Projected likelihood contrasts for testing homogeneity in finite mixture models with nuisance parameters

📝 Abstract

This paper develops a test for homogeneity in finite mixture models where the mixing proportions are known a priori (taken to be 0.5) and a common nuisance parameter is present. Statistical tests based on the notion of Projected Likelihood Contrasts (PLC) are considered. The PLC is a slight modification of the usual likelihood ratio statistic or the Wilk’s $\Lambda$ and is similar in spirit to the Rao’s score test. Theoretical investigations have been carried out to understand the large sample statistical properties of these tests. Simulation studies have been carried out to understand the behavior of the null distribution of the PLC statistic in the case of Gaussian mixtures with unknown means (common variance as nuisance parameter) and unknown variances (common mean as nuisance parameter). The results are in conformity with the theoretical results obtained. Power functions of these tests have been evaluated based on simulations from Gaussian mixtures.

💡 Analysis

This paper develops a test for homogeneity in finite mixture models where the mixing proportions are known a priori (taken to be 0.5) and a common nuisance parameter is present. Statistical tests based on the notion of Projected Likelihood Contrasts (PLC) are considered. The PLC is a slight modification of the usual likelihood ratio statistic or the Wilk’s $\Lambda$ and is similar in spirit to the Rao’s score test. Theoretical investigations have been carried out to understand the large sample statistical properties of these tests. Simulation studies have been carried out to understand the behavior of the null distribution of the PLC statistic in the case of Gaussian mixtures with unknown means (common variance as nuisance parameter) and unknown variances (common mean as nuisance parameter). The results are in conformity with the theoretical results obtained. Power functions of these tests have been evaluated based on simulations from Gaussian mixtures.

📄 Content

arXiv:0805.2460v1 [math.ST] 16 May 2008 IMS Collections Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen Vol. 1 (2008) 272–281 c⃝Institute of Mathematical Statistics, 2008 DOI: 10.1214/193940307000000194 Projected likelihood contrasts for testing homogeneity in finite mixture models with nuisance parameters Debapriya Sengupta∗1 and Rahul Mazumder2 Indian Statistical Institute Abstract: This paper develops a test for homogeneity in finite mixture mod- els where the mixing proportions are known a priori (taken to be 0.5) and a common nuisance parameter is present. Statistical tests based on the no- tion of Projected Likelihood Contrasts (PLC) are considered. The PLC is a slight modification of the usual likelihood ratio statistic or the Wilk’s Λ and is similar in spirit to the Rao’s score test. Theoretical investigations have been carried out to understand the large sample statistical properties of these tests. Simulation studies have been carried out to understand the behavior of the null distribution of the PLC statistic in the case of Gaussian mixtures with unknown means (common variance as nuisance parameter) and unknown vari- ances (common mean as nuisance parameter). The results are in conformity with the theoretical results obtained. Power functions of these tests have been evaluated based on simulations from Gaussian mixtures.

1. Introduction Finite mixture models are often used to understand whether the data comes from a heterogeneous or a homogeneous population. In particular, consider the case of a mixture of two populations with the mixing proportions known (Goffinet et al. [7]). We are interested to know whether the data is sampled from a proper mixture of two distributions or a single distribution. In particular, consider a mixture family g, with generating population densities given by M0 = {f(·|θ, η) : θ ∈Θ, η ∈E}, where θ is the main parameter of interest and η is the common nuisance parameter. We assume that the mixing proportion is known a priori to be 0.5. The mixture model then becomes (1.1) g(z|θ1, θ2, η) = 0.5 f(z|θ1, η) + 0.5 f(z|θ2, η). The null hypothesis for homogeneity is, θ1 = θ2. In several practical examples (for example, arising in speech analysis and non- parametric regression methodology) detection of the location of discontinuity in the local mean or the local variance (or local amplitude) are of interest (Figure 1). The theoretical results developed in this paper can be used in such problems. Fig- ure 1 demonstrates several scenarios of signals being scanned through a running ∗Supported in part by Grant No. 12(30)/04-IRSD, DIT, Govt. of India. 1Bayesian and Interdisciplinary Research Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India, e-mail: dps@isical.ac.in 2C. V. Raman Hall, 205 B. T. Road, Kolkata 700108, India, e-mail: rahul.mazumder@gmail.com AMS 2000 subject classifications: Primary 62G08, 60G35; secondary 60J55. Keywords and phrases: Gaussian mixture models, projected likelihood contrast. 272 Projected likelihood contrasts 273 Fig 1. Left column shows time plots of data with solid vertical lines marking the windows con- sidered. The top two panels indicate a simulated noisy signal (with additive Gaussian noise) with mean function having a jump discontinuity. The bottom panels describe a portion of digitized speech waveform. In the right column three fitted densities of y-values: nonparametric kernel smoothed density (solid line), single component Gaussian fit (dashed line) and mixture of two Gaussian fit with equal mixing weights (curve indicated by +), are shown corresponding to the frames indicated in the left column. window of specified bandwidth. When the center of the window is placed at points of discontinuity the raw signal values (y-axis) will have a distribution which can be adequately modeled by (1.1). This basic idea has been explored by Hall and Titterington [8] in the context of edge and peak preserving smoothers. A brief list of references dealing with the study of mixture distributions and properties of the Likelihood Ratio Test (LRT) tests are provided below. In Tit- terington et al. [13], McLachlan and Basford [11] and Lindsay [10] one may find extensive discussions about the background of finite mixture models. The asymp- totic distributions of the LRT in mixture models have been studied in Bickel and Chernoff[1], Chernoffand Lander [5], Ghosh and Sen [6], Lemdani and Pons [9]. Different modifications of LRT tests in mixture models are proposed and studied by Chen et al. [4] and Self and Liang [12]. In this paper we introduce a concept of Projected Likelihood Contrasts (PLC), a modified version of the LRT test or the Wilks’ Λ (Wilks [14]) statistic, which we motivate as follows. Consider i.i.d. observations Z1, Z2, . . . , ZN generated by some element of the class of densities g given by (1.1). The likelihood under the full mixture model is given by (1.2) LN(θ1, θ2, η) = N X i=1 log g(Zi|θ1,

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