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多元统计分析中的特征值分布AI 解读

Ten invariant multivariate testing problems involving the real, complex, or quaternion structure of covariance matrices are considered. In each problem the maximal invariant statistic and its distribution are described, as well as the maximum likelihood estimators and likelihood ratio test statistics. These results are obtained by means of a new, unified method based on invariance arguments.

多元关联与预测的图论度量AI 解读

Interpoint-distance-based graphs can be used to define measures of association that extend Kendall's notion of a generalized correlation coefficient. We present particular statistics that provide distribution-free tests of independence sensitive to alternatives involving non-monotonic relationships. Moreover, since ordering plays no essential role, the ideas are fully applicable in a multivariate setting. We also define an asymmetric coefficient measuring the extent to which…

离散指数族多参数估计中改进估计量的构造AI 解读

This paper extends and unifies the theory of simultaneous estimation for the discrete exponential family. We discuss construction of estimators which theoretically dominate the uniformly minimum variance unbiased estimator (UMVUE) under a weighted squared error loss function, and show by means of computer simulation results that new simultaneous Poisson means estimators perform more favorably than those previously proposed. Our improved estimators shift the UMVUE towards a…

The normal, Poisson, gamma, binomial, negative binomial, and NEFGHS distributions are the six univariate natural exponential families (NEF) with quadratic variance functions (QVF). This sequel to Morris (1982) treats certain statistical topics that can be handled within this unified NEF-QVF formulation, including unbiased estimation, Bhattacharyya and Cramer-Rao lower bounds, conditional distributions and moments, quadratic regression, conjugate prior distributions, moments…

自回归参数的固定精度估计AI 解读

For a first order non-explosive autoregressive process with unknown parameter $\beta \in \lbrack -1, 1 \rbrack$, it is shown that if data are collected according to a particular stopping rule, the least squares estimator of $\beta$ is asymptotically normally distributed uniformly in $\beta$. In the case of normal residuals, the stopping rule may be interpreted as sampling until the observed Fisher information reaches a preassigned level. The situation is contrasted with the…

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"…

用一组标量偏自相关系数刻画协方差矩阵AI 解读

It has been shown that the autocovariance matrices of a stationary multivariate time series can be uniquely characterized by a sequence of the normalized partial autocorrelation matrices having singular values less than one. In this note, we show that the same autocovariance matrices can be also uniquely characterized by a set of sequences of scalar partial autocorrelation coefficients whose magnitudes are all less than one.

一阶移动平均参数最小二乘估计的一致性AI 解读

The consistency of the least squares estimator of the parameter of the first order moving average time series is proven for the parameter in the interval $\\lbrack -1, 1 \\rbrack$.

位置参数 L 估计权重函数的极小极大选择准则AI 解读

Let $X_1, \cdots, X_n$ be independent with common distribution $F$ symmetric about $\mu$. Let $T_n = n^{-1} \sum^n_{i = 1} J(i/(n + 1))X_{in}$ be an $L$-estimate of $\mu$ based on a weight function $J$ and the order statistics $X_{1n} \leq \cdots \leq X_{nn}$ of $X_1, \cdots, X_n$. Under very general regularity conditions $n^{1/2}T_n$ has asymptotic variance $\sigma^2(J, F)$. A weight function $J_0$ is found that minimizes the maximum of $\sigma^2(J, F)/s^2(F)$, whenever $s(F)$…

有限总体二次函数估计量的性质：批次方法AI 解读

Polynomial finite population functions can be expressed as totals over derived populations of batches, or ordered sequences of units from the original population. This paper extends the results of Godambe and Godambe and Joshi on nonexistence of best unbiased estimators and admissibility of the Horvitz-Thompson estimator to the real batch total case. The admissibility results are only partly extendible; an example is given to show that Horvitz-Thompson type estimators of the…

随机删失数据下“新优于旧”寿命性质的检验AI 解读

A life distribution $F$, with survival function $\bar{F} \equiv 1 - F$, is new better than used (NBU) if $\bar{F}(x + y) \leq \bar{F}(x)\bar{F}(y)$ for all $x, y \geq 0$. We propose a test of $H_0 : F$ is exponential, versus $H_1 : F$ is NBU, but not exponential, based on a randomly censored sample of size $n$ from $F$. Our test statistic is $J^c_n = \int \int \bar{F}_n(x + y) dF_n(x) dF_n(y)$, where $F_n$ is the Kaplan-Meier estimator. Under mild regularity on the amount of…

随机化设计的极小极大性：若干一般结果AI 解读

In many design settings where model violations are present, a "stochastic" minimaxity for many standard randomization procedures is demonstrated. This result requires no special analytic properties of the loss function and estimators. Next, under the squared loss and with the restriction to the use of linear estimators, a recipe for finding a randomized strategy is given. As a special case, randomizing an $A$-optimal design in the standard manner and using the least squares…

回归中一步变量选择的最优性质AI 解读

We consider the selection of the best subset of independent variables of a fixed size for possible inclusion in a regression model. The classical procedures (largest $\mathbb{R}^2$ to enter) are shown to be uniformly invariant Bayes in the sense of Paulson (1952) and Kudo (1956).

大型稀疏列联表平滑的惩罚函数方法AI 解读

Probabilities in a large sparse contingency table are estimated by maximizing the likelihood modified by a roughness penalty. It is shown that if certain smoothness criteria on the underlying probability vector are met, the estimator proposed is consistent in a one-dimensional table under a sparse asymptotic framework. Suggestions are made for techniques to apply the estimator in practice, and generalization to higher dimensional tables is considered.

有限取值参数估计的收敛速度：错误概率能有多小？AI 解读

In this paper we investigate the rates of convergence to zero of error probabilities in the estimation problem of a finite-valued parameter. It is shown that if a consistent estimator attains Bahadur's bound for the probability of incorrect decision at some parametric point then the error probability does not tend to zero exponentially fast for some other value of the parameter. We evaluate the minimal possible rate of convergence of this probability at a fixed parametric…

卡方检验准则的 BAN 估计量AI 解读

Wijsman (1959a) developed the theory of BAN estimators of a parameter $\beta$ under some fairly general conditions assuming that $n^{1/2}(y_n - g(\beta)) \rightarrow_L N_s(0, \sigma(\beta))$. The present article considers the complementary, but somewhat more general, approach under the constraint equation model that restricts the parameter $\mu$ so that $f(\mu) = 0$ under general conditions requiring $n^{1/2}(y_n - \mu) \rightarrow_L N_s(0, \sigma^\ast(\mu))$. At the same…

高斯 ARMA 过程估计量的二阶渐近效率AI 解读

In this paper we investigate an optimal property of maximum likelihood and quasi-maximum likelihood estimators of Gaussian autoregressive moving average processes by the second order approximation of the sampling distribution. It is shown that appropriate modifications of these estimators for Gaussian ARMA processes are second order asymptotically efficient if efficiency is measured by the degree of concentration of the sampling distribution up to second order. This concept…

平滑样条：回归、导数与反卷积AI 解读

The statistical properties of a cubic smoothing spline and its derivative are analyzed. It is shown that unless unnatural boundary conditions hold, the integrated squared bias is dominated by local effects near the boundary. Similar effects are shown to occur in the regularized solution of a translation-kernel integral equation. These results are derived by developing a Fourier representation for a smoothing spline.

三阶段假设检验的渐近效率AI 解读

Multi-stage hypothesis tests are studied as competitors of sequential tests. A class of three-stage tests for the one-dimensional exponential family is shown to be asymptotically efficient, whereas two-stage tests are not. Moreover, in order to be asymptotically optimal, three-stage tests must mimic the behavior of sequential tests. Similar results are obtained for the problem of testing two simple hypotheses.

融贯且连续的推断AI 解读

If the sampling model is a continuous function of the parameter and if either the parameter space or the observation space is compact, then a coherent inference which is a continuous function of the observation must be the posterior of a proper, countably additive prior.

广义尼罗河问题：参数函数的稳健置信集AI 解读

The pivotal model is described and applied to the estimation of parametric functions $\phi(\theta)$. This leads to equations of the form $H(x; \theta) = G\{p(x, \theta)\}$. These can be solved directly or by the use of differential equations. Examples include various parametric functions $\phi(\theta, \sigma)$ in a general location-scale distribution $f(p), p = (x - \theta)/\sigma$ and in two location-scale distributions. The latter case includes the ratio of the two scale…

利用汇总数据估计马尔可夫概率模型参数AI 解读

In this paper we consider the problem of estimating the parameters of a Markov model using so-called macro data. It will be shown that the stochastic process of the macro data is a Markov chain, which uniquely determines the probability structure of the underlying Markov model. A conditional least squares estimator exists under very weak conditions and this estimator is strongly consistent as time tends to infinity. Moreover this estimator is shown to be asymptotically…

拟似然函数AI 解读

The connection between quasi-likelihood functions, exponential family models and nonlinear weighted least squares is examined. Consistency and asymptotic normality of the parameter estimates are discussed under second moment assumptions. The parameter estimates are shown to satisfy a property of asymptotic optimality similar in spirit to, but more general than, the corresponding optimal property of Gauss-Markov estimators.

Cox 回归模型中回归与生存参数估计的渐近联合分布AI 解读

In this paper it is shown that the Cox likelihood (Cox, 1972) may be treated as a standard likelihood, in the sense that its maximizer $\hat{\beta}$ is asymptotically normally distributed with asymptotic covariance matrix equal to $-\{E \partial^2 \log L (\beta)/\partial\beta\partial\beta'\}^{-1}$. In the process, an asymptotic representation of the score function is obtained in terms of functions of the independent observations. This representation may have some uses in…