Abstract

It is well known that the classical quantile enjoys many desirable prop

erties and has solid practical applications under absolutely continuous populations. In the discrete setting, however, most of these properties fail to hold, such

as the consistency and asymptotic normality of the sample classical quantile,

which in turn may cause inaccurate or even misleading conclusions in practical

inferential problems. In this article, we introduce two new definitions of quantile, called the upper- and lower-quantiles, constructed via linear interpolation

technique to eliminate discontinuities of classical quantile function under discrete

populations. The sample counterparts of upper- and lower-quantiles enjoy appealing theoretical properties, such as consistency and asymptotic normality, for

both discrete and absolutely continuous populations. More importantly, these

tools could provide valid inferences for statistical problems involving quantiles in

discrete populations, which is firmly supported by both theoretical proofs and

listed in the alphabetic order.

empirical applications. All theoretical results presented in this article are validated through extensive simulation studies, and potential applications of the new

concepts are demonstrated with an illustrative example.

Key words and phrases: Confidence or prediction interval; Discrete population; Upper- and lower-quantiles

Information

Preprint No.SS-2025-0436
Manuscript IDSS-2025-0436
Complete AuthorsXiaolin Chen, Ruiyun Zhang, Min-Ge Xie, Xiaodong Yan
Corresponding AuthorsXiaolin Chen
Emailsxlchen@amss.ac.cn

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Acknowledgments

Chen’s research is supported by the National Social Science Fund of China

Supplementary Materials

The Supplementary Material includes the bootstrap procedure, further discussion about upper- and lower-quantiles, the technical proofs, and addi-

tional numerical experiments.

Supplementary materials are available for download.