Second Order Expansions for Sample Median with Random Sample Size

Abstract

In the paper, second-order Chebyshev-Edgeworth expansions are proved for the sample median when the sample size has negative binomial or discrete Pareto-like distributions. The limiting distributions of the scaled sample median depend not only on the sample size distribution but also on the chosen scaling factor. The limiting distributions are the generalized Laplace, the normal and the scaled Student distributions, depending on the random, non-random or mixed scaling factor. Second order Cornish-Fisher expansions are also derived and the negative moments of the random sample sizes are calculated.