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August 2000 A likelihood ratio test for $MTP_2$ within binary variables
Francesco Bartolucci, Antonio Forcina
Ann. Statist. 28(4): 1206-1218 (August 2000). DOI: 10.1214/aos/1015956713

Abstract

Multivariate Totally Positive $(MTP_2)$ binary distributions have been studied in many fields, such as statistical mechanics, computer storage and latent variable models. We show that $MTP_2$ is equivalent to the requirement that the parameters of a saturated log-linear model belong to a convex cone, and we provide a Fisher-scoring algorithm for maximum likelihood estimation.We also show that the asymptotic distribution of the log-likelihood ratio is a mixture of chi-squares (a distribution known as chi-bar-squared in the literature on order restricted inference); for this we derive tight bounds which turn out to have very simple forms. A potential application of this method is for Item Response Theory (IRT) models, which are used in educational assessment to analyse the responses of a group of subjects to a collection of questions (items): an important issue within IRT is whether the joint distribution of the manifest variables is compatible with a single latent variable representation satisfying local independence and monotonicity which, in turn, imply that the joint distribution of item responses is $MTP_2$.

Citation

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Francesco Bartolucci. Antonio Forcina. "A likelihood ratio test for $MTP_2$ within binary variables." Ann. Statist. 28 (4) 1206 - 1218, August 2000. https://doi.org/10.1214/aos/1015956713

Information

Published: August 2000
First available in Project Euclid: 12 March 2002

zbMATH: 1105.62351
MathSciNet: MR1811325
Digital Object Identifier: 10.1214/aos/1015956713

Subjects:
Primary: 62H20
Secondary: 62G10 , 62H15 , 62H17

Keywords: Chi-bar-squared distribution , conditional association , item response models , order-restricted inference , stochastic ordering

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.28 • No. 4 • August 2000
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