Asymptotic Analysis of Tail Probabilities Based on the Computation of Moments

Abstract

Choudhury and Lucantoni recently developed an algorithm for calculating moments of a probability distribution by numerically inverting its moment generating function. They also showed that high-order moments can be used to calculate asymptotic parameters of the complementary cumulative distribution function when an asymptotic form is assumed, such as $F^c(x) \sim \alpha x^\beta e^{-\eta x}$ as $x \rightarrow \infty$. Moment-based algorithms for computing asymptotic parameters are especially useful when the transforms are not available explicitly as in models of busy periods or polling systems. Here we provide additional theoretical support for this moment-based algorithm for computing asymptotic parameters and new refined estimators for the case $\beta \neq 0$. The new refined estimators converge much faster (as a function of moment order) than the previous estimators, which means that fewer moments are needed, thereby speeding up the algorithm. We also show how to compute all the parameters in a multiterm asymptote of the form $F^c(x) \sim \sum^m_{k = 1} \alpha_k x^{\beta - k + 1} e^{-\eta x}$. We identify conditions under which the estimators converge to the asymptotic parameters and we determine rates of convergence, focusing especially on the case $\beta \neq 0$. Even when $\beta = 0$, we show that it is necessary to assume the asymptotic form for the complementary distribution function; the asymptotic form is not implied by convergence of the moment-based estimators alone. In order to get good estimators of the asymptotic decay rate $\eta$ and the asymptotic power $\beta$ when $\beta \neq 0$, a multiple-term asymptotic expansion is required. Such asymptotic expansions typically hold when $\beta \neq 0$, corresponding to the dominant singularity of the transform being a multiple pole ($\beta$ a positive integer) or an algebraic singularity (branch point, $\beta$ noninteger). We also show how to modify the moment generating function in order to calculate asymptotic parameters when all moments do not exist (the case $\eta = 0)$.