One-sided Multivariate Tests for High Dimensional Data

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For a multivariate normal population with size larger than dimension , n > p ,

Kudo (1963), Shorack (1967) and Perlman (1969) derived the likelihood ratio tests of

the null hypothesis that the mean vector is zero with a one-sided alternative for a

known covariance matrix, a partially known covariance matrix and a completely

unknown covariance matrix, respectively. Because these tests may be tedious to use,

Tang et al.(1989) developed approximate likelihood ratio tests and Follmann (1996)

proposed one-sided modifications of the usual omnibus chi-squared test and

Hotelling’s T2 test. Chongcharoen et al.(2002) considered a modification of

Follmann's test (the new test) to include information of off diagonal of covariance

matrix , which adjusts for possibly unequal variances. For the non-normal population,

Boyett and Shuster (1977) proposed a nonparametric one-sided test and

Chongcharoen et al. (2002) used their technique to develop nonparametric versions of

Perlman’s test, Follmann’s test, the new test and the Tang-Gnecco-Geller test. Also

Chongcharoen et al. (2002) considered known and partially known covariance

matrices. Chongcharoen (2009) studied the powers of these one-sided tests for an

unknown covariance matrices. In some situations, there are no longer data for n > p .

That is, when the number n of available observations is smaller than the dimension


of the observed vectors. For example, the data comes from DNA micro arrays

where thousands of gene expression levels are measured on relatively few subjects.

The one-sided multivariate tests as above are no longer valid for this kind of data. The

proposed tests are tests for one-sided multivariate tests with n < p provided

reasonable type I error rate for one-sided covariance structures. Their powers are

compared for alternatives. An example for using these proposed tests on DNA micro

arrays is given. However, the methodology is valid for any application which involves

high–dimensional data.

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