A discrete analogue of odd Weibull-G family of distributions: properties, classical and Bayesian estimation with applications to count data

In the statistical literature, several discrete distributions have been developed so far. However, in this progressive technological era, the data generated from different fields is getting complicated day by day, making it difficult to analyze this real data through the various discrete distributions available in the existing literature. In this context, we have proposed a new flexible family of discrete models named discrete odd Weibull-G (DOW-G) family. Its several impressive distributional characteristics are derived. A key feature of the proposed family is its failure rate function that can take a variety of shapes for distinct values of the unknown parameters, like decreasing, increasing, constant, J-, and bathtub-shaped. Furthermore, the presented family not only adequately captures the skewed and symmetric data sets, but it can also provide a better fit to equi-, over-, under-dispersed data. After producing the general class, two particular distributions of the DOW-G family are extensively studied. The parameters estimation of the proposed family, are explored by the method of maximum likelihood and Bayesian approach. A compact Monte Carlo simulation study is performed to assess the behavior of the estimation methods. Finally, we have explained the usefulness of the proposed family by using two different real data sets.


Introduction
The Weibull (W) model is a well-known continuous distribution and has been used extensively over the last several decades for modelling data in many areas, especially in engineering, reliability, and biological fields. It is commonly applied for modellig monotone hazard rates. The cumulative distribution function (CDF) of W distribution can be presented as where λ and β are the scale and shape parameters, respectively. Because of the flexibility of the W distribution, many statisticians proposed several modifications by introducing one or more parameters to the baseline model in Equation (1). For instance, Lai et al. [43], Pal et al. [52], Lee et al. [44], Silva et al. [62], Pinho et al. [55], Almalki and Yuan [7], Sarhan and Apaloo [61], Nofal et al. [51], Al-Babtain et al. [4], Abd EL-Baset and Ghazal [1], and references cited therein. Alzaatreh et al. [8] developed a new approach in which they used a general form to produce a new family, called the transformed-transformer family. Bourguignon et al. [12] utilized this technique to introduce a new flexible family of continuous distributions and it is known as odd Weibull-G (OW-G) family. The CDF of OW-G family is given by where is a vector of model parameters (1 × j; j = 1, 2, 3, . . . , k) and G(x; ) is the CDF of the baseline distribution. For more details about the generator G(x; )/(1 − G(x; )), see Cooray [17]. Due to the flexibility of this generator, several authors used it for producing flexible families which can be utilized for modelling different types of data. For instance, Tahir et al. [63], Korkmaz and Genç [40], Korkmaz [37], Hamedani et al. [30], Korkmaz et al. [38], El-Morshedy and Eliwa [24], Korkmaz et al. [38], Reyad et al. [58], Bakouch et al. [9], Nascimento et al. [47], Alizadeh et al. [5,6], Korkmaz et al. [39], Afify and Alizadeh [2], Cordeiro et al. [16], and references cited therein. The concept of discretization generally emerges when it becomes impossible or inconvenient to record the lifetime of a product/device on a continuous scale. These circumstances may appear when the life length needs to be recorded on a discrete scale rather than on a continuous analogue. For example, in survival analysis, the survival times for those having the diseases like lung cancer, or the period from remission to relapse may be recorded as a number of days/weeks; in engineering systems, the number of successful cycle prior to the failure when device work in cycle, the number of times a device is switched on/off. Moreover, in many practical problems, the count phenomenon occurs as, for example, the number of occurrences of earthquakes in a calendar year, the number of absences, the number of accidents, the number of kinds of species in ecology, the number of insurance claims, and so on. Therefore, we can easily infer that it is rational and appropriate to model such situations by a suitable discrete distribution.
In view of the existing literature, we found that although several discrete models have been proposed over the past few decades, we are still facing a lack of more admirable discrete distributions that adequately capture the diversity of real data sets. Such phenomenon motivates us to provide a flexible family of discrete models for analyzing a broad spectrum of discrete real-world data sets. Therefore, in present article, we propose a family of discrete uni-variate distributions with two additional parameters using survival discretization method. The main objectives of proposing DOW-G family are as follows: • To produce discrete models that not only fit a positively skewed, a negatively skewed, or a symmetric shaped data set, they are also capable enough for fitting equi-, under-, and over-dispersed real data. • To develop discrete distributions whose failure rate functions can take diverse shapes for different values of parameters (eg. increasing, decreasing, constant, J-and bathtubshaped). • To generate models for modelling probability distribution of count data.
• To produce consistently superior fits than other developed discrete distributions with the same baseline model and other popular discrete distributions in the existing literature.
The remaining parts of this article are as follows: Section 2 introduces the DOW-G family. Some statistical characteristics of the DOW-G family are derived in Section 3. Two special models of the proposed family are extensively studied in Section 4. The method of maximum likelihood and the Bayesian approach is used to estimate the unknown parameters of the DOW-G family in Section 5. An extensive simulation study is conducted to investigate the behaviour of different estimation methods in Section 6. Section 7 contains two applications to real data sets. Finally, some important remarks about the presented study are discussed in Section 8.

Quantile function (QF)
Under the DOW-G family, the qth QF, say x q , is the solution of F X (x q ; p, β, ) − q = 0; x q > 0, then where q ∈ (0, 1) and G −1 denotes the baseline QF. By putting q = 0.5, we can obtain the median of the proposed family. To study the effect of shape parameters on skewness and kurtosis, one can use

Moments, dispersion index, skewness and kurtosis
Suppose a RV X ∼ DOW-G(p, β, ), then the rth moment of the RV X can be proposed by where μ r = ∞ x=0 x r f x (x; p, β, ). Using the expression of Equation (9), the mean (μ 1 ) and variance can be, respectively, formulated as The dispersion index (DsI) is defined as the variance / |mean|, it determines that whether a given distribution is suitable for equi-, under-or over-dispersed data sets. The DsI is widely used in ecology as a standard measure for measuring repulsion (under dispersion) or clustering (over dispersion). If DsI < 1 (DsI > 1) the distribution is under-dispersed (over-dispersed), whereas the distribution is equi-dispersed at DsI = 1. The moment generating function (MGF) of the proposed family can be formulated as The first four partial derivatives of M X (t), with respect to t at t = 0, produce the first four raw moments about the origin, i.e. μ r = d r dt r M X (t)| t=0 . Furthermore, by using Equations (9) or (11), the skewness and kurtosis based on moments can be computed as skewness = (μ 3 − 3μ 2 μ 1 + 2μ 3 1 )/(variance) 3/2 and kurtosis = (μ 4 − 4μ 3 μ 1 + 6μ 2 μ 2 1 − 3μ 4 1 )/(variance) 2 .

Mean of excess over threshold and tail value at risk
Considerable attention has been paid to measuring operational risk over the past few decades. In this segment, we obtain few risk measures such as mean excess over the threshold (MEOT) and tail value at risk (TVAR) for the DOW-G family. The MEOT of the RV X following DOW-G family is given as Based on an arbitrary choice of G(x; ), we can get m(0) ≤ m(x), then the DOW-G family belongs to a class of discrete distributions having new worse than used in expectation, for more detail refer Marshall and Proschan [45]. Hence, the DOW-G family plays an vital role in the reliability theory. Regarding the TVAR, for any quantile x q , the TVAR of DOW-G family can be derived as

Rényi and Shannon entropies
The amount of uncertainty associated with a RV X is referred to as Entropy. It is widely applicable in many areas such as information theory, econometrics, survival analysis, computer science and quantum information, for more details, see Rényi [57]. The Rényi and Shannon entropies of the RV X can be formulated as and x ∈ N 0 , (15) respectively, where δ ∈ (0, ∞) and δ = 1. It is notable that the Shannon entropy can be derived as a particular case of the Rényi entropy if δ → 1, i.e. I(X) = lim δ=1 I δ (X).

Order statistics and L-moment statistics
Order statistics (ORST) play an important role in various fields of statistical theory and practice. Suppose X 1 , X 2 , . . .,X n be a random sample (RS) from the DOW-G(p, β, ), and let X 1:n , X 2:n , . . . , X n:n be their corresponding ORST. Then, the CDF of the ith ORST X i:n for an integer value of x is proposed as where The PMF of the ith ORST can be formulated as The uth moment of Z i:n is given by L-moments (L-MS) are linear combinations of ORSTs and are used to study the shape of a probability distribution. Hosking and Wallis [32] has proposed L-MS to summaries theoretical distribution and observed samples. Let X(i|n) be ith largest observations based on a sample of size n, then the L-MS is * Using Equation (19), we get *

Special models
In this segment, we study two particular distributions of the DOW-G family to demonstrate its viability. Some other main objectives of providing these models are: to evaluate the aforementioned characteristics for particular models of the presented family; to investigate the behaviour of various estimation methods for estimation of unknown parameters (in Section 6); to express the practicality of the developed family in real data analysis through two special models (in Section 7).
Regarding the rth moment, it is difficult to express it in an explicit form, and consequently, Maple software is utilized to interpret some of the descriptive statistics of the DOWGeo model. Table 1 lists some descriptive statistics using the DOWGeo model for different values of the parameters p, β and θ .
From Table 1, it can be easily observed that the DOWGeo model is appropriate of modelling over-and (under)-dispersed data; for some selected values of the model parameters DsI= 1, so it can be used as an alternative model to Poisson distribution; it is suitable for modelling negative and positive skewed as well as symmetric data sets; and DOWGeo model can be used to analyze either leptokurtic (kurtosis > 3) or platykurtic (kurtosis < 3) data sets. Tables 2-4 reports some numerical computations of MEOT, TVAR and Rényi entropy at x = 5 based on DOWGeo parameters when δ = 2 and q = 0.5.
From Tables 2-4, it is clearly seen that: the MEOT and Rényi entropy increase whereas the TVAR can take bathtub-shaped for some fixed values of β and θ with p −→ 1; the MEOT and Rényi entropy increase whereas the TVAR have inverse-J-shaped for some fixed values of β and p with θ −→ 1; and the MEOT and Rényi entropy decrease whereas the TVAR increases for some fixed values of θ and p with β grows.

The DOW-Inverse Weibull (DOWIW) distribution
Consider the CDF of the inverse Weibull (IW) model. Then, the PMF of the DOWIW distribution can be defined as where 0 < p, θ < 1, α > 0 and β > 0. Figures 3 and 4 show the PMF (a : 2) of the DOWIW distribution, respectively. The shape of the PMF is only unimodal, whereas the HRF can be increasing, increasing-constant-, or decreasing. Like the DOWGeo model, the DOWIW distribution can be applied to model skewed data set. Moreover, it is appropriate for modelling over-, equi-or under-dispersed data sets.

Maximum likelihood estimation
In this segment, unknown parameters of the DOW-G family are estimated by maximum likelihood estimation (MLE). Let X 1 , X 2 , . . . , X n be a RS from the proposed family. Then, the likelihood function (l) can be expressed as  Furthermore, the log-likelihood function (L) can be obtained as and where [ (·; j )] j = ∂ (·; j ) ∂ j ; j = 1, 2, 3, . . . , k. By putting Equations (24)-(26) equals to zero and solving them, immediately produces the MLEs of unknown parameters of the DOW-G family. Since the analytical solution to the above normal equations is difficult to obtain, therefore, we can solve these equations through an iterative method such as Newton-Raphson.

Bayesian estimation
When the experimenter has some prior knowledge (in the form of prior distribution) regarding the unknown parameters involved in the model, Bayesian estimation becomes crucial. In such a context, we discuss the estimation of unknown parameters of the DOW-G family under the Bayesian framework. Here, we assume the independent prior densities as p ∼ Beta(a 1 , b 1 ), β ∼ Gamma(a 2 , b 2 ), and j ∼ d j ( j , τ j ); j = 1, 2, . . . , k, where d j ( j , τ j ) and τ j are the prior density and set of associated hyperparameters (i.e. parameters involved in the prior distribution), respectively, for the j th member of the vector parameter . Then, the joint prior distribution of (p, β, is where, a 1 , b 1 , a 2 , b 2 > 0, and the domain of hyperparameters in τ j depends on the arbitrary choice of prior density d j . If we choose a 1 , b 1 , a 2 , b 2 and τ j such that Equation (27) has minimal or no effect on posterior distribution, we can perform Bayesian study under noninformative priors.
Loss functions (LFs) have an important role in statistical decision inference and Bayesian theory. Therefore, in order to develop the Bayes estimators (BEs) of unknown parameters (or any of their functions), we ought to consider the question of which type of LF will be used. Here, we use a popular LF, named as a squared error loss function (SELF). It is widely applicable due to its symmetric nature (i.e. it equally penalized negative and positive error). The SELF has the following form whereξ is any estimate of the parameter ξ , then the BE of ξ under SELF isξ * = E ξ (ξ |x), here the expectation is carried out under the posterior distribution of the unknown parameter ξ . Thus, in our case, the BE of g(p, β, ) (any function of parameters p, β, ) with SELF, can be derived aŝ Since the analytical solution of the expectation in Equation (29) is difficult to derive due to the non-closure form of the joint posterior density (28), therefore we need numerical approximation methods. Here, we utilize two important Markov Chain Monte Carlo (MCMC) approaches, known as Gibbs sampling [25] and Metropolis-Hastings (MH) algorithm [31,46] to generate the RSs from the posterior density and computing posterior quantities of interest. The work of the Gibbs algorithm is to decompose the joint density (28) into full conditional posterior densities of p, β, , so that these densities can be further used to simulate posterior RSs and to achieve the BEs of the unknown parameters.
To implement the Gibbs algorithm, the unnormalized conditional posterior densities of the parameters p, β, and j ; j = 1, 2, . . . , k are as follows where, l(x; p, β, ) is the likelihood function in Equation (22). Since the posterior distributions (30)-(32) are difficult to simulate directly by conventional techniques of generating RSs, therefore, we employ MH algorithm to generate posterior RSs for every element of the parameter vector = (p, β, 1 , 2 , . . . , k ) from their respective densities given in Equations (30)- (32). The detailed steps of MH with in Gibbs algorithm for obtaining BE of l (lth element of ), l = 1, 2, . . . , k + 2, are as follows: Step I. Initialize with some starting guess of l as (0) l and set i = 1.
Step II. Obtain the proposal point * (i) l from the proposal distribution q( l )) and u from Uniform distribution U(0, 1). Here, we utilizeˆ , 1), then Step IV. Set i = i + 1.
Step VI. Under SELF, the BE of l , sayˆ l is obtained asˆ l = 1 where m 0 is the burn-in-iterations of the Markov Chain.

A Monte Carlo simulation study
This section is devoted for an extensive simulation study to examine the behaviour of the various estimation methods discussed in Section 5. For this purpose, we generate samples of different sizes, i.e. n = 25, 50, and 100 from DOWGeo and DOWIW distributions. To draw the required RSs, we assume the parametric values of DOWGeo distribution as (0.5, 0.7, 0.8) and (0.9, 1.3, 0.5), while for DOWIW distribution, the model parameters are taken to be as (0.6, 0.8, 0.3, 1.2) and (0.9, 1.5, 0.5, 0.7). In classical setup, we compute the MLEs of the unknown parameters with their associated standard errors (Ses). The MLEs and Ses of the unknown parameters of DOWGeo distribution are presented in Table 5, whereas  Table 6 holds the MLEs and Ses in case of DOWIW distribution.
For Bayesian analysis of DOWGeo distribution, Beta(a 1 , b 1 ), Gamma(a 2 , b 2 ), and Beta(a 3 , b 3 ) are utilized as the independent informative priors for the parameters p, β,  and θ, respectively, while in case of DOWIW distribution, the variability of the unknown parameters p, β, θ, and α is measured by assuming independent informative priors as Beta(a 4 , b 4 ), Gamma(a 5 , b 5 ), Beta(a 6 , b 6 ), and Gamma(a 7 , b 7 ), respectively. In these prior densities, the hyperparameters have been selected such that the mean of the prior density is approximately equal to the corresponding predetermined value of the parameter. Under this setting, in order to calculate the Bayes estimates of unknown parameters, we generate 51,000 items for each parameter by utilizing MH within Gibbs sampling. The starting 1000 burn-in iterations for each of the chains have been discarded for removing the effects of initial values of the parameters. Also, every 25th observation is stored to neutralize the auto-correlation of successive draws. The convergence diagnostics of each generated chain is investigated by Geweke's [26] test at a 95% credibility level. This diagnostics is also examined by plotting posterior densities, MCMC runs, and auto-correlation functions. The above plots can be seen in Figures S1, S2, S3, and S4 for DOWGeo(0.5, 0.7, 0.8), DOWGeo (0.9, 1.3, 0.5), DOWIW(0.6, 0.8, 0.3, 1.2), and DOWIW (0.9, 1.5, 0.5, 0.7) distributions, respectively. After the convergence diagnostics, we have used these generated values to obtain Bayes estimates with their associated posterior standard errors (PSes) under SELF. The Bayes estimates and their PSes for the unknown parameters of DOWGeo distribution are given in Table 5, whereas Table 6 consists the Bayes estimates and PSes for the parameters of DOWIW distribution. All numerical computations are performed using open-source software R.
From this simulation study, we have observed that for each pair of sample size and true parametric value, both estimation methods work satisfactorily, but Bayesian method outperforms the MLE with respect to the estimation errors. Also, it is noticed that the estimation error associated with an estimate for both estimation procedures, tends to decrease as we increase the sample size.

Real data illustration
In this segment, we illustrate the applications of the DOWGeo and DOWIW models by using two real data sets. The fitting of the models are compared using some well-known measures, namely, −L, Akaike information criterion (AkIC), correct Akaike information  criterion (CAkIC), Hannan-Quinn information criterion (HQIC), Chi-square (χ 2 ) with degree of freedom (DF) and the associated P-value. The competitive distributions are provided in Table 7.

Data set I: count cysts of kidneys using steroids
This data set is taken from the study of Chan et al. [15]. Here, we examine the fitting capability of the DOWGeo model with some other competitive distributions like: GGeo, Geo, DR, DIR, DIW, NeBi, Poi, DFx-I, DLi-I, DLi-II, DLi-III, DLogL, DB, DLo, and DPa. The MLEs with their associated standard errors (Se) are tabulated in Table 8, whereas Tables 9  and 10 list the goodness of fit statistics (GOFS). In Tables 9 and 10, ExFr and ObFr represent the expected and observed frequencies, respectively. From Tables 9 and 10 it is noted that, the DPa, GGeo, NeBi, DLogL, DB, DLo and DIW distributions work quite satisfactory for analyzing data set I besides the DOWGeo distribution under condition (P-value > 0.05). Depending on −L, AkIC, CAkIC, HQIC, χ 2 and  P-value, it is easily visible that the DOWGeo model provides the best fit among all other tested models since it has the smallest value of −L, AkIC, CAkIC, HQIC, and χ 2 , whereas it achieve the highest P-value associated to the χ 2 test. Figure S5 depicts the profile of L for each parameter based on data set I, which indicates that the estimators of the parameters are unimodal. Figures 5 and S6 support the results of Tables 9 and 10, and they conclude that data set I follows the DOWGeo, DPa, GGeo, NeBi, DLogL, DBX-II, DLo and DIW models. But, the DOWGeo model is better than other rival models. Table 11 lists some theoretical and empirical descriptive statistics for data set I. From Table 11, it is clear that the theoretical measures are closed to empirical ones. Most of the distribution is at the left and platykurtic. Further, the data considered herein suffering from over-dispersion phenomena.

Data set II: COVID-19 in South Korea
This data set can be viewed at (https://www.worldometers.info/coronavirus/country/ south-korea/) and consists the daily new deaths in South Korea for COVID-19 from 15   Table 12, whereas Table 13 lists the GOFS.
Based on the fitting measures in Table 13, we can announce that the DOWIW model is the best fitted model. Figure S7 shows the profiles of L for each parameter based on data set II, and it shows that every estimator has a unimodal function. Figures 6 and S8 support the results of Table 13, and we can say that the data set II follows the DOWIW model. Some theoretical and empirical descriptive statistics based on data set II can be viewed in Table 14.
From Table 14, the empirical mean, variance, DsI and skewness are closed to theoretical ones. Data set II suffering from over-dispersion phenomena, and most of the distribution is at the left with platykurtic.

Bayesian estimation for real data sets I and II
Here, we compute the BEs with their PSes for the unknown parameters of the DOWGeo and DOWIW distributions using a similar manner discussed in Section 5. Since there is no prior information available regarding the unknown parameters of the special distributions, therefore, we have used non-informative priors for the unknown parameters of the models under study. The calculated estimates with the related estimation errors of DOWGeo and DOWIW distributions are jointly given in Table 15.
From the comparison of Bayes estimates (in Table 15) and MLEs (in Tables 8 and 12), we can easily observe that the result obtained in case of simulation study do hold in case of real-life data sets as well.

Concluding remarks
In the present research article, we have introduced a new family of discrete distributions called DOW-G family. Its several important statistical characteristics have been investigated. Two particular distribution of the proposed family are studied in detail. These special models provides good flexibility in terms of shapes for the PDFs. Further, we have noticed that the proposed family can model a positively skewed, a negatively skewed or a symmetric shaped data set. In addition, the DOW-G family can be used quite effectively for modelling a wide variety of failure data because its HRF can take diverse shapes (increasing, decreasing, constant, J-, and bathtub-shaped). Moreover, it is suitable for modelling under-, equiand over-dispersed data set. In classical and non-classical setups, the method of maximum likelihood and Bayesian approach have been utilized to estimate the unknown parameters of particular models.
An extensive Monte Carlo simulation analysis has been conducted to evaluate the behaviour of above stated estimation methods. The results of simulation studies declare that these two estimation procedures perform quite satisfactorily in estimating unknown parameters of the model, but the Bayesian method dominates the method of maximum likelihood in terms of estimation errors. The flexibility of the DOW-G family has also been exemplified by using two distinctive real data sets. Hence, it is reasonable to say that the special distributions of the developed family can serve as an alternative model to the existing discrete models for modelling count or failure data in various fields including reliability, insurance, medicine, economics, and demography, etc.
It is worth noting that the above research can be expanded into several dimensions, for example-with different types of censoring schemes, we can study the particulars models of the proposed family (see Tyagi et al. [65]); In reliability inference, we can analyze stressstrength reliability or load share systems under special distributions of the DOW-G family. Also, various characteristics of order statistics from the proposed family could be explored and to infer the bi-variate data, a bi-variate discrete family may also be studied.