A semi-deterministic generalization of a \((2 \times 2)\) contingency table. (English. Russian original) Zbl 0852.62054

Seneta, E.

Theory Probab. Math. Stat. 50, 117-121 (1995); translation from Teor. Jmovirn. Mat. Stat. 50, 114-118 (1994).

Summary: In each of a sequence of \(m\) multinomial trials, an individual of type \(i\) can be selected with probability \(p_i\), \(i = 1,2,3,4\). For the \(k\)-th individual so selected \((k = 1,2, \dots, m)\), \((w_k - 1)\) further individuals of the same type are added, so the final number of individuals is \(N = \sum^m_{k = 1} w_k\). The usual asymptotic distribution as \(m \to \infty\) for the chi-square goodness-of-fit statistic with a more general multiplicative norming than \(N\) is obtained under the usual assumption \(p_1 p_4 - p_2 p_3 = 0\), providing a certain condition of the \(w_k\)’s is satisfied. The necessity of the condition and asymptotic power of the test are also discussed.

MSC:

62H17	Contingency tables
62E20	Asymptotic distribution theory in statistics
62G10	Nonparametric hypothesis testing

Keywords:

multinomial trials; chi-square goodness-of-fit statistic; asymptotic power