Estimation of the Derivative of the Logarithm of a Density

Abstract

Suppose $g$ is a probability density on $R = (-\infty, +\infty)$ with a continuous derivative and with $I(g) = \int (g'/g)^2g < + \infty$. Suppose $\{\varepsilon_n\}$ is a sequence of positive numbers converging to zero and $V_i, Z_i$ are random variables, $Z_i \rightarrow 0$ with probability one and $V_n$ is conditionally (given $V_1, \cdots, V_{n-1}, Z_1, \cdots, Z_n$) distributed according to $g$. Estimates $h_n$ of $g'/g$ are constructed, which are based on $V_1 + Z_1, \cdots, V_n + Z_n$ and have the following properties. For almost all $\omega, h_n(\omega, \bullet) \rightarrow g'/g$ on $\{t; g(t) > 0\}$. If $n\varepsilon_n \uparrow + \infty, \sum n^{-1}\varepsilon_n^{-1}|Z_n| < + \infty$ a.e. then for almost all $\omega, h_n(\omega, \bullet) \rightarrow g'/g$ in $L_2(g)$ and $\int h_n^2 dG_n \rightarrow I(g)$ where $G_n$ is the empirical distribution function of $V_1 + Z_1, \cdots, V_n + Z_n$. The results on the pointwise and $L_2(g)$ convergences hold also if $h_n$ are replaced by $h_n(\omega, \nu + \eta_n(\nu, \omega))$ provided $\eta_n$ are small and preserve the measurability of the estimators.