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- W2000704862 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 $nvarepsilon_n uparrow + infty, sum n^{-1}varepsilon_n^{-1}|Z_n| < + infty$ 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." @default.
- W2000704862 created "2016-06-24" @default.
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- W2000704862 date "1973-05-01" @default.
- W2000704862 modified "2023-09-25" @default.
- W2000704862 title "Estimation of the Derivative of the Logarithm of a Density" @default.
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- W2000704862 doi "https://doi.org/10.1214/aos/1176342424" @default.
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