Dispersion and variability orders of mixture exponential distributions and their sample spacings, and some associated characterizations

Suppose we have two sets of exponential random variables with the first set having a common hazard rate and the second having different hazard rate parameters. In this work, we then discuss star, convex transform, right spread and dispersive orders between these two sets of random variables. The results enable us to compare sample spacings in terms of dispersion and variability orders. We also establish some characterizations results between an exponential random variable and a mixture of exponential random variables, and apply these results for the comparisons of sample spacings for illustration. These characterization results complete and extend some of the known results in the literature.