Abstract
Testing the number of components in a finite mixture is considered one of the challenging problems. In this paper, exponential finite mixtures are used to determine the number of components in a finite mixture. A sequential testing procedure is adopted based on the likelihood ratio test (LRT) statistic. The distribution of the test statistic under the null hypothesis is obtained using a resampling technique based on B bootstrap samples. The quantiles of the distribution of the test statistic are evaluated from the B bootstrap samples. The performance of the test is examined through the empirical power and application on two real datasets. The proposed procedure is not only used for testing the number of components but also for estimating the optimal number of components in a finite exponential mixture distribution. The innovation of this paper is the sequential test, which tests the more general hypothesis of a finite exponential mixture of k components versus a mixture of k + 1 components. The special case of testing an exponential mixture of one component versus two components is the one commonly used in the literature.
Highlights
Inference on the number of components can be conducted through statistical tests such as likelihood ratio tests
Feng and McCulloch [13] discussed the idea of bootstrapping the likelihood ratio test (LRT). e general method for determining the number of components is based on the LRT. e LRT statistic is used as appropriate test statistic for testing hypotheses. e test statistic is defined as −2lnλ, where λ represents the ratio between the maximized likelihood functions under the null hypothesis (H0) and the alternative hypothesis (H1), respectively, (L0)&(L1)
We use the mix tools package for R, which provides a set of functions to analyze a variety of finite mixture models. e repmix package is used to generate a random sample for a mixture of univariate exponential distributions. en, we require the MLE of the mixing distribution
Summary
Moisheer [16]; who used a mixture of two inverse Weibull distributions. e discrete Poisson distribution was used by Karlis and Xekalaki [17]. Some criteria are used to choose the number of components in finite mixtures models such as McLachlan and Peel [11]. We use a sequential test to specify the number of components in the finite exponential mixture by using a resampling procedure called bootstrap. See McLachlan and Peel [11]
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