Abstract
Problem statement: For a multivariate normal population with size smaller than dimension, n<p, the likelihood ratio tests of the null hypothesis that the mean vector was zero with a one-sided alternative were no longer valid because they involved with sample covariance matrix which was singular. Approach: The test statistics for one-sided multivariate hypotheses with n<p were proposed. Results: The simulation study showed that the proposed tests provided reasonable type I error rate for one-sided covariance structures. They also give good powers. The application of these tests was given by testing of one-sided hypotheses on DNA micro array data. Conclusion: Under that there have no such other tests available at present for this kind of hypothesis testing with n<p yet, the proposed tests are good ones. However, the methodology is valid for any one-sided hypotheses application which involves high-dimensional data.
Highlights
Suppose one uses a matched-pair design to covariance matrix and for a partially known covariance compare the multivariate responses of two treatments. matrix, they compared the powers of these tests with
V, Kudo (1963); Shorack (1967) and Perlman (1969) For example, the data come from DNA micro arrays derived the likelihood ratio test of H0 versus H1-H0 for where thousands of gene expression levels are the cases in which V is known, known up to a measured in relatively few subjects
An example: The proposed tests are applied to an example of DNA micro array data which the data are 8280 (p) gene expression information on 110 childhoods suffering from acute lymphoblastic leukemia
Summary
Suppose one uses a matched-pair design to covariance matrix and for a partially known covariance compare the multivariate responses of two treatments. matrix, they compared the powers of these tests with. If the weights wi = ni / σi are equal, the correlation matrix for X of simple order restriction is Eq 2: are proposed by several researchers recently such as: Dempster (1958; 1960) proposed a test for testing the mean difference of two independent samples and developed its approximate distribution. SD > zα or SD < −zα where, normal dzisα2tirsibuthtieon.(1T−hα2e)ythalsqouashnotiwleedofthatht ethisstatnedstarids an invariant test under the group of scalar transformations Xi → TXi , where T=daig (t1, t2,...,tp) and t1, t2 ,...tp are nonzero constants They claimed by simulation that for all the components of the random vector are independent, that is, the covariance matrix is a diagonal matrix, their test has the attained significance level given in Table 1 in their study reasonably well in all cases.
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