SOME STATISTICAL PROPERTIES OF ESTIMATORS OF DENSITY AND DISTRIBUTION FUNCTIONS
Hajime Yamato · Bulletin of Mathematical Statistics · 1972
and introduction.Let X1, X2, ••• , X, be a random sample of size n from a population with an unknown probability density function f(x).The estimator of the form f(x) = 1 " -E w n(x-X,) of the unknown density f(x) based on this sample, where wn(y)> 0 n on R1 and .1 w"(y)dy =1, is shown to be not unbiased for any probability density function.For the class of all continuous probability density functions, the estimator fn(x) is asymptotically unbiased if and only if the sequence of functions { wn(y)dy} converges to the unit distribution function except for the origin.Furthermore for this class the consistency and the asymptotic normality of the estimator fn(x) is discussed.In case f(x) is symmetric around zero, we propose an estimator f,i(x)= 1 -2 { f n(x)+ f"(-x)} .Then the variance of the proposed estimator fn(x) is asymptotically half of the variance of the estimator fm(x) at a non-zero point x of continuity.We also consider the integration of our estimator Fn(x) as an estimator of the distribution function, which is compared with the empirical distribution function Pnqx).We propose an estimator P"(x)=-1-{F,(x)+1-F"(-x)1 and the corrected empirical distribution function P,,*(x)= -1-{F:(x)+1-F,*(-x-0)1 for all (absolutely) continuous symmetric distribution functions.The mean square error of P:(x) is smaller than half of the mean square error of F,*(x) for x with F(x) 0 or 1, and the estimator P"(x) is asymptotically at least as good as F"(x) .The density estimator of the form :was introduced by Rosenblatt [8], and several authors have discussed the statistical properties of the estimator f,(x).Concerning the unbiasedness of density estimators, the following result is obtained by Rosenblatt [8] : let a function S(y: x1, ••• , x") .0 be Borel measurable in (y, x1, ••• , x") and symmetric in (x1, --• , x").Then there are