Approximating the moments and distribution of the likelihood ratio statistic for multinomial goodness of fit

Paul J. Smith; Donald S. Rae; Ronald W. Manderscheid; Sam Silbergeld

doi:10.1080/01621459.1981.10477715

Approximating the moments and distribution of the likelihood ratio statistic for multinomial goodness of fit

Paul J. Smith
, Donald S. Rae
, Ronald W. Manderscheid
, Sam Silbergeld

Research output: Contribution to journal › Article › peer-review

Abstract

Approximations were derived for the mean and variance of G², the likelihood ratio statistic for testing goodness of fit in a k cell multinomial distribution. These approximate moments, accurate to O(N⁻³), may be used in fitting the distribution of G². Extensive numerical studies of the exact and approximate distributions were performed in the special case of equiprobability. The asymptotic chi-squared distribution was found to fit poorly for k ≥ 4. Satisfactory results were obtained using a multiplicative correction to the chi-squared fit. More sophisticated procedures using the beta distribution produced increased accuracy, but at the cost of excessive computational labor.

Original language	English (US)
Pages (from-to)	737-740
Number of pages	4
Journal	Journal of the American Statistical Association
Volume	76
Issue number	375
DOIs	https://doi.org/10.1080/01621459.1981.10477715
State	Published - Sep 1981
Externally published	Yes

Keywords

Asymptotic moment expansions
Beta distribution
Chi-squared distribution
Entropy statistic
Equiprobability
Information statistic

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1080/01621459.1981.10477715

Cite this

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title = "Approximating the moments and distribution of the likelihood ratio statistic for multinomial goodness of fit",

abstract = "Approximations were derived for the mean and variance of G2, the likelihood ratio statistic for testing goodness of fit in a k cell multinomial distribution. These approximate moments, accurate to O(N−3), may be used in fitting the distribution of G2. Extensive numerical studies of the exact and approximate distributions were performed in the special case of equiprobability. The asymptotic chi-squared distribution was found to fit poorly for k ≥ 4. Satisfactory results were obtained using a multiplicative correction to the chi-squared fit. More sophisticated procedures using the beta distribution produced increased accuracy, but at the cost of excessive computational labor.",

keywords = "Asymptotic moment expansions, Beta distribution, Chi-squared distribution, Entropy statistic, Equiprobability, Information statistic",

author = "Smith, \{Paul J.\} and Rae, \{Donald S.\} and Manderscheid, \{Ronald W.\} and Sam Silbergeld",

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T1 - Approximating the moments and distribution of the likelihood ratio statistic for multinomial goodness of fit

AU - Smith, Paul J.

AU - Rae, Donald S.

AU - Manderscheid, Ronald W.

AU - Silbergeld, Sam

PY - 1981/9

Y1 - 1981/9

N2 - Approximations were derived for the mean and variance of G2, the likelihood ratio statistic for testing goodness of fit in a k cell multinomial distribution. These approximate moments, accurate to O(N−3), may be used in fitting the distribution of G2. Extensive numerical studies of the exact and approximate distributions were performed in the special case of equiprobability. The asymptotic chi-squared distribution was found to fit poorly for k ≥ 4. Satisfactory results were obtained using a multiplicative correction to the chi-squared fit. More sophisticated procedures using the beta distribution produced increased accuracy, but at the cost of excessive computational labor.

AB - Approximations were derived for the mean and variance of G2, the likelihood ratio statistic for testing goodness of fit in a k cell multinomial distribution. These approximate moments, accurate to O(N−3), may be used in fitting the distribution of G2. Extensive numerical studies of the exact and approximate distributions were performed in the special case of equiprobability. The asymptotic chi-squared distribution was found to fit poorly for k ≥ 4. Satisfactory results were obtained using a multiplicative correction to the chi-squared fit. More sophisticated procedures using the beta distribution produced increased accuracy, but at the cost of excessive computational labor.

KW - Asymptotic moment expansions

KW - Beta distribution

KW - Chi-squared distribution

KW - Entropy statistic

KW - Equiprobability

KW - Information statistic

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DO - 10.1080/01621459.1981.10477715

M3 - Article

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EP - 740

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

IS - 375

ER -

Approximating the moments and distribution of the likelihood ratio statistic for multinomial goodness of fit

Abstract

Keywords

ASJC Scopus subject areas

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