Research Article Volume 5 Issue 6
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
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
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
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.
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) e−θx.x>0. θ>0.f(x:θ)= θ2θ2+1 (θ+x) e−θx.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 e−θx,x>0.θ>0.= θ2+1+ θxθ2+1 e−θx,x>0.θ>0. (2.2)
S(x+1)= θ2+1+ θ(x+1)θ2+1 e−θ(x+1),x>0.θ>0.S(x+1)= θ2+1+ θ(x+1)θ2+1 e−θ(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− e−θ)−θ e−θθ2+1 e−θx,x=0,1,2,3....(θ2+1+ θx)(1− e−θ)−θ e−θθ2+1 e−θx,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− e−θ)−θ(θ2+1)(1− e−θt)+θ(1− e−θ)(θ2+1)(1−e−θt)2GD(t)=(θ2+1)(1− e−θ)−θ(θ2+1)(1− e−θt)+θ(1− e−θ)(θ2+1)(1−e−θt)2
Proposition 2: The cumulative distribution of DS distribution is given by
F(x)=(θ2+1)−(θ2+1+θ (x+1))e−θ(x+1)(θ2+1)F(x)=(θ2+1)−(θ2+1+θ (x+1))e−θ(x+1)(θ2+1)
The survival function of DS distribution has obtained as
SD(x)=(θ2+1+θ (x+1))e−θ(x+1)(θ2+1)SD(x)=(θ2+1+θ (x+1))e−θ(x+1)(θ2+1)
The failure hazard rate may be obtained as
rD(x)=(θ2+1+ θx)(1− e−θ)−θ e−θ(θ2+1+θ x)rD(x)=(θ2+1+ θx)(1− e−θ)−θ e−θ(θ2+1+θ x)
The reversed failure rate
r*(x)=[(θ2+1+ θx)(1− e−θ)−θ e−θ]e−θx(θ2+1)−(θ2+1+θ (x+1))e−θ(x+1)r∗(x)=[(θ2+1+ θx)(1− e−θ)−θ e−θ]e−θx(θ2+1)−(θ2+1+θ (x+1))e−θ(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))e−θ(θ2+1+θ (x+2))]=log[(θ2+1+θ (x+1))e−θ(θ2+1+θ (x+2))]
The Proportions of probabilities is given by
fD(x+1;θ)fD(x;θ)=e−θ[1+θ(1− e−θ)(θ2+1+ θx)(1− e−θ)−θ e−θ]fD(x+1;θ)fD(x;θ)=e−θ[1+θ(1− e−θ)(θ2+1+ θx)(1− e−θ)−θ e−θ]
Probability recurrence relation:
Probability recurrence relation of DS distribution may be obtained as
Pr+2= e−θ(2Pr+1− e−θPr)r=1,2,3,…Pr+2= e−θ(2Pr+1− e−θPr)r=1,2,3,… (2.5)
Where P0=(θ2+1)(1− e−θ)−θ e−θθ2+1P0=(θ2+1)(1− e−θ)−θ e−θθ2+1 , and
P1=(θ2+1+ θ)(1− e−θ)−θ e−θθ2+1 e−θP1=(θ2+1+ θ)(1− e−θ)−θ e−θθ2+1 e−θ (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− e−θ)−θ(θ2+1)(1−e−θ− e−θt)+θ(1− e−θ)(θ2+1)(1−e−θ− e−θt)2(θ2+1)(1− e−θ)−θ(θ2+1)(1−e−θ− e−θt)+θ(1− e−θ)(θ2+1)(1−e−θ− e−θt)2 . (2.7)
First four factorial moments may be obtained as
μ'[1]= e−θ[(θ2+1)(1−e−θ)+θ](θ2+1)(1−e−θ)2
μ'[2]=2 e−2θ[(θ2+1)(1−e−θ)+2θ](θ2+1)(1−e−θ)3
μ'[3]=6 e−3θ[(θ2+1)(1−e−θ)+3θ](θ2+1)(1−e−θ)4
μ'[4]=24 e−4θ[(θ2+1)(1−e−θ)+4θ](θ2+1)(1−e−θ)4
Proposition 3: The general form of factorial moment may also be written as
μ'[r]=r! e−θr[(θ2+1)(1−e−θ)+θr](θ2+1)(1−e−θ)r+1 (2.8)
Hence, mean and variance may be obtained as
Mean = e−θ[(θ2+1)(1−e−θ)+θ](θ2+1)(1−e−θ)2 , and
Variance= e−θ[(θ2+1)2(1−e−θ)2+(θ2+1)(1−e−θ)θ−e−θθ2](θ2+1)2(1−e−θ)4 respectively.
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)=Px1−P0 (3.0)
Where Px denotes the pmf of discrete Shanker distribution.
Hence,fz(x;θ)=(θ2+1+ θx)(1− e−θ)−θ e−θ(θ2+θ+1) e−θ(x−1),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)(1−e−θ)−θe−θ}(1−te−θ)+θ(1−e−θ) ]((θ2+θ+1)(1−te−θ)2 (3.2)
Probability recurrence relation ZTDS distribution may obtained as
Pzr=e−θ[2Pzr−1−e−θPzr−2]r=2,3,4,….
Where
Pz1=(θ2+1+ θ)(1− e−θ)−θ e−θ(θ2+θ+1)
and
Pz2=(θ2+1+2θ)(1− e−θ)−θ e−θ(θ2+θ+1) e−θ
(3.3)
Proposition 4: The cumulative distribution of ZTDS distribution is given by
Fz(x)=(θ2+θ+1)−(θ2+θ+θx+1)e−θx(θ2+θ+1)
The survival function of ZTDS distribution is given by
Sz(x)=(θ2+θ+θx+1))e−θx(θ2+θ+1)
The Failure hazared rate may be obtained as
rz(x)=P(X=x)P(X≥x−1) ,
=(θ2+1+ θx)(1− e−θ)−θ e−θ(θ2+1+θx) .
The reversed failure rate
r*z(x)=P(X=x)P(X≤x)
=(θ2+1+ θx)(1− e−θ)−θ e−θ(θ2+1+θx) .
The second rate of failure is obtained as
r*z(x)=P(X=x)P(X≤x)
=[(θ2+1+ θx)(1− e−θ)−θ e−θ]e−θ(x−1)(θ2+θ+1)−(θ2+θ+1+θx))e−θx
The proportions of probabilities is given by
fz(x+1;θ)fz(x;θ)=e−θ[1+θ(1− e−θ)(θ2+1+ θx)(1− e−θ)−θ e−θ]
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)(1−e−θ)−θe−θ}(1−t−te−θ)+θ(1−e−θ) ]((θ2+θ+1)(1−t−te−θ)2 (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−θη'[r−1]−r(r−1)−e−θη'[r−2]] , r≥2 (3.5)
where η
η'[1]= [(θ2+1)(1−e−θ)+θ](θ2+θ+1)(1−e−θ)2
η'[2]= 2e−θ[(θ2+1)(1−e−θ)+2θ](θ2+θ+1)(1−e−θ)3 (3.6)
Variance σZ2 of ZTDS distribution may be obtained as
σZ2= [(θ2+1)2(1−e−θ)2+5(θ2+1)(1−e−θ)e−θθ+4θ2e−θ](θ2+θ+1)2(1−e−θ)4
Proposition 5: The general form of factorial moment may be written as
η'[r]=r! e−θ(r−1)[(θ2+1)(1−e−θ)+θr](θ2+θ+1)(1−e−θ)r+1 . (3.7)
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)=1−e−θ−θe−θθ2+θ+1−fo ,
f/(θ)=e−θ+e−θ(θ3+2θ2+θ−1)(θ2+θ+1)2 based on relative frequency fo .
Similarly, function of θ may be written as
f(θ)=1−e−θ−θe−θθ2+θ+1−μ ,
f/(θ)=e−θ+e−θ(θ3+2θ2+θ−1)(θ2+θ+1)2 , based on mean μ
Newton- raphson iterative method
θn+1=θn−f(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
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
None.
None.
©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.
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