Characterization of Generalized Uniform Distribution through Expectation

Normally the mass of a root has a uniform distribution but some have different uniform distributions named Generalized Uniform Distribution (GUD). The characterization result based on expectation of function of random variable has been obtained for generalized uniform distribution. Applications are given for illustrative purpose including a special case of uniform distribution.


Introduction
Normally the mass of a root has a uniform distribution.Plant develops into the reproductive phase of growth; a mat of smaller roots grows near the surface to a depth of approximately 1/6-th of maximum depth achieve (see G. Ooms and K. L. Moore [1]).Dixit [2] studied the problem of efficient estimation of parameters of a uniform distribution in the presence of outliers.He assumed that a set of random variables 1 2 , , , n X X X  represents the masses of roots where out of n-random variables some of these roots (say k) have different masses; therefore, those masses have different uniform distributions with unknown parameters and these k observations are distributed with Generalize Uniform Distribution (GUD) with probability density function (pdf) ( ) where a b −∞ ≤ < ≤ ∞ are known constants; X α is positive absolutely continuous function and ( ) In this paper the problem of characterization of GUD with pdf given in (1.1) has been studied and the characterization also holds for uniform distribution on interval ( ) , a θ when 0 α = .Various approaches were used to characterize uniform distribution; few of them have used coefficient of correlation of smaller and the larger of a random sample of size two; Bartoszyn'ski [4], Terreel [5], Lopez-Bldzquez [6] as Kent [7], have used independence of sample mean and variance; Lin [8], Too [9], Arnold [10], Driscoll [11], Shimizu [12], and Abdelhamid [13] have used moment conditions, n-fold convolution modulo one and inequalities of Chernoff-type were also used (see Chow [14] and Sumrita [15]).
In contrast to all above brief research background and application of characterization of member of Pearson family, this research does not provide unified approach to characterized generalized uniform.
The aim of the present research note is to give a path breaking new characterization for generalized uniform distribution through expectation of function of random variable, ( ) X φ using identity and equality of expectation of function of random variable.Characterization theorem was derived in Section 2 with method for characterization as remark and Section 3 devoted to applications for illustrative purpose including special case of uniform distribution.

Characterization
Theorem 2.1.Let X be a continuous random variable (rv) with distribution function ( ) Proof: Given ( ) Differentiating with respect to θ on both sides of (2.2) and replacing X for θ after simplification one gets which establishes necessity of (2.1).Conversely given (2.1), let ( ) Since 0 a = , the following identity holds ( ) ( ) with respect to x and simplifying after tacking as one factor one gets (2.5) as ( ) ( ) ( ) Substituting derivative of ( ) where ( ) 3) and by uniqueness theorem from (2.4) and (2.7) ( ) Since ( ) a α + = is satisfy only when range of X is truncated by θ from right and integrating (2.8) on the interval ( ) , a θ on both sides, one gets ( ) ; k x θ de- rived in (2.8) as ( ) Hence ( ) ; k x θ derived in (2.9) reduces to ( ) ; f x θ defined in (1.1) which establishes sufficiency of (2.1).

Examples
Using method describe in remark 2.1 Generalize Uniform Distribution (GUD) through expectation of non-constant function of random variable such as mean, th r raw moment, e θ , e θ − , th p quantile, distribution function, reliability function and hazard function is given to illustrate application and significant of unified approach of characterization result (2.1) of theorem 2.1.

; f x θ is characterized. Example 3 . 3 .
In context of remark 2.2 uniform distribution with pdf given in (2.13) characterized through

Example 3 . 3 .
The pdf ( ) ; f x θ defined in (2.13) can be characterized through non-constant function such as > in the same setup and showed that the UMVUE is better than MLE when one parameter of GUD is known, where as both parameters of the GUD are unknown, [ ]