaiaadwgaju aGpaWaaWbaaSqabeaajugWa8qacqGHsislcqaH4oqCaaqcfa4aamWa aOWdaeaajuaGpeWaaeWaaOWdaeaajugib8qacqaH4oqCjuaGpaWaaW baaSqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaaGymaaGccaGL OaGaayzkaaqcfa4damaaCaaaleqabaqcLbmapeGaaGOmaaaajuaGda qadaGcpaqaaKqzGeWdbiaaigdacqGHsislcaWGLbWcpaWaaWbaaeqa baqcLbmapeGaeyOeI0IaeqiUdehaaaGccaGLOaGaayzkaaWcpaWaaW baaeqabaqcLbmapeGaaGOmaaaajugibiabgUcaRKqbaoaabmaak8aa baqcLbsapeGaeqiUde3cpaWaaWbaaeqabaqcLbmapeGaaGOmaaaaju gibiabgUcaRiaaigdaaOGaayjkaiaawMcaaKqbaoaabmaak8aabaqc LbsapeGaaGymaiabgkHiTiaadwgajuaGpaWaaWbaaSqabeaajugWa8 qacqGHsislcqaH4oqCaaaakiaawIcacaGLPaaajugibiabeI7aXjab gkHiTiaadwgajuaGpaWaaWbaaSqabeaajugWa8qacqGHsislcqaH4o qCaaqcLbsacqaH4oqCjuaGpaWaaWbaaSqabeaajugWa8qacaaIYaaa aaGccaGLBbGaayzxaaaapaqaaKqba+qadaqadaGcpaqaaKqzGeWdbi abeI7aXLqba+aadaahaaWcbeqaaKqzadWdbiaaikdaaaqcLbsacqGH RaWkcaaIXaaakiaawIcacaGLPaaajuaGpaWaaWbaaSqabeaajugWa8 qacaaIYaaaaKqbaoaabmaak8aabaqcLbsapeGaaGymaiabgkHiTiaa dwgal8aadaahaaqabeaajugWa8qacqGHsislcqaH4oqCaaaakiaawI cacaGLPaaajuaGpaWaaWbaaSqabOqaaKqzadWdbiaaisdaaaaaaaaa @9CB7@  respectively.

Zero truncated discrete shanker (ZTDS) distribution

Zero- truncated distributions are applicable for the situations when the data to be modeled originate from a generating mechanism that structurally excludes zero counts. The discrete Shanker distribution must be adjusted to count for the missing zeros. Here the zero-truncated discrete Shanker distribution has been derived.

 The pmf f z ( x;θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGMbqcfa4damaaBaaaleaajugWa8qacaWG6baal8aabeaa juaGpeWaaeWaaOWdaeaajugib8qacaWG4bGaai4oaiabeI7aXbGcca GLOaGaayzkaaaaaa@4100@ of Zero-truncated DS distribution has been derived as

f z ( x;θ )= P x 1 P 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGMbWcpaWaaSbaaeaajugWa8qacaWG6baal8aabeaajuaG peWaaeWaaOWdaeaajugib8qacaWG4bGaai4oaiabeI7aXbGccaGLOa GaayzkaaqcLbsacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiaadcfa juaGpaWaaSbaaSqaaKqzadWdbiaadIhaaSWdaeqaaaGcbaqcLbsape GaaGymaiabgkHiTiaadcfal8aadaWgaaqaaKqzadWdbiaaicdaaSWd aeqaaaaaaaa@4CC4@                                                                                                                 (3.0)

Where P x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGqbWdamaaBaaabaqcLbmapeGaamiEaaqcfa4daeqaaaaa @3A82@ denotes the pmf of discrete Shanker distribution.

Hence, f z ( x;θ )= ( θ 2 +1+ θx)( 1  e θ )θ  e θ ( θ 2 +θ+1)   e θ( x1 ) ,x=1, 2,3, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGibGaamyzaiaad6gacaWGJbGaamyzaiaacYcacaWGMbWc paWaaSbaaeaajugWa8qacaWG6baal8aabeaajuaGpeWaaeWaaOWdae aajugib8qacaWG4bGaai4oaiabeI7aXbGccaGLOaGaayzkaaqcLbsa cqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiaacIcacqaH4oqCjuaGpa WaaWbaaSqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaaGymaiab gUcaRiaacckacqaH4oqCcaWG4bGaaiykaKqbaoaabmaak8aabaqcLb sapeGaaGymaiabgkHiTiaacckacaWGLbWcpaWaaWbaaeqabaqcLbma peGaeyOeI0IaeqiUdehaaaGccaGLOaGaayzkaaqcLbsacqGHsislcq aH4oqCcaGGGcGaamyzaSWdamaaCaaabeqaaKqzadWdbiabgkHiTiab eI7aXbaaaOWdaeaajugib8qacaGGOaGaeqiUde3cpaWaaWbaaOqabe aajugWa8qacaaIYaaaaKqzGeGaey4kaSIaeqiUdeNaey4kaSIaaGym aiaacMcaaaGaaiiOaiaadwgal8aadaahaaqabeaajugWa8qacqGHsi slcqaH4oqClmaabmaapaqaaKqzadWdbiaadIhacqGHsislcaaIXaaa liaawIcacaGLPaaaaaqcLbsacaGGSaGaamiEaiabg2da9iaaigdaca GGSaGaaiiOaiaaikdacaGGSaGaaG4maiaacYcacqGHMacVaaa@8A0A@                           (3.1)

Probability recurrence relation for ZTDS distribution

The pgf G z ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGhbWcpaWaaSbaaeaajugWa8qacaWG6baal8aabeaajuaG peWaaeWaaOWdaeaajugib8qacaWG0baakiaawIcacaGLPaaaaaa@3DDB@ of zero-truncated DS distribution may be obtained as

G z ( t )= x=1 t x f z ( x;θ ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGhbWcpaWaaSbaaeaajugWa8qacaWG6baal8aabeaajuaG peWaaeWaaOWdaeaajugib8qacaWG0baakiaawIcacaGLPaaajugibi abg2da9KqbaoaaqadakeaajugibiaadshaaSqaaKqzadGaamiEaiab g2da9iaaigdaaSqaaKqzadGaeyOhIukajugibiabggHiLdqcfa4dam aaCaaaleqabaqcLbmapeGaamiEaaaajugibiaadAgajuaGpaWaaSba aOqaaKqzadWdbiaadQhaaSWdaeqaaKqba+qadaqadaGcpaqaaKqzGe WdbiaadIhacaGG7aGaeqiUdehakiaawIcacaGLPaaajugibiaacYca aaa@59E2@

= t[ { ( θ 2 +1 )( 1 e θ )θ e θ }( 1t e θ )+θ( 1 e θ )  ] ( ( θ 2 +θ+1 ) ( 1t e θ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 tcfaieaaaaaaaaa8qadaWcaaGcpaqaaKqzGeWdbiaadshajuaGdaWa daGcpaqaaKqba+qadaGadaGcpaqaaKqba+qadaqadaGcpaqaaKqzGe WdbiabeI7aXLqba+aadaahaaWcbeqaaKqzadWdbiaaikdaaaqcLbsa cqGHRaWkcaaIXaaakiaawIcacaGLPaaajuaGdaqadaGcpaqaaKqzGe WdbiaaigdacqGHsislcaWGLbWcpaWaaWbaaeqabaqcLbmapeGaeyOe I0IaeqiUdehaaaGccaGLOaGaayzkaaqcLbsacqGHsislcqaH4oqCca WGLbWcpaWaaWbaaeqabaqcLbmapeGaeyOeI0IaeqiUdehaaaGccaGL 7bGaayzFaaqcfa4aaeWaaOWdaeaajugib8qacaaIXaGaeyOeI0Iaam iDaiaadwgajuaGpaWaaWbaaSqabeaajugWa8qacqGHsislcqaH4oqC aaaakiaawIcacaGLPaaajugibiabgUcaRiabeI7aXLqbaoaabmaak8 aabaqcLbsapeGaaGymaiabgkHiTiaadwgajuaGpaWaaWbaaSqabeaa jugWa8qacqGHsislcqaH4oqCaaaakiaawIcacaGLPaaajugibiaacc kaaOGaay5waiaaw2faaaWdaeaajuaGpeWaaeWaaOWdaeaajugib8qa caGGOaGaeqiUdexcfa4damaaCaaaleqabaqcLbmapeGaaGOmaaaaju gibiabgUcaRiabeI7aXjabgUcaRiaaigdaaOGaayjkaiaawMcaaKqb aoaabmaak8aabaqcLbsapeGaaGymaiabgkHiTiaadshacaWGLbqcfa 4damaaCaaaleqabaqcLbmapeGaeyOeI0IaeqiUdehaaaGccaGLOaGa ayzkaaqcfa4damaaCaaaleqabaqcLbmapeGaaGOmaaaaaaaaaa@8F89@                                                      (3.2)

Probability recurrence relation ZTDS distribution may obtained as

P z r = e θ [ 2 P z r1 e θ P z r2 ]r=2,3,4,. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGqbWcpaWaaWbaaeqabaqcLbmapeGaamOEaaaajuaGpaWa aSbaaSqaaKqzadWdbiaadkhaaSWdaeqaaKqzGeWdbiabg2da9iaadw gal8aadaahaaqabeaajugWa8qacqGHsislcqaH4oqCaaqcfa4aamWa aOWdaeaajugib8qacaaIYaGaamiuaSWdamaaCaaabeqaaKqzadWdbi aadQhaaaWcpaWaaSbaaeaajugWa8qacaWGYbGaeyOeI0IaaGymaaWc paqabaqcLbsapeGaeyOeI0IaamyzaKqba+aadaahaaWcbeqaaKqzad WdbiabgkHiTiabeI7aXbaajugibiaadcfajuaGpaWaaWbaaSqabeaa jugWa8qacaWG6baaaSWdamaaBaaabaqcLbmapeGaamOCaiabgkHiTi aaikdaaSWdaeqaaaGcpeGaay5waiaaw2faaKqzGeGaamOCaiabg2da 9iaaikdacaGGSaGaaG4maiaacYcacaaI0aGaaiilaiabgAci8kaac6 caaaa@680D@

Where P z 1 = ( θ 2 +1+ θ)( 1  e θ )θ  e θ ( θ 2 +θ+1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadcfal8aadaahaaqabeaajugWa8qacaWG6baaaSWd amaaBaaabaqcLbmapeGaaGymaaWcpaqabaqcLbsapeGaeyypa0tcfa 4aaSaaaOWdaeaajugib8qacaGGOaGaeqiUde3cpaWaaWbaaeqabaqc LbmapeGaaGOmaaaajugibiabgUcaRiaaigdacqGHRaWkcaGGGcGaeq iUdeNaaiykaKqbaoaabmaak8aabaqcLbsapeGaaGymaiabgkHiTiaa cckacaWGLbWcpaWaaWbaaKGbagqabaqcLbmapeGaeyOeI0IaeqiUde haaaGccaGLOaGaayzkaaqcLbsacqGHsislcqaH4oqCcaGGGcGaamyz aSWdamaaCaaabeqaaKqzadWdbiabgkHiTiabeI7aXbaaaOWdaeaaju gib8qacaGGOaGaeqiUde3cpaWaaWbaaeqabaqcLbmapeGaaGOmaaaa jugibiabgUcaRiabeI7aXjabgUcaRiaaigdacaGGPaaaaaaa@69D8@ and P z 2 = ( θ 2 +1+2θ)( 1  e θ )θ  e θ ( θ 2 +θ+1)   e θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadcfal8aadaahaaqabeaajugWa8qacaWG6baaaSWd amaaBaaabaqcLbmapeGaaGOmaaWcpaqabaqcLbsapeGaeyypa0tcfa 4aaSaaaOWdaeaajugib8qacaGGOaGaeqiUde3cpaWaaWbaaeqabaqc LbmapeGaaGOmaaaajugibiabgUcaRiaaigdacqGHRaWkcaaIYaGaeq iUdeNaaiykaKqbaoaabmaak8aabaqcLbsapeGaaGymaiabgkHiTiaa cckacaWGLbWcpaWaaWbaaeqabaqcLbmapeGaeyOeI0IaeqiUdehaaa GccaGLOaGaayzkaaqcLbsacqGHsislcqaH4oqCcaGGGcGaamyzaSWd amaaCaaajyaGbeqaaKqzadWdbiabgkHiTiabeI7aXbaaaOWdaeaaju gib8qacaGGOaGaeqiUdexcfa4damaaCaaaleqabaqcLbmapeGaaGOm aaaajugibiabgUcaRiabeI7aXjabgUcaRiaaigdacaGGPaaaaiaacc kacaWGLbWcpaWaaWbaaeqabaqcLbmapeGaeyOeI0IaeqiUdehaaaaa @702B@                 (3.3)

Proposition 4: The cumulative distribution of ZTDS distribution is given by

F z ( x )= ( θ 2 +θ+1 )( θ 2 +θ+θx+1 ) e θx ( θ 2 +θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadAeal8aadaWgaaqaaKqzadWdbiaadQhaaSWdaeqa aKqba+qadaqadaGcpaqaaKqzGeWdbiaadIhaaOGaayjkaiaawMcaaK qzGeGaeyypa0tcfa4aaSaaaOWdaeaajuaGpeWaaeWaaOWdaeaajugi b8qacqaH4oqCl8aadaahaaqabeaajugWa8qacaaIYaaaaKqzGeGaey 4kaSIaeqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaaqcLbsacqGH sisljuaGdaqadaGcpaqaaKqzGeWdbiabeI7aXTWdamaaCaaabeqaaK qzadWdbiaaikdaaaqcLbsacqGHRaWkcqaH4oqCcqGHRaWkcqaH4oqC caWG4bGaey4kaSIaaGymaaGccaGLOaGaayzkaaqcLbsacaWGLbqcfa 4damaaCaaaleqabaqcLbmapeGaeyOeI0IaeqiUdeNaamiEaaaaaOWd aeaajuaGpeWaaeWaaOWdaeaajugib8qacqaH4oqCl8aadaahaaqabe aajugWa8qacaaIYaaaaKqzGeGaey4kaSIaeqiUdeNaey4kaSIaaGym aaGccaGLOaGaayzkaaaaaaaa@6F51@

The survival function of ZTDS distribution is given by

S z ( x )= ( θ 2 +θ+θx+1 )) e θx ( θ 2 +θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadofal8aadaWgaaqaaKqzadWdbiaadQhaaSWdaeqa aKqba+qadaqadaGcpaqaaKqzGeWdbiaadIhaaOGaayjkaiaawMcaaK qzGeGaeyypa0tcfa4aaSaaaOWdaeaajuaGpeWaaeWaaOWdaeaajugi b8qacqaH4oqCl8aadaahaaqabeaajugWa8qacaaIYaaaaKqzGeGaey 4kaSIaeqiUdeNaey4kaSIaeqiUdeNaamiEaiabgUcaRiaaigdaaOGa ayjkaiaawMcaaKqzGeGaaiykaiaadwgajuaGpaWaaWbaaSqabeaaju gWa8qacqGHsislcqaH4oqCcaWG4baaaaGcpaqaaKqba+qadaqadaGc paqaaKqzGeWdbiabeI7aXTWdamaaCaaabeqaaKqzadWdbiaaikdaaa qcLbsacqGHRaWkcqaH4oqCcqGHRaWkcaaIXaaakiaawIcacaGLPaaa aaaaaa@6306@

The Failure hazared rate may be obtained as

r z ( x )= P( X=x ) P( Xx1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadkhal8aadaWgaaqaaKqzadWdbiaadQhaaSWdaeqa aKqba+qadaqadaGcpaqaaKqzGeWdbiaadIhaaOGaayjkaiaawMcaaK qzGeGaeyypa0tcfa4aaSaaaOWdaeaajugib8qacaWGqbqcfa4aaeWa aOWdaeaajugib8qacaWGybGaeyypa0JaamiEaaGccaGLOaGaayzkaa aapaqaaKqzGeWdbiaadcfajuaGdaqadaGcpaqaaKqzGeWdbiaadIfa cqGHLjYScaWG4bGaeyOeI0IaaGymaaGccaGLOaGaayzkaaaaaaaa@5147@ ,

= ( θ 2 +1+ θx)( 1  e θ )θ  e θ ( θ 2 +1+θx ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabg2da9Kqbaoaalaaak8aabaqcLbsapeGaaiikaiab eI7aXTWdamaaCaaabeqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkca aIXaGaey4kaSIaaiiOaiabeI7aXjaadIhacaGGPaqcfa4aaeWaaOWd aeaajugib8qacaaIXaGaeyOeI0IaaiiOaiaadwgal8aadaahaaqabe aajugWa8qacqGHsislcqaH4oqCaaaakiaawIcacaGLPaaajugibiab gkHiTiabeI7aXjaacckacaWGLbqcfa4damaaCaaaleqabaqcLbmape GaeyOeI0IaeqiUdehaaaGcpaqaaKqba+qadaqadaGcpaqaaKqzGeWd biabeI7aXTWdamaaCaaabeqaaKqzadWdbiaaikdaaaqcLbsacqGHRa WkcaaIXaGaey4kaSIaeqiUdeNaamiEaaGccaGLOaGaayzkaaaaaaaa @6688@ .

The reversed failure rate

r z * ( x )= P( X=x ) P( Xx ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadkhal8aadaqhaaqaaKqzadWdbiaadQhaaSWdaeaa jugWa8qacaGGQaaaaKqbaoaabmaak8aabaqcLbsapeGaamiEaaGcca GLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiaa dcfajuaGdaqadaGcpaqaaKqzGeWdbiaadIfacqGH9aqpcaWG4baaki aawIcacaGLPaaaa8aabaqcLbsapeGaamiuaKqbaoaabmaak8aabaqc LbsapeGaamiwaiabgsMiJkaadIhaaOGaayjkaiaawMcaaaaaaaa@516C@

= ( θ 2 +1+ θx)( 1  e θ )θ  e θ ( θ 2 +1+θx ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabg2da9Kqbaoaalaaak8aabaqcLbsapeGaaiikaiab eI7aXTWdamaaCaaabeqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkca aIXaGaey4kaSIaaiiOaiabeI7aXjaadIhacaGGPaqcfa4aaeWaaOWd aeaajugib8qacaaIXaGaeyOeI0IaaiiOaiaadwgal8aadaahaaqabe aajugWa8qacqGHsislcqaH4oqCaaaakiaawIcacaGLPaaajugibiab gkHiTiabeI7aXjaacckacaWGLbqcfa4damaaCaaaleqabaqcLbmape GaeyOeI0IaeqiUdehaaaGcpaqaaKqba+qadaqadaGcpaqaaKqzGeWd biabeI7aXTWdamaaCaaabeqaaKqzadWdbiaaikdaaaqcLbsacqGHRa WkcaaIXaGaey4kaSIaeqiUdeNaamiEaaGccaGLOaGaayzkaaaaaaaa @6688@ .

The second rate of failure is obtained as

r z * ( x )= P( X=x ) P( Xx ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadkhal8aadaqhaaqaaKqzadWdbiaadQhaaSWdaeaa jugWa8qacaGGQaaaaKqbaoaabmaak8aabaqcLbsapeGaamiEaaGcca GLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiaa dcfajuaGdaqadaGcpaqaaKqzGeWdbiaadIfacqGH9aqpcaWG4baaki aawIcacaGLPaaaa8aabaqcLbsapeGaamiuaKqbaoaabmaak8aabaqc LbsapeGaamiwaiabgsMiJkaadIhaaOGaayjkaiaawMcaaaaaaaa@516B@

= [( θ 2 +1+ θx)( 1  e θ )θ  e θ ] e θ( x1 ) ( θ 2 +θ+1 )( θ 2 +θ+1+θx )) e θx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqGH9aqpjuaGdaWcaaGcpaqaaKqzGeWdbiaacUfacaGGOaGa eqiUde3cpaWaaWbaaeqabaqcLbmapeGaaGOmaaaajugibiabgUcaRi aaigdacqGHRaWkcaGGGcGaeqiUdeNaamiEaiaacMcajuaGdaqadaGc paqaaKqzGeWdbiaaigdacqGHsislcaGGGcGaamyzaSWdamaaCaaabe qaaKqzadWdbiabgkHiTiabeI7aXbaaaOGaayjkaiaawMcaaKqzGeGa eyOeI0IaeqiUdeNaaiiOaiaadwgal8aadaahaaqabeaajugWa8qacq GHsislcqaH4oqCaaqcLbsacaGGDbGaamyzaKqba+aadaahaaWcbeqa aKqzadWdbiabgkHiTiabeI7aXTWaaeWaa8aabaqcLbmapeGaamiEai abgkHiTiaaigdaaSGaayjkaiaawMcaaaaaaOWdaeaajuaGpeWaaeWa aOWdaeaajugib8qacqaH4oqCjuaGpaWaaWbaaSqabeaajugWa8qaca aIYaaaaKqzGeGaey4kaSIaeqiUdeNaey4kaSIaaGymaaGccaGLOaGa ayzkaaqcLbsacqGHsisljuaGdaqadaGcpaqaaKqzGeWdbiabeI7aXT WdamaaCaaabeqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcqaH4oqC cqGHRaWkcaaIXaGaey4kaSIaeqiUdeNaamiEaaGccaGLOaGaayzkaa qcLbsacaGGPaGaamyzaKqba+aadaahaaWcbeqaaKqzadWdbiabgkHi TiabeI7aXjaadIhaaaaaaaaa@8B47@

The proportions of probabilities is given by

f z ( x+1;θ ) f z ( x;θ ) = e θ [ 1+ θ( 1  e θ ) ( θ 2 +1+ θx)( 1  e θ )θ  e θ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcfaieaa aaaaaaa8qadaWcaaGcpaqaaKqzGeWdbiaadAgal8aadaWgaaqaaKqz adWdbiaadQhaaSWdaeqaaKqba+qadaqadaGcpaqaaKqzGeWdbiaadI hacqGHRaWkcaaIXaGaai4oaiabeI7aXbGccaGLOaGaayzkaaaapaqa aKqzGeWdbiaadAgal8aadaWgaaqaaKqzadWdbiaadQhaaSWdaeqaaK qba+qadaqadaGcpaqaaKqzGeWdbiaadIhacaGG7aGaeqiUdehakiaa wIcacaGLPaaaaaqcLbsacqGH9aqpcaWGLbWcpaWaaWbaaeqabaqcLb mapeGaeyOeI0IaeqiUdehaaKqbaoaadmaak8aabaqcLbsapeGaaGym aiabgUcaRKqbaoaalaaak8aabaqcLbsapeGaeqiUdexcfa4aaeWaaO Wdaeaajugib8qacaaIXaGaeyOeI0IaaiiOaiaadwgajuaGpaWaaWba aSqabeaajugWa8qacqGHsislcqaH4oqCaaaakiaawIcacaGLPaaaa8 aabaqcLbsapeGaaiikaiabeI7aXTWdamaaCaaabeqaaKqzadWdbiaa ikdaaaqcLbsacqGHRaWkcaaIXaGaey4kaSIaaiiOaiabeI7aXjaadI hacaGGPaqcfa4aaeWaaOWdaeaajugib8qacaaIXaGaeyOeI0IaaiiO aiaadwgal8aadaahaaqabeaajugWa8qacqGHsislcqaH4oqCaaaaki aawIcacaGLPaaajugibiabgkHiTiabeI7aXjaacckacaWGLbqcfa4d amaaCaaaleqabaqcLbmapeGaeyOeI0IaeqiUdehaaaaaaOGaay5wai aaw2faaaaa@88AE@

Factorial moment recurrence relation for ZTDS distribution 

Factorial moment generating function M z ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWcpaWaaSbaaKqbagaajugWa8qacaWG6baajuaGpaqa baWdbmaabmaapaqaa8qacaWG0baacaGLOaGaayzkaaaaaa@3DCB@  of ZTDS distribution may be obtained as

M z ( t )= ( 1+t )[ { ( θ 2 +1 )( 1 e θ )θ e θ }( 1tt e θ )+θ( 1 e θ )  ] ( ( θ 2 +θ+1 ) ( 1tt e θ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaad2eal8aadaWgaaqaaKqzadWdbiaadQhaaSWdaeqa aKqba+qadaqadaGcpaqaaKqzGeWdbiaadshaaOGaayjkaiaawMcaaK qzGeGaeyypa0tcfa4aaSaaaOWdaeaajuaGpeWaaeWaaOWdaeaajugi b8qacaaIXaGaey4kaSIaamiDaaGccaGLOaGaayzkaaqcfa4aamWaaO WdaeaajuaGpeWaaiWaaOWdaeaajuaGpeWaaeWaaOWdaeaajugib8qa cqaH4oqCl8aadaahaaqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaS IaaGymaaGccaGLOaGaayzkaaqcfa4aaeWaaOWdaeaajugib8qacaaI XaGaeyOeI0IaamyzaSWdamaaCaaabeqaaKqzadWdbiabgkHiTiabeI 7aXbaaaOGaayjkaiaawMcaaKqzGeGaeyOeI0IaeqiUdeNaamyzaKqb a+aadaahaaWcbeqaaKqzadWdbiabgkHiTiabeI7aXbaaaOGaay5Eai aaw2haaKqbaoaabmaak8aabaqcLbsapeGaaGymaiabgkHiTiaadsha cqGHsislcaWG0bGaamyzaSWdamaaCaaabeqaaKqzadWdbiabgkHiTi abeI7aXbaaaOGaayjkaiaawMcaaKqzGeGaey4kaSIaeqiUdexcfa4a aeWaaOWdaeaajugib8qacaaIXaGaeyOeI0IaamyzaKqba+aadaahaa WcbeqaaKqzadWdbiabgkHiTiabeI7aXbaaaOGaayjkaiaawMcaaKqz GeGaaiiOaaGccaGLBbGaayzxaaaapaqaaKqba+qadaqadaGcpaqaaK qzGeWdbiaacIcacqaH4oqCl8aadaahaaqabeaajugWa8qacaaIYaaa aKqzGeGaey4kaSIaeqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaa qcfa4aaeWaaOWdaeaajugib8qacaaIXaGaeyOeI0IaamiDaiabgkHi TiaadshacaWGLbWcpaWaaWbaaeqabaqcLbmapeGaeyOeI0IaeqiUde haaaGccaGLOaGaayzkaaqcfa4damaaCaaaleqabaqcLbmapeGaaGOm aaaaaaaaaa@9D88@                                      (3.4)

Factorial moment recurrence relation of ZTDS distribution may be obtained as

η [ r ] ' = e θ ( 1 e θ ) 2 [ 2( 1 e θ )r e θ η [ r1 ] ' r( r1 ) e θ η [ r2 ] ' ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabeE7aOTWdamaaDaaabaWdbmaadmaapaqaaKqzadWd biaadkhaaSGaay5waiaaw2faaaWdaeaajugWa8qacaGGNaaaaKqzGe Gaeyypa0tcfa4aaSaaaOWdaeaajugib8qacaWGLbWcpaWaaWbaaOqa beaajugWa8qacqGHsislcqaH4oqCaaaak8aabaqcfa4dbmaabmaak8 aabaqcLbsapeGaaGymaiabgkHiTiaadwgal8aadaahaaqabeaajugW a8qacqGHsislcqaH4oqCaaaakiaawIcacaGLPaaajuaGpaWaaWbaaS qabeaajugWa8qacaaIYaaaaaaajuaGdaWadaGcpaqaaKqzGeWdbiaa ikdajuaGdaqadaGcpaqaaKqzGeWdbiaaigdacqGHsislcaWGLbqcfa 4damaaCaaaleqabaqcLbmapeGaeyOeI0IaeqiUdehaaaGccaGLOaGa ayzkaaqcLbsacaWGYbGaeyOeI0IaamyzaSWdamaaCaaabeqaaKqzad WdbiabgkHiTiabeI7aXbaajugibiabeE7aOTWdamaaDaaabaWdbmaa dmaapaqaaKqzadWdbiaadkhacqGHsislcaaIXaaaliaawUfacaGLDb aaa8aabaqcLbmapeGaai4jaaaajugibiabgkHiTiaadkhajuaGdaqa daGcpaqaaKqzGeWdbiaadkhacqGHsislcaaIXaaakiaawIcacaGLPa aajugibiabgkHiTiaadwgal8aadaahaaqabeaajugWa8qacqGHsisl cqaH4oqCaaqcLbsacqaH3oaAl8aadaqhaaqaa8qadaWadaWdaeaaju gWa8qacaWGYbGaeyOeI0IaaGOmaaWccaGLBbGaayzxaaaapaqaaKqz adWdbiaacEcaaaaakiaawUfacaGLDbaaaaa@8C2B@ , r2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadkhacqGHLjYScaaIYaaaaa@3A35@                     (3.5)

where η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabeE7aObaa@3868@

η [ 1 ] ' =  [ ( θ 2 +1 )( 1 e θ )+θ ] ( θ 2 +θ+1 ) ( 1 e θ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabeE7aOTWdamaaDaaakeaal8qadaWadaGcpaqaaKqz adWdbiaaigdaaOGaay5waiaaw2faaaWdaeaajugWa8qacaGGNaaaaK qzGeGaeyypa0tcfa4aaSaaaOWdaeaajugib8qacaGGGcqcfa4aamWa aOWdaeaajuaGpeWaaeWaaOWdaeaajugib8qacqaH4oqCl8aadaahaa GcbeqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcaaIXaaakiaawIca caGLPaaajuaGdaqadaGcpaqaaKqzGeWdbiaaigdacqGHsislcaWGLb qcfa4damaaCaaakeqabaqcLbmapeGaeyOeI0IaeqiUdehaaaGccaGL OaGaayzkaaqcLbsacqGHRaWkcqaH4oqCaOGaay5waiaaw2faaaWdae aajuaGpeWaaeWaaOWdaeaajugib8qacqaH4oqCl8aadaahaaGcbeqa aKqzadWdbiaaikdaaaqcLbsacqGHRaWkcqaH4oqCcqGHRaWkcaaIXa aakiaawIcacaGLPaaajuaGdaqadaGcpaqaaKqzGeWdbiaaigdacqGH sislcaWGLbWcpaWaaWbaaOqabeaajugWa8qacqGHsislcqaH4oqCaa aakiaawIcacaGLPaaajuaGpaWaaWbaaOqabeaajugWa8qacaaIYaaa aaaaaaa@73CE@

η [ 2 ] ' =  2 e θ [ ( θ 2 +1 )( 1 e θ )+2θ ] ( θ 2 +θ+1 ) ( 1 e θ ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabeE7aOTWdamaaDaaabaWdbmaadmaapaqaaKqzadWd biaaikdaaSGaay5waiaaw2faaaWdaeaajugWa8qacaGGNaaaaKqzGe Gaeyypa0tcfa4aaSaaaOWdaeaajugib8qacaGGGcGaaGOmaiaadwga l8aadaahaaqabeaajugWa8qacqGHsislcqaH4oqCaaqcfa4aamWaaO WdaeaajuaGpeWaaeWaaOWdaeaajugib8qacqaH4oqCjuaGpaWaaWba aSqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaaGymaaGccaGLOa Gaayzkaaqcfa4aaeWaaOWdaeaajugib8qacaaIXaGaeyOeI0Iaamyz aKqba+aadaahaaWcbeqaaKqzadWdbiabgkHiTiabeI7aXbaaaOGaay jkaiaawMcaaKqzGeGaey4kaSIaaGOmaiabeI7aXbGccaGLBbGaayzx aaaapaqaaKqba+qadaqadaGcpaqaaKqzGeWdbiabeI7aXTWdamaaCa aabeqaaKqzadWdbiaaikdaaaqcLbsacqGHRaWkcqaH4oqCcqGHRaWk caaIXaaakiaawIcacaGLPaaajuaGdaqadaGcpaqaaKqzGeWdbiaaig dacqGHsislcaWGLbWcpaWaaWbaaeqabaqcLbmapeGaeyOeI0IaeqiU dehaaaGccaGLOaGaayzkaaqcfa4damaaCaaaleqabaqcLbmapeGaaG 4maaaaaaaaaa@7AA4@                  (3.6)

Variance σ Z 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabeo8aZTWdamaaBaaabaqcLbmapeGaamOwaaWcpaqa baWaaWbaaeqabaqcLbmapeGaaGOmaaaaaaa@3D0D@  of ZTDS distribution may be obtained as

σ Z 2 =  [ ( θ 2 +1 ) 2 ( 1 e θ ) 2 +5( θ 2 +1 )( 1 e θ ) e θ θ+4 θ 2 e θ ] ( θ 2 +θ+1 ) 2 ( 1 e θ ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHdpWCl8aadaWgaaqaaKqzadWdbiaadQfaaSWdaeqaamaa CaaabeqaaKqzadWdbiaaikdaaaqcLbsacqGH9aqpjuaGdaWcaaGcpa qaaKqzGeWdbiaacckajuaGdaWadaGcpaqaaKqba+qadaqadaGcpaqa aKqzGeWdbiabeI7aXTWdamaaCaaabeqaaKqzadWdbiaaikdaaaqcLb sacqGHRaWkcaaIXaaakiaawIcacaGLPaaal8aadaahaaqabeaajugW a8qacaaIYaaaaKqbaoaabmaak8aabaqcLbsapeGaaGymaiabgkHiTi aadwgajuaGpaWaaWbaaSqabeaajugWa8qacqGHsislcqaH4oqCaaaa kiaawIcacaGLPaaajuaGpaWaaWbaaSqabeaajugWa8qacaaIYaaaaK qzGeGaey4kaSIaaGynaKqbaoaabmaak8aabaqcLbsapeGaeqiUdexc fa4damaaCaaaleqabaqcLbmapeGaaGOmaaaajugibiabgUcaRiaaig daaOGaayjkaiaawMcaaKqbaoaabmaak8aabaqcLbsapeGaaGymaiab gkHiTiaadwgal8aadaahaaqabeaajugWa8qacqGHsislcqaH4oqCaa aakiaawIcacaGLPaaajugibiaadwgal8aadaahaaqabeaajugWa8qa cqGHsislcqaH4oqCaaqcLbsacqaH4oqCcqGHRaWkcaaI0aGaeqiUde 3cpaWaaWbaaeqabaqcLbmapeGaaGOmaaaajugibiaadwgajuaGpaWa aWbaaSqabeaajugWa8qacqGHsislcqaH4oqCaaaakiaawUfacaGLDb aaa8aabaqcfa4dbmaabmaak8aabaqcLbsapeGaeqiUdexcfa4damaa CaaaleqabaqcLbmapeGaaGOmaaaajugibiabgUcaRiabeI7aXjabgU caRiaaigdaaOGaayjkaiaawMcaaKqba+aadaahaaWcbeqaaKqzadWd biaaikdaaaqcfa4aaeWaaOWdaeaajugib8qacaaIXaGaeyOeI0Iaam yzaKqba+aadaahaaWcbeqaaKqzadWdbiabgkHiTiabeI7aXbaaaOGa ayjkaiaawMcaaKqba+aadaahaaWcbeqaaKqzadWdbiaaisdaaaaaaa aa@A047@

Proposition 5: The general form of factorial moment may be written as

η [ r ] ' = r!  e θ( r1 ) [ ( θ 2 +1 )( 1 e θ )+θr ] ( θ 2 +θ+1 ) ( 1 e θ ) r+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabeE7aOTWdamaaDaaabaWdbmaadmaapaqaaKqzadWd biaadkhaaSGaay5waiaaw2faaaWdaeaajugWa8qacaGGNaaaaKqzGe Gaeyypa0tcfa4aaSaaaOWdaeaajugib8qacaWGYbGaaiyiaiaaccka caWGLbqcfa4damaaCaaaleqabaqcLbmapeGaeyOeI0IaeqiUde3cda qadaWdaeaajugWa8qacaWGYbGaeyOeI0IaaGymaaWccaGLOaGaayzk aaaaaKqbaoaadmaak8aabaqcfa4dbmaabmaak8aabaqcLbsapeGaeq iUdexcfa4damaaCaaaleqabaqcLbmapeGaaGOmaaaajugibiabgUca RiaaigdaaOGaayjkaiaawMcaaKqbaoaabmaak8aabaqcLbsapeGaaG ymaiabgkHiTiaadwgal8aadaahaaqabeaajugWa8qacqGHsislcqaH 4oqCaaaakiaawIcacaGLPaaajugibiabgUcaRiabeI7aXjaadkhaaO Gaay5waiaaw2faaaWdaeaajuaGpeWaaeWaaOWdaeaajugib8qacqaH 4oqCl8aadaahaaqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaeq iUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaaqcfa4aaeWaaOWdaeaa jugib8qacaaIXaGaeyOeI0IaamyzaKqba+aadaahaaWcbeqaaKqzad WdbiabgkHiTiabeI7aXbaaaOGaayjkaiaawMcaaKqba+aadaahaaWc beqaaKqzadWdbiaadkhacqGHRaWkcaaIXaaaaaaaaaa@83EA@ .                       (3.7)

Method of estimation

The parameter θ of ZTDS distribution has been estimated using Newton-Rapson iterative method, selecting appropriate initial guest value θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabeI7aXTWdamaaBaaabaqcLbmapeGaaGimaaWcpaqa baaaaa@3ABF@  for θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH4oqCaaa@385B@ , where the function of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH4oqCaaa@385B@  may be written as

f( θ )=1 e θ θ e θ θ 2 +θ+1 f o MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadAgajuaGdaqadaGcpaqaaKqzGeWdbiabeI7aXbGc caGLOaGaayzkaaqcLbsacqGH9aqpcaaIXaGaeyOeI0IaamyzaSWdam aaCaaabeqaaKqzadWdbiabgkHiTiabeI7aXbaajugibiabgkHiTKqb aoaalaaak8aabaqcLbsapeGaeqiUdeNaamyzaSWdamaaCaaabeqaaK qzadWdbiabgkHiTiabeI7aXbaaaOWdaeaajugib8qacqaH4oqCjuaG paWaaWbaaSqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaeqiUde Naey4kaSIaaGymaaaacqGHsislcaWGMbWcpaWaaSbaaeaajugWa8qa caWGVbaal8aabeaaaaa@5C5E@ ,

f / ( θ )= e θ + e θ ( θ 3 +2 θ 2 +θ1 ) ( θ 2 +θ+1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadAgal8aadaahaaqabeaajugWa8qacaGGVaaaaKqb aoaabmaak8aabaqcLbsapeGaeqiUdehakiaawIcacaGLPaaajugibi abg2da9iaadwgal8aadaahaaqabeaajugWa8qacqGHsislcqaH4oqC aaqcLbsacqGHRaWkjuaGdaWcaaGcpaqaaKqzGeWdbiaadwgal8aada ahaaqabeaajugWa8qacqGHsislcqaH4oqCaaqcfa4aaeWaaOWdaeaa jugib8qacqaH4oqCl8aadaahaaqabeaajugWa8qacaaIZaaaaKqzGe Gaey4kaSIaaGOmaiabeI7aXTWdamaaCaaajyaGbeqaaKqzadWdbiaa ikdaaaqcLbsacqGHRaWkcqaH4oqCcqGHsislcaaIXaaakiaawIcaca GLPaaaa8aabaqcfa4dbmaabmaak8aabaqcLbsapeGaeqiUde3cpaWa aWbaaeqabaqcLbmapeGaaGOmaaaajugibiabgUcaRiabeI7aXjabgU caRiaaigdaaOGaayjkaiaawMcaaSWdamaaCaaabeqaaKqzadWdbiaa ikdaaaaaaaaa@6CE7@ based on relative frequency f o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadAgal8aadaWgaaqaaKqzadWdbiaad+gaaSWdaeqa aaaa@3A2D@ .

Similarly, function of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH4oqCaaa@385B@  may be written as

f( θ )=1 e θ θ e θ θ 2 +θ+1 μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadAgajuaGdaqadaGcpaqaaKqzGeWdbiabeI7aXbGc caGLOaGaayzkaaqcLbsacqGH9aqpcaaIXaGaeyOeI0IaamyzaSWdam aaCaaabeqaaKqzadWdbiabgkHiTiabeI7aXbaajugibiabgkHiTKqb aoaalaaak8aabaqcLbsapeGaeqiUdeNaamyzaSWdamaaCaaabeqaaK qzadWdbiabgkHiTiabeI7aXbaaaOWdaeaajugib8qacqaH4oqCl8aa daahaaqabeaajugWa8qacaaIYaaaaKqzGeGaey4kaSIaeqiUdeNaey 4kaSIaaGymaaaacqGHsislcqaH8oqBaaa@5A14@ ,

f / ( θ )= e θ + e θ ( θ 3 +2 θ 2 +θ1 ) ( θ 2 +θ+1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiaadAgajuaGpaWaaWbaaSqabeaajugWa8qacaGGVaaa aKqbaoaabmaak8aabaqcLbsapeGaeqiUdehakiaawIcacaGLPaaaju gibiabg2da9iaadwgal8aadaahaaqabeaajugWa8qacqGHsislcqaH 4oqCaaqcLbsacqGHRaWkjuaGdaWcaaGcpaqaaKqzGeWdbiaadwgaju aGpaWaaWbaaSqabeaajugWa8qacqGHsislcqaH4oqCaaqcfa4aaeWa aOWdaeaajugib8qacqaH4oqCjuaGpaWaaWbaaSqabeaajugWa8qaca aIZaaaaKqzGeGaey4kaSIaaGOmaiabeI7aXTWdamaaCaaabeqaaKqz adWdbiaaikdaaaqcLbsacqGHRaWkcqaH4oqCcqGHsislcaaIXaaaki aawIcacaGLPaaaa8aabaqcfa4dbmaabmaak8aabaqcLbsapeGaeqiU dexcfa4damaaCaaaleqabaqcLbmapeGaaGOmaaaajugibiabgUcaRi abeI7aXjabgUcaRiaaigdaaOGaayjkaiaawMcaaSWdamaaCaaabeqa aKqzadWdbiaaikdaaaaaaaaa@6E91@ , based on mean μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsacq aH8oqBaaa@3852@

Newton- raphson iterative method

θ n+1 = θ n f( θ ) f / ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabeI7aXTWdamaaBaaabaqcLbmapeGaamOBaiabgUca RiaaigdaaSWdaeqaaKqzGeWdbiabg2da9iabeI7aXTWdamaaBaaaba qcLbmapeGaamOBaaWcpaqabaqcLbsapeGaeyOeI0scfa4aaSaaaOWd aeaajugib8qacaWGMbqcfa4aaeWaaOWdaeaajugib8qacqaH4oqCaO GaayjkaiaawMcaaaWdaeaajugib8qacaWGMbWcpaWaaWbaaeqabaqc LbmapeGaai4laaaajuaGdaqadaGcpaqaaKqzGeWdbiabeI7aXbGcca GLOaGaayzkaaaaaaaa@5327@ , n=0, 1, 2, …, where θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabeI7aXTWdamaaBaaabaqcLbmapeGaaGimaaWcpaqa baaaaa@3ABF@  is the initial guest value.

Replacing θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabeI7aXTWdamaaBaaabaqcLbmapeGaaGimaaWcpaqa baaaaa@3ABF@  by θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaayyqcLbsaqa aaaaaaaaWdbiabeI7aXTWdamaaBaaabaqcLbmapeGaaGymaaWcpaqa baaaaa@3AC0@  and repeating the process till it converse. (Balagurusamy).12

Goodness of fit

In this section, an attempt has been made to test the suitability of ZTDS distribution. Eight data sets, which are used by Shanker et. al.13 have been used for a comparative study(Tables 1-8).

No. of neonatal death

Observed No. of Mothers

Expected Frequency

ZTDS

ZTP

ZTPL

1

409

399.7

399.7

408.1

2

88

102.3

102.3

89.4

3

19

17.5

17.5

19.3

4

5

2.2

2.2

4.1

5

1

0.3

0.3

1.1

Total

522

522

522.2

522

Estimate θ

1.7914

0.512047

4.199697

X2

0.181

3.464

0.145

d.f.

2

1

2

p- value

0.9137

0.0627

0.9301

Table 1 Number of mothers in rural area having at least one live birth and neonatal death

No. of Neonatal Death

Observed No. of Mothers

Expected Frequency

ZTDS

ZTP

ZTPL

1

71

71

66.5

72.3

2

32

29.43

35.1

28.4

3

7

12.3

10.9

10.9

4

5

4.11

3.3

4.1

5

3

2.2

0.8

2.2

Total

118

118

118

118

Estimate θ

1.2053

1.055102

2.049609

x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEaK qbaoaaCaaabeqaaKqzadGaaGOmaaaaaaa@3A11@

2.289

0.696

2.274

d.f.

3

1

2

p- value

0.5147

0.4041

0.3208

Table 2 The number of estate area having at least one live birth and one neonatal death

No. of Neonatal Death

Observed No. of Mothers

Expected Frequency

ZTDS

ZTP

ZTPL

1

176

176

164.3

171.6

2

44

50.13

61.2

51.3

3

16

13.35

15.2

15

4

6

3.41

2.8

4.3

5

2

1.11

0.5

1.7

Total

244

244

244

244

Estimate θ

15,499

0.744522

2.209422

x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEaK qbaoaaCaaabeqaaKqzadGaaGOmaaaaaaa@3A11@

2.852

7.301

1.882

d.f.

2

1

2

Table 3 Number of mothers in urban area with at least two live births by the number of infant and child deaths

No. of Neonatal death

Observed No. of Mothers

Expected Frequency

ZTDS

ZTP

ZTPL

1

745

744.97

708.9

738.1

2

212

215.02

255.1

214.8

3

50

58.01

61.2

61.3

4

21

15

11

17.2

5

7

3.77

1.6

4.8

6

3

1.33

0.2

1.8

Total

1038

1,038

1038

1038

Estimate θ

1.5376

0.719783

3.007722

x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEaK qbaoaaCaaabeqaaKqzadGaaGOmaaaaaaa@3A11@

8.256

37.046

4.773

d.f.

4

2

3

p-value

0.0826

0

0.1892

Table 4 Number of mothers in rural area with at least two live birth by the numbers of infant and child deaths

No. of Neonatal death

Observed No. of Mothers

Expected Frequency

ZTDS

ZTP

ZTPL

1

683

683.04

659

674.4

2

145

150.81

177.4

154.1

3

29

31.36

31.8

34.6

4

11

6.27

4.3

7.7

5

5

1.22

2.2

Total

873

873

873

873

Estimate θ

2

0.538402

4.00231

x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEaK qbaoaaCaaabeqaaKqzadGaaGOmaaaaaaa@3A11@

10.022

8.718

5.31

d.f.

3

1

2

p- value

0.0184

0.0031

0.0703

Table 5 Number of literate mothers with at least one live birth by the number of infant deaths

No. of neonatal death

Observed No. Of mothers

Expected Frequency

ZTDS

ZTP

ZTPL

1

89

89

76.8

83.4

2

25

31.26

39.9

32.3

3

11

10.2

13.8

12.2

4

6

3.18

3.6

4.5

5

3

0.96

0.7

1.6

6

1

0.4

0.2

0.9

Total

135

135

135

135

Estimate θ

1.3568

1.038289

2.089084

x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEaK qbaoaaCaaabeqaaKqzadGaaGOmaaaaaaa@3A11@

3.912

7.90

3.428

d.f.

2

1

2

p- value

3.912

7.9

3.428

Table 6 Number of mothers having experienced at least one child death

No. of Neonatal Death

Observed No. of Mothers

Expected Frequency

ZTDS

ZTP

ZTPL

1

567

567.04

545.8

561.4

2

135

138.37

162.5

139.7

3

28

31.71

32.3

34.2

4

11

6.98

4.8

8.2

5

5

1.9

0.6

2.6

Total

746

746

746

746

Estimate θ

2

0.595415

3.625737

6.227

26.855

3.839

d.f.

3

2

2

p-value

0.1012

0

0.1467

Table 7 Number of mothers having at least one neonatal death

No. of Neonatal Death

Observed No. of Mothers

Expected Frequency

ZTDS

ZTP

ZTPL

1

38

38

28.7

36.1

2

17

21.32

25.7

20.5

3

10

10.9

15.3

11.2

4

9

5.28

6.9

3.1

5

3

2.47

2.5

1.6

6

2

1.13

0.7

0.8

7

1

1

0.2

0.8

8

0

0.39

0.1

Total

80

80

80

80

Estimate θ

0.9316

1.791615

1.185582

x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEaK qbaoaaCaaabeqaaKqzadGaaGOmaaaaaaa@3A11@

3.753

9.827

2.467

d.f.

3

2

3

p- value

3

2

3

Table 8 Number of European red mites on apple leaves, reported by German

Conclusion

The discrete Shanker distribution has been introduced by discretizing the continuous Shanker distribution. Zero- truncated discrete Shanker (ZTDS) distribution have also been investigated. The parameter of the distribution has been estimated using Newton – Raphson iterative method. The application of ZTDS distribution to eight sets of data covering demography, biological sciences and social sciences have been studied. A comparative study has been made with ZTP and ZTPL distributions of Shanker et al.13 It is observed that in most cases ZTPL gives much closer fits than ZTP distribution. It is also observed that ZTDS gives very closer fit to ZTPL and in some cases ZTDS gives better fit than ZTPL distribution.14-18

Acknowledgments

None.

Conflicts of interest

None.

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