Abstract

We use discrete fractional calculus for showing the existence of delta and nabla discrete distributions and then apply time scales for definitions of delta and nabla discrete fractional Weibull distributions. Also, we study the Bayesian estimation of the functions of parameters of these distributions.

1. 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 [1–9]. 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.

2. 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 is an arbitrary nonempty closed subset of the real numbers The most well-known examples are and The forward (backward) jump operator is defined by , where and A point is said to be right-dense if and (left-dense if and ) and right-scattered if (left-scattered if ). The forward (backward) graininess function is defined by . More generally, we denote all , , and with

Definition 1. A function is called regulated if its right-sided limits exist at all right-dense points in and its left-sided limits exist at all left-dense points in

Definition 2. A function is called rd-continuous (ld-continuous) if it is continuous at right-dense (left-dense) points in and its left-sided (right-sided) limits exist at left-dense (right-dense) points in

The set is derived from the time scale as follows: If has a left-scattered maximum (right-scattered minimum) , then Otherwise,

Definition 3. A function is said to be delta (nabla) differentiable at a point if there exists a number with the property that, given any , there exists a neighborhood of such that for all

For a function it is possible to introduce a derivative and an integral in such a manner that and in the case and and in the case , where the forward and backward difference operators are defined by and , respectively. Also, we define the iterated operators and for

Definition 4. A function is called -regressive (-regressive) provided that for all

The set of all -regressive and rd-continuous (regressive and ld-continuous) functions forms an Abelian group under the circle plus addition defined by for all The additive inverse of is defined by for all

For real numbers and we denote and .

Theorem 5. Let and be a fixed point. Then the delta (nabla) exponential function is the unique solution of the initial value problem

If , when , where and , it is easy to see that and if , where “” is ordinary exponential function. Moreover, in the special case, . More generally, we will denote all , , and with

Definition 6. The delta (nabla) Taylor monomials are the functions , and are defined recursively as follows:

We consider three cases for the time scale .

(a) If , then and the Taylor monomials can be written explicitly as For each , define the th Taylor monomial to be and denoted the special gamma function.

In this paper, we only consider the special case, as Taylor monomial (tm).

(b) If , then and the Taylor monomials can be written explicitly aswhere and product is zero when 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 , define the delta th Taylor monomial to beIn this paper, we only consider the special case as delta Taylor monomial (dtm).

(c) If , then and the Taylor monomials can be written explicitly aswhere More generally, for any , the rising function is defined as and The rising and falling are related by

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.

For each define the nabla th Taylor monomial to beIn this paper, we only consider the special case as nabla Taylor monomial (ntm).

More generally, we will denote all , , and with

Let be a real number and The delta Riemann right fractional sum of order is defined by Abdeljawad [13] asWe define the nabla Riemann right fractional sum of order asThe delta Riemann right fractional difference of order is defined by Abdeljawad [13] as for and , where is the greatest integer less than Also, the nabla Riemann right fractional difference of order is defined by for

In [14], authors have obtained the following alternative definition for delta Riemann right fractional differenceSimilarly, we can prove the following formula for nabla Riemann right fractional difference:For an introduction to discrete fractional calculus, the reader is referred to [15–18].

3. 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, , coincides with Riemann-Liouville right fractional integral of the pdf at the origin for and Marchaud fractional derivative of the pdf at the origin for ; 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, , coincides with delta Riemann right fractional sum of the pmf at for and delta Riemann right fractional difference of the pmf at for ; that is, where is the delta Riemann right fractional sum, while is the delta Riemann right fractional difference.

Proof. For , substitute in expression (13) and also, for and , in expression (17).

Here, considering the limits of summation we can define the discrete distributions with the support 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, , where and

Theorem 8. We suppose that is a discrete random variable. The expectation of the ntm function, , coincides with nabla Riemann right fractional sum of the pmf at for and nabla Riemann right fractional difference of the pmf at for ; that is,where is the nabla Riemann right fractional sum, while is the nabla Riemann right fractional difference.

Proof. For , substitute in expression (14) and also for and , in expression (18).

Then, considering the limits of summation in recent theorem we can define the discrete distributions with support 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, , where and

4. 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.

4.1. 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 bywhere .

Now we show that

For this purpose, by using Theorems and (integration by substitution) from [21] and considering and under the substitution , we have

(i) Particular Cases. (a) For in (28) reduces to a one-parameter nabla discrete fractional Weibull with pmfObviously, this is the pmf of geometric distribution (the number of independent trials required for first success) or nabla discrete exponential distribution.

(b) For in (28) is a new delta discrete distribution with pmfIf we substitute , (31) and (32) are given byrespectively. 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 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 respectively.

To continue, we use the beta type I, beta type II, and Kummer-beta distributions. These distributions can be found in [22–25].

(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 and This prior density is known as the Kummer-beta density and denoted by The posterior probability density function of , corresponding to , is given by is a natural conjugate prior density. Note that, for , the above prior density simplifies to a beta type I density with parameters and Under the squared error loss function (SELF) given by , where is a function of and is a decision, the Bayesian estimate of , corresponding to posterior density , is given by