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经验似然比置信区域AI 解读

An empirical likelihood ratio function is defined and used to obtain confidence regions for vector valued statistical functionals. The result is a nonparametric version of Wilks' theorem and a multivariate generalization of work by Owen. Cornish-Fisher expansions show that the empirical likelihood intervals for a one dimensional mean are less adversely affected by skewness than are those based on Student's $t$ statistic. An effective method is presented for computing…

通过清空盒子构造 πPS 抽样设计AI 解读

We present a simple but universal technique for the construction of $\pi PS$ sampling designs. A tool that is used in the construction consists of playing a game in which objects are removed from $N$ boxes, $n$ at a time, and at most one from each box at a time. Necessary and sufficient conditions on $N, n$ and the contents of the boxes are established such that all boxes can be emptied by this process. It is shown that every $\pi PS$ design can be derived from such a game.…

极值分布指数的矩估计量AI 解读

We extend Hill's well-known estimator for the index of a distribution function with regularly varying tail to an estimate for the index of an extreme-value distribution. Consistency and asymptotic normality are proved. The estimator is used for high quantile and endpoint estimation.

极值指数估计与高分位数估计AI 解读

This paper consists of two parts. An easy proof is given for the weak consistency of Pickands' estimate for the main parameter of an extreme-value distribution. Moreover, further natural conditions are given for strong consistency and for asymptotic normality of the estimate. Next a large quantile of a distribution is estimated by a combination of extreme or intermediate order statistics. This leads to an asymptotic confidence interval.

爆炸性自回归中残差经验过程的弱收敛AI 解读

This paper proves the weak convergence of the residual empirical process in an explosive autoregression model to the Brownian bridge. As an application the Kolmogorov-Smirnov goodness-of-fit test for testing that the errors have a specified distribution is shown to be asymptotically distribution-free.

若干长程依赖序列的经验过程及其在U统计量中的应用AI 解读

Let $(X_j)^\infty_{j = 1}$ be a stationary, mean-zero Gaussian process with covariances $r(k) = EX_{k + 1} X_1$ satisfying $r(0) = 1$ and $r(k) = k^{-D}L(k)$ where $D$ is small and $L$ is slowly varying at infinity. Consider the two-parameter empirical process for $G(X_j),$ $\bigg\{F_N(x, t) = \frac{1}{N} \sum^{\lbrack Nt \rbrack}_{j = 1} \lbrack 1\{G(X_j) \leq x\} - P(G(X_1) \leq x) \rbrack; // -\infty < x < + \infty, 0 \leq t \leq 1\bigg\},$ where $G$ is any measurable…

Lᵣ可微分布族中Cramér–Rao下界的达到AI 解读

A rigorous proof is presented that global attainment of the Cramer-Rao bound is possible only if the underlying family of distributions is exponential. The proof is placed in the context of $\mathbb{L}_r(P_\vartheta)$-differentiability, requiring strong differentiability in $\mathbb{L}_r(P_\vartheta)$ of the $r$th root of the likelihood ratio relative to $P_\vartheta$.

一类对数似然比统计量与其渐近卡方分布之间的随机不等式AI 解读

For decomposable covariance selection models, stochastic inequalities which relate the null distribution of the log-likelihood ratio statistic to its asymptotic $\chi^2$ distribution are obtained. The implications are twofold: First, the null distribution of the log-likelihood ratio statistic is seen to be stochastically larger than its asymptotic $\chi^2$ distribution. Extremely large samples apart, for the $\chi^2$ approximation to be valid, a deflation of the…

列联表卡方独立性检验的检验水平界AI 解读

Bounds are obtained on the limiting size of the level-$\alpha \chi^2$-test of independence in a contingency table, as the sample size increases. The situations considered include (a) sampling with one or both sets of marginal totals random, (b) performing the test with or without the continuity correction and (c) with or without conditioning on the event $\mathscr{E}_k$ that the minimum estimated expected cell count is greater than a given $k \geq 0$. Bounds for both the…

三维列联表严格可折叠性的交并检验AI 解读

The structure of the strict collapsibility model in a three-dimensional contingency table is investigated with the goal of developing more powerful tests. After showing that strict collapsibility can be written as a finite union of distinct models, an intersection union test is proposed which is at least as powerful asymptotically as the likelihood ratio test. Intersection union tests for strong collapsibility and marginal independence are also suggested.

多变量位置与协方差的 S 估计和 M 估计之间的关系AI 解读

We discuss the relation between $S$-estimators and $M$-estimators of multivariate location and covariance. As in the case of the estimation of a multiple regression parameter, $S$-estimators are shown to satisfy first-order conditions of $M$-estimators. We show that the influence function IF $(\mathbf{x; S}, F)$ of $S$-functionals exists and is the same as that of corresponding $M$-functionals. Also, we show that $S$-estimators have a limiting normal distribution which is…

凹性与估计AI 解读

Simplified conditions are given for consistency and asymptotic normality of $M$-estimates derived by maximization of averages of independent identically distributed random concave functions. Applications are made to maximum likelihood estimation.

极小极大偏差稳健回归AI 解读

This paper considers the problem of minimizing the maximum asymptotic bias of regression estimates over $\varepsilon$-contamination neighborhoods for the joint distribution of the response and carriers. Two classes of estimates are treated: (i) $M$-estimates with bounded function $\rho$ applied to the scaled residuals, using a very general class of scale estimates, and (ii) bounded influence function type generalized $M$-estimates. Estimates in the first class are obtained…

一个回归型估计问题AI 解读

Let $X_1, \cdots, X_n$ be random vectors that take values in a compact set in $R^d, d = 1, 2$. Let $Y_1, \cdots, Y_n$ be random variables (the responses) which conditionally on $X_1 = x_1, \cdots, X_n = x_n$ are independent with densities $f(y \mid x_i, \theta(x_i)), i = 1, \cdots, n$. Assuming that $\theta$ lies in a sup-norm compact space $\Theta$ of real-valued functions, an $L_1$-consistent estimator (of $\theta$) is constructed via empirical measures. The rate of…

回归函数的非参数估计AI 解读

A data dependent method of estimating a regression function is proposed here. The model is allowed to be heteroscedastic. Applications to piecewise polynomial, spline, orthogonal series, kernel and nearest neighbor methods are discussed.

对称单峰密度及其众数的贝叶斯方法AI 解读

Bayes solutions for i.i.d. sampling from a decreasing density on the half line and for i.i.d. sampling from a symmetric unimodal density with an unknown mode are given. Posterior quantities are obtained as finite sums; Monte Carlo methods based on sampling finite Markov chains are developed for their evaluation.

In this paper we shall investigate some nonasymptotic properties of the Grenander estimator of a decreasing density $f$. This estimator is defined as the slope of the smallest concave majorant of the empirical c.d.f. It will be proved that its risk, measured with $\mathbb{L}^1$-loss, is bounded by some functional depending on $f$ and the number $n$ of observations. For classes of uniformly bounded densities with a common compact support, upper bounds for the functional are…

删失下 Kaplan–Meier 及其他最大似然估计量的可容许性AI 解读

For the nonparametric estimation of a survival function when censoring is present, the Kaplan-Meier estimator is often used. The admissibility of this estimator and other related maximum likelihood estimators is demonstrated. This is done by reducing the problem to one involving just the multinomial distribution and then using the stepwise Bayes technique to prove admissibility.

具有乘法结构的计数过程模型估计AI 解读

It is assumed that we observe one realization of an $r$-dimensional counting process with intensities that are products of a predictable weight process, a common function of time and parameters $\beta_i, i = 1, \cdots, r$, which distinguish the components. Provided the realization observed brings increasing information on $\beta$ as the observed time grows, strong consistency of a partial ML estimator is proved. For such realizations it is also proved that the estimate,…

爆炸性自回归过程的 BootstrapAI 解读

Asymptotic validity of the bootstrap is established for the least squares estimate of the parameter of an explosive first-order autoregressive process. It is noted that nonnormal limit distributions are obtained for both the traditional and the bootstrap estimates. The theoretical bootstrap validity results are supported by appropriate simulation.

回归模型自助法的 Edgeworth 展开AI 解读

The asymptotic performance of the bootstrap in linear regression models is studied. Edgeworth expansions show that asymptotically, the bootstrap is always at least as good as, and in some cases better than, the classical normal approximation. The performances of both the bootstrap and the normal approximation depend on the rate of increase in the elements of the design matrix.

经验函数的自助法AI 解读

We develop the complete bootstrapped parallel to the asymptotic theory of weighted empirical and quantile processes. Utilizing this parallel theory, we present a general body of techniques to establish the asymptotic validity of the bootstrap method of constructing confidence bands for statistical functions. These techniques are demonstrated to be applicable to the construction of asymptotic bootstrap confidence bands for a variety of concrete functions.

步态分析与 BootstrapAI 解读

This paper is about random coefficient trigonometric regression models and their use in gait analysis. Here gait analysis means free-speed walking on a level surface. Our study focuses on bootstrap-based prediction regions for the angular rotation curves of test children, when the relevant training data are gathered from normal children of comparable ages. Considerations that led to our choice of model and use of the bootstrap are given. Prediction regions and empirical…

自相似过程的有效参数估计AI 解读

Asymptotic normality of the maximum likelihood estimator for the parameters of a long range dependent Gaussian process is proved. Furthermore, the limit of the Fisher information matrix is derived for such processes which implies efficiency of the estimator and of an approximate maximum likelihood estimator studied by Fox and Taqqu. The results are derived by using asymptotic properties of Toeplitz matrices and an equicontinuity property of quadratic forms.