摘要:
本文在椭球等高分布假定下,讨论了二次型X'AX（A为对称阵）的非中心Cochran定理。主要结果如下: 若X～ECn（μ,Ln;g）,g（x）>0为x的连续函数,且X有有限的2n阶矩。Ai,i=1,2,…,m为n×n对称阵。A=∑Ai,λ1,…,λk互不相同且非零。考虑下面的条件: (a) $\mathrm{X}^{\prime} A_i \mathrm{X} \stackrel{d}{=} \sum_{j=1}^k \lambda_j y_{i j},\left(y_{i 1}, \cdots, y_{i k}\right)^{\prime} \sim G x^2\left(n_{i 1}, \cdots, n_{i k} ; \delta_{i 1}^2, \cdots, \delta_{i k}^2 ; g\right) j=1, \cdots, m$.(b) $\left(\mathrm{X}^{\prime} A_1 \mathrm{X}, \cdots, \mathrm{X}^{\prime} A_m \mathrm{X}\right) \stackrel{d}{=}\left(\sum_{j=1}^k \lambda_j z_j, \cdots, \sum_{g=(m-1) b+1}^{m i k} \lambda_{f-(m-1) k} z_j\right)$$$\left(\varepsilon_1, \cdots, z_{m k}\right)^{\prime} \sim G \chi^2\left(n_{11}, \cdots, n_{1 k}, n_{21}, \cdots, n_{m k} ; \delta_{11}^2, \cdots \delta_{1 k}^2, \delta_{21}^2, \cdots, \delta_{m k}^2 ; g\right)$$(c) $\mathrm{X}^{\prime} A \mathrm{X} \underset{\int_j}{d} \sum_{j=1}^k \lambda_f y_j,\left(y_1, \cdots, y_k\right)^{\prime} \sim G \chi^2\left(n_1, \cdots, n_k ; \delta_1^2, \cdots, \delta_k^2 ; g\right)$(d) $r(A)=\Sigma r\left(A_i\right)=\Sigma \Sigma r\left(A_i E_j\right), A=\Sigma \lambda_j E_j, E_j^2=E_g, E_g E_y=0, j \neq j^{\prime}=1, \cdots, k$,(e) $k$ 个等式 $n_j=\Sigma n_{i j}$ 中至少掎 $k-1$ 个成立. 则(I) $(a),(b) \Rightarrow(c),(d),(e)$,(II) (a), (c), (e) $\Rightarrow(b),(d)$,(III) $(b),(c) \Rightarrow(a),(d),(b)$,(IV) $(c),(d) \Rightarrow(u),(b),(c)$.
Abstract:
Let X～ECn（μ, In,g）, where g（·） is a positive and continuous function. In this paper, non-central Cochran’s theorem in the quadratic form of X’AX, where A is a symmetry matrix, is discussed.
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