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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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