On the Increasing Convex Order of Relative Spacings of Order Statistics

Relative spacings are relative differences between order statistics. In this context, we extend previous results concerning the increasing convex order of relative spacings of two distributions from the case of consecutive spacings to general spacings. The sufficient conditions are given in terms of the expected proportional shortfall order. As an application, we compare relative deprivation within some parametric families of income distributions

Dependence Among Order Statistics for time-transformed exponential models

Let X1, ..., Xn be a random vector distributed according to a time-transformed exponential model. This is a special class of exchangeable models, which, in particular, includes multivariate distributions with Schur-constant survival functions. Let for 1 i n, Xi:n denote the corresponding ith-order statistic. We consider the problem of comparing the strength of dependence between any pair of Xi’s with that of the corresponding order statistics. It is in particular proved that for m = 2, ..., n, the dependence of X2:m on X1:m is more than that of X2 on X1 according to more stochastic increasingness (positive monotone regression) order, which in turn implies that X1:m, X2:mº is more concordant than X1, X2. It will be interesting to examine whether these results can be extended to other exchangeable models

On the range of heterogeneous samples

Let Rn be the range of a random sample X1,...,Xn of exponential random variables with hazard rate [lambda]. Let Sn be the range of another collection Y1,...,Yn of mutually independent exponential random variables with hazard rates [lambda]1,...,[lambda]n whose average is [lambda]. Finally, let r and s denote the reversed hazard rates of Rn and Sn, respectively. It is shown here that the mapping t|->s(t)/r(t) is increasing on (0,[infinity]) and that as a result, Rn=X(n)-X(1) is smaller than Sn=Y(n)-Y(1) in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X(n) is seen to be more stochastically increasing in X(1) than Y(n) is in Y(1). In other words, the pair (X(1),X(n)) is more dependent than the pair (Y(1),Y(n)) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X1,...,Xn form a random sample from a continuous distribution while Y1,...,Yn are mutually independent lifetimes with proportional hazard rates.Cebysev's sum inequality Copula Dispersive ordering Exponential distribution Likelihood ratio ordering Monotone regression dependence Order statistics Proportional hazards model Range