Submit manuscript...
eISSN: 2378-315X

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

Download PDF

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)S(x) , survival function; r(x)r(x) , failure hazard rate; r(x)r(x) reversed failure rate; fz(x;θ):fz(x;θ): pmf of DS distribution; fD(x;θ)fD(x;θ) , pmf of ZTDS distribution; η'[r]η'[r] : rthrth factorial moment of ZTDS distribution; μ'[r]μ'[r] : rthrth raw moment of DS distribution; PrPr rthrth Probability of DS distribution; PzrPzr rthrth 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)= θ2θ2+1 (θ+x) eθx.x>0. θ>0.f(x:θ)= θ2θ2+1 (θ+x) eθx.x>0. θ>0.                                         (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)=xf(x:θ)dxS(x)=xf(x:θ)dx

= θ2+1+ θxθ2+1 eθx,x>0.θ>0.= θ2+1+ θxθ2+1 eθx,x>0.θ>0.                                                 (2.2)

S(x+1)= θ2+1+ θ(x+1)θ2+1 eθ(x+1),x>0.θ>0.S(x+1)= θ2+1+ θ(x+1)θ2+1 eθ(x+1),x>0.θ>0.                       (2.3)

Where f(x;θ)f(x;θ) denotes the pdf of Shanker distribution.

The pmf of discrete Shanker distribution fD(x;θ)fD(x;θ)  may be obtained as

fD(x;θ)=S(x)S(x+1)fD(x;θ)=S(x)S(x+1)

= (θ2+1+ θx)(1 eθ)θ eθθ2+1 eθx,x=0,1,2,3....(θ2+1+ θx)(1 eθ)θ eθθ2+1 eθx,x=0,1,2,3....                      (2.4)

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

GD(t)=(θ2+1)(1 eθ)θ(θ2+1)(1 eθt)+θ(1 eθ)(θ2+1)(1eθt)2GD(t)=(θ2+1)(1 eθ)θ(θ2+1)(1 eθt)+θ(1 eθ)(θ2+1)(1eθt)2

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

F(x)=(θ2+1)(θ2+1+θ (x+1))eθ(x+1)(θ2+1)F(x)=(θ2+1)(θ2+1+θ (x+1))eθ(x+1)(θ2+1)

The survival function of DS distribution has obtained as

SD(x)=(θ2+1+θ (x+1))eθ(x+1)(θ2+1)SD(x)=(θ2+1+θ (x+1))eθ(x+1)(θ2+1)

The failure hazard rate may be obtained as

rD(x)=(θ2+1+ θx)(1 eθ)θ eθ(θ2+1+θ x)rD(x)=(θ2+1+ θx)(1 eθ)θ eθ(θ2+1+θ x)

The reversed failure rate

r*(x)=[(θ2+1+ θx)(1 eθ)θ eθ]eθx(θ2+1)(θ2+1+θ (x+1))eθ(x+1)r(x)=[(θ2+1+ θx)(1 eθ)θ eθ]eθx(θ2+1)(θ2+1+θ (x+1))eθ(x+1)

The second rate of failure is obtained as

r*(x)=log[s(x)s(x+1)]r(x)=log[s(x)s(x+1)]

=log[(θ2+1+θ (x+1))eθ(θ2+1+θ (x+2))]=log[(θ2+1+θ (x+1))eθ(θ2+1+θ (x+2))]

The Proportions of probabilities is given by

fD(x+1;θ)fD(x;θ)=eθ[1+θ(1 eθ)(θ2+1+ θx)(1 eθ)θ eθ]fD(x+1;θ)fD(x;θ)=eθ[1+θ(1 eθ)(θ2+1+ θx)(1 eθ)θ eθ]

Probability recurrence relation:

 Probability recurrence relation of DS distribution may be obtained as

Pr+2= eθ(2Pr+1 eθPr)r=1,2,3,Pr+2= eθ(2Pr+1 eθPr)r=1,2,3,                                                                                (2.5)

Where P0=(θ2+1)(1 eθ)θ eθθ2+1P0=(θ2+1)(1 eθ)θ eθθ2+1 , and

P1=(θ2+1+ θ)(1 eθ)θ eθθ2+1 eθP1=(θ2+1+ θ)(1 eθ)θ eθθ2+1 eθ                                                                   (2.6)

 Here PrPr  denotes Pr(X= r).

Factorial moment recurrence relation

Factorial moment generating function (fmgf) may be obtained as

MD(t)=MD(t)= (θ2+1)(1 eθ)θ(θ2+1)(1eθ eθt)+θ(1 eθ)(θ2+1)(1eθ eθt)2(θ2+1)(1 eθ)θ(θ2+1)(1eθ eθt)+θ(1 eθ)(θ2+1)(1eθ eθt)2 .                                                     (2.7)

First four factorial moments may be obtained as

μ'[1]= eθ[(θ2+1)(1eθ)+θ](θ2+1)(1eθ)2

μ'[2]=2 e2θ[(θ2+1)(1eθ)+2θ](θ2+1)(1eθ)3

μ'[3]=6 e3θ[(θ2+1)(1eθ)+3θ](θ2+1)(1eθ)4

μ'[4]=24 e4θ[(θ2+1)(1eθ)+4θ](θ2+1)(1eθ)4

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

μ'[r]=r! eθr[(θ2+1)(1eθ)+θr](θ2+1)(1eθ)r+1                                                      (2.8)

Hence, mean and variance may be obtained as

Mean = eθ[(θ2+1)(1eθ)+θ](θ2+1)(1eθ)2 , and

Variance= eθ[(θ2+1)2(1eθ)2+(θ2+1)(1eθ)θeθθ2](θ2+1)2(1eθ)4  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 fz(x;θ) of Zero-truncated DS distribution has been derived as

fz(x;θ)=Px1P0                                                                                                                 (3.0)

Where Px denotes the pmf of discrete Shanker distribution.

Hence,fz(x;θ)=(θ2+1+ θx)(1 eθ)θ eθ(θ2+θ+1) eθ(x1),x=1, 2,3,                           (3.1)

Probability recurrence relation for ZTDS distribution

The pgf Gz(t) of zero-truncated DS distribution may be obtained as

Gz(t)=x=1txfz(x;θ),

=t[{(θ2+1)(1eθ)θeθ}(1teθ)+θ(1eθ) ]((θ2+θ+1)(1teθ)2                                                      (3.2)

Probability recurrence relation ZTDS distribution may obtained as

Pzr=eθ[2Pzr1eθPzr2]r=2,3,4,.

Where Pz1=(θ2+1+ θ)(1 eθ)θ eθ(θ2+θ+1) and Pz2=(θ2+1+2θ)(1 eθ)θ eθ(θ2+θ+1) eθ                 (3.3)

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

Fz(x)=(θ2+θ+1)(θ2+θ+θx+1)eθx(θ2+θ+1)

The survival function of ZTDS distribution is given by

Sz(x)=(θ2+θ+θx+1))eθx(θ2+θ+1)

The Failure hazared rate may be obtained as

rz(x)=P(X=x)P(Xx1) ,

=(θ2+1+ θx)(1 eθ)θ eθ(θ2+1+θx) .

The reversed failure rate

r*z(x)=P(X=x)P(Xx)

=(θ2+1+ θx)(1 eθ)θ eθ(θ2+1+θx) .

The second rate of failure is obtained as

r*z(x)=P(X=x)P(Xx)

=[(θ2+1+ θx)(1 eθ)θ eθ]eθ(x1)(θ2+θ+1)(θ2+θ+1+θx))eθx

The proportions of probabilities is given by

fz(x+1;θ)fz(x;θ)=eθ[1+θ(1 eθ)(θ2+1+ θx)(1 eθ)θ eθ]

Factorial moment recurrence relation for ZTDS distribution 

Factorial moment generating function Mz(t)  of ZTDS distribution may be obtained as

Mz(t)=(1+t)[{(θ2+1)(1eθ)θeθ}(1tteθ)+θ(1eθ) ]((θ2+θ+1)(1tteθ)2                                      (3.4)

Factorial moment recurrence relation of ZTDS distribution may be obtained as

η'[r]=eθ(1eθ)2[2(1eθ)reθη'[r1]r(r1)eθη'[r2]] , r2                     (3.5)

where η

η'[1]= [(θ2+1)(1eθ)+θ](θ2+θ+1)(1eθ)2

η'[2]= 2eθ[(θ2+1)(1eθ)+2θ](θ2+θ+1)(1eθ)3                  (3.6)

Variance σZ2  of ZTDS distribution may be obtained as

σZ2= [(θ2+1)2(1eθ)2+5(θ2+1)(1eθ)eθθ+4θ2eθ](θ2+θ+1)2(1eθ)4

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

η'[r]=r! eθ(r1)[(θ2+1)(1eθ)+θr](θ2+θ+1)(1eθ)r+1 .                       (3.7)

Method of estimation

The parameter θ of ZTDS distribution has been estimated using Newton-Rapson iterative method, selecting appropriate initial guest value θ0  for θ , where the function of θ  may be written as

f(θ)=1eθθeθθ2+θ+1fo ,

f/(θ)=eθ+eθ(θ3+2θ2+θ1)(θ2+θ+1)2 based on relative frequency fo .

Similarly, function of θ  may be written as

f(θ)=1eθθeθθ2+θ+1μ ,

f/(θ)=eθ+eθ(θ3+2θ2+θ1)(θ2+θ+1)2 , based on mean μ

Newton- raphson iterative method

θn+1=θnf(θ)f/(θ) , n=0, 1, 2, …, where θ0  is the initial guest value.

Replacing θ0  by θ1  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

x2

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

x2

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

x2

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

x2

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

x2

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

x2

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.

References

  1. Nakagawa T, Osaki S. The discrete Weibull distribution. IEE Trans Reliab.1975;24(5):300–301.
  2. Roy D. Discrete Rayleigh distribution. IEE Trans Reliab. 2004;53(2):255–260.
  3. Ghitany ME, Atiech B, Nadarajah S. Lindley distribution and its applications. Mathematics and Computers in Simulation. 2008a;78(4):493–506.
  4. Krishna H, Pundir PS. Discrete Burr and discrete Pareto distributions. Stat Methodol. 2009;6:177–188.
  5. Gomez-Deniz. Another generalization of the geometric distribution. 2010;19(2):399–415.
  6. Marshall AW, Olkin I. A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika. 1997;84(3):641–652.
  7. Borah M, Saikia KR, Hazarika J. A study on two parameter discrete quasi- Lindley distribution and its derived distributions. International Journal of Mathematical Archive. 2015;6(12):149–156.
  8. Borah M, Saikia KR, Hazarika J. Discrete Janardan distribution and Its Applications. (Accepted for publication in the Journal of the Ethiopian Statistical Association). 2016.
  9. Borah M, Saikia KR. Certain properties of discrete Sushila distribution and its applications. International Journal of Scientific Research. 2016;5(6):490–498.
  10. Dutta P, Borah M. Zero-Modified Poisson-Lindley distribution, (Accepted for publication). 2014b.
  11. Shanker R, Fesshaye H, Selvaraj S. On Zero-Truncation of Poisson and Poisson-Lindley Distributions and Their Applications. Biom Biostat Int J. 2015;2(6):00045.
  12. Balagurusamy E. Numerical Methods, Tata Mc Graw – Hill Education Private Limited, New Delhi, India. 1999.
  13. Shanker R. Shanker distribution and its Applications, International Journal of Statistics and Applications. 2015;5(6):338–348.
  14. Ghitany ME, Al-Mutairi DK. Estimation methods for the discrete Poisson-Lindley distribution. J Stat Comput Simul. 2009;79(1):1–9.
  15. Lindley DV. Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society, Series B. 1958;20(1):102–107.
  16. Sankaran M. The discrete Poisson-Lindley distribution. Biometrics. 1970;26(1):145–149.
  17. Shanker R, Mishra. A two parameter Poisson- Lindley distribution. International journal of Statistics and Systems. 2014;9(1):79–85.
  18. Shanker R, Sharma, S, Shanker R. A discrete two-parameter Poisson-Lindley distribution. Journal of the Ethiopian Statistical Association. 2012;21:15–22.
Creative Commons Attribution License

©2017 Borah, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.