On some non parametric estimators of the quantile density function for a stationary associated process

Abstract

In this article, we consider smooth estimators for the quantile density function (qdf) for a sequence $\{X_n,n\ge 1\}$ of stationary non negative associated random variables with a common marginal distribution function. The qdf is given by $q(u)=Q^{\prime }(u),u\in (0,1)$, $Q(u)$ representing the corresponding quantile function. The smooth estimators of $q(u)$ considered here are adapted from those of $Q(u)$ considered in Chaubey, Dewan, and Li (2021). A few asymptotic properties of these estimators are established parallel to those in the i.i.d. case. A numerical study comparing the mean squared errors of various estimators indicates the advantages and a few limitations of various estimators. The smoothing parameter is selected based on the BCV and RLCV (a variation of likelihood cross-validation) criteria. It is concluded, based on the numerical studies, that the RLCV criterion may produce over-smoothing, hence BCV criterion may be preferable. The numerical studies also suggest that, overall, the estimator proposed by Soni, Dewan, and Jain (2012) seems to have some advantage over the other estimators considered in this article.

Keywords:

• Associated sequence
• kernel smoothing
• quantile function
• quantile density function

Acknowledgments

The first author would like to acknowledge the partial support for this research from NSERC of Canada through a Discovery Grant as well as the facilities provided at the Indian Statistical Institute, Delhi Center while he was on sabbatical leave. The second author thanks the Science and Engineering Research Board (SERB), Department of Science and Technology for its research grant. The authors are also thankful to the reviewers for their constructive comments that have improved the presentation. The support from the Hainan Normal University through the research grant NSFC:11261016 to Jun Li is also acknowledged.

Disclosure statement

The authors declare that there are no relevant financial or non- financial competing interests to report.