The Student’s [formula omitted] approximation to distributions of pivotal statistics from ranked set samples

Abstract

For data collected using ranked set sampling, pivotal statistics are commonly used in inferential procedures for testing the population mean, and their asymptotic distributions are used as surrogates of the underlying true distributions. However, the sample size of a ranked set sample (RSS) is often small so that the distribution of its pivotal statistic can deviate much from the limiting normality. In this paper, we propose to approximate the distribution by the Student’s t distribution, of which the number of degrees of freedom (DF) is estimated from the data. We consider three estimators of the DF, two based on the Welch-type approximation (Welch, 1947) and the third simply given by the difference between the total sample size and the set size of the RSS. We numerically compare the corresponding approximate t distributions with the asymptotic normal distribution via simulation in two aspects: (i) the approximation error; and (ii) the coverage probability. We also apply the proposed Student’s t approximation to tree height data in Platt et al. (1988) and Chen et al. (2003). Our results show that all the three approximate t distributions seem to be consistently better than the asymptotic normal distribution for the RSS under both perfect and imperfect ranking. We further give recommendations to practitioners based on the relative performance of the three DF estimators.