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New Discrete Lifetime Distribution with Applications to Count Data Journal of Statistical Theory and Applications Vol. 20(2), June 2021, pp. 304–317 DOI: https://doi.org/10.2991/jsta.d.210203.001; ISSN 1538-7887 https://www.atlantis-press.com/journals/jsta New Discrete Lifetime Distribution with Applications to Count Data Beih S. El-Desouky*, Rabab S. Gomaa, Alia M. Magar Department of Mathematics, Faculty of Science, Mansoura University, Egypt ARTICLEINFO A BSTRACT Article History In this paper, we present a new class of distribution called generalized Hermite–Genocchi distribution (GHGD). This model is Received 09 Aug 2020 obtained by compounding generalized Hermite–Genocchi polynomials given by Gould and Hopper with powers series distri- Accepted 25 Jan 2021 bution. 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. Keywords Generalized Hermite distribution Hermite polynomials Genocchi polynomials Hermite–Genocchi polynomials Discrete distribution Reliability © 2021 The Authors. Published by Atlantis Press B.V. 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] [ ] n ∑m yk xn*mk H ; .x; y/ = n@ ; .n g 0; m Ë N/: n m @. * /@ k=0 k n mk For more details, see [2], [3] and [5]. Gupta and Jain [9] extended the Hermite distribution (HD) of the generalized HD defined by [ ] n ∑m a.n*mj/bj P.Y = n/ = e*.a+b/ ; . * /@ @ j=0 n mj j where a g 0; b g 0 and m Ë N: 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 [7–9]. 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. H I Çr ∑r x P.X/ = B .훼 / i H ; x + 훾; 훽 : (1.1) i*1 n m i i=1 i=1 *Corresponding author. Email: [email protected] B. S. El-Desouky et al. / Journal of Statistical Theory and Applications 20(2) 304–317 305 훽; 훾 g 0; r g 1; m Ë N; where H I ∑∞ Çr ∑r l 1 .r/ 훾; 훽 훼 훼 i ; l ; ; 훾; ; ; 훽 ; =H Mn . ; r/ = . i*1/ Hn;m i + B l ;l ; ;l 1 2 § r=0 i=1 i=1 .r/ 훾; 훽 훼 훼 훼 ; 훼 ; ; 훼 ; < 훼 < : HMn . ; r/ is convergent and positive for r = . 0 1 § r*1/ 0 r 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 Z+ ∪ ^0` is said to follow GHGD with three parameters, that is GHG.훼; 훽; 훾/, if its probability mass function can be written as h H ; .훾; 훽/ n m ; n 훼 훽; 훾 x = 0 n G. ; / P.X = x; 훼; 훽; 훾/ = l (2.1) n 훼x 훾; 훽 Hn;m .x + / n ; x > 0; j G.훼; 훽; 훾/ where 훽 g 0 is scale parameter, 훾 g 0 is shape parameter, 0 < 훼 < 1 is shape parameter, m Ë N, ∑∞ 훼 훽; 훾 훼l l 훾; 훽 ; G. ; / = Hn;m . + / l=0 and [ ] n ∑m 훽k n@ n*mk H ; .x + 훾; 훽/ = .x + 훾/ : n m @ . * /@ k=0 k n mk G.훼; 훽; 훾/ is convergent and positive for 0 < 훼 < 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 휶; 휷 and 휸 are present in Figure 1 Three examples in Figure 1 showing effects of scale and shape parameters. 2.1.2. Cumulative distribution function The cumulative distribution function (cdf) of GHGD is given by F.x/ = P.X f x/ = 1 * P.X > x/ 훼x+1G.훼; 훽; x + 훾 + 1/ = 1 * : G.훼; 훽; 훾/ Figure 2 showing shape and behavior of Cdf plots of GHG distribution with several values of parameters 훼, 훽 and 훾. 306 B. S. El-Desouky et al. / Journal of Statistical Theory and Applications 20(2) 304–317 Figure 1 Shape and behavior of pmf plots of GHGD with serval values of parameters 훼; 훽 and 훾. Figure 2 Cdf of GHGD for different values of 훼; 훽 and 훾. 2.1.3. Moments and related measures The moment-generating function of GHGD is given by tx MX.t/ = E.e / G.훼et; 훽; 훾/ = : G.훼; 훽; 훾/ B. S. El-Desouky et al. / Journal of Statistical Theory and Applications 20(2) 304–317 307 The r * th factorial moments 휇 is given by [r] 휇 [r] = E[.X/r] ³∞ 훼x 훾; 훽 .x/r Hn;m .x + / = = x 0 : G.훼; 훽; 훾/ 휇¨ The r * th moments r is given by 휇¨ r r = E.X / ³∞ r 훼x 훾; 훽 x Hn;m .x + / = = x 0 : G.훼; 훽; 훾/ The mean and variance are given, respectively, by ∑∞ 훼x 훾; 훽 x Hn;m.x + / E.X/ = x=0 : G.훼;훽,훾/ b∑∞ c 2 훼x 훾; 훽 f x Hn;m.x + /g f x=0 g Var.X/ = f G.훼;훽,훾/ g f g d e 2 b∑∞ c 훼x 훾; 훽 f x Hn;m.x + /g *f x=0 g : f G.훼;훽,훾/ g f g d e The plots in Figure 3, it is apparent that both mean and variance of GHGD have bounds. 2.1.4. Over-dispersion The over-dispersion (OD) index of GHGD is given by 휎2 OD = 휇 ³∞ ³∞ 2 훼x 훾; 훽 훼x 훾; 훽 x Hn;m .x + / x Hn;m .x + / (2.2) = = = x 0 * x 0 : ³∞ G.훼; 훽; 훾/ 훼x 훾; 훽 x Hn;m .x + / x=0 From Figure 3 and Eq. (2.2), we can obtain the following corollary: Corollary 2.2. 1. OD = .>/.</ 1if and only if훼 = .>/.</ 0:4, 훽 = .>/.</ 0:3and훾 = .>/.</ 1: 2. GHGD is no over-dispersion, over-dispersion and under-dispersion for 훼 = .>/.</ 0:4, 훽 = .>/.</ 0:3 and 훾 = .>/.</ 1; respectively. We obtained that numerically. 308 B. S. El-Desouky et al. / Journal of Statistical Theory and Applications 20(2) 304–317 2.1.5. Surprise index The surprise index (SI) of GHGD is given by E.P.X = x; 훼; 훽; 훾// SI = P.X = x; 훼; 훽; 훾/ H I 2x 2 ³∞ 훼 .H ; .x + 훾; 훽// ( ) = n m _ 훼x + 훾; 훽 : 훼 훽; 훾 Hn;m .x / x=0 G. ; / From Figure 4 for various value of 훼; 훽 and 훾; where 훼; 훽 and 훾; decreases, large values of x become more surprising. 2.1.6. Generating function The probability-generating function of GHGD is given by ∑ X x GX.t/ = E.t / = P.X = x/ t x=0 G.훼t; 훽; 훾/ = : G.훼; 훽; 훾/ Figure 3 Plots the mean and variance of GHGD with serval values of parameters 훼; 훽 and 훾. Figure 4 log(SI)’s for GHGD. B. S. El-Desouky et al. / Journal of Statistical Theory and Applications 20(2) 304–317 309 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 [11–14]. Theorem 3.1. The GHG distribution is log-concave. Proof. Consider the function 훼 H ; .x + 훾 + 1; 훽/ ; 훼; 훽; 훾 P.X = x + 1/ n m : b.x / = = 훾; 훽 P.X = x/ Hn;m.x + / Its derivative is given by ` a ∑k=0 * * r 훽k n@ .x + 훾 + 1/n mk 1 s 훼 r ; s r n k@ .n * mk * 1/@ s [ ] d b.x; 훼; 훽; 훾/ p m q = 훾; 훽 dx .Hn;m.x + // ` a r ∑k=0 훽k 훾 n*mk*1 s 훼 ; ; ; ; 훾; ; ; ; ; 훽; ; ; n@ .x + / rHn;m . x + + 1 / s r n k@ .n * mk * 1/@ s [ ] p m q * < 0: 훾; 훽 2 .Hn;m.x + // Note that b.x; 훼; 훽; 훾/ is decreasing function in x for 0 < 훼 < 1, 훽 > 0 and 훾 > 0 thus, the G.훼; 훽; 훾/ 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.
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