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A result on hypothesis testing for a multivariate normal distribution when some observations are missing
Authors:Arthur Cohen
Institution:Department of Statistics, Rutgers University USA
Abstract:Let Ui = (Xi, Yi), i = 1, 2,…, n, be a random sample from a bivariate normal distribution with mean μ = (μx, μy) and covariance matrix
/></figure>. Let <em>X</em><sub><em>i</em></sub>, <em>i</em> = <em>n</em> + 1,…, <em>N</em> represent additional independent observations on the <em>X</em> population. Consider the hypothesis testing problem <em>H</em><sub>0</sub> : <em>μ</em> = 0 vs. <em>H</em><sub>1</sub> : <em>μ</em> ≠ 0. We prove that Hotelling's <em>T</em><sup>2</sup> test, which uses (<em>X</em><sub><em>i</em></sub>, <em>Y</em><sub><em>i</em></sub>), <em>i</em> = 1, 2,…, <em>n</em> (and discards <em>X</em><sub><em>i</em></sub>, <em>i</em> = <em>n</em> + 1,…, <em>N</em>) is an admissible test. In addition, and from a practical point of view, the proof will enable us to identify the region of the parameter space where the <em>T<sup>2</sup></em>-test cannot be beaten. A similar result is also proved for the problem of testing <em>μ</em><sub><em>x</em></sub> ? <em>μ</em><sub><em>y</em></sub> = 0. A Bayes test and other competitors which are similar tests are discussed.</td>
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Keywords:62H15  62C15  Hypothesis testing  multivariate normal distribution  missing values  admissibility  Bayes test  similar test
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