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摘要暂缺

存在缺失数据时二元正态分布的最大似然估计AI 解读

The maximum likelihood estimators (m.l.e.) are obtained for the parameters of a bivariate normal distribution with equal variances when some of the observations are missing on one of the variables. The likelihood equation for estimating $\rho$, the correlation coefficient, may have multiple roots but a result proved here provides a unique root which is the m.l.e. of $\rho$. The problem of estimating the difference $\delta$ of the two means is also considered and it is shown…

最小二乘与最佳线性无偏估计之差的欧氏范数新界AI 解读

A new bound is established for the Euclidean norm of the difference between the least squares estimator and the best linear unbiased estimator of the vector of expectations in the general linear model. The bound is valid regardless of the rank of the dispersion matrix and is expressed in substantially simpler terms than the bounds given earlier by Haberman and by Baksalary and Kala.

乘积极限估计量收敛速度的一个注记AI 解读

In this note, we give a lemma which shows that the expected squared difference between the Bayes estimator with a Dirichlet process prior and the Kaplan-Meier product limit (PL) estimator for a survival function based on censored data is $O(n^{-2})$. This lemma, together with already proven pointwise consistency properties of the Bayes estimator, is used to establish two properties of the PL estimator; namely, the mean square consistency of the PL estimator with rate $O(n^{-1})$…

用非参数贝叶斯模型逼近参数贝叶斯模型AI 解读

Let $\tau$ be a prior distribution over the parameter space $\Theta$ for a given parametric model $P_\theta, \theta \in \Theta$. For the sample space $\mathscr{X}$ (over which $P_\theta$'s are probability measures) belonging to a general class of topological spaces, which include the usual Euclidean spaces, it is shown that this parametric Bayes model can be approximated by a nonparametric Bayes model of the form of a mixture of Dirichlet processes prior, so that (i) the…

II 型删失数据的卡方拟合优度检验AI 解读

The theory of general chi-square statistics for testing fit to parametric families of distributions is extended to samples censored at sample quantiles. Data-dependent cells with sample quantiles as cell boundaries are employed. Asymptotic distribution theory is given for statistics in which unknown parameters are estimated by estimators asymptotically equivalent to linear combinations of functions of order statistics. Emphasis is placed on obtaining statistics having a…

统计操作的条件独立理论AI 解读

A general calculus of conditional independence is developed, suitable for application to a wide range of statistical concepts such as sufficiency, parameter-identification, adequacy and ancillarity. A vehicle for this theory is the statistical operation, a structure-preserving map between statistical spaces. Concepts such as completeness and identifiability of mixtures arise naturally and play an important part. Some general theorems are exemplified by applications to…

独立正态均值估计中 Berger 现象的两个例子AI 解读

Two examples are presented. In each, $p$ independent normal random variables having unit variance are observed. It is desired to estimate the unknown means, $\theta_i$, and the loss is of the form $L(\theta, a) = (\Sigma^p_{i=1} \nu(\theta_i))^{-1} \Sigma^p_{i=1}\nu(\theta_i)(\theta_i - a_i)^2$. The usual estimator, $\delta_0(x) = x$, is minimax with constant risk. In the first example $\nu(t) = e^{rt}$. It is shown that when $r \neq 0, \delta_0$ is inadmissible if and only…

连续指数族中不可容许估计量的改进及伽马尺度参数联合估计AI 解读

A general technique is developed for improving upon inadmissible estimators of natural parameters (or integral powers thereof) from continuous exponential families. The technique is to reduce the problem to the study of a differential inequality. Typical differential inequalities are presented and solved. Explicit results are given for the simultaneous estimation of gamma scale parameters (and their inverses) for a variety of natural loss functions. Surprising behavior is…

列联表的马尔可夫场与对数线性交互模型AI 解读

We use a close connection between the theory of Markov fields and that of log-linear interaction models for contingency tables to define and investigate a new class of models for such tables, graphical models. These models are hierarchical models that can be represented by a simple, undirected graph on as many vertices as the dimension of the corresponding table. Further all these models can be given an interpretation in terms of conditional independence and the…

最大似然估计对任意有界碗形损失的二阶效率AI 解读

Let $X_1, X_2, \cdots$ be a sequence of i.i.d. random variables, each having density $f(x, \theta_0)$ where $\{f(x, \theta)\}$ is a family of densities with respect to a dominating measure $\mu$. Suppose $n^{\frac{1}{2}}(\hat{\theta} - \theta)$ and $n^{\frac{1}{2}}(T - \theta)$, where $\hat{\theta}$ is the mle and $T$ is any other efficient estimate, have Edgeworth expansions up to $o(n^{-1})$ uniformly in a compact neighbourhood of $\theta_0$. Then (under certain regularity…

最大似然与最小Logit卡方估计量的n⁻²阶均方误差AI 解读

The $n^{-2}$ order mean squared errors of the maximum likelihood and the minimum chi-square estimator of the logit regression model are derived and the latter is shown to be superior for many parameter values considered. The maximum likelihood is shown to be better if the bias of each estimator is corrected to the order of $n^{-1}$; however, the difference is shown to be negligibly small in many practical situations.

最小卡方，而非最大似然！AI 解读

The sovereignty of MLE is questioned. Minimum $\chi^2_\lambda$ yields the same estimating equations as MLE. For many cases, as illustrated in presented examples, and further algorithmic exploration in progress may show that for all cases, minimum $\chi^2_\lambda$ estimates are available. In this sense minimum $\chi^2$ is the basic principle of estimation. The criterion of asymptotic sufficiency which has been called "second order efficiency" is rejected as a criterion of…

序约束统计推断算法与柯西中值性质AI 解读

Most algorithms in order restricted statistical inference express the estimates in terms of certain summary statistics computed from pooled samples. These algorithms may or may not yield optimal estimates depending on whether or not the Cauchy mean value property holds strictly for the summary statistics. In this paper a minimum lower sets algorithm, which holds generally, is described and used to prove the optimality of estimates described by a max-min formula.

多元正态协方差矩阵的经验贝叶斯估计AI 解读

Let $\mathbf{S}_{p \times p}$ have a Wishart distribution with scale matrix $\Sigma$ and $k$ degrees of freedom. Estimators of $\Sigma$ are given for each of the loss functions $L_1(\hat{\Sigma}, \Sigma) = \operatorname{tr} (\hat{\Sigma}\Sigma^{-1}) - \log \det (\hat{\Sigma}\Sigma^{-1}) - p$ and $L_2(\hat{\Sigma}, \Sigma) = \operatorname{tr} (\hat{\Sigma}\Sigma^{-1} - I)^2$. The obvious estimators of $\Sigma$ are the scalar multiples of $\mathbf{S}$, i.e., $a\mathbf{S}$ where…

摘要暂缺

This paper deals with the asymptotic properties of so-called autoregressive integrated moving average processes. Moreover, it is shown that least squares estimates of the parameters of a Gaussian autoregressive integrated process are consistent and also best asymptotically normal.

测量误差模型中若干估计量的性质AI 解读

The limiting behavior of estimators for several errors-in-variables models is investigated. It is assumed that an estimator of the covariance matrix of the measurement error is available. Models are delineated on the basis of the prior knowledge of the error structure. In all cases the limiting distribution of the estimators, standardized by $n^{\frac{1}{2}}$, is normal. Modifications of the estimators that guarantee finite moments and improve the small sample behavior of…

序贯假设检验的完全类AI 解读

We consider problems of sequential testing when the loss function is the sum of a component due to an error in the terminal decision and a cost of observation component. In all cases we establish a characterization of a complete class or an essentially complete class. In order to obtain such results for testing a null hypothesis against an alternative hypothesis we establish complete class results for testing the closure of the null hypothesis against the closure of the…

概率密度的自适应正交级数估计AI 解读

Given a sample set $X_1, \cdots, X_N$ of independent identically distributed real-valued random variables, each with the unknown probability density function $f(\cdot)$, the problem considered is to estimate $f$ from the sample set. The function $f$ is assumed to be in $L_2(a, b); f$ is not assumed to be in any parametric family. This paper constructs an adaptive "two-pass" solution to the problem: in a preprocessing step (the first pass), a preliminary rough estimate of $f$…

密度估计偏差的 L₂ 范数渐近分布AI 解读

Let $(\phi_\nu)^\infty_{\nu=0}$ be a complete orthonormal system of functions on a interval $\lbrack a, b\rbrack$ and let $w$ be a function defined on $\mathbb{R}$ with support $\lbrack a, b\rbrack$ and strictly positive on $(a, b)$. Let $(x_j)^\infty_{j=1}$ be i.i.d.rv's absolutely continuous with density function $f$ with respect to Lebesgue measure $\mu$ on $\mathbb{R}$. Let $f_n$ be the estimate of $f$ defined for $x\in (a, b)$ by $f_n(x) = \Sigma^m_{\nu=0}a_\nu(m)\hat{d}_\nu\phi_\nu(x)/w(x)$…

符号秩次序统计量的几乎处处线性定理AI 解读

Almost sure asymptotic linearity of signed rank order statistic in shift parameter is studied under suitable regularity conditions and the results are extended to stationary $\phi$-mixing processes as well.

U 统计量的 Edgeworth 展开AI 解读

It is shown that, under some regularity conditions on the kernel, a one-sample $U$-statistic with kernel of degree two admits an asymptotic expansion with remainder term $o(N^{-1})$.

广义回归残差的经验分布函数AI 解读

Generalised residuals, as defined by Cox and Snell, may be thought of as residuals from generalised regression. Under regularity conditions on the regression and on the estimator of the unknown parameters, the asymptotic behaviour of the empirical distribution of these residuals is determined. The addition of a random adjustment to the maximum likelihood estimator leads to the familiar Brownian bridge as the limit.