Exponential and Hypoexponential Distributions: Some Characterizations

The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some $n\ge 2$, $X_1, X_2,\,\ldots,\,X_n$ are independent copies of a random variable $X$ with unknown distribution $F$ and a specific linear combination of $X_j$'s has hypoexponential distribution, then $F$ is exponential. Thus, we obtain new characterizations of the exponential distribution. As corollaries of the main results, we extend some previous characterizations established recently by Arnold and Villase\~{n}or (2013) for a particular convolution of two random variables.


Introduction and Main Results
Sums of exponentially distributed random variables play a central role in many stochastic models of real-world phenomena.Hypoexponential distribution is the convolution of k exponential distributions each with their own rate λi, the rate of the i th exponential distribution.As an example, consider the distribution of the time to absorption of a finite state Markov process.If we have a k + 1 state process, where the first k states are transient and the state k + 1 is an absorbing state, then the time from the start of the process until the absorbing state is reached is phase-type distributed.This becomes the hypoexponential if we start in state 1 and move skip-free from state i to i + 1 with rate λi until state k transitions with rate λ k to the absorbing state k + 1.
The distribution of the sum Sn := Z1 + Z2 + . . .+ Zn, where λi for i = 1, . . ., n are not all identical, is called (general) hypoexponential distribution (see [1,2]).It is absolutely continuous and we denote by gn its density.It is called the hypoexponetial distribution as it has a coefficient of variation less than one, compared to the hyper-exponential distribution which has coefficient of variation greater than one and the exponential distribution which has coefficient of variation of one.In this paper, we deal with a particular case of the hypoexponential distribution when all λi are distinct, i.e., λi = λj when i = j.In this case, it is known ([3], p. 311; [11], Chapter 1, Problem 12)) that Here the weight ℓj is defined as Please note that ℓj := ℓj (0), where ℓ1(x), . . ., ℓn(x) are identified (see [4]) as the Lagrange basis polynomials associated with the points λ1, . . ., λn.The convolution density gn in (1) is the weighted average of the values of the densities of Z1, Z2, . . ., Zn, where the weights ℓj sum to 1 (see [4]).Notice, however, since the weights can be both positive or negative, gn is not a "usual" mixture of densities.If we place λj 's in increasing or decreasing order, then the corresponding coefficients ℓj's alternate in sign.
Consider the Laplace transforms ϕi(t) := E[e −tZ i ], t ≥ 0, i = 1, 2, . . ., n.They are well-defined and will play a key role in the proofs of the main results.
To begin with, let us look at the case when all Zi's are identically distributed, i.e., λi = λ for i = 1, 2, . . ., n, so we can use ϕ for the common Laplace transform.The sum Sn = Z1 + Z2 + . . .+ Zn has Erlang distribution whose Laplace transform φ, because of the independence, is expressed as follows: If we go in the opposite direction, assuming that Sn has Erlang distribution with Laplace transform φ, then we conclude that ϕi(t) = λ(λ + t) −1 for each i = 1, 2, . . ., n, which in turn implies that Zi ∼ Exp(λ).By words, if Zi are independent and identically distributed random variables and their sum has Erlang distribution, then the common distribution is exponential.
Does a similar characterization hold when the rate parameters λi are all different?The answer to this question is not obvious.It is our goal in this paper to show that the answer is positive.
The studies of characterization properties of exponential distributions are abundant.Comprehensive surveys can be found in [5,6,7,8].More recently, Arnold and Villaseñor [9] obtained a series of exponential characterizations involving sums of two random variables and conjectured possible extensions for sums of more than two variables (see also [10]).Corollary 1 below extends the characterizations in [9,10] to sums of n variables, for any fixed n ≥ 2.
The exponential distribution has the striking property that if λ = 1 (unit exponential), then the density f equals the survival function (the tail of the cumulative distribution function) F = 1 − F .Therefore, in case of unit exponential distribution, (2) can be written as follows: (6) We will show that ( 6) is a sufficient condition for X1, X2, . . ., Xn to be unit exponential.
We organize the rest of the paper as follows.Section 2 contains preliminaries needed in the proofs of the theorems.The proofs themselves are given in Section 3. We discuss the findings in the concluding Section 4.
In addition to ( 9), we will need some properties of Lagrange basis polynomials ℓj collected below.
According to Proposition 5 in [13] we obtain, for n ≥ 2 and k ≥ 1, the following chain of relations: Clearly, the equality in (10) holds if and only if k = 1.The proof is complete.
The properties in Lemma 2 can be easily verified, as an illustration, for n = 2, k = 1, and k = 2. Indeed, .

Proofs of the Characterization Theorems
In the proofs of both theorems we follow the four-step scheme.
• Assume the characterization property where ℓj is given in (3).
Dividing both sides of this equation by ϕ(µ1t)ϕ(µ2t) where ψ := 1/ϕ.Consider the series which, as a consequence of Cramér's condition for ϕ, is convergent in a proper neighborhood of t = 0. To prove the theorem, it is sufficient to show that We will prove that ( 12) implies ( 14) by showing that the coefficients {a k } ∞ k=0 in ( 13) satisfy a0 = 1, a1 = λ −1 > 0, and By (12) we have H(t) ≡ 1 and therefore h0 = 1 and h k = 0 for all k ≥ 1.
In this paper, we dealt with the situation where the rate parameters λi are all distinct from each other.The other extreme case of equal λi's is trivial.The obtained characterization seems of interest on its own, but it can also serve as a basis for further investigations of intermediate cases of mixed type with some ties and at least two distinct parameters (see [2]).Of certain interest is also the case where not all weights µi's are positive (see [1]).
k, 1 ≤ k ≤ n − 1, where the equality holds if and only if k = 1.
proved that if X1 and X2 are two independent and non-negative random variables with common density f and E[X1] < Exp(λ) for some λ > 0. Motivated by this result, we extended it in two directions considering: (i) arbitrary number n ≥ 2 of independent identically distributed non-negative random variables and (ii) linear combination of independent variables with arbitrary positive and distinct coefficients µ1, µ2, . . ., µn.Namely, our main result is that Sn = µ1X1+µ2X2+. ..+µnXn has density gn(x) =