Characterization of Generalized Uniform Distribution through Expectation ()
1. 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 
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)
(1.1)
where 
are known constants; 
is positive absolutely continuous function and 
is everywhere differentiable function. Since derivative of 
being positive, range is truncated by 
from right
.
Dixt [3] obtained Maximum Likelihood Estimator (MLE) and the Uniformly Minimum Variance (UMVU) estimator of reliability functional, 
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, 
is estimated by using mixture estimate and is consistent.
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 
when
. 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, 
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.
2. Characterization
Theorem 2.1. Let 
be a continuous random variable (rv) with distribution function 
having pdf
. Assume that 
is continuous on the interval
, where
. Let 
be a differentiable functions of 
on the interval
, where 
and more over 
be nonconstant. Then 
is the pdf of Generalize Uniform Distribution (GUD) defined in (1.1) if and only if
(2.1)
Proof: Given 
defined in (1.1), if 
is such that 
where 
is differentiable function then
(2.2)
Differentiating with respect to 
on both sides of (2.2) and replacing X for 
after simplification one gets
(2.3)
which establishes necessity of (2.1). Conversely given (2.1), let 
be the pdf of rv X such that
(2.4)
Since
, the following identity holds
(2.5)
Differentiating 
with respect to x and simplifying after tacking 
as one factor one gets (2.5) as
(2.6)
Substituting derivative of 
in (2.6) it reduces to
(2.7)
where 
is derived in (2.3) and by uniqueness theorem from (2.4) and (2.7)
(2.8)
Since 
is decreasing function for 
and 
is satisfy only when range of X is truncated by 
from right and integrating (2.8) on the interval 
on both sides, one gets 
derived in (2.8) as
(2.9)
and
Hence 
derived in (2.9) reduces to 
defined in (1.1) which establishes sufficiency of (2.1).
Remark 2.1. Using 
derived in (2.3), the 
given in (1.1) can be determined by
(2.10)
and pdf is given by
(2.11)
where 
is increasing function for 
with 
such that it satisfies
(2.12)
Remark 2.2. If 
characterization theorem 2.1 also holds for uniform distribution with pdf
(2.13)
3. Examples
Using method describe in remark 2.1 Generalize Uniform Distribution (GUD) through expectation of non-constant function of random variable such as mean, 
raw moment, 
, 
, 
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.
Example 3.1. Characterization of Generalize Uniform Distribution (GUD) through hazard function
therefore
From (2.3) one gets 
as
and using (2.10) of remark 2.1 the
By characterization method describe in remark 2.1, if
then
and substituting 
as appeared in (2.12) for (2.11),
is characterized.
Example 3.2. The characterization of 
defined in (2.1) through non constant function such as
(3.1)
and using
(3.2)
and defining 
given in (2.10) and substituting 
as appeared in (2.12) for (2.11), 
is characterized.
Example 3.3. In context of remark 2.2 uniform distribution with pdf given in (2.13) characterized through 
quantile
therefore
.
From (2.3) one gets 
as
and using (2.10) of remark 2.1 the
.
By characterization method describe in remark 2.1, if
then
Example 3.3. The pdf 
defined in (2.13) can be characterized through non-constant function such as
and using
and defining 
given in (2.10) and substituting 
as appeared in (2.12) for (2.11), 
is characterized.
4. Conclusion
To characterize pdf defined in (1.1) one needs any arbitrary non-constant function of X which should be differentiable and integrable only.
Acknowledgements
This work is supported by UGC Major Research Project No: F.No.42-39/2013 (SR), dated 12-3-2013.