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定号二次型联合分布的近似AI 解读

Multidimensional Wilson-Hilferty transformations support Gaussian approximations to certain joint distributions of quadratic forms in jointly Gaussian variates. Central and noncentral joint distributions are studied and applications are noted. Parameters of the approximating distributions are given up to terms of order o(v −2) in the degrees of freedom v. Numerical studies validate using these approximations over a range of parameters for an essential subclass of the…

定号二次型联合分布的近似AI 解读

Multidimensional Wilson-Hilferty transformations support Gaussian approximations to certain joint distributions of quadratic forms in jointly Gaussian variates. Central and noncentral joint distributions are studied and applications are noted. Parameters of the approximating distributions are given up to terms of order o(v −2) in the degrees of freedom v. Numerical studies validate using these approximations over a range of parameters for an essential subclass of the…

依赖情形下 Friedman 检验的效率AI 解读

Efficiencies of Friedman's χ2 r test are studied when observations within blocks are independent and when they are exchangeably dependent. At local alternatives the limiting power is found to increase monotonically with (a) the canonical correlations of distributions having expansions in canonical form, (b) the stochastically decreasing character of the mixing distributions in certain exchangeable sequences, and (c) an index of peakedness in a class of distributions…

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结构对称性下矩阵分布所诱导的不变分布AI 解读

Summary Let Y(n x m) be a random matrix in the space F n x m of real matrices having location-scale parameters (θ, Ω) such that (i) θ belongs to a linear subspace M of F n x m; (ii) the distribution L(Y) exhibits a type of structural symmetry; and (iii) a certain transformation T(Y) is translation-invariant with respect to M and is invariant under changes of scale. Then the distribution L{T(Y)} is invariant for all underlying symmetric distributions L(Y). The types of…

无矩条件下的线性模型AI 解读

In models y = Xβ + e having spherical errors without moments, the usual estimator β is shown to be (a) median-unbiased for β, (b) most concentrated around β among all median-unbiased linear estimates, and (c) consistent for β in a sequence of identical but dependent experiments possibly having Cauchy errors. It is shown for all spherical error laws that the joint null distributions of variance ratios have exactly their normal theory forms, and for unimodal error laws that…

无矩条件下的线性模型AI 解读

In models y = Xβ + e having spherical errors without moments, the usual estimator β is shown to be (a) median-unbiased for β, (b) most concentrated around β among all median-unbiased linear estimates, and (c) consistent for β in a sequence of identical but dependent experiments possibly having Cauchy errors. It is shown for all spherical error laws that the joint null distributions of variance ratios have exactly their normal theory forms, and for unimodal error laws that…

若干变分结果及其在多重推断中的应用AI 解读

Let $(\mathbf{M}, \mathbf{T}, \mathbf{S})$ be random matrices such that $\mathbf{M}$ and $\mathbf{S}$ are Hermitian positive definite almost everywhere. Let $\mathbf{M}_{(t)} = \lbrack m_{ij}; 1 \leqq i, j \leqq t\rbrack, \mathbf{S}_{(t)} = \lbrack s_{ij}; 1 \leqq i, j \leqq t\rbrack$ and $\mathbf{T}_{(r,s)} = \lbrack t_{ij}; 1 \leqq i \leqq r, 1 \leqq j \leqq s\rbrack$, and define $Q(r, s) = P\lbrack G((\mathbf{M}_{(r)})^{-\frac{1}{2}}\mathbf{T}_{(r,s)}(\mathbf{S}_{(s)}) ^{-\frac{1}{2}}) \leqq c\rbrack$…

线性假设检验的正态理论近似AI 解读

Let $\{\mathbf{y}_1, \cdots, \mathbf{y}_N\}$ be $N$ independent repetitions of an experiment for which $\mathscr{E}\mathbf{y} = \mathbf{X\beta}$ and $\mathscr{E}(\mathbf{y} - \mathbf{X\beta})(\mathbf{y} - \mathbf{X\beta} ' = \sigma^2\mathbf{I}$, where $\mathbf{X}$ is nonstochastic of full rank and $\mathbf{\beta'} = \lbrack \mathbf{\beta_1'}, \cdot, \mathbf{\beta}_r'\rbrack$ and $\sigma^2$ are unknown parameters. We investigate some large-sample properties of $N^{\frac{1}{2}} (\hat{\mathbf{\beta}}_N - \mathbf{\beta})$…

定号二次型分布的高斯近似AI 解读

Let Qk = Σk j-1 cj(xj + aj)2 be a definite quadratic form in independent standardized Gaussian variables, xj, EQk = 01. The normalizing transformation (Qk/01)h is investigated, where h is determined by the first three moments of Qk. A new Gaussian approximation to the noncentral chi-square (x2) distribution is found for which the coefficient of skewness is smaller than a cube root transformation in the literature. Our transformation specializes to the cube root…

一类二元卡方分布的不等式AI 解读

The inequality is shown to hold when the joint distribution of U 1 and U 2 is a member of a class of bivariate Chi-Square distributions given. The relevance of this result to the construction of two-sided simultaneous confidence intervals for variances is noted.

方差的同时置信区间AI 解读

Given a sample of n vector observations from a multivariate normal population, Anderson and Roy-Gnanadesikan have given for the variances a set of confidence intervals which are approximate in that a lower bound only is known for the joint confidence coefficient. In the present study, exact procedures are developed in terms of multivariate Chi-Square distributions, and more general approximate procedures are given via Bonferroni's inequality. A numerical investigation…