$L$-Statistics in Complex Survey Problems

Abstract

We study linear combinations of order statistics ($L$-statistics) in survey problems with a stratified multistage sampling design. Two general types of $L$-statistics are considered: smooth $L$-statistics with weights generated by a smooth function and nonsmooth $L$-statistics (sample quantiles). The trimmed sample mean, the decile mean and variance, the sample Lorenz curve and the sample Gini's family parameters are examples of smooth $L$-statistics or functions of smooth $L$-statistics used in survey problems. It is shown that under weak conditions the smooth $L$-statistics are asymptotically normal and their asymptotic variances can be consistently estimated by jackknifing. For the sample quantiles, their asymptotic normality requires more conditions on the finite population distribution functions. Consistent estimators for the asymptotic variances of the sample quantiles are derived. Asymptotic validity of the Woodruff's confidence intervals for population quantiles is also proved.