A Class of Weibull Mixtured Distributions

We have derived a class of mixture distributions which we call weibull mixtures of distributions. Estimation of unknown parameters along with some properties of these distributions are also prescribed. Citation: Adnan MAS, Kiser H (2012) A Class of Weibull Mixtured Distributions. J Biomet Biostat 3:135. doi:10.4172/2155-6180.1000135 J Biomet Biostat ISSN:2155-6180 JBMBS, an open access journal Page 2 of 6 Volume 3 • Issue 2 • 1000135 Definition 3.3 If a random variable X has the density function 2 1 (log ) 2 2 1 1 0 2 (log ) ( ; , ) ; 0 1 2 2 b x r b ar r e x f x a b abr e dr x x r − ∞ − − + = >   +     ∫ (3.5) then it is said to have a weibull mixture of Lognormal distribution with parameters a,b since 0 ( ; , ) 1 f x a b dx ∞ = ∫ (3.6) Various moments of the distribution are given in the next theorem. Theorem 3.2 If X is a weibull mixture of lognormal variable with parameters a,b then its characteristic function is given by 1 1 0 2 1 ( ) 1 2 2 b b ar x r t abr e r φ ∞ − − + =   +     ∫ 2 1 2 0 ( ) ! k k k it e k ∞ = ∑ 1 2 2 2 0 2 1 2 2 2 m r r m m r k m dr m + − =     +         ∑ (3.7) and the sth moment about origin is 2 1 2 1 2 2 0 0 2 1 . 2 2 1 2 2 b s r b ar r m m r m r e abr e s m dr m r ∞ − − − = −     +           +     ∑ ∫ Definition 3.4 A random variable X having the density function ( ) 1 1 0 ( ; , , , ) ; 0 b r x r b ar e x f x a b abr e dr x r α β α β α β α + − + − ∞ − − = > + ∫ (3.8) is defined a weibull mixture of Gamma distribution with parameters a,b,α and β whereas 0 ( ; , , , ) 1. f x a b dx α β ∞ = ∫ (3.9) The characteristic function and moments are followed by the next theorem. Theorem 3.3 If X denotes a weibull mixture of gamma variate with parameters a,b,α and β then its characteristic function is obtain as ln 1 1 0 ( ) 1 b it ar r b x it t ab r e dr α β φ β −   − − −   ∞ −     = −     ∫ (3.10) 1 1 1 1 , b Mean a b α β −     = + +           2 1 2 1 2 1 1 2 1 1 1 1 , b b b Variance a a a b b b α β − − −               = + + + + − +                      


Preliminaries
A mixture distribution is a weighted average of probability distribution of positive weights that sum to one. The distributions thus mixed are called the components of the mixture. The weights themselves comprise a probability distribution called the mixing distribution. Because of this property of weights, a mixture is in particular again a probability distribution. Mixtures occur most commonly when the parameter θ of a family of distributions, given by the density by the density function f(x, θ), is itself subject to the change variation. The mixing distribution g(x; θ)is then a probability distribution on the parameter of the distributions. The general formula for the finite mixture is

Main Results
Here in this paper, we define the weibull mixtures of some well known distributions such as normal, lognormal, gamma, exponential, beta, rectangular, erlang, chi-square, t and F distributions. Then some characteristics of these distributions such as characteristic functions, moments, and shape chaacteristics are also obtained. The main results of the paper are presented in form of definitions and theorems. Comparison of the probability density functions and the first two moments are prescribed in the tertiary section.

Definition 3.1
A random variable X is said to have a weibull mixtured distribution if its probability density function is defined as Where g(x,α) is a probability density function. The name of weibull mixture distribution comes from the fact that the distribution (3.1) is the weighted average of g(x,α) with weights equal to the ordinates of weibull distribution.

Definition 3.2
If X follows a weibull mixture of Normal distribution with parameters a and b, then the density function is given by with parameters a and b such that The characteristic function and moments of the same distribution are presented in the theorem below.

Theorem 3.1
If X has a weibull mixture of normal distributions with parameters a and b then its characteristic function is represented as  (3.4) and the 2s th moment about origin is and (2s+1) th moment about origin is zero. Mean = 0,

Definition 3.3
If a random variable X has the density function then it is said to have a weibull mixture of Lognormal distribution with parameters a,b since Various moments of the distribution are given in the next theorem.

Theorem 3.2
If X is a weibull mixture of lognormal variable with parameters a,b then its characteristic function is given by and the s th moment about origin is  The characteristic function and moments are followed by the next theorem.

Theorem 3.3
If X denotes a weibull mixture of gamma variate with parameters a,b,α and β then its characteristic function is obtain as Remark: ϕx(t),μ 1 ,μ 2 ,μ 3 ,μ 4 ,β 1 and β 2 are true for Gamma distribution with parameters α and β when a = b = 0. For α = 1, weibull mixture of Gamma distribution should be equivalent to weibull mixture of Exponential distribution. As such we also derived the weibull mixture of Exponential distribution.

Estimates of parameters by the method of moments:
Let X 1 , X 2 , X 3 ,……..,X m be a random sample from the distribution (3.8). We assume that parameters a,b and β are known. Then the distribution contains only one unknown parameter α. We have Hence by the method of moments, we get,

Definition 3.5
A random variable X having the density function ( )
Parameter estimation: If X 1 , X 2 , X 3 ,……..,X m be a random sample drawn from the distribution (3.12) and parameters a,b are assumed known, then the distribution contains only one unknown parameter α. So,

Definition 3.6
If a random variable X has the density function The characteristic function as well as the moments is stated in the following theorem.

Theorem 3.5
If X has weibull mixture of erlang distributions with parameters a,b, α and β then its characteristic function is given by Remark: a = b = 0 provides all the values of ϕx(t),μ 1 ,μ 2 ,μ 3 ,μ 4 ,β 1 and β 2 to be true for Erlang distribution with parameters α and β.
Estimating parameters: For a random sample X 1 , X 2 , X 3 ,……..,X m from the distribution (3.16), we assume that parameters a,b and β are known and α unknown parameter. Here, . We obtain . Therefore,

Definition 3.7
A random variable X having the density function

Theorem 3.6
If X follows a weibull mixture of rectangular distribution with parameters a,b and m then its characteristic function is obtained as

Definition 3.8
A random variable X having the density function

Theorem 3.7
If X follows weibull mixture of beta distributions of first kind with parameters a,b, α and β, then its s th moment about origin is given by

Definition 3.9
A random variable X having the density function Next theorem presents some properties of the same distribution.

Theorem 3.8
If X follows weibull mixture of beta distribution of second kind with parameters a,b, α and β then its s th moment about origin is given by

Definition 3.10
A random variable X 2 with the density function If X 2 has weibull mixture chi-square distribution with parameters a,b and n then its characteristic function is expressed as Remark: Setting a=b=0 we find that all the values of ϕ x (t),μ 1 ,μ 2 ,μ 3 ,μ 4 ,β 1 and β 2 are true for Chi-square distribution with parameters n.
Parameter estimation: Let X 1 , X 2 , X 3 ,……..,X m be a random sample from the distribution (3.28). We assume that parameters a and b are known and n is unknown. Now, As such,

Definition 3.11
If t as a random variable has the density function The following theorem expresses here some of the properties of the distribution.

Theorem 3.10
If t is weibull mixture of t distribution with parameters a,b and n then the 2s th moment about origin is given by Weibull mixtured Gamma Weibull mixtured Exponential Weibull mixtured Erlang Weibull mixtured Chi-square Weibull mixtured t 0 Weibull mixtured F