Confidence intervals for functions of quantiles using linear combinations of order statistics

SETH MICHAEL STEINBERG. Confidence Intervals for Functions of Quantiles Using Linear Combinations of Order Statistics. (Under the direction of C.E. DAVIS) Estimators for quantiles based on linear combinations of order statistics have been proposed by Harrell and Davis (1982) and Kaigh and Lachenbruch (1982). Both estimators have been demonstrated to be at least as efficient for small sample point estimation as an ordinary sample quantile estimator based on one or two order statistics. Distribution free confidence intervals for quantiles can be constructed using either of the two approaches. By means of a simulation study, these confidence intervals have been compared with several other methods of constructing confidence intervals for quantiles in small samples. For the median, the Kaigh and Lachenbruch method performed the best overall. For other quantiles, no method performed better than the method which uses pairs of order statistics. The interquantile difference is often useful as a measure of dispersion. Both the Harrell-Davis and Kaigh-Lachenbruch estimators are modified to estimate interquantile differences. Theoretical developments needed to establish large-sample use of the normal distribution for these estimators are presented. Both of these methods are used to form pivotal quantities with asymptotic normal distributions, and thus are readily used for construction of confidence intervals. The poi nt estimators of i nterquantil e di fference are compared