Asymptotic distributions of estimators of the derivative of a density and the score function (Q1066587)

To estimate an unknown density f and its derivative f' from a real random sample the second author [Commun. Stat. 2, 493-506 (1973; Zbl 0283.62040)] and \textit{B. K. Kim} and the second author [Commun. Stat., Theory Methods A9, 697-709 (1980; Zbl 0437.62043)] proposed simple functions \(f_ n\) and \(g_ n\) of the order statistics (n is the sample size). In this paper the authors prove, under different assumptions, the asymptotic normality of \(g_ n(x)\) and \(g_ n(x)/f_ n(x)\) as estimates of f'(x) and f'(x)/f(x), respectively.