On a Simple Estimate of the Reciprocal of the Density Function

摘要: 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 first derivative in neighborhood $p$th population quantile $\nu_p(= F^{-1} (p))$. In order to convert median or any other "quick estimator" [1] into test we must estimate its variance, for large samples asymptotic variance which depends on $1/f(\nu_p)$. Siddiqui [4] proposed estimator $S_{mn} = n(2m)^{-1}(x_{\lbrack np\rbrack+m} - x_{\lbrack np\rbrack-m+1})$ $1/f(\nu_p)$, showed it is asymptotically normally distributed and suggested that $m$ chosen $n^{\frac{1}{2}}$. this note show value minimizing mean square error (AMSE) $n^{\frac{1}{5}}$ (yielding AMSE $n^{-\frac{4}{5}}$). Our analysis similar Rosenblatt's [2] study simple function.