Bayesian Estimation in Delta and Nabla Discrete Fractional Weibull Distributions

One of the active areas of research in statistics is to model discrete life time data by developing discretized version of suitable continuous lifetime distributions. The discretization of a continuous distribution using different methods has attracted renewed attention of researchers in the last few years; for example, see [1–9]. Recently, these different methods are classified based on different criteria of discretization in detail by Chakraborty (see [10]). In this article, we present a newmethod for discretization of most of continuous distributions, where their pdfs consist of the monomial Taylor and exponential function. As an example, we do discretization for Weibull distribution. Our discretization method in comparison with prior methods for discretization of continuous distributions has two main advantages. First, for a given continuous distribution, it is possible to generate two types (delta and nabla types) of corresponding discrete distributions. Second, the main result of this paper is a unification of the continuous distributions and their corresponding discrete distributions, which is at the same time a distribution to the case of so-called time scale. We use discrete fractional calculus for showing the existence of delta and nabla discrete distributions and then apply time scales for definition of delta and nabla discrete distributions and as a unification theory under which continuous and discrete distributions are subsumed. Finally, we study the Bayesian estimation of functions of parameters of these distributions. 2. Preliminaries


Introduction
One of the active areas of research in statistics is to model discrete life time data by developing discretized version of suitable continuous lifetime distributions.The discretization of a continuous distribution using different methods has attracted renewed attention of researchers in the last few years; for example, see [1][2][3][4][5][6][7][8][9].Recently, these different methods are classified based on different criteria of discretization in detail by Chakraborty (see [10]).
In this article, we present a new method for discretization of most of continuous distributions, where their pdfs consist of the monomial Taylor and exponential function.As an example, we do discretization for Weibull distribution.Our discretization method in comparison with prior methods for discretization of continuous distributions has two main advantages.First, for a given continuous distribution, it is possible to generate two types (delta and nabla types) of corresponding discrete distributions.Second, the main result of this paper is a unification of the continuous distributions and their corresponding discrete distributions, which is at the same time a distribution to the case of so-called time scale.We use discrete fractional calculus for showing the existence of delta and nabla discrete distributions and then apply time scales for definition of delta and nabla discrete distributions and as a unification theory under which continuous and discrete distributions are subsumed.Finally, we study the Bayesian estimation of functions of parameters of these distributions.

Preliminaries
In this section, we provide a collection of definitions and related results which are essential and will be used in the next discussions.As mentioned in [11,12] the definitions and theorem are as follows.
A time scale T is an arbitrary nonempty closed subset of the real numbers R.
Definition 6.The delta (nabla) Taylor monomials are the functions ℎ  : T ×T → R,  ∈ N 0 , and are defined recursively as follows: We consider three cases for the time scale T.
(a) If T = R, then () = () =  and the Taylor monomials can be written explicitly as For each  ∈ R \ {−N}, define the th Taylor monomial to be and Γ denoted the special gamma function.
(b) If T = Z, then () =  + 1 and the Taylor monomials can be written explicitly as where and product is zero when  + 1 −  = 0 for some .More generally, for  arbitrary define where the convention of that division at pole yields zero.This generalized falling function allows us to extend (7) to define a general Taylor monomial that will serve us well in the probability distributions setting.
For each  ∈ R \ {−N}, define the delta th Taylor monomial to be In this paper, we only consider the special case ℎ  () fl ℎ  (, 0) =   /Γ( + 1) as delta Taylor monomial (dtm).(c) If T = Z, then () =  − 1 and the Taylor monomials can be written explicitly as where More generally, for any  ∈ R \ {−N}, the  rising function is defined as   = Γ( + )/Γ() and 0  = 0.The rising   and falling   are related by   = ( +  − 1)  .This rising function allows us to extend (10) in order to define a general Taylor monomial that will serve us well in the probability distributions setting.
Let  be a real number and  :  N → R. The delta Riemann right fractional sum of order  > 0 is defined by Abdeljawad [13] as We define the nabla Riemann right fractional sum of order  > 0 as The delta Riemann right fractional difference of order  > 0 is defined by Abdeljawad [13] as for  ∈ −(−) N and  = [] + 1, where [] is the greatest integer less than .Also, the nabla Riemann right fractional difference of order  > 0 is defined by for  ∈ − N .
In [14], authors have obtained the following alternative definition for delta Riemann right fractional difference Similarly, we can prove the following formula for nabla Riemann right fractional difference: For an introduction to discrete fractional calculus, the reader is referred to [15][16][17][18].

Generating Discrete Distributions by Discrete Fractional Calculus
The following results show the relationship between continuous and discrete fractional calculus and statistics and also allow us to define different types of discrete distributions.Suppose that  is a positive continuous random variable.The expectation of the tm function, ℎ −1 (), coincides with Riemann-Liouville right fractional integral of the pdf at the origin for  > 0 and Marchaud fractional derivative of the pdf at the origin for −1 <  < 0; that is, we have where is the Riemann-Liouville right fractional integral, while is the Marchaud fractional derivative [19].The definitions of fractional operators can be found in [20].
It can be seen that the limits of the above integrals are equal to the support of random variable .Considering this point, we present the following theorems for discrete random variable .
Theorem 7. Suppose that  is a discrete random variable.The expectation of the dtm function, ℎ −1 (), coincides with delta Riemann right fractional sum of the pmf at −1 for  > 0 and delta Riemann right fractional difference of the pmf at −1 for  < 0,  ∉ {−N}; that is, where is the delta Riemann right fractional sum, while is the delta Riemann right fractional difference.
Here, considering the limits of summation we can define the discrete distributions with the support N −1 or a finite subset of it.In this case, we will call  delta discrete random variable.In this work, we will define the delta discrete fractional Weibull distribution.Another example is the delta discrete uniform distribution, DU{ − 1, , . . .,  + }, where  ∈ R and  ∈ N −1 .
Theorem 8. We suppose that  is a discrete random variable.The expectation of the ntm function, ℎ −1 (), coincides with nabla Riemann right fractional sum of the pmf at 1 for  > 0 and nabla Riemann right fractional difference of the pmf at 1 for  < 0,  ∉ {−N}; that is, where is the nabla Riemann right fractional sum, while is the nabla Riemann right fractional difference.
Then, considering the limits of summation in recent theorem we can define the discrete distributions with support N 1 or a finite subset of it.In this case, we will call  nabla discrete random variable.In this work, we will define the nabla discrete fractional Weibull distribution.Another example is the nabla discrete uniform distribution, DU{1, 2, . . .,  −  + 1}, where  ∈ R and  ∈  N .

The Delta and Nabla Discrete Fractional Weibull Distributions
In this section, we will introduce delta and nabla discrete fractional Weibull distributions, by substituting continuous Taylor monomial and exponential functions with their corresponding discrete types (on the discrete time scale) in continuous Weibull distribution.

The Nabla Discrete Fractional Weibull Distribution
Definition 9.It is said that the random variable  has a nabla discrete fractional Weibull distribution with (, ) parameters, denoted by  ∇ (, ), if its pmf is given by where  > 0, 0 <  < 1.

Now we show that
For this purpose, by using Theorems 4.3 and 3.8 (integration by substitution) from [21] and considering (  ) ∇ =  −1 and under the substitution  =   , we have Pr respectively.It can be seen that ( 33) is the same geometric distribution (the number of failures for first success).Then (34), as a general case of (33), is a type of negative binomial distribution.
(c) For  = 2,  ∇ (, ) in ( 28) is a nabla discrete distribution with pmf where we will call it nabla discrete Rayleigh distribution.
(ii) Statistical Properties.If  ∼  ∇ (, ), then the survival function, the hazard function, and the mean of random variable  are given by To continue, we use the beta type I, beta type II, and Kummer-beta distributions.These distributions can be found in [22][23][24][25].
(iii) Bayesian Estimation in Nabla Discrete Fractional Weibull Distribution.In this section, we study the Bayesian estimation of functions of parameter  of nabla discrete fractional Weibull distribution.The likelihood function of , in this case, is given by We take a prior distribution given below: where B(, ) = Γ()Γ()/Γ( + ) and This prior density is known as the Kummer-beta density and denoted by KB(, , ).The posterior probability density function of , corresponding to (), is given by () is a natural conjugate prior density.Note that, for  = 0, the above prior density simplifies to a beta type I density with parameters  and .Under the squared error loss function (SELF) given by L((), ) = (() − ) 2 , where () is a function of  and  is a decision, the Bayesian estimate γB of () =   , corresponding to posterior density ( | ), is given by Similarly, under the weighted squared error loss function (WSELF) given by L((), ) = ()(() − ) 2 , where () is a function of , the Bayesian estimate γ of (), for two different forms of (), is given below.
(a) Let () =  −2 .The Bayesian estimate γ of (), known as the minimum expected loss (MEL) estimate, for () =   , corresponding to posterior density ( | ), is given by This loss function was used by Tummala and Sathe [26] for estimating the reliability of certain life time distributions and by Zellner and Park [27] for estimating functions of parameters of some econometric models.

Now we show that
For this purpose, we apply Theorem 5.40 (change of variable) from [12].Considering (  ) Δ =  −1 and then under the substitution  =   , we have respectively.It can be seen that ( 51) is the same geometric distribution (the number of independent trials required for first success).Then (52), as a general case of (51), is a type of negative binomial distribution.
where we will call it delta discrete Rayleigh distribution.