Some Characterizations of Exponential Distribution

There are some characterizations of the exponential distribution based on the relation of the maximum of two observations expressed as linear combination of the two observations. In this paper some generalizations of this known characterization of the exponential distribution using the relations between the maximum and minimum of n (≥ 2) independent and identically distributed random variables having absolutely continuous (with respect to Lebesgue measure) distribution function will be presented. Mathematical Subject Classification (2010): 60E05; 62E10; 62H10.


Introduction
Many times the researcher wants to verify whether the data that she/he has obtained belong to a certain family of distribution.For that purpose, the researcher has to rely on the characterization of the assumed distribution.Hence the characterization of distributions becomes important and essential.The univariate exponential distribution is the most commonly used distribution in modeling reliability and life testing analysis.There are many characterizations of the exponential distribution using ordered random variables.Ferguson (1967) characterized the exponential distribution using the regression properties of the first two order statistics.There are many generalizations of this characterization by using order statistics and record values.For details, see Ahsanullah et al. (2013) and David and Nagaraja (2009).Recently Arnold and Villasenor (2013) characterized the exponential distribution by using the identical distribution of  1 +  2 2 and max ( 1 ,  2 ) assuming  1 and  2 as identically distributed absolutely continuous (with respect to Lebesgue measure) non-negative random variables with the restriction that their distribution function is infinitely differentiable.In this paper some generalizations of this characterization of the exponential distribution based on the relation between maximum and minimum of  (≥ 2) independent and identically distributed continuous random variables are derived.

The Main Results
We shall present two interesting characterizations in this paper.To prove the first characterization, we shall need the following lemma.

Proof:
We shall prove by induction.
Since H(x) = F(x)f(x), we have H(0) = 0. Routine differentiation with respect to x and simplifying yields the following results: Similarly, we have It is easy to see that we can write Then, we have This completes the proof of Lemma 2.1.▄ Theorem 2.1.Suppose X, X 1 and X 2 are independent and identically distributed random variables with cumulative distribution function (cdf) F(x) with F(0) = 0 and F(0) > 0 for all x > 0. Assume that F(x) is absolutely continuous (with respect to Lebesgue measure) and infinitely differentiable with dF(x) dx = f(x) and f(0) > 0. Let Z = max (X 1 , X 2 ) and W = min (X 1 , X 2 ).Then Z = d W + X, where = d denotes equality in distribution and X is independent of W, if and only if F(x) = 1 − e −λx , x ≥ 0 and λ is an arbitrary positive real number.

Proof:
Necessity: Assume that F(x) = 1 − e −λx , x ≥ 0, λ > 0. Let f 1 (x) and f 2 (x) be the pdfs of Z and W + X respectively.Then it is easy to see that (2.1) and x 0 e −λ(x−y) e −λy dy Thus, Z = d W + X .
On differentiating (k + 2) times both sides of (2.3) with respect to  and putting  = 0, we obtain The LHS of Eqn 2.4 is given by The RHS of Eqn 2.4 is given by Equating both sides we get Hence, Or equivalently, Repeating the same procedure, we obtain Hence, Expanding () in Taylor series, we have Thus, This completes the proof of Theorem 2.1.▄ Remark 2.1: In Theorem 2.1 above, the equality of distribution can be replaced by the equality in expectation.
The following theorem characterizes the exponential distribution using the distributional relation between the maximum and minimum of  (> 2) random variables.
We have (, 0) = 0 and if ℎ() is monotonically increasing then for (2.5) to be true, we must have (, ) = 0 for all  and almost all .
Since (, ) = 0 for all  and almost all , then we must have   (, ) = 0. Thus ℎ() is constant.Since the hazard rate is a constant, it follows that  is exponential.If ℎ() is monotonically decreasing, then a similar arguments leads to the same conclusion.This completes the proof.▄