Abstract

In many practical sampling situations, units are selected with probabilities proportional to predetermined weight functions that depend on their underlying values. A classical example, discussed by Cox, arises in textile manufacturing, where fibers are sampled with probabilities proportional to their lengths. Similar selection mechanisms appear in wildlife surveys, in which larger animals or groups are more likely to be observed, as well as in customer satisfaction surveys conducted in shopping malls, where individuals who spend more time shopping have a higher chance of being included in the sample.
　This type of selection mechanism is commonly referred to as length-biased sampling, or more generally as size-biased sampling, and it is widely encountered across a variety of scientific and socio-economic applications. Under the assumption of independent and identically distributed observations, extensive theoretical work has been devoted to the development and analysis of estimators within the length-biased sampling framework.
　The present study considers length-biased sampling from a finite population and places particular emphasis on the estimation of quantiles. Quantiles play a central role in statistical analysis as robust measures of location that are less sensitive to outliers than the mean. They are especially useful in the analysis of skewed or heavy-tailed socio-economic data, and quantiles, as well as functions thereof, are routinely published in official statistics.
Within this context, we investigate the asymptotic properties of the sample quantile estimator under length-biased sampling from a finite population. In particular, we derive a Bahadur representation for the quantile estimator, which provides a precise stochastic approximation and serves as a fundamental tool for establishing asymptotic distributions and assessing the estimator’s accuracy. The results contribute to a deeper theoretical understanding of quantile estimation under informative sampling designs and offer a foundation for further methodological developments and practical applications.