### 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

### Cite this

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*Communications in Statistics - Theory and Methods*, vol. 27, no. 8, pp. 2075-2082. https://doi.org/10.1080/03610929808832210

**Limiting conditional distribution for tests of independence in the two way table.** / Kang, Seung Ho; Klotz, Jerome.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Limiting conditional distribution for tests of independence in the two way table

AU - Kang, Seung Ho

AU - Klotz, Jerome

PY - 1998/1/1

Y1 - 1998/1/1

N2 - 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 χ2 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 Nrc(0, Λ). We then establish the two statistics have a limiting conditional distribution equivalent to ZT Λ-Z, which has a central χv2, distribution with degrees of freedom v = (r - 1)(c - 1) = rank(Λ).

AB - 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 χ2 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 Nrc(0, Λ). We then establish the two statistics have a limiting conditional distribution equivalent to ZT Λ-Z, which has a central χv2, distribution with degrees of freedom v = (r - 1)(c - 1) = rank(Λ).

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UR - http://www.scopus.com/inward/citedby.url?scp=0041972057&partnerID=8YFLogxK

U2 - 10.1080/03610929808832210

DO - 10.1080/03610929808832210

M3 - Article

VL - 27

SP - 2075

EP - 2082

JO - Communications in Statistics - Theory and Methods

JF - Communications in Statistics - Theory and Methods

SN - 0361-0926

IS - 8

ER -