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Approximate distribution of the maximum of \(c-1\) \(\chi^2\)-statistics \((2\times 2)\) derived from \(2\times c\) contingency table. (English) Zbl 0252.62016
Summary: In a \(2\times c\) contingency table, let \(\chi_1^2\) be the \(\chi^2\)-statistic for \(2\times 2\) table composed of 1st column vs. the sum of 2nd, 3rd, …and \(c\)th column. Let \(\chi_2^2\) be the \(\chi^2\)-statistic for \(2\times 2\) table composed of 1st plus 2nd columns vs. the sum of 3rd, 4th,…and \(c\)th columns. Finally in this way \(\chi_{c-1}^2\) can be defined by the \(\chi^2\)-statistic for \(2\times 2\) table composed of the sum of the first \(c-1\) columns vs. \(c\)th column. In this paper, it is shown that the asymptotic distribution of \(T=\max\{\chi_1^2,\dots, \chi_{c-1}^2\}\) is expressed in terms of the multivariate normal probability of the \(c-1\)-dimensional cube for large sample sizes. Application to the procedure by M. Otake and S. Jablon [Meeting Japan. Stat. Soc. 1971] for regrouping a \(2\times c\) table, where the columns are ordered with respect to a numerical variable, is stated in relation to Leukemia data at ABCC.

MSC:
62E20 Asymptotic distribution theory in statistics
62G99 Nonparametric inference
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