Discrete Burr Type Iii Distribution

In this paper, the discrete Burr type III distribution is introduced using the general approach of discretizing a continuous distribution and proposed it as a suitable lifetime model. The equivalence of continuous and discrete Burr type III distribution is established. Some important distributional properties and estimat ion of the parameters, reliability, failure rate and the second rate of failure functions are discussed based on the maximum likelihood method and Bayesian approach.


Intro duction
An important aspect of lifetime analysis is to find a lifetime d istribution that can adequately describe the ageing behavior of the device concerned. Most of the lifet imes are continuous in nature and hence many continuous life distributions have been proposed in literature. On the other hand, discrete failure data are arising in several common situations for examp le: · Reports on field failure are collected weekly, monthly and the observations are the number of failures, without a specification of the failure times.
· A piece of equip ment operates in cycles and experimenter observes the number of cycles successfully completed prior to failure. A frequently referred examp le is copier whose life length would be the total number of copies it produces. Another example is the number o f on/off cycles of a switch before failure occurs, see Lai and Xie [1].
In the last two decades, standard discrete distributions like geometric and negative binomial have been employed to model life time data. Usually, if the discrete model is used with lifet ime data, it is a mu ltino mial distribution. This arises because effectively the continuous data have been grouped, see Lawless [2]. However, there is a need to find mo re plausible discrete lifetime d istributions to fit to various types of lifet ime data. For this purpose, discretizing popular continuous lifetime d istributions can be helpful in this manner, since, it effects on speed, accuracy and understandability of the generated data using these discrete lifetime models.
= θ x . The interests in discrete failure data came relat ively late in comparison to its continuous analogue. The subject matter has to some extent been neglected. It was only briefly mentioned by few scientists. Khan, Khalique and Abouammoh [3], d iscussed two discrete Weibull distributions (type I and type II), and suggested a simp le method to estimate the unknown parameters for one of them, since the usual methods of estimat ion are not easy to apply. Kulasekera [4] presented approximate maximu m likelihood estimators of the parameters of a discrete Weibull distribution under censoring.
A discrete analogue of the normal distribution was obtained [5], that is characterized by maximu m entropy, specified mean and variance, and integer support on (−∞, ∞). Szab lowski [6], introduced new natural parameters in a formula defining a family of discrete normal distributions, where one of the parameters is closely related to the expectation and the other to the variance of that family. The discrete version of the normal and Ray leigh distributions were also proposed by Roy [7], [8] respectively. The discrete Weibull models were obtained [9], in order to model the number of cycles to failu re when components are subjected to cyclical loading. In addition, some d istributional properties for these models were p resented.
A discrete version of the Laplce (double exponential) distribution was derived by Inusah and Ko zubowski [10], and discussed some of its statistical propert ies and statistical issues of estimation under the discrete Laplace model. The discrete Burr type XII and Pareto distribution were obtained [11], using the general approach of d iscretizing and then, some important distributional properties and estimation of reliab ility characteristics were proposed.
A discrete inverse Weibull distribution was proposed [12], which is a discrete version of the continuous inverse Weibull variable, defined as X -1 where X denotes the continuous Weibull random variable. The d iscrete version of Lindley distribution was introduced [13], by discretizing the continuous failure model of the Lindley d istribution. Also, a compound discrete Lindley distribution in closed form is obtained after revising some of its properties.
A discrete generalized exponential distribution of a second type (DGE 2 (α,ρ)), was presented [14], which can be considered as another generalizat ion of the geo metric distribution.
A discrete analog of the generalized exponential distribution (DGE(α,ρ)) was presented [15], which can be viewed as another generalization of the geo metric distribution, and some of its distributional and mo ment properties were discussed. Burr type III distribution proposed as a lifetime model, see [16], the author discussed the distributional and the reliability properties of BurrIII(c, k).
In this paper, a discrete analogue of the BurrIII(c, k) distribution is introduced, since, it plays an important role in environment and other allied sciences. It is called discrete Burr type III d istribution denoted by dBurrIII(c, θ). This distribution is suggested as a suitable lifet ime model to fit a range of d iscrete lifet ime data. The rest of the paper is organized as follows: In Section 2, BurrIII(c, k) distribution is given with its reliab ility characteristics. The discrete analogue of BurrIII(c, k) distribution is developed with its distributional properties and reliability characteristics along with a graphical description. In Section 3, some important results on dBurrIII(c, θ ) are proved. The maximu m likelihood (ML) and Bayes estimat ions in dBurrIII(c, θ) are illustrated in detail through a simulation studies in Section 4.

Continuous Burr Type III Distri bution
A lifetime rv X fo llo ws the Burr type III distribution BurrIII(c, k) if its pdf is given by the corresponding survival function (RF), failure rate function (HRF) and the second rate of failu re function (SHRF) are respectively given by and

Discrete Burr III Distributi on
Based on the reliability function of continuous BurrIII rv X, wh ich is given by (3), the R ( x ) for dBurrIII(c, θ ) distribution at integer points of X, is given by (6) Here, note that R ( x ) is same for BurrIII(c, k) distribution and dBurrIII(c, θ) distribution at the integer points of x. Also, it is a positively skewed distribution. Now, by using (1), the p mf of the discrete Burr type III distribution with the parameters c and θ, dBurrIII ( c, θ ) , can be define as The HRF or the failure rate function is given by to study the behaviors of this function see Fig.(3).

The Second Rate of Failure * ( ) is Given by
For discrete distributions, failure rate h(x) is a conditional probability with un ity as its upper bound. It was pointed out that calling this the failure rate function might add to the confusion that is already common in industry that failure rate and failu re probability are so metimes mixed -up [9]. To solve this problem they introduced second rate of failu re h * ( x ) with the same monotonicity as h ( x ) .

The Moment of dBurrIII(c, ) is Given by
.
(10) In particular: (i) The mean of lifet ime of dBurrIII(c , ) can be obtained by using (10) as follows (11) (ii) The second mo ment is given by can be obtained by using (11) and (12) as follows Obviously, fro m Fig.(1), the mean of d BurrIII(c, θ) is decreasing. Also, the variance of dBurrIII(c, θ) is decreasing and it's noticeable fro m the graph that the mean is decreasing faster than the variance. Although the variance for dBurrIII(c, θ) tends to increase at the beginning but after that it adopts the same behavior as the mean.
In particular, the mean of lifet ime μ of dBurrIII(c, θ) can be obtained by using the first derivative of (18), wh ich is known as the first factorial mo ment and it is given by It is clear that the second factorial mo ment can be obtained by getting the second derivative of (18) as follows (20) More generally, the r th factorial mo ment is given by − θ log (1+x −c ) ] ; r = 1, 2, … Fro m (19) and (20) the variance V ( c, θ ) of d BurrIII(c, θ) is given by

The Characteristic Function ( ) for dBurrIII(c, ) is Given by
which is clearly the same result in (10).

Graphical Description
The curves of two populations of dBurrIII(c, θ) are plotted in Fig.(2), the first curve p1(x) when ( θ = .75 and c = 1) and the the second one , p2 ( x ) when(θ = .25 and c = .5). The curves of the corresponding failure rate function and the second rate of failu re function of dBurrIII(c , θ ) are illustrated in Fig.(3) and Fig.(4), respectively. Fig.(3) demonstrate some of the possible shapes of h(x) for selected values of θ where (c=.1), the first curve h1(x) at (θ = .2) and the second one, h2(x) at (θ = 2). It is obvious that h(x) is a decreasing function. In Fig.(4)

Result (2 )
Let X i ′ s (i = 1, 2, … n) be non-negative independently and identically d istributed ( iid ) integer valued rv′s and Y = min 1 ≤i≤n Then, Y is d BurrIII (c , θ n ) if and only if X i is dBurrIII(c, θ).

Result (3 )
If X non-negative rv and (t) is a positive nu mber. Then, X t = [X t ]~dBurrIII � c t , θ� if and only if X~BurrIII ( c, k ) .

Result (5 )
That is RF of geo metric rv. Thus, Y~ Geo (

Esti mation of the Parameters Based on the ML Method
Let n ite ms be put on test and their lifetimes are recorded as X 1 , X 2 , … , X n . If these X i 's are assumed to be iid rv's following dBurrIII(c, θ), their likelihood function is given by (23) and (23) can be rewritten as follo ws where ∅ ( , c ) = log [ . Now, to find the two log-likelihood equations we need first to obtain the log-likelihood function which is given by

Case I ( is Known and is Un known)
In this case, the MLE of the unknown parameter θ is θ � , that is the solution of the following likelihood equation, with an observed sample this equation can be solved using an iterative nu merical method.
The solution of this equation will p rovide the M LE of θ by using numerical co mputation.
The M LE`s of the reliability, the failure rate and the second rate of failure functions are obtained based on the invariance property of the ML, respectively as follows

Case II ( are Un known)
In this case, the solution of the follo wing likelihood equations provide the MLE`s of the unknown parameters θ and c, which are denoted by θ � and c �, respectively. With an observed sample these equations can be solved using an iterative numerical method.
So those, the first derivatives with respect to θ and c, of the log-likelihood equation (25), are g iven by By using numerical co mputation, the solution of these normal equations will provide the MLE of θ and c.
The M LE`s of the reliability, the failure rate and the second rate of failure functions are obtained based on the invariance property of the ML, respectively as follows

Case I ( is Known and is Un known)
Assume that the prior knowledge of θ is adequately represented by beta distribution with parameters (a) and (1) then the pdf of the prior density of θ is given by ; where δ = (a + ∑ w 1 ) and τ = a + ∑ w 2 . (29) So, fro m (28) and (29), | distributed as followes: Assuming a squared-error loss function and informat ive prior, the Bayes estimate of the parameter θ is given by Based on binary loss function, which is the mode of π�θ|x�, the highest post estimate (HPE) of the parameter θ is given by (32) Assuming a modified LINEX loss function, the Bayes estimate of the parameter θ is given by Now, fro m (30) considering a function of θ where (θ ω = z) in order to get a well known form for the posterior density of the new parameter θ ω , then π�z|x� will be at the . ( Based on binary loss function, which is the mode of π�z|x�, the highest post estimate (HPE) of the parameter θ ω is given by . (36) Assuming a mod ified LINEX loss function, the bayes estimate of the parameter θ ω is given by Also, under General entropy loss function, the bayes estimate of the parameter θ ω is given by Assume that: c is distributed as a non-informat ive prior, θ is distributed as beta distribution where, c and θ are independent.
The joint prior density of c and θ can be written as follows: π ( c, θ ) ∝