On a Simple Estimate of the Reciprocal of the Density Function

Abstract

Let $x_1 < x_2 < \cdots < x_n$ be an ordered random sample of size $n$ from the absolutely continuous cdf $F(x)$ with positive density $f(x)$ having a continuous first derivative in a neighborhood of the $p$th population quantile $\nu_p(= F^{-1} (p))$. In order to convert the median or any other "quick estimator" [1] into a test we must estimate its variance, or for large samples its asymptotic variance which depends on $1/f(\nu_p)$. Siddiqui [4] proposed the estimator $S_{mn} = n(2m)^{-1}(x_{\lbrack np\rbrack+m} - x_{\lbrack np\rbrack-m+1})$ for $1/f(\nu_p)$, showed it is asymptotically normally distributed and suggested that $m$ be chosen to be of order $n^{\frac{1}{2}}$. In this note we show that the value of $m$ minimizing the asymptotic mean square error (AMSE) is of order $n^{\frac{1}{5}}$ (yielding an AMSE of order $n^{-\frac{4}{5}}$). Our analysis is similar to Rosenblatt's [2] study of a simple estimate of the density function.