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保序众数回归的最大似然估计AI 解读

For each $t$ in an index set $T$, let $P_t$ be a probability measure with mode $M(t)$. In this paper we consider a maximum likelihood nonparametric estimator $\hat{M}_n(t)$ of $M(t)$ subject to the constraint that $\hat{M}_n(\cdot)$ be isotonic with respect to an order on $T$. The estimator is a solution to a minimization problem with zero-one loss. The estimator is not a max-min or min-max representation of "naive" estimators. Naive modal estimators are used but they are…

多元 M 估计的小样本渐近展开AI 解读

Asymptotic expansions are derived for the densities of multivariate $M$-estimates. The expansion is based on a saddlepoint technique and yields good accuracy in the tails for small sample sizes. Numerical results are given for robust estimation of location and scale and these are compared with known Monte Carlo results.

论自适应估计AI 解读

We simplify a general heuristic necessary condition of Stein's for adaptive estimation of a Euclidean parameter in the presence of an infinite dimensional shape nuisance parameter and other Euclidean nuisance parameters. We derive sufficient conditions and apply them in the construction of adaptive estimates for the parameters of linear models and multivariate elliptic distributions. We conclude with a review of issues in adaptive estimation.

自相关、自回归与自回归逼近AI 解读

Theorems are proved relating to the rate of almost sure convergence of autocovariances, and hence autocorrelations, to their true values. These rates are uniform in the lag up to some order $P(T)$, increasing with $T$. The key assumption is that the process is stationary and the best linear predictor is the best predictor. In particular for an ARMA process and $P(T) = O\{(\ln T)^a\}, a < \infty$, the rate is $O\{(\ln \ln T/T)^{1/2}\}$. These results are used to discuss…

L 估计、R 估计与极小极大性质AI 解读

Let $\{X_i\}$ be a sample from $F(x - \theta)$ where $F$ is in a class $\mathscr{F}$ of symmetric distributions on the line and $\theta$ is the location parameter to be estimated. Huber has shown that maximum likelihood estimation has a minimax property over a convex $\mathscr{F}$. Here a simple convex $\mathscr{F}$ is given for which neither $L$- nor $R$-estimation has the minimax property. In particular, this example shows that a recent assertion concerning $L$-estimation…

Schur 凹随机变量的矩形与椭球概率不等式AI 解读

It is shown that if the density $f(\mathbf{x})$ of $\mathbf{X} = (X_1, \cdots, X_n)$ is Schur-concave, then (1) $P(|X_i| \leq a_i, i = 1, \cdots, n)$ is a Schur-concave function of $(\phi(a_1), \cdots, \phi(a_n))$, and (2) $P\{\Sigma(X_i/a_i)^2 \leq 1\}$ is a Schur-concave function of $(\phi(a^2_1), \cdots, \phi(a^2_n))$, where $\phi; \lbrack 0, \infty) \rightarrow \lbrack 0, \infty)$ is any increasing and convex function. By letting $\phi(a) = a$, (1) implies that $P(|X_i| \leq a_i, i = 1, \cdots, n) \leq P(|X_i| \leq \bar{a}, i = 1, \cdots, n)$…

平稳时间序列的一个刻画问题AI 解读

If a strictly stationary process $\{Z_k\}$ has residuals $Z_{k+1} - \sum^k_{j=1} a_{k,j}Z_j$ independent of $(Z_1, \cdots, Z_k)$ for all $k \geq m$, it is shown that the process is Gaussian or degenerate or $m$-step Markovian. Generalized (nonlinear) autoregressive stationary processes are defined and partially characterized.

一个有用的经验贝叶斯恒等式AI 解读

For any decision problem, one wishes to find that estimator which minimizes the expected loss. If the loss function is squared error, then the estimator is the mean of the Bayes posterior distribution. Unfortunately the prior distribution may be unknown, but in certain situations empirical Bayes methods can circumvent this problem by using past observations to estimate either the prior or the Bayes estimate directly. Empirical Bayes methods are particularly appealing when…

加权 L² 分位数距离估计量的渐近性质AI 解读

The asymptotic properties of a family of minimum quantile function distance estimators are considered. These procedures take as the parameter estimates that vector which minimizes a weighted $L^2$ distance between the empirical quantile function and an assumed parametric family of quantile functions. Regularity conditions needed for these estimators to be consistent and asymptotically normal are presented. For single parameter families of distributions, the optimal form of…

存在长度偏倚时的非参数估计AI 解读

We derive the nonparametric maximum likelihood estimate, $\hat{F}$ say, of a lifetime distribution $F$ on the basis of two independent samples, one a sample of size $m$ from $F$ and the other a sample of size $n$ from the length-biased distribution of $F$, i.e. from $G_F(x) = \int^x_0 u dF(u)/\mu, \mu = \int^\infty_0 x dF(x)$. We further show that $(m + n)^{1/2}(\hat{F} - F)$ converges weakly to a pinned Gaussian process with a simple covariance function, when $m + n \rightarrow \infty$…

若干非参数分布族的无偏估计AI 解读

This paper is concerned with the theory of unbiased estimation for nonparametric families of distributions subject to "generalised moment" restrictions. Necessary and sufficient conditions under which symmetric statistics are unique minimum variance unbiased estimators of their expectations are obtained, and some new boundedly complete families of distributions are exhibited.

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,…

Chernoff 效率与亏损AI 解读

In 1952 Chernoff introduced a measure of asymptotic efficiency for tests. Comparison in the sense of Chernoff is concerned with fixed alternatives. In contrast to Bahadur's approach, where the probabilities of first and second kind are treated in an unbalanced way, in Chernoff's approach both probabilities go to zero. For the calculation of Chernoff efficiencies one has to develop large deviation theorems both under the null hypothesis and under the alternative hypothesis.…

非参数检验问题中的H₀邻接性与样本Pitman效率AI 解读

The concept of $H_0$-contiguity is studied for certain nonparametric testing problems. Furthermore, it is shown that a meaningful sample definition of Pitman efficiency is possible under $H_0$-contiguity, and this definition turns out to coincide with the usual efficiency concept based on comparing asymptotic distributions.

由连续分段线性函数生成的秩检验AI 解读

The two-sample problem of testing against location shift is fundamental to much of the theory of rank tests. Generally testing and estimation is carried out with a fixed (non-random) set of scores for the ranks. However Beran (1974), following ideas of Stein and Hajek, developed a notable class of adaptive estimators. When used in testing, these give asymptotically efficient tests, regardless of the underlying distribution. These ideas are used here to focus attention upon…

分布端点的估计AI 解读

We propose a method of estimating the endpoint, $\theta$, of a distribution when only limited information is available about the behaviour of the distribution in the neighbourhood of $\theta$. By using increasing numbers of extreme order statistics we obtain an estimator which improves on earlier estimators based on only a bounded number of extremes. In a certain particular model our estimator is equal to a maximum likelihood estimator, but it is robust against departures…

置信集合估计中的充分性与不变性AI 解读

This paper describes how sufficiency and invariance considerations can be applied in problems of confidence set estimation to reduce the class of set estimators under investigation. Let $X$ be a random variable taking values in $\mathscr{X}$ with distribution $P_\theta, \theta \in \Theta$, and suppose a confidence set is desired for $\gamma = \gamma(\theta)$, where $\gamma$ takes values in $\Gamma$. The main tools used are (i) the representation of randomized set estimators…

非线性变量测量误差模型的估计AI 解读

An estimation procedure is presented for the coefficients of the nonlinear functional relation, where observations are subject to measurement error. The distributional properties of the estimators are derived, and a consistent estimator of the covariance matrix is given. In deriving the results it is assumed that the covariance matrix of the observational errors is known and that this covariance matrix is $o(n^{-1/3})$, where $n$ is the index of the sequence of estimators.

多维线性回归的最优稳健设计AI 解读

The model $E(y \mid x) = \theta_0 + \sum^k_{i=1} \theta_ix_i + \psi(\mathbf{x})$ is considered, where $\psi(\mathbf{x})$ is an unknown contamination with $| \psi(\mathbf{x})|$ bounded by given $\varphi(\mathbf{x})$. Optimal designs are studied in terms of least squares estimation and a family of minimax criteria. In particular, analogs of D-, A- and E-optimal designs are studied in the general case of an arbitrary $k$. Some commonly used integer designs are considered and…

某些统计量中以估计量替代参数的渐近影响AI 解读

In a variety of statistical problems, one is interested in the limiting distribution of statistics $\hat{T}_n = T_n(y_1, y_2, \cdots, y_n; \hat{\lambda}_n)$, where $\hat{\lambda}_n$ is an estimator of a parameter in the distribution of the $y_i$ and where the limiting distribution of $T_n = T_n(y_1, y_2, \cdots, y_n; \lambda)$ is relatively easy to find. For cases in which the limiting distribution of $T_n$ is normal with mean independent of $\lambda$, a useful method is…

含估计参数的统计量的渐近正态性AI 解读

Often a statistic of interest would take the form of a member of a common family, except that some vital parameter is unknown and must be estimated. This paper describes methods for showing the asymptotic normality of such statistics with estimated parameters. Whether or not the limiting distribution is affected by the estimator is primarily a question of whether or not the limiting mean (derived by replacing the estimator by a mathematical variable) has a nonzero derivative…

关于不一致的 M 估计量AI 解读

If $M$ is not convex, and the underlying density is multi-modal, $M$-estimators can be inconsistent. Examples are given, as well as some positive results.

有限阶自回归过程参数的 M 估计递归计算AI 解读

Stochastic approximation methods are used to generate a sequence of "$M$-estimates" for the unknown parameters of an autoregressive process of known, finite order which may have heavy-tailed innovations. Weak dependence properties, which can be demonstrated for many autoregressive processes, are used in the proof that the sequence converges almost surely to the parameters. A brief Monte Carlo study verifies that bounded influence functions provide protection for recursive…

异方差线性模型中的稳健估计AI 解读

We consider a heteroscedastic linear model in which the variances are given by a parametric function of the mean responses and a parameter $\theta$. We propose robust estimates for the regression parameter $\beta$ and show that, as long as a reasonable starting estimate of $\theta$ is available, our estimates of $\beta$ are asymptotically equivalent to the natural estimate obtained with known variances. A particular method for estimating $\theta$ is proposed and shown by…