Computing the asymptotic expansion of the median of the erlang distribution

Abstract

We describe an algorithm for computing a large number of coefficients in the asymptotic expansion of the median of the Erlang distribution. In particular, in this paper we present the values of the first sixty coefficients which allow us to assess the importance of the higher-order terms in the behavior of the partial sums of that asymptotic expansion. As a consequence, we provide tight bounds for the median of the Erlang distribution and we also see that a conjecture concerning the complete monotonicity of a sequence of medians of the Erlang distributions is supported by numerical results.