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

In this article, we consider smooth estimators for the quantile density function (qdf) for a sequence {Xn,n≥1} of stationary non negative associated random variables with a common marginal distribution function. The qdf is given by q(u)=Q′(u),u∈(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.