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Wishart 矩阵特征根的正相依性AI 解读

It is shown that the characteristic roots of a Wishart matrix (identity covariance matrix) and the roots of $S_1 S_2^{-1}$ and $S_1(S_1 + S_2)^{-1}$ where $S_1, S_2$ are independent $p \times p$ Wishart matrices with the same covariance matrix, satisfy certain types of dependency relationships. That is, it is shown that these roots are (a) positive orthant dependent, (b) associated, (c) stochastically increasing in sequence, and (d) positively likelihood ratio dependent. An…

一种二元 t 分布及其应用AI 解读

A particular bivariate t-distribution is defined and investigated. Included is a discussion of how to compute probabilities for this distribution. Applications involve the construction of double sample tests for hypotheses pertaining to the mean of a normal distribution with unknown variance. Some bounds are given for the error incurred when the two component variables are provisionally assumed to be independent.

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两阶段卡方拟合优度检验AI 解读

Consider sampling in two stages for testing a simple hypothesis about a population whose elements may be classified into a finite number of classes. The joint asymptotic distribution of the two Pearson X2 statistics based on the first and combined samples is derived and used to construct a two-stage test. The advantage of this test relative to the conventional one-stage test is discussed in terms of its asymptotic power. Tables of critical values for 1 percent and 5 percent…

关于一种两阶段抽样检验的札记AI 解读

A double sample test for a hypothesis concerning the mean μ of a normal random variable with unknown variance σ2 is presented in this note. This test is an alternative to the usual single sample t test and is made by taking samples at two stages. After the first sample has been observed the hypothesis is rejected, accepted or a second sample is taken. This double sample test is a modification of a double sample test suggested by D. Owen. However, for the test being suggested…

关于双样本检验的一则札记AI 解读

A double sample test for a hypothesis concerning the mean μ of a normal random variable with unknown variance σ2 is presented in this note. This test is an alternative to the usual single sample t test and is made by taking samples at two stages. After the first sample has been observed the hypothesis is rejected, accepted or a second sample is taken. This double sample test is a modification of a double sample test suggested by D. Owen. However, for the test being suggested…