Asymptotic Distribution of The $\chi ^2 $ Criterion when the Number of Observations and Number of Groups Increase Simultaneously

摘要: The expression\[ \chi ^2 = \mathop \sum \limits_{i 1}^{s + 1} \frac{n}{{p_i }}\left( {\frac{{m_i }}{n} - p} \right)^2 \] is used in estimating the divergence between given probabilities $p_1 ,p_2 , \ldots ,p_{s $ for possible outcomes of test and relative frequencies ${{m_1 } / n},{{m_2 n}, ,{{m_{s n}$ with which these out-comes appear n independent tests. Let \[ \begin{gathered} F(x) P\left\{ {\chi < x} \right\},\quad \Phi (u) \frac{1}{{\sqrt {2\pi }}\int_{ \infty }^u {e^{ t^{{2 2}} dt, \hfill \\ K_s (x)\left\{ \frac{1} {{2^{{s \Gamma \left( {\frac{s} {2}} \right)}}\int_0^\infty {y^{{s {2 1}}} e^{{{ y} dy,\quad x \geqq 0,} 0\qquad,x 0. \end{gathered} \right. following theorems are proved: Theorem 1. If\[ {\min }\limits_{1 \leqq i s (np_i ) \to \]for simultaneous unlimited increase ofnands, then\[ F(s u\sqrt {2s} \]...