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AMP 链图模型的马尔可夫等价类刻画AI 解读

Chain graphs (CG) (= adicyclic graphs) use undirected and directed edges to represent both structural and associative dependences. Like acyclic directed graphs (ADGs), the CG associated with a statistical Markov model may not be unique, so CGs fall into Markov equivalence classes, which may be superexponentially large, leading to unidentifiability and computational inefficiency in model search and selection. It is shown here that, under the Andersson–Madigan–Perlman (AMP)…

AMP 链图马尔可夫模型的分离与完备性AI 解读

Pearl ’s well-known $d$-separation criterion for an acyclic directed graph (ADG) is a pathwise separation criterion that can be used to efficiently identify all valid conditional independence relations in the Markov model determined by the graph. This paper introduces $p$-separation, a pathwise separation criterion that efficiently identifies all valid conditional independences under the Andersson–Madigan–Perlman (AMP) alternative Markov property for chain graphs ( =…

多元正态分布中条件独立性的格模型AI 解读

The lattice conditional independence model $\mathbf{N}(\mathscr{K})$ is defined to be the set of all normal distributions on $\mathbb{R}^I$ such that for every pair $L, M \in \mathscr{K}, x_L$ and $x_M$ are conditionally independent given $x_{L \cap M}$. Here $\mathscr{K}$ is a ring of subsets of the finite index set $I$ and, for $K \in \mathscr{K}, x_K$ is the coordinate projection of $x \in \mathbb{R}^I$ onto $\mathbb{R}^K$. Statistical properties of $\mathbf{N}(\mathscr{K})$…

多个协方差矩阵及多元正态总体相等的似然比检验无偏性AI 解读

Random samples of size $N_\alpha$ are taken from the $p$-variate normal populations $N_p(\mu^{(\alpha)}, \Sigma^{(\alpha)}), 1 \leqslant \alpha \leqslant k$, with $\mu^{(\alpha)}$ and $\Sigma^{(\alpha)}$ unknown. Bartlett's modification of the likelihood ratio test (LRT) for the hypothesis $H_1:\Sigma^{(1)} = \cdots = \Sigma^{(k)}$ rejects $H_1$ for large values of $|S|^n/\Pi|S^{(\alpha)}|^{n_\alpha}$, where $S = \Sigma S^{(\alpha)}, n_\alpha = N_\alpha - 1, n = \Sigma n_i$,…

利用协变量的均值不变检验AI 解读

We consider the problem of testing a hypothesis about the means of a subset of the components of a multivariate normal distribution with unknown covariance matrix, when the means of a second subset (the covariates) are known. Because of the possible correlation between the two subsets, information provided by the second subset can be useful for inferences about the means of the first subset. In this paper attention is restricted to the class of procedures invariant under the…

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关于条件分布中代入操作的一则札记AI 解读

The following proposition is sometimes used in distribution theory: for each fixed $z$ suppose that $T(X, z)$ has the distribution $Q$ and is independent of $Y$; then $T(X, Z(Y))$ has the distribution $Q$ and is independent of $Y$. An example is presented to show this result is false in general. Additional conditions under which the proposition becomes valid are presented.

The main result of the current research describes a monotonicity property of certain invariant tests for the multivariate analysis of variance problem. Suppose $X: r \times p$ has a normal distribution, $EX = \Theta$ and the rows of $X$ are independent, each with unknown covariance matrix $\Sigma: p \times p$. Let $S = p \times p$ have a Wishart distribution $W(\Sigma, p, n), S$ independent of $X$. If $K$ is the acceptance region of an invariant test for the null hypothesis $\Theta = 0$…

非中心 F 检验的功效：增加变量对 Hotelling T² 检验的影响AI 解读

We show that the power of the noncentral F-test increases with the number of denominator d.f. and decreases with the numerator d.f. Hence, increasing the dimension of the observed vectors may decrease the power of Hotelling's T 2-test for means, unless the additional non-centrality provided by the extra variates is sufficiently large. A function determining the exact increase in noncentrality required for increased power is studied and tabulated We present procedures…

广义样本协方差矩阵的非奇异性AI 解读

Let $X = (X_1, \cdots, X_n)$ where the $X_i: p \times 1$ are independent random vectors, and let $A: n \times n$ be positive semi-definite symmetric. This paper establishes necessary and sufficient conditions that the random matrix $XAX'$ be positive definite w.p.1. The results are applied to cases where $A$ has a particular form or $X_1, \cdots, X_n$ are i.i.d. In particular, it is shown that in the i.i.d. case, the sample covariance matrix $\sigma(X_i - \bar{X})(X_i - \bar{X})'$…