### Abstract

For testing the hypothesis of independence in the r x c two way contingency table, we consider the likelihood ratio statistic -2 ln(λ) and the χ^{2} statistic conditionally given the sufficient statistic of row and column margin totals. We use Stirling's formula in the conditional characteristic function of normalized cell counts to give a direct proof that the limiting null conditional distribution is multivariate normal Z : script N_{rc}(0, Λ). We then establish the two statistics have a limiting conditional distribution equivalent to Z^{T} Λ^{-}Z, which has a central χ_{v}^{2}, distribution with degrees of freedom v = (r - 1)(c - 1) = rank(Λ).

Original language | English |
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Pages (from-to) | 2075-2082 |

Number of pages | 8 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 27 |

Issue number | 8 |

DOIs | |

Publication status | Published - 1998 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability