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$$L$$-statistics in complex survey problems. (English) Zbl 0807.62007
Summary: 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 R. S. Woodruff’s [J. Am. Stat. Assoc. 47, 635-646 (1952)] confidence intervals for population quantiles is also proved.

##### MSC:
 62D05 Sampling theory, sample surveys 62G30 Order statistics; empirical distribution functions 62G09 Nonparametric statistical resampling methods
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