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广义最小二乘协方差矩阵的最小上界及其在 SUR 与异方差模型中的应用AI 解读

In a general normal regression model, this paper first derives the least upper bound (LUB) for the covariance matrix of a generalized least squares estimator (GLSE) relative to the covariance matrix of the Gauss-Markov estimator. Second the result is applied to the (unrestricted) Zellner estimator in an N-equation seemingly unrelated regression (SUR) model and to the GLSE in a heteroscedastic model.

摘要暂缺

含辅助统计量模型中的等变估计AI 解读

This paper reformulates a result of Hora and Buehler on best equivariant estimators to treat a model admitting an ancillary statistic. The approach itself was established by Pitman, Girshick and Savage and Kiefer, and expanded by Zidek. The model considered in this paper is assumed to be generated as an orbit under a group acting on the parameter space. The general result obtained here is applied to a model in the Nile problem, a model with a known variation coefficient, a…

广义最小二乘估计量风险上界的一种方法AI 解读

First, an approach to an upper bound for the risk matrix of GLSE's is established when the information on the parameter space of the structural parameter in the covariance matrix of the error can be utilized. Second, this result is applied to regression with (i) serial correlation and (ii) heteroscedastic covariance structure. In the heteroscedastic regression, the problem of estimating the common mean of two normal populations is studied in detail.

Gauss–Markov 定理的一个非线性版本AI 解读

This article gives a nonlinear version of the Gauss—Markov theorem. It is shown that in a linear regression model y = Xβ + u, the lower bound for the risk matrix of a nonlinear estimator of β belonging to a certain class is the covariance matrix of the Gauss—Markov estimator, provided the distribution of error term u belongs to the class of elliptically symmetric distributions with second moments.

Gauss–Markov 定理的非线性版本AI 解读

This article gives a nonlinear version of the Gauss—Markov theorem. It is shown that in a linear regression model y = Xβ + u, the lower bound for the risk matrix of a nonlinear estimator of β belonging to a certain class is the covariance matrix of the Gauss—Markov estimator, provided the distribution of error term u belongs to the class of elliptically symmetric distributions with second moments.

SUR 模型中 Zellner 估计量与异方差模型中两阶段 Aitken 估计量的协方差界AI 解读

This article gives a bound for the covariance matrix of Zellner's estimator with the unrestricted sample covariance matrix in the two equations seemingly unrelated regression (SUR) model, and a bound for the covariance matrix of the two-stage Aitken estimator (2SAE) in a heteroscedastic model with two distinct variances. Some efficiencies of these estimators relative to the ordinary least squares estimator (OLSE) are also considered.

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.

两个似不相关回归方程之间的独立性检验AI 解读

When the error terms in two different regression equations are correlated, Zellner proposed an alternative estimator for the coefficients of each equation based on an estimated covariance matrix between the two error terms. However, since an estimated covariance matrix is used, the OLSE seems better than Zellner's estimator when the correlation of the two equations is close enough to zero. This paper considers the problem of testing the independence between two regression…

Hotelling T² 检验的一项稳健性性质AI 解读

This paper shows that Hotelling's $T^2$-test for testing $\mu = 0$ in the one-sample problem is robust against departures from normality in the following sense. It is still UMPI in a broad class of distributions, and the null distribution under any member of the class is the same as that under normality.

最小二乘回归中序列相关的局部稳健检验AI 解读

Kariya and Eaton and Kariya studied a robustness property of the usual tests for serial correlation against departure from normality. When the results were applied to a regression model $y = X \beta + u(X: nxk),$ it was assumed that the column space of $X$ is spanned by some $k$ latent vectors of the covariance matrix of error term $u.$ In this paper we delete this assumption and in a much broader class of distributions derive a locally best invariant test for a one-sided…

一般多元方差分析问题AI 解读

This paper treats a generalization of the classical MANOVA testing problem. The problem is reduced via invariance considerations and a new test statistic is proposed. This new test is shown to be a unique locally best invariant test and locally minimax.

序列相关检验的一项稳健性性质AI 解读

This paper shows that the UMP or UMPU tests for serial correlation, derived under the assumption of a normal distribution, are quite robust against departure from normality. In fact, the tests are still UMP or UMPU in much broader classes of distributions and the null distributions remain unchanged under these classes. The results will be applied to a linear model.

球对称性的稳健检验AI 解读

Let $R^n$ be Euclidean $n$-space and let $O(n)$ be the group of $n \times n$ orthogonal matrices. Consider $\mathscr{F}_0 = \{f\mid f$ is a density on $R^n, f(x) = f(gx), x \in R^n, g \in O(n)\}$, and let $Q = \{q\mid q: \lbrack 0, \infty) \rightarrow \lbrack 0, \infty), q$ is nonincreasing, $\int_{R^n} q(\| x \|^2) dx = 1\}$. If $\Sigma$ is an $n \times n$ positive definite matrix, set $\mathscr{F}_1(\Sigma) = \{f|f(x) = \mid \Sigma|^{-\frac{1}{2}}q(x'\Sigma^{-1}x), q \in Q\}$…