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K 归一化平方误差损失下多个泊松参数的同时估计AI 解读

In this study, we consider the simultaneous estimation of the parameters of the distributions of $p$ independent Poisson random variables using the loss function $L_k(\lambda, \hat{\lambda}) = \sum (\lambda_i - \hat{\lambda}_i)^2/\lambda^k_i$ for a given positive integer $k$. New estimators are derived, which include the minimax estimators proposed by Clevenson and Zidek (1975), as special cases. The case when more than one observation is taken from some of the variables is…

多元正态均值最小最大估计量的选择AI 解读

The problem of estimating a $p$-variate normal mean under arbitrary quadratic loss when $p \geq 3$ is considered. Any estimator having uniformly smaller risk than the maximum likelihood estimator $\delta^0$ will have significantly smaller risk only in a fairly small region of the parameter space. A relatively simple minimax estimator is developed which allows the user to select the region in which significant improvement over $\delta^0$ is to be achieved. Since the desired…

P值的渐近对数正态性AI 解读

Sufficient conditions for asymptotic lognormality of exact and approximate, unconditional and conditional $P$-values are established. It is pointed out that the mean, which is half the Bahadur slope, and the standard deviation of the asymptotic distribution of the log transformed $P$-value together, but not the mean alone, permit approximation of both the level and power of the test. This provides a method of discriminating between tests that have Bahadur efficiency one. The…

最大凸下包络斜率估计量的渐近分布AI 解读

Isotonic estimation involves the estimation of a function which is known to be increasing with respect to a specified partial order. For the case of a linear order, a general theorem is given which simplifies and extends the techniques of Prakasa Rao and Brunk. Sufficient conditions for a specified limit distribution to obtain are expressed in terms of a local condition and a global condition. It is shown that the rate of convergence depends on the order of the first…

随机回归模型的最小二乘估计及其在动态系统辨识与控制中的应用AI 解读

Strong consistency and asymptotic normality of least squares estimates in stochastic regression models are established under certain weak assumptions on the stochastic regressors and errors. We discuss applications of these results to interval estimation of the regression parameters and to recursive on-line identification and control schemes for linear dynamic systems.

发现新物种概率的估计AI 解读

We search a population by selecting one member at a time with replacement and observing the species of each selected member. We are interested in predicting the conditional probability of discovering a new species in the next selection after $n$ observations. The existence and asymptotic behavior of the uniformly minimum variance unbiased estimator of the unconditional probability is investigated under the condition that the species are uniformly distributed. We also compare…

二阶可容许性的充要条件及其在Berkson生物测定问题中的应用AI 解读

A theorem is proved which gives a necessary and sufficient condition for improving, up to $o(n^{-2})$, the mean squared error of the maximum likelihood estimate $\hat{\theta}$ by using an estimate of the form $\hat{\theta} + d(\hat{\theta})/n$. An application is made to a bioassay problem of Berkson.

非参数判别的渐近误分类概率AI 解读

Let $(X, Y), (X_1, Y_1), \cdots, (X_n, Y_n)$ be independent identically distributed random vectors from $R^d \times \{0, 1\}$, and let $\hat{Y}$ be the $k$-nearest neighbor estimate of $Y$ from $X$ and the $(X_i, Y_i)$'s. We show that for all distributions of $(X, Y)$, the limit of $L_n = P(\hat{Y} \neq Y)$ exists and satisfies $\lim_{n\rightarrow\infty} L_n \leq (1 + a_k)R^\ast, a_k = \frac{\alpha \sqrt k}{k - 3.25}\big(1 + \frac{\beta}{\sqrt{k-3}}\big), k \text{odd}, k \geq 5,$…

非参数回归函数估计的几乎处处收敛AI 解读

Let $(X, Y), (X_1, Y_1), \cdots, (X_n, Y_n)$ be independent identically distributed random vectors from $R^d \times R$, and let $E(|Y|^p) < \infty$ for some $p \geq 1$. We wish to estimate the regression function $m(x) = E(Y \mid X = x)$ by $m_n(x)$, a function of $x$ and $(X_1, Y_1), \cdots, (X_n, Y_n)$. For large classes of kernel estimates and nearest neighbor estimates, sufficient conditions are given for $E\{|m_n(x) - m(x)|^p\} \rightarrow 0$ as $n \rightarrow \infty$,…

参数空间受限时正态均值的极小极大估计AI 解读

If $X$ is a $N(\theta, 1)$ random variable, let $\rho (m)$ be the minimax risk for estimation with quadratic loss subject to $|\theta| \leq m$. Then $\rho (m) = 1 - \pi^2/m^2 + o(m^{-2})$. We exhibit estimates which are asymptotically minimax to this order as well as approximations to the least favorable prior distributions. The approximate least favorable distributions (correct to order $m^{-2}$) have density $m^{-1} \cos^2 \big(\frac{\pi}{2m} s\big), |s| \leq m$ rather…

有限样本空间统计问题的完全类定理AI 解读

This paper contains a complete class theorem (Theorem 3.2) which applies to most statistical estimation problems having a finite sample space. This theorem also applies to many other statistical problems with finite sample spaces. The description of this complete class involves a stepwise algorithm. At each step of the process it is necessary to construct the Bayes procedures in a suitably modified version of the original problem. The complete class is a minimal complete…

随机右删失数据的回归分析AI 解读

This paper proposes a new estimator of the parameter vector in a linear regression model when the observations are randomly censored on the right and when the error distribution is unknown. This estimator is explicitly defined and easily computable. The paper contains sufficient conditions under which this estimator is mean square consistent and asymptotically normal. A numerical example is given.

多元检验的稳健性AI 解读

This paper gives necessary and sufficient conditions for the null distribution of a test statistic to remain the same in the class of left $\mathscr{O}(n)$-invariant distributions and in the class of elliptically symmetric distributions. Secondly, it is shown that in certain special cases, the usual MANOVA tests are still uniformly most powerful invariant in a class of left $\mathscr{O}(n)$-invariant distributions including elliptically symmetric distributions.

协方差矩阵的不变检验AI 解读

Minimal complete classes of invariant tests are presented for modifications of the problem of testing the independence of $Y$ and $X$, where $(Y, X) \equiv (Y, X_1, \cdots, X_p)$ is a multivariate normal random vector. One modification involves having extra independent observations on $Y$. Others involve extra variates $Z \equiv (Z_1, \cdots, Z_q)$ such that $(Y,X,Z)$ is multivariate normal. Among other results, locally most powerful invariant tests and asymptotically most…

多项总体间随机序关系的似然比检验AI 解读

Likelihood ratio tests concerning the parameters of two multinomial populations are discussed. A stochastic ordering restriction is considered as a one sided alternative to equality. The one and two sample tests for equality versus stochastic ordering and stochastic ordering versus all alternatives are derived and their large sample distributions are obtained. The large sample distributions are mixtures of chi-squared distributions. The tests developed provide discrete…

均值序贯估计中三阶段抽样的渐近理论AI 解读

We describe the asymptotic theory of triple sampling as it pertains to the estimation of a mean. We obtain limit theorems for the case of the normal distribution. Our results show that triple sampling combines the simplicity of Stein's double sampling technique with the efficiency of the fully sequential Anscombe-Chow-Robbins procedure.

回归模型的自助法AI 解读

The regression and correlation models are considered. It is shown that the bootstrap approximation to the distribution of the least squares estimates is valid, and some error bounds are given.

Bootstrap 的若干渐近理论AI 解读

Efron's "bootstrap" method of distribution approximation is shown to be asymptotically valid in a large number of situations, including $t$-statistics, the empirical and quantile processes, and von Mises functionals. Some counter-examples are also given, to show that the approximation does not always succeed.

基于典型相关与线性乘线性交互的双向列联表独立性检验AI 解读

Tests for independence of rows and columns in an $r \times s$ contingency table are developed from canonical correlation analysis and from models of linear-by-linear interaction. The resulting test statistics are asymptotically equivalent under the null hypothesis. They are consistent and asymptotically unbiased. Approximate critical values are available from existing tables. The proposed tests are most appropriate when the matrix of joint probabilities is well approximated…

通过参数变换改善非线性最小二乘的近似置信域AI 解读

In a previous paper, it was shown that parameter-effects nonlinearities of a nonlinear regression model-experimental design-parameterization combination can be quantified by means of a parameter-effects curvature array $A$ based on second derivatives of the model function. In this paper, the individual terms of $A$ are interpreted and local compensation methods are suggested. A method of computing the parameter-effects array corresponding to a transformed set of parameters…

特定例子中 Efron–Hinkley 与似然比辅助统计量的比较AI 解读

Two approximate ancillaries, the Efron-Hinkley ancillary and the (signed) likelihood ratio, are compared in a specific example by means of their level curves, their marginal densities and the conditional and unconditional likelihood functions. It is shown that in the present model the latter of the two ancillaries is the better. It is almost exactly distribution constant and has more stable conditional likelihood functions.

Efron bootstrap 的渐近准确性AI 解读

In the non-lattice case it is shown that the bootstrap approximation of the distribution of the standardized sample mean is asymptotically more accurate than approximation by the limiting normal distribution. The exact convergence rate of the bootstrap approximation of the distributions of sample quantiles is obtained. A few other convergence rates regarding the bootstrap method are also studied.

多元正态分布均值的估计AI 解读

Estimation of the means of independent normal random variables is considered, using sum of squared errors as loss. An unbiased estimate of risk is obtained for an arbitrary estimate, and certain special classes of estimates are then discussed. The results are applied to smoothing by use of moving averages and to trimmed analogs of the James-Stein estimate. A suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate.

Identifiability of finite mixtures of von Mises distributions is proved.