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

Abstract

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