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摘要暂缺

有偏抽样模型中经验分布的大样本理论AI 解读

Vardi (1985a) introduced an $s$-sample model for biased sampling, gave conditions which guarantee the existence and uniqueness of the nonparametric maximum likelihood estimator $\mathbb{G}_n$ of the common underlying distribution $G$ and discussed numerical methods for calculating the estimator. Here we examine the large sample behavior of the NPMLE $\mathbb{G}_n$, including results on uniform consistency of $\mathbb{G}_n$, convergence of $\sqrt n (\mathbb{G}_n - G)$ to a…

摘要暂缺

乘积极限估计量的重度删失极限定理AI 解读

A key identity for the product-limit estimator due to Aalen and Johansen (1978) and Gill (1980) is shown to be a consequence of the exponential formula of Doleans-Dade (1970). The basic counting processes in the censored data problem are shown to converge jointly to Poisson processes under "heavy-censoring": $G_n \rightarrow_d \delta_0$, but $n(1 - G_n) \rightarrow \alpha$ where $G_n$ is the censoring distribution. The Poisson limit theorem for counting processes implies…

参数—非参数模型中的信息与渐近效率AI 解读

Asymptotic lower bounds for estimation of the parameters of models with both parametric and nonparametric components are given in the form of representation theorems (for regular estimates) and asymptotic minimax bounds. The methods used involve: (i) the notion of a "Hellinger-differentiable (root-) density", where part of the differentiation is with respect to the nonparametric part of the model, to obtain appropriate scores; and (ii) calculation of the "effective score"…

乘积极限估计量的渐近最优性AI 解读

The product limit estimator due to Kaplan and Meier (1958) is well-known to be the nonparametric maximum likelihood estimator of a distribution function based on censored data. It is shown here that the product limit estimator is an asymptotically optimal estimator in two senses: in the sense of a Hajek-Beran type representation theorem for regular estimators; and in an asymptotic minimax sense similar to the classical result for the uncensored case due to Dvoretzky, Kiefer,…

删失数据下生存曲线的置信带AI 解读

For arbitrarily right-censored data, the Kaplan-Meier product-limit estimator Ŝ 0N provides a nonparametric estimate of the survival function S 0 = 1 − F 0 . We provide large-sample simultaneous confidence bands for S 0 , centred at Ŝ 0N . The derivation uses the weak convergence of N½ { Ŝ 0N ( t ) − S 0 ( t )}, on a finite interval, to a Gaussian process, a theorem of Breslow & Crowley (1974), and transforms both the time and space axes of the limiting process to achieve a…

与经验分布函数线性界有关的分布AI 解读

$X_1, \cdots, X_n$ are i.i.d. Uniform (0, 1) rv's with empirical df $\Gamma_n$ and order statistics $0 < U_1 < \cdots < U_n < 1.$ Define random variables $U_\ast, i_\ast$ (for $n \geqq 2$) by $\max_{1\leqq i \leqq n - 1} \frac{U_{i + 1}}{i} = \frac{U_{i_\ast} + 1}{i_\ast}, U_\ast = U_{i_\ast + 1};$ $i_\ast + 1$ is the (random) index of the order statistic at which the maximum is achieved and $U_\ast$ is the value of that order statistic. The distributions of $(U_\ast, i_\ast)$…

次序统计量函数的重对数律AI 解读

A general law of the iterated logarithm for linear combinations of order statistics is proved. The key tools are (1) iterated logarithm convergence of the uniform empirical process $U_n$ in $\rho_q$-metrics due to B. R. James and (2) almost sure "nearly linear" bounds for the empirical distribution function. A law of the iterated logarithm for the quantile process is also established.

Glivenko–Cantelli定理与次序统计量函数的强大数定律AI 解读

A strengthened version of the Glivenko-Cantelli theorem for the uniform empirical distribution function is proved. The strengthened Glivenko-Cantelli theorem is used to establish strong laws of large numbers for linear functions of order statistics.