Journal of Statistical Theory and Applications

Volume 20, Issue 2, June 2021, Pages 304 - 317

New Discrete Lifetime Distribution with Applications to Count Data

Authors
Beih S. El-Desouky*, Rabab S. Gomaa, Alia M. Magar
Department of Mathematics, Faculty of Science, Mansoura University, Egypt
*Corresponding author. Email: b_desouky@yahoo.com
Corresponding Author
Beih S. El-Desouky
Received 9 August 2020, Accepted 25 January 2021, Available Online 22 February 2021.
DOI
https://doi.org/10.2991/jsta.d.210203.001How to use a DOI?
Keywords
Generalized Hermite distribution, Hermite polynomials, Genocchi polynomials, Hermite–Genocchi polynomials, Discrete distribution, Reliability
Abstract

In this paper, we present a new class of distribution called generalized Hermite–Genocchi distribution (GHGD). This model is obtained by compounding generalized Hermite–Genocchi polynomials given by Gould and Hopper with powers series distribution. Statistical properties and reliability characteristics are studied. The model has been applied to several real data. Finally, a simulation study is performed to assess the performance of the model.

Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

In this paper, we introduced a new discrete distribution based on the generalized Hermite polynomials given by, see [1]

Hn,m(x,y)=n!k=0[nm]ykxnmkk!(nmk)!,  (n0,m).

For more details, see [24].

Gupta and Jain [5] extended the Hermite distribution (HD) of the generalized HD defined by

P(Y=n)=e(a+b)j=0[nm]a(nmj)bj(nmj)!j!,
where a0,b0 and m.

The distribution has been applied to the frequency of bacteria in leucoytes and frequency of larvae in corn plants [6].

Moreover, there are a lot of popular statistical distributions that have specific applications, but sometimes, observable data contain distinct features not shown by these classic distributions. So to overcome these limitations, researchers often develop new distributions so that these new distributions can be used in these cases where the classical distributions don't provide any suitable fit. There are many techniques with which we can get new distribution, for more details see [79].

Recently, El-Desouky et al. [10] introduced a new generalized Hermite–Genocchi distribution (GHGD). By compounding (1) and powers series distribution defined new multivariate distribution called GHGD.

P(X_)=Bi=1r(αi1)xiHn,m(i=1rxi+γ,β).(1.1)
β,γ0;r1,m,
where
1B=HMn(r)(γ,β;α¯r)=1,2,,r=0i=1r(αi1)iHn,mi=1ri+γ,β,

HMn(r)(γ,β;α¯r) is convergent and positive for α¯r=(α0,α1,,αr1),0<αr<1.

The paper is organized as follows: In Section 2, when set r=1 in (1.1), we introduce a new univariate discrete distribution and discuss mathematical and statistical properties of the model. In Section 3, we introduce monotonic properties. In Section 4, reliability characteristics are obtained. In Section 5, moment and maximum likelihood estimates of unknown parameters are presented and simulation study is performed. In Section 6, we apply the new model to real data sets to illustrate the usefulness and applicability of the model. Graphical assesment of goodness of fit of the model based on empirical probability generating function is presented. Finally, in Section 7, conclusion and remarks are given.

2. GENERALIZED HERMITE–GENOCCHI DISTRIBUTION

Definition 2.1.

A discrete random variable X taking value in the set +{0} is said to follow GHGD with three parameters, that is GHG(α;β,γ), if its probability mass function can be written as

P(X=x;α,β,γ)={Hn,m(γ,β)G(α;β,γ),x=0αxHn,m(x+γ,β)G(α;β,γ),x>0,(2.1)
where β0 is scale parameter, γ0 is shape parameter, 0<α<1 is shape parameter, m,
G(α;β,γ)==0αHn,m(+γ,β),
and
Hn,m(x+γ,β)=k=0[nm]βkk!n!(nmk)!(x+γ)nmk.

G(α;β,γ) is convergent and positive for 0<α<1.

2.1. Structural Properties of GHGD Model

2.1.1. Shape and behavior of pmf plots of GHG distribution with serval values of parameters α,β and γ are present in Figure 1

Three examples in Figure 1 showing effects of scale and shape parameters.

Figure 1

Shape and behavior of pmf plots of GHGD with serval values of parameters α,β and γ.

2.1.2. Cumulative distribution function

The cumulative distribution function (cdf) of GHGD is given by

F(x)=P(Xx)=1P(X>x)=1αx+1G(α;β,x+γ+1)G(α;β,γ).

Figure 2 showing shape and behavior of Cdf plots of GHG distribution with several values of parameters α, β and γ.

Figure 2

Cdf of GHGD for different values of α,β and γ.

2.1.3. Moments and related measures

The moment-generating function of GHGD is given by

MX(t)=E(etx)=G(αet;β,γ)G(α;β,γ).

The rth factorial moments μ[r] is given by

μ[r]=E[(X)r]=x=0(x)r_αxHn,m(x+γ,β)G(α;β,γ).

The rth moments μr is given by

μr=E(Xr)=x=0xrαxHn,m(x+γ,β)G(α;β,γ).

The mean and variance are given, respectively, by

E(X)=x=0xαx  Hn,m(x+γ,β)G(α;β,γ).Var(X)=[x=0x2αx  Hn,m(x+γ,β)G(α;β,γ)][x=0xαx  Hn,m(x+γ,β)G(α;β,γ)]2.

The plots in Figure 3, it is apparent that both mean and variance of GHGD have bounds.

Figure 3

Plots the mean and variance of GHGD with serval values of parameters α,β and γ.

2.1.4. Over-dispersion

The over-dispersion (OD) index of GHGD is given by

OD=σ2μ=x=0x2αxHn,m(x+γ,β)x=0xαxHn,m(x+γ,β)x=0xαxHn,m(x+γ,β)G(α;β,γ).(2.2)

From Figure 3 and Eq. (2.2), we can obtain the following corollary:

Corollary 2.2.

  1. OD =(>)(<)1 if and only if α=(>)(<)0.4, β=(>)(<)0.3 and γ=(>)(<)1.

  2. GHGD is no over-dispersion, over-dispersion and under-dispersion for α=(>)(<)0.4, β=(>)(<)0.3 and γ=(>)(<)1, respectively.

We obtained that numerically.

2.1.5. Surprise index

The surprise index (SI) of GHGD is given by

SI=E(P(X=x,α;β,γ))P(X=x,α;β,γ)=(x=0α2x(Hn,m(x+γ,β))2G(α;β,γ))/(αxHn,m(x+γ,β)).

From Figure 4 for various value of α,βandγ, where α,βandγ, decreases, large values of x become more surprising.

Figure 4

log(SI)'s for GHGD.

2.1.6. Generating function

The probability-generating function of GHGD is given by

GX(t)=E(tX)=x=0P(X=x)tx=G(αt;β,γ)G(α;β,γ).

3. MONOTONIC PROPERTIES

Log-concavity is an essential property of the probability distribution. Characteristics such as reliability function, failure rate, mean residual and moment of log-concave probability have specific properties see [1114].

Theorem 3.1.

The GHG distribution is log-concave.

Proof.

Consider the function

b(x,α,β,γ)=P(X=x+1)P(X=x)=αHn,m(x+γ+1,β)Hn,m(x+γ,β).

Its derivative is given by

db(x,α,β,γ)dx=α[nm]k=0βkn!(x+γ+1)nmk1k!(nmk1)!(Hn,m(x+γ,β))                           αHn,m(x+γ+1,β)[nm]k=0βkn!(x+γ)nmk1k!(nmk1)!(Hn,m(x+γ,β))2<0.

Note that b(x,α,β,γ) is decreasing function in x for 0<α<1, β>0 and γ>0 thus, the G(α,β,γ) is log-concave. The behavior of GHG distribution can be illustrated as in Figure 1.

Corollary 3.2.

As a direct consequence of log-concavity, see [11], the following results hold for GHG distribution:

  1. It is strongly unimodal.

  2. It has all moments.

  3. It has an increasing failure rate distribution.

  4. It has monotonically decreasing mean residual function.

  5. It remains log-concave if truncated.

  6. It gives unimodal and log-concave distribution when convoluted with any other discrete distribution.

4. RELIABILITY PROPERTIES

The survival function of GHGD is given by

S(x)=P(X>x)=1P(Xx)=αx+1G(α;β,x+γ+1)G(α;β,γ).(4.1)

In Figure 5, shape and behaviour of survival function plots of GHG distribution with several values of parameters α, β and γ.

Figure 5

Survival function for GHGD.

Also, the hazard rate function is given by

h(x)=P(X=x)S(x)=Hn,m(x+γ,β)αG(α;β,γ+x+1).(4.2)

The failure rate is increasing, see (Theorem 3.1) and (Corollary 3.2) and Figure 6.

Figure 6

Hazard function for GHGD.

The mean residual life (MRL) of the GHGD is given by

μ(x)=E(Xx|Xx)=1S(x1)y=xS(y1)=y=xαyG(α;β,γ+y)αxG(α;β,γ+x).

The mean time to failure (MTTF) of GHGD is given by

μ=x=0S(x)=x=0αx+1G(α;β,x+γ+1)G(α;β,γ).

The reversed hazard rate is given by

h(x)=P(X=t)P(Xt)=αxHn,m(γ+x,β)G(α;β,γ)αx+1G(α;β,γ+x+1).

The shape and behavior of reversed hazard rate GHG distribution with several values of parameters α, β and γ, see Figure 7.

Figure 7

Reversed hazard function for GHGD.

Definition 4.1. [13]

A discrete distribution of nonnegative random variable is said to be

  • New better (worse) than used, denoted by NBU(NWU) if

    S(x+y)()S(x)S(y).

  • New better (worse) than used in expectation, denoted by NBUE(NWUE) if

    j=0S(t+j)()j=0S(j)S(t).

Corollary 4.2.

As a result of IFR, see [13], the following results hold:

  1. GHGD is IFRA

  2. GHGD is NBU

  3. GHGD is DMRL

  4. GHGD is NBUE.

5. PARAMETER ESTIMATION AND SIMULATION

5.1. Maximum Likelihood Estimators

Let x=(x1,x2,,xn) be a random sample of size n drawn from GHGD. Then, the likelihood function of vector (α,β,γ) is given by

L(α,β,γ)=i=1NαxiHn,m(xi+γ,β)G(α;β,γ)=(1G(α;β,γ))Nαi=1Nxii=1NHn,m(xi+γ,β).

The log-likelihood function can be written as

logL=Nlog(=0αk=0[nm]βkn!k!(nmk)!(+γ)nmk)+i=1Nxilogα+i=1Nlog(k=0[nm]βkn!k!(nmk)!(xi+γ)nmk).(5.1)

Computing the first partial derivatives of (5.1) with respect to α,β and γ, we get

αlogL=NG(α;β,γ)=0α1Hn,m(+γ,β)+i=1Nxiα=i=1NxiαNαE(X).(5.2)
βlogL=NG(α;β,γ)=0αk=0[nm]kβk1n!k!(nmk)!(+γ)nmk+i=1N1Hn,m(xi+γ,β)k=0[nm]kβk1n!k!(nmk)!(xi+γ)nmk.(5.3)
γlogL=NG(α;β,γ)=0αk=0[nm]βkn!k!(nmk)!(nmk)(+γ)nmk1+i=1N1Hn,m(xi+γ,β)k=0[nm]βkn!(nmk)k!(nmk)!(xi+γ)nmk1.(5.4)

Equating the Equations (5.25.4) to zero and solving them with the help of R software, the MLES can be obtained. We notice that, these equations cannot solve analytically, there is an alternative procedure like Newton-Raphson is required to solve them numerically.

5.2. Simulation

In this section, we evaluate MLE performance to sample n. Evaluation based on simulation study described in the following steps:

  1. Generate 1000 samples with n=50,100,500,800 and 1000 from GHGD.

  2. Calculate MLES for 1000 sampls.

  3. Calculating absolute bias, standard errors and mean square errors (MSE).

The results obtained in Table 1.

n Parameter MLE Standard Error Abs. Bias MSE
50 α=0.2 0.2187 0.011 0.0187 0.0005
β=0.05 0.0809 0.057 0.0309 0.0038
γ=0.3 0.2462 0.1434 0.0538 0.0212
100 α=0.2 0.2118 0.0137 0.0118 0.0003
β=0.05 0.0573 0.018 0.0073 0.0003
γ=0.3 0.3169 0.123 0.0169 0.0124
500 α=0.2 0.2086 0.0063 0.0086 0.0001
β=0.05 0.0447 0.0069 0.0053 0.00007
γ=0.3 0.2947 0.0414 0.0053 0.0017
800 α=0.2 0.2079 0.0059 0.0079 0.00009
β=0.05 0.0455 0.004 0.0045 0.00004
γ=0.3 0.3051 0.0408 0.0051 0.0016
1000 α=0.2 0.2065 0.0056 0.0065 0.00007
β=0.05 0.0456 0.0045 0.0044 0.000039
γ=0.3 0.3003 0.0348 0.0003 0.00121
Table 1

Result from the simulated data.

It can be seen that

  1. The bias values decrease as n.

  2. MSEs decrease as n. This shows the consistency of the estimators.

  3. The MLE method performs well for the parameters.

6. DATA ANALYSIS

In this section, we explain the empirical importance of GHGD using real data applications. The fitted model is compared using χ2 statistic, Akaike information criterion (AIC), Bayesian information criterion (BIC) and correct Akaike information criterion (AICc).

6.1. Data Set 1

This data represents counts of cysts in embryonic mouse kidneys which subjected to steroids, taken from McElduff et al. [15] and [16]. We compare the fits of GHGD with HD, zero-inflated Poisson distribution (ZIPD), negative binomial distribution (NBD), zero-inflated negative binomial distribution (ZINBD), zero-inflated generalized Poisson distribution (ZIGPD) and zero-inflated Hermite distribution (ZIHD). The MLES and goodness of fit are presented in Table 2.

Count Observed Frequency HD ZIPD NBD ZIGPD ZINBD ZIGH GHGD
0 65 17.938 60.87 20.87 85.113 63.7 65.02 64.86
1 14 24 3.5 24.6 8.64 5.5 11 14.16
2 10 24 7.5 20.973 6.307 6.02 9 7
3 6 19 9.5 15.57 5.9 6.2 8 6.52
4 4 12 9.6 10.71 4.99 5 5.5 5.36
5 2 6.77 7.8 7.02 0.05 6 2.5 3.9
6 2 3.3 5.44 4.449 4.53×107 4.2 2.52 2.77
7 2 1.48 3 2.74 2.03×1012 3.17 2.05 1.97
8 1 0.612 1.5 1.7 1.20×1019 2.27 0.9 1.4
9 1 0.5 1.02 1.25 6.76×1029 1.5 0.61 0.98
10 1 0.5 0.27 0.58 2.642×1040 3.04 0.6 0.68
11 2 0.7 0.7 0.34 5.57×1054 2.5 1.7 0.48
12 1 0.2 0.5 0.198 5.05×1070 1.9 1.6 0.92
Total 111 111 111 111 111 111 111 111
df 4 5 4 1 4 3 2
Estimates of the parameter λ=1.5 λ=4.0 λ=2.25 λ=1.05 λ=3.85 λ=1.15 α=0.203
θ=0.4 ω=0.54 θ=2.48 ω=0.56 ω=0.56 ω=0.53 β=1.79
θ=1.35 θ=4.33 θ=1.01 γ=0.182
χ2 value 154.39 27.66 117.43 34.07 22.62 2.32 1.77
P value 0.0001 0.0001 0.0001 0.0001 0.0001 0.0914 0.8804
AIC 476.238 383.634 450.82 3238.76 371.02 368.4 353.287
BIC 477.368 384.7 451.95 3240.45 372.71 370.24 361.415
AICc 473.83 384.83 448.42 3236.09 368.35 365.22 353.511
Table 2

Distribution of the counts of cysts from 111 steroid-treated kidneys [15] and the expected frequencies computed using HD, ZIPD, NBD, ZIGPD, ZINBD, ZIHD and GHGD.

From the plots of the log-likelihood function of α,β and γ in Figure 8a–8c, we observe that the likelihood equations have a unique solution.

Figure 8

The profiles of the log-likelihood function of α,β and γ.

6.2. Data Set 2

This data represents the distribution of mistakes in copying groups of random digits, see [17]. We compare the fits of GHGD with hyper-Poisson distribution (HPD), zero- inflated Poisson distribution (ZIPD), zero-inflated Conway–Maxwell–Poisson distribution (ZICMPD),ZINBD, ZIGPD and zero-inflated hyper-Poisson distribution (ZIHPD). The MLES and goodness of fit are presented in Table 3.

Count Observed Frequency HPD ZIPD ZIHPD ZICMPD ZINBD ZIGPD GHGD
0 35 24.41 41.1937 36.84 40.6 43.69 41.999 34.67
1 11 21.09 9.039 7.5 5.4 8.74 7.98 10.5
2 8 9.69 6.24 8.5 5.7 5.5 7.981 8.5
3 4 3.07 3.018 5.01 5.1 1.51 1.98 4.59
4 2 0.74 0.05093 2.05 3.2 0.56 0.06 1.74
Total 60 60 60 60 60 60 60 60
df 1 1 1 1 1 1 1
Estimates of the parameter λ=1.23 λ=1.45 λ=0.63 λ=2.3 λ=0.54 λ=2.0 α=0.0382
θ=1.02 ω=0.579 ω=0.601 ω=0.7 ω=0.2 ω=0.55 β=0.0006
θ=1.23 θ=0.7 θ=0.5 θ=0.00000035 γ=0.845
χ2 value 11.168 6.53 1.968 8.09 11.25 67.07 0.074
P value 0.0008 0.0106 0.1607 0.051 0.0008 0.0001 0.9948
AIC 224.34 181.87 169.233 206.22 206.244 301.45 149.57
BIC 221.74 179.27 165.332 205.13 205.072 300.82 155.853
AICc 223.746 181.272 170.033 224.30 230.24 325.45 149.998
Table 3

Distribution of mistakes in copying groups of random digits [17] and the expected frequencies computed using HPd, ZIPD, ZIHPD, ZICMPD, ZINBD, ZIGPD and GHGD distribution.

From the plots of the log-likelihood function of α,β and γ in Figure 9a–9c, we observe that the likelihood equations have a unique solution.

Figure 9

The profiles of the log-likelihood function of α,β and γ.

6.3. Data Set 3

This data represents counts of Collenbola microarthropods in 200 samples of forest soil, see [18,19]. We compare the fits of GHGD with (HPD), (ZIPD), (ZICMPD),(ZINBD), (ZIGPD) and (ZIHPD). The MLES and goodness of fit are presented in Table 4.

Count Observed Frequency HPD ZIPD ZIHPD ZICMPD ZINBD ZIGPD GHGD
0 122 135.133 134.46 118.5 129.6 133.79 157.33 120.09
1 40 54 28.7 36.56 40 41.2 25.75 39.85
2 14 7.31 21.1 23.24 24 17.2 9.5 18.52
3 16 1.58 11.05 14.25 5.2 5.61 5.5 13.07
4 4 1.5 3.64 5.5 0.5 1.72 1.35 5.97
5 2 0.74 1.05 1.95 0.8 0.48 0.57 2.5
Total 200 200 200 200 200 200 200 200
df 1 2 2 1 1 1 2
Estimates of the parameter λ=2.5 λ=1.45 λ=0.25 λ=3.95 λ=4.76 λ=0.73 α=0.075
θ=0.2 ω=0.578 ω=0.55 ω=0.60 ω=0.37 ω=0.65 β=0.0011
θ=1.02 θ=2.74 θ=0.81 θ=0.000127 γ=0.331
χ2 value 158.27 12.6 4.36 90.07 36.37 57.60 1.817
P value 0.0001 0.0018 0.1130 0.0001 0.0001 0.0001 0.7694
AIC 1228.03 621.6 582.99 744.7 660.2 1298.1 474.151
BIC 1227.6 621.18 582.37 744.15 659.5 1297.5 484.046
AICc 1229.3 625.6 591.9 750.77 669.2 1307.1 474.273
Table 4

Distribution of the counts of Collenbola microarthropods in 200 samples of fort soil [19] and the expected frequencies computed using HPD, ZIPD, ZIHPD, ZICMPD, ZINBD, ZIGPD and GHGD distribution.

From the plots of the log-likelihood function of α,β and γ in Figure 10a–10c, we observe that the likelihood equations have a unique solution.

Figure 10

The profiles of the log-likelihood function of α,β and γ.

6.4. Graphical Assesment of Goodness of Fit

Plotting both the empirical probability generating function (EPGF) and log pgf's on the same graph allows us to compare the fit of a number of discrete distributions using only one plot, see [20].

The log of the EPGF of data set 1 is plotted in Figure 11. The EPGF is shown as black line, whilst a series of distributions fitted to data. The GHGD pgf shown by the red line, indicates that the GHGD is a good fit to the data.

Figure 11

EPGF plot of counts of cysts from 111 steroid-treated kidneys with fitted it log pgf's for the Hermite distribution, zero-inated Poisson distribution, negative binomial distribution, zero-inated negative binomial distribution and generalized Hermite–Genocchi distribution.

The log of the EPGF of data set 2 is plotted in Figure 12. The EPGF is shown as black line, whilst a series of distributions fitted to data. The GHGD pgf shown by the red line, indicates that the GHGD is a good fit to the data.

Figure 12

EPGF plot of the distribution of mistakes in copying groups of random digits with fitted it log pgf's for the hyper-Poisson distribution, zero-inflated Poisson distribution, zero-inflated negative binomial distribution, zero-inflated hyper-Poisson distribution and generalized Hermite–Genocchi distribution.

The log of the EPGF of data set 3 is plotted in Figure 13. The EPGF is shown as black line, whilst a series of distributions fitted to data. The GHGD pgf shown by the red line, indicates that the GHGD is a good fit to the data.

Figure 13

EPGF plot of counts of Collenbola microarthropods of forest soil with fitted it log pgf's for the hyper-Poisson distribution, zero-inflated Poisson distribution, zero-inflated negative binomial distribution, zero-inflated hyper-Poisson distribution and generalized Hermite–Genocchi distribution.

7. CONCLUSION

A new three parameters discrete distribution is proposed and its important monotonic and reliability concepts are introduced. The model proposed parameters are estimated by Maximum likelihood and the simulation study is performed to establish the accuracy of the maximum likelihood estimators. Applications of the new model in the analysis of three real-life data are presented. We show by three applications of the real data that the proposed distribution can yield better fits than some other distributions.

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

AUTHORS' CONTRIBUTIONS

All authors have read and agreed to the published version of the manuscript.

ACKNOWLEDGMENTS

The author would like to thank the Editor-in-Chief, and the anonymous referees for their careful reading and constructive comments and suggestions which greatly improved the presentation of the paper.

REFERENCES

10.B.S. El-Desouky, R.S. Gomaa, and A.M. Magar, An Extension of Apostol Type of Hermite-Genocchi Polynomials and their Probabilistic Representation, FILO-MAT.
Journal
Journal of Statistical Theory and Applications
Volume-Issue
20 - 2
Pages
304 - 317
Publication Date
2021/02/22
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
https://doi.org/10.2991/jsta.d.210203.001How to use a DOI?
Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Beih S. El-Desouky
AU  - Rabab S. Gomaa
AU  - Alia M. Magar
PY  - 2021
DA  - 2021/02/22
TI  - New Discrete Lifetime Distribution with Applications to Count Data
JO  - Journal of Statistical Theory and Applications
SP  - 304
EP  - 317
VL  - 20
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.210203.001
DO  - https://doi.org/10.2991/jsta.d.210203.001
ID  - El-Desouky2021
ER  -