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In this paper, the sample range from a heterogeneous exponential sample is shown to be larger than that from a homogeneous exponential sample in the sense of the star ordering. Then, by using this result, some equivalent characterizations of stochastic comparisons of sample ranges with respect to various stochastic orders are established. In this process, two open problems mentioned in Mao and Hu (2010) [16] are solved. The main results established here extend and strengthen several known results in the literature including those of Khaledi and Kochar (2000) [8] , Zhao and Li (2009) [22] and Genest et al. (2009) [7] .

A sufficient condition for comparing convolutions of heterogeneous exponential random variables in terms of right spread order is established. As a consequence, it is shown that a convolution of heterogeneous independent exponential random variables is more skewed than that of homogeneous exponential random variables in the sense of NBUE order. This gives a new insight into the distribution theory of convolutions of independent random variables. A sufficient condition is also derived for comparing such convolutions in terms of Lorenz order.

Let R"n be the range of a random sample X"1,...,X"n of exponential random variables with hazard rate @l. Let S"n be the range of another collection Y"1,...,Y"n of mutually independent exponential random variables with hazard rates @l"1,...,@l"n whose average is @l. Finally, let r and s denote the reversed hazard rates of R"n and S"n, respectively. It is shown here that the mapping [email protected]?s(t)/r(t) is increasing on (0,~) and that as a result, R"n=X"("n")-X"("1") is smaller than S"n=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 X"1,...,X"n form a random sample from a continuous distribution while Y"1,...,Y"n are mutually independent lifetimes with proportional hazard rates.

In this paper, we introduce a new copula-based dependence order to compare the relative degree of dependence between two pairs of random variables. Relationship of the new order to the existing dependence orders is investigated. In particular, the new ordering is stronger than the partial ordering, more monotone regression dependence as developed by Avérous et al. [J. Avérous, C. Genest, S.C. Kochar, On dependence structure of order statistics, Journal of Multivariate Analysis 94 (2005) 159–171]. Applications of this partial order to order statistics, k-record values and frailty models are given.