Two new estimators of distribution functions

A.K.Md.E. Saleh and P.J. Farrell (2009). Two new estimators of distribution functions. Journal of Statistical Research, Vol. 43, No. 1, pp. 109-115.

Abstract

This paper considers the estimation of a distribution function $F_{X}(x)$ based on a random sample $X_{1}, X_{2}, \ldots, X_{n}$ when the sample is suspected to come from a close-by distribution $F_{0}(x)$ . Two new estimators, namely $F_{n}^{PT}(x)$ and $F_{n}^{S}(x)$ are defined and compared with the ``empirical distribution function'', $F_{n}(x)$ , under local departure; that is $F_{X}(x) = F_{0}(x) + n^{-1/2} \delta$ , where $\max_{x} \vert F_{X}(x) - F_{0}(x) \vert \le n^{-1/2} \delta$ . In this case, we show that $F_{n}^{S}(x)$ is superior to $F_{n}^{PT}(x)$ in the neighbourhood of $F_{0}(x)$ .