Approximating Noncentral Chi-Squared to the Moments and Distribution of the Likelihood Ratio Statistic for Multinomial Goodness of Fit

Björn Holmquist; Anna Sjöström; Sultana Nasrin

doi:10.1007/978-3-030-56773-6_11

Approximating Noncentral Chi-Squared to the Moments and Distribution of the Likelihood Ratio Statistic for Multinomial Goodness of Fit

Björn Holmquist, Anna Sjöström, Sultana Nasrin

Department of Statistics

Research output: Chapter in Book/Report/Conference proceeding › Book chapter › Research › peer-review

Abstract

The chi-square distribution is often assumed to hold for the asymptotic distribution of two times the log likelihood ratio statistic under the null hypothesis. Approximations are derived for the mean and variance of G2, the likelihood ratio statistic for testing goodness of fit in a s category multinomial distribution. The first two moments of G2 are used to fit the distribution of G2 to a noncentral chi-square distribution. The fit is generally better than earlier attempts to fit to scaled versions of asymptotic central chi-square distributions. The results enlighten the complex role of the dimension of the multivariate variable in relation to the sample size, for asymptotic likelihood ratio distribution results to hold.

Original language	English
Title of host publication	Recent Developments in Multivariate and Random Matrix Analysis
Subtitle of host publication	Festschrift in Honour of Dietrich von Rosen
Editors	Thomas Holgersson, Martin Singull
Publisher	Springer Nature
ISBN (Electronic)	978-3-030-56773-6
ISBN (Print)	978-3-030-56772-9
DOIs	https://doi.org/10.1007/978-3-030-56773-6_11
Publication status	Published - 2020

Subject classification (UKÄ)

Probability Theory and Statistics

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10.1007/978-3-030-56773-6_11

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Holmquist, B., Sjöström, A., & Nasrin, S. (2020). Approximating Noncentral Chi-Squared to the Moments and Distribution of the Likelihood Ratio Statistic for Multinomial Goodness of Fit. In T. Holgersson, & M. Singull (Eds.), Recent Developments in Multivariate and Random Matrix Analysis: Festschrift in Honour of Dietrich von Rosen Springer Nature. https://doi.org/10.1007/978-3-030-56773-6_11

Holmquist, Björn ; Sjöström, Anna ; Nasrin, Sultana. / Approximating Noncentral Chi-Squared to the Moments and Distribution of the Likelihood Ratio Statistic for Multinomial Goodness of Fit. Recent Developments in Multivariate and Random Matrix Analysis: Festschrift in Honour of Dietrich von Rosen. editor / Thomas Holgersson ; Martin Singull. Springer Nature, 2020.

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abstract = "The chi-square distribution is often assumed to hold for the asymptotic distribution of two times the log likelihood ratio statistic under the null hypothesis. Approximations are derived for the mean and variance of G2, the likelihood ratio statistic for testing goodness of fit in a s category multinomial distribution. The first two moments of G2 are used to fit the distribution of G2 to a noncentral chi-square distribution. The fit is generally better than earlier attempts to fit to scaled versions of asymptotic central chi-square distributions. The results enlighten the complex role of the dimension of the multivariate variable in relation to the sample size, for asymptotic likelihood ratio distribution results to hold.",

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Holmquist, B, Sjöström, A & Nasrin, S 2020, Approximating Noncentral Chi-Squared to the Moments and Distribution of the Likelihood Ratio Statistic for Multinomial Goodness of Fit. in T Holgersson & M Singull (eds), Recent Developments in Multivariate and Random Matrix Analysis: Festschrift in Honour of Dietrich von Rosen. Springer Nature. https://doi.org/10.1007/978-3-030-56773-6_11

Approximating Noncentral Chi-Squared to the Moments and Distribution of the Likelihood Ratio Statistic for Multinomial Goodness of Fit. / Holmquist, Björn; Sjöström, Anna; Nasrin, Sultana.
Recent Developments in Multivariate and Random Matrix Analysis: Festschrift in Honour of Dietrich von Rosen. ed. / Thomas Holgersson; Martin Singull. Springer Nature, 2020.

Research output: Chapter in Book/Report/Conference proceeding › Book chapter › Research › peer-review

TY - CHAP

T1 - Approximating Noncentral Chi-Squared to the Moments and Distribution of the Likelihood Ratio Statistic for Multinomial Goodness of Fit

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AU - Sjöström, Anna

AU - Nasrin, Sultana

PY - 2020

Y1 - 2020

N2 - The chi-square distribution is often assumed to hold for the asymptotic distribution of two times the log likelihood ratio statistic under the null hypothesis. Approximations are derived for the mean and variance of G2, the likelihood ratio statistic for testing goodness of fit in a s category multinomial distribution. The first two moments of G2 are used to fit the distribution of G2 to a noncentral chi-square distribution. The fit is generally better than earlier attempts to fit to scaled versions of asymptotic central chi-square distributions. The results enlighten the complex role of the dimension of the multivariate variable in relation to the sample size, for asymptotic likelihood ratio distribution results to hold.

AB - The chi-square distribution is often assumed to hold for the asymptotic distribution of two times the log likelihood ratio statistic under the null hypothesis. Approximations are derived for the mean and variance of G2, the likelihood ratio statistic for testing goodness of fit in a s category multinomial distribution. The first two moments of G2 are used to fit the distribution of G2 to a noncentral chi-square distribution. The fit is generally better than earlier attempts to fit to scaled versions of asymptotic central chi-square distributions. The results enlighten the complex role of the dimension of the multivariate variable in relation to the sample size, for asymptotic likelihood ratio distribution results to hold.

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DO - 10.1007/978-3-030-56773-6_11

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SN - 978-3-030-56772-9

BT - Recent Developments in Multivariate and Random Matrix Analysis

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PB - Springer Nature

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Holmquist B, Sjöström A, Nasrin S. Approximating Noncentral Chi-Squared to the Moments and Distribution of the Likelihood Ratio Statistic for Multinomial Goodness of Fit. In Holgersson T, Singull M, editors, Recent Developments in Multivariate and Random Matrix Analysis: Festschrift in Honour of Dietrich von Rosen. Springer Nature. 2020 doi: 10.1007/978-3-030-56773-6_11

Approximating Noncentral Chi-Squared to the Moments and Distribution of the Likelihood Ratio Statistic for Multinomial Goodness of Fit

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