Equilibrium Distributions and Discrete Schur-constant Models

This paper introduces Schur-constant equilibrium distribution models of dimension n for arithmetic non-negative random variables. Such a model is defined through the (several orders) equilibrium distributions of a univariate survival function. First, the bivariate case is considered and analyzed in depth, stressing the main characteristics of the Poisson case. The analysis is then extended to the multivariate case. Several properties are derived, including the implicit correlation and the distribution of the sum.


Introduction
Schur-constant models for discrete survival data have been studied by several authors including Castañer et al. (2015), Lefèvre et al. (2017), and Ta and Van (2017). In this case the discrete setting means that it is valued at N 0 = {0, 1, ...}. There is also a vaste literature regarding Schur-constant models for continuous survival data valued in R + ; let us mention, among others, Caramellino and Spizzichino (1994), Chi et al. (2009), andNelsen (2005). Castañer et al. (2015) discuss the properties of a Schur-constant vector (X 1 , . . . , X n ) generated by the survival function of an univariate random variable (r.v.). In that paper, it is M. Mercè Claramunt mmclaramunt@ub.edu Anna Castañer acastaner@ub.edu shown that this survival function has to be n-monotone. The present paper aims to develop a class of Schur-constant models generated by an admissible univariate survival function. The admissibility condition is the existence of the (n − 1) − th equilibrium distribution of the univariate r.v. Because of it, this new model is called Schur-constant (multivariate) equilibrium distribution model. Some properties of the bivariate continuous case have been previously studied by Nair and Sankaran (2014). For the sake of completeness, we include in this introduction some definitions and well-known results.
Let (X 1 , . . . , X n ) be a vector of n (≥ 2) arithmetic non-negative random variables, called lifetimes. It is said to have a Schur-constant joint survival function if for all (x 1 , . . . , x n ) ∈ N n 0 , P (X 1 ≥ x 1 , . . . , X n ≥ x n ) = S(x 1 + . . . + x n ), where S is an admissible function from N 0 to [0, 1]. In fact, S is the survival function of each of the marginal r.v. X i and it has to be n-monotone on N 0 . A function f (x): N 0 → R is said to be n-monotone if it satisfies A general representation (Castañer et al. 2015) valid for any discrete Schur-constant model, putting a b = 0 when a < b, is: where the variable Z is distributed as X 1 + . . . + X n , i.e., with a probability mass function (p.m.f.) given by As a particular case, for n = 2, The p.m.f. of any subvector in (X 1 , . . . , X n ) can be obtained from the j − th iterated forward difference of S, (1.6) For bivariate Schur-constant vectors we can obtain an easy expression of the p.m.f. of the marginal. This is the next Lemma 1.1, which will be used in the next section of the paper. Lemma 1.1 If the vector (X 1 , X 2 ) is Schur-constant, the probability mass function of X 1 fulfills Proof The survival function of X 1 , for n = 2, is obtained from (1.3) when x 2 = 0, From definition, P (X 1 = x) = − S(x). Then, considering (1.8), (1.9) From (1.9), for x = 0, P (X 1 = 0) = E 1 Z+1 . Now, let us apply again the definition of the survival function so that 2 S(x) = P (X 1 = x) − P (X 1 = x + 1). Using this expression and (1.5), the desired result (1.7) is obtained.
On the basis of these definitions, we develop our analysis. The paper is organized as follows. In Section 2, we introduce the discrete Schur-constant bivariate equilibrium distribution model and provide its characteristics. We prove that in the Poisson bivariate Schur-constant equilibrium distribution model, Z is also Poisson distributed. In Section 3, we extend the analysis to the multivariate case. The paper ends with some concluding remarks.

Schur-constant Bivariate Equilibrium Distribution Model
In renewal theory, the asymptotic distribution of the age or residual life in a renewal process is known as the univariate equilibrium distribution (Cox 1962). Let X be a non negative discrete random variable representing the lifetime of a component with finite mean μ and survival function S(x) = P (X ≥ x), x ≥ 0. The (first order) equilibrium distribution is defined through the probability mass function, as (2.10) or, via the survival function, as (2.11) Expression (2.11) is well defined as survival function as long as it is a decreasing function and S * (0) = 1. This latter condition can be easily checked taking into account that E(X) = ∞ h=1 S(h). Much attention has been paid to the equilibrium distributions (of first and higher orders) associated with a given distribution function, but most studies are for continuous univariate random variables (see for instance Deshpande et al. (1986), Sunoj (2004) and the references therein). Regarding the discrete univariate case, two papers should be mentioned: Li (2011) and Willmot et al. (2005). In addition, Lefèvre and Loisel (2010) introduce a specific discrete version of the stationary-excess operator for discrete non-negative random variables. This discrete stationary-excess operator H maps any non-negative discrete random variable X to an associated discrete non-negative random variable X H whose probability mass function is Notice that the equilibrium distribution can be also obtained by applying this discrete stationary-excess operator.
In this section, we discuss a (first order) bivariate equilibrium distribution of discrete random variables. It represents in fact a family of discrete Schur-constant distributions, that we call Schur-constant bivariate equilibrium distributions. Although it has been recently analyzed by Gupta (2012) and Nair and Sankaran (2014) in a continuous setting, as far as we know, the bivariate discrete case has not previously been evaluated. The asymptotic joint distribution of age and residual life is defined by the survival function (2.12) Now we prove, in the next proposition, that this survival function corresponds to a Schurconstant survival function.

Proposition 2.1
The random vector (X * , Y * ) with survival function G given by (2.12) is Schur-constant and its marginal distribution is the equilibrium distribution of X given by (2.11).
Proof It is enough to put y = 0 in expression (2.12) to check that X * is the equilibrium distribution of X. With respect to the Schur-constancy property, from (2.12), we obtain (2.14) Then, from (2.13) and (2.14), S * (x) is always 2-monotone.
Notice that, from Proposition 2.1, the survival function S * (x), given by (2.11) is always 2-monotone although S(x) may not be. Now we derive several useful formulas. As (X * , Y * ) is Schur-constant, (2.15) and the p.m.f of the sum Z = X * + Y * , by (1.5) and (2.14), is In fact, Z turns out to be the (first) length-biased type transform of X (as defined in Lefèvre and Loisel 2013). The basic characteristics of (X * , Y * ) are available in terms of the expectations of simple functions of X and this is our next proposition.
Proposition 2.2 Let (X * , Y * ) be as in Proposition 2.1. Then, Proof Considering (2.10), we have It is a simple exercise that ( 2.20) Substituting this in (2.19), we obtain (2.17). Let us derive now a formula for the second ordinary moment of X * , We know that Considering this, summation by parts yields (2.22) By (2.17) and (2.22), we obtain formula (2.18). Castañer et al. (2015) presented simple formulas for the Pearson correlation coefficient, ρ, between any two variables in a Schur-constant vector. One of them allows to compute ρ in terms of the expectation and the variance of one of the marginal random variables. But in these Schur-constant discrete bivariate equilibrium distributions, it is convenient to relate ρ to the main characteristics of the original variable X, the building block of the model.

23)
and, in terms of the ordinary moments of X, (2.24) Proof Formula (2.23) is direct from Proposition 6.2. in Castañer et al. (2015). Let us now establish (2.24). We could evaluate E(X * Y * ) directly using (2.15). However, a simpler method consists in using (2.23) in which E(X * ) is substituted by (2.17) and V (X * ) is substituted by (2.18). Simple algebra yields the desired formula.
Then, we are ready to introduce the concept of a generator of a Schur-constant equilibrium distribution model. A non-negative discrete random variable X with equilibrium distribution (2.15) is the generator of a Schur-constant equilibrium distribution (X 1 , X 2 ) when P (X 1 ≥ x 1 , X 2 ≥ x 2 ) = S * (x 1 + x 2 ). Now, we apply these results to two specific models generated by well-known distributions, the Poisson one and the geometric one.

Poisson Schur-constant Bivariate Equilibrium Model
Let us consider that the Schur-constant equilibrium distribution model (X 1 , X 2 ) is generated by a r.v. X that follows a Poisson distribution with parameter λ; i.e., P (X = k) = e −λ λ k k! . This model is an interesting and particular one, because of its properties. Using (2.16), it is immediate to conclude that Z = X 1 +X 2 also follows a Poisson distribution of parameter λ, Proof It is well known from Panjer (1981) that the Poisson distribution is the only one that fulfills a recursive equation on its probabilities of the form The distribution of X 1 can be found using the property that X 1 is the equilibrium distribution of X. Since X is Poisson distributed, S(x) = ∞ h=x e −λ λ h h! and then being (· , ·) the incomplete Gamma function. The same result can be obtained using (1.7) and Proposition 2.4. The correlation coefficient between X 1 and X 2 in this model, computed using (2.24), is ρ = −λ 6+λ .

Geometric Schur-constant Bivariate Equilibrium Model
Let us consider a geometric r.v. X with survival function S(x) = q x . We use X as a generator of a Schur-constant equilibrium vector (X 1 , X 2 ). Then, from Proposition 2.1the r.v. X 1 is the equilibrium distribution of X. Taking into account that the mean of the generator X is μ = q/(1 − q), from (2.11), We immediately see that X 1 = d X. The variance of the generator X is V (X) = q/(1 − q) 2 , so the Pearson correlation coefficient ρ computed from (2.23) is equal to 0. This last result is an obvious one because X 1 and X 2 are independent r.v. (Castañer et al. 2015).

Schur-constant Multivariate Equilibrium Distribution Model
In this section we generalize the results of the previous section. The first order equilibrium distribution of a discrete random variable X has been used in Section 2 to define a special class of bivariate Schur-constant models. In the present section, this analysis is extended to the general multivariate case through the use of the n − th order equilibrium distribution. Let X be a non negative discrete random variable with finite mean μ and survival function S(x). The second order equilibrium distribution can be defined by its survival function (3.28) or, alternatively, by its p.m.f being μ 1:1 the first order moment of X * (the first order equilibrium distribution of X). Likewise, we can define recursively the n − th order equilibrium distribution, and p n * (x) = S (n−1) * (x + 1) μ n−1:1 , for x ∈ N 0 , (3.31) being μ i:1 < ∞ the first order moment of the i − th order equilibrium distribution of X, with μ 0:1 = μ, and S (0) * (x) = S(x). Following the same procedure as in Section 2, let us define the random vector (X 1 , . . . , X n ) with survival function P (X 1 ≥ x 1 , . . . , X n ≥ x n ) = S (n−1) * (x 1 + . . . + x n ).
(3.32) Our next proposition is the parallel of Proposition 2.1 in the bivariate case. Now, we prove that the survival function (3.32) corresponds to a Schur-constant survival function.
Proof We know from Proposition 2.1 that it is true for n = 2, because S * (x) is 2-monotone.
A Schur-constant vector of n components can be defined using a survival function not necessarily n-monotone, using the previous proposition; we say then that such vector follows a Schur-constant multivariate equilibrium distribution model. We can also extend the concept of a generator that has been previously defined for the bivariate models. A non-negative random variable X with (n − 1) − th order equilibrium distribution (3.30) is the generator of a Schur-constant multivariate equilibrium distribution (X 1 , . . . , X n ) if P (X 1 ≥ x 1 , . . . , X n ≥ x n ) = S (n−1) * (x 1 +. . .+x n ). By exchangeability, all of the X i 's have the same ordinary moments, and the same Pearson correlation coefficient. In this model we are able to obtain all of them from the ordinary moments of the original distribution X, as long as X i is the (n − 1) − th order equilibrium distribution of X. Let us first establish the following lemma.
Lemma 3.1 The set of discrete functions F (x) such that F (x) = x n−1 is given by P n (x)+C, C being a constant and P n (x) a polynomial of degree n, such that P n (x) = n r=0 a r (n)x r with a 0 (n) = 0 and a r (n), for r = 1, . . . , n, that can be computed recursively, a n (n) = 1 n , Proof Let us consider F (x) = P n (x) + C being P n (x) = n r=0 a r (n)x r whit a 0 (n) = 0, a polynomial of degree n on x. The difference of any of these functions is exactly the same and equals Isolating a n−h+1 (n) from (3.37) the statement of the lemma is obtained.
Proof By definition, x j S (i−1) * (x + 1). (3.39) From Lemma 3.1, j +1 r=1 a r (j + 1)x r = x j , where the coefficients a r (·) are given by expression (3.34). Therefore, the sum in (3.39) can be computed by applying summation by parts, (3.40) In Li (2011) an equivalent expression relating the n − th factorial moments is found.
As an example, Tables 2 and 3 include the values of the Pearson correlation coefficient in a Schur-constant multivariate equilibrium distribution model built from a Poisson distributed random variable of mean λ, as a function of n, the number of elements of the random vector. One interesting property of these models is that the probability mass function of Z = X 1 + . . . + X n is easily calculated from the survival function of X. This is our next proposition.

Conclusions
Discrete Schur-constant models (of dimension n) can be constructed using univariate survival functions that are n-monotone. This paper introduces a class of Schur-constant models that are generated by a wider class of univariate survival functions, i.e., those survival functions such that their (n − 1) − th equilibrium distribution exists.