Characterization of Power-Function Distribution through Expectation ()
1. Introduction
Several characterizations of power function distribution have been made notably by Fisz [1], Basu [2], Govindarajulu [3] and Dallas [4] using independence of suitable function of order statistics and distributional properties of transformation of exponential variable.
Other attempts were made for the characterization of exponential and related distributions assuming linear relation of conditional expectation by Beg [5], characterization based on record valves by Nagraja [6], characterization of some types of distributions using recurrence relations between expectations of function of order statistics by Alli [7], characterization results on exponential and related distributions by Tavangar [8], and characterization continuous distributions through lower record statistics by Faizan [9] included the characterization of power function distribution.
Direct characterization for power function distribution has been given in Arslan [10] who used the product of order statistics [contraction is a particular case of product of order statistics which has interesting applications such as in economic modeling and reliability see Alamatsaz [11], Kotz [12] and Alzaid [13]] where as Moothathu [14] used Lorenz curve. [Graph of fraction of total income owned by lowest pth fraction of the population is Lorenz curve of distribution of income [15].
This research note provides the characterization based on identity of distribution and equality of expectation of function of random variable for power-function distribution with the probability density function (p.d.f.)
(1.1)
where 
are known constants, 
is positive absolutely continuous function and 
is everywhere differentiable function. Since derivative of 
being positive and since range is truncated by 
from right
.
The aim of the present research note is to give the new characterization through the expectation of function 
for the power function distribution. Examples are given for the illustrative purpose.
2. Characterization
Theorem 2.1 Let X be a random variable with distribution function F. Assume that F is continuous on the interval, 
where
. Let 
and 
be two distinct differentiable and integrable functions of 
on the interval 
where 
and moreover 
be non constant. Then 
is the p.d.f. of power function distribution 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 (2.2) with respect to 
on both sides and replacing 
for 
and simplifying one gets
(2.3)
which establishes necessity of (2.1). Conversely given (2.1), let 
be such that
(2.4)
Since 
the following identity holds:
(2.5)
Differentiating integrand of (2.5) and tacking 
as one factor one gets (2.5) as
(2.6)
where 
is function of 
derived in (2.3). From (2.4) and (2.6) by uniqueness theorem
(2.7)
Since 
is decreasing function with 
and since
, integrating (2.7) on both sides one gets
(2.8)
Substituting 
in (2.7), 
reduces to
defined in (1.1), which establishes sufficiency of (2.1).
Note: Author does not claim the relations between f and g in the preceding analysis.
Remark 2.1 Using 
derived in (2.3), 
given in (1.1) can be determined by
(2.9)
and p.d.f. is given by
(2.10)
where 
is increasing function in the interval 
for 
with 
such that it satisfies
3. Illustrative Examples
Example 1 Using method described in the remark characterization of power function distribution through survival function quantile; 
is illustrated.
Example 2 The p.d.f. 
defined in (1.1) can be characterized through non constant functions of 
such as
by using
and defining 
given in (2.9) and using 
as appeared in (2.11) for (2.10).
4. Conclusion
To characterize the p.d.f. defined in (1.1), one needs any arbitrary non constant function of 
which should be differentiable and integrable only.
NOTES