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On testing the number of components in finite mixture models with known relevant component distributions
Author(s) -
Chen Jiahua,
Cheng Ping
Publication year - 1997
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315786
Subject(s) - mathematics , asymptotic distribution , likelihood ratio test , test statistic , statistic , statistics , null distribution , sample size determination , null hypothesis , statistical hypothesis testing , asymptotically optimal algorithm , component (thermodynamics) , variance (accounting) , delta method , mathematical optimization , physics , thermodynamics , accounting , estimator , business
The limiting distribution of the log‐likelihood‐ratio statistic for testing the number of components in finite mixture models can be very complex. We propose two alternative methods. One method is generalized from a locally most powerful test. The test statistic is asymptotically normal, but its asymptotic variance depends on the true null distribution. Another method is to use a bootstrap log‐likelihood‐ratio statistic which has a uniform limiting distribution in [0,1]. When tested against local alternatives, both methods have the same power asymptotically. Simulation results indicate that the asymptotic results become applicable when the sample size reaches 200 for the bootstrap log‐likelihood‐ratio test, but the generalized locally most powerful test needs larger sample sizes. In addition, the asymptotic variance of the locally most powerful test statistic must be estimated from the data. The bootstrap method avoids this problem, but needs more computational effort. The user may choose the bootstrap method and let the computer do the extra work, or choose the locally most powerful test and spend quite some time to derive the asymptotic variance for the given model.