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上确界误差下回归函数的多项式估计AI 解读

Regression with the error measured by the supremum norm is considered. Analytic functions on [0, 1] and functions with a bounded $r$th derivative are considered as functions to be estimated. It is assumed that the experimenter chooses the points at which the observations are taken. Polynomial and piecewise polynomial estimates are considered. Asymptotic and nonasymptotic bounds for the error are obtained.

渐近有效的递归估计AI 解读

Stochastic approximation procedures were shown by Sakrison to become asymptotically efficient estimators when used to minimize the Kullback-Leibler information, if certain conditions hold. Further results in this direction were obtained by Nevel'son and Has'minskij. This paper gives, first, alternative conditions for convergence and, secondly, shows that, under weaker conditions, asymptotic optimality is obtained by a modified stochastic approximation procedure. The modified…

论 Cramér–Rao 不等式AI 解读

Sufficient conditions for the Cramer-Rao inequality are formulated which do not impose any requirement on the estimator and which are independent of the choice of the densities. A hereditary property is proved for these conditions and the attainability of the lower bound is studied.

渐近有效的随机逼近：RM 情形AI 解读

Anbar (1971) and, independently, Abdelhamid (1971) have shown that if the density $g$ of the errors of estimates of function values is known, a transformation of observations leads to stochastic approximation methods which under mild conditions produce asymptotically efficient estimators (the first author considers the RM case, the second the RM and KW cases). This paper treats the RM case and shows that the same asymptotic result can be achieved without the knowledge of the…

密度对数导数的估计AI 解读

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 $V_1 + Z_1, \cdots, V_n + Z_n$…

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