Bernard Garel

Likelihood ratio test for univariate Gaussian mixture

We consider Gaussian mixtures and particularly the problem of testing homogeneity, that is testing no mixture, against a mixture with two components. Seven distinct cases are addressed, corresponding to the possible restrictions on the parameters. For each case, we give a statistic that we claim to be the likelihood ratio test statistic. The proof is given in a simple case. With the help of a bound for the maximum of a Gaussian process we calculate the percentile points. The results are illustrated by simulation.

EditorialAdvances in Mixture Models

The importance of mixture distributions is not only remarked by a number of recent books on mixtures including Lindsay (1995), Böhning (2000), McLachlan and Peel (2000) and Frühwirth-Schnatter (2006) which update previous books by Everitt and Hand (1981), Titterington et al. (1985) and McLachlan and Basford (1988). Also, a diversity of publications on mixtures appeared in this journal since 2003 (which we take here as a milestone with the appearance of the first special issue on mixtures) including Hazan et al. (2003), Benton and Krishnamoorthy (2003), Woodward and Sain (2003), Besbeas and Morgan (2004), Jamshidian (2004), Hürlimann (2004), Bohacek and Rozovskii (2004), Tao et al. (2004), Vaz de Melo Mendes and Lopes (2006), Agresti et al. (2006), Bartolucci and Scaccia (2006), D’Elia and Piccolo (2005), Neerchal and Morel (2005), Klar and Meintanis (2005), Bocci et al. (2006), Hu and Sung (2006), Seidel et al. (2006), Nadarajah (2006), Almhana et al. (2006), Congdon (2006), Priebe et al. (2006), and Li and Zha (2006). In the following we give a brief introduction to the papers contributing novels aspects in this Special Issue. These come from a diversity of areas as different as capture–recapture modelling, likelihood based cluster analysis, semiparametric mixture modelling in microarray data, latent class analysis or integer lifetime data analysis—just to mention a few. Mixture models are frequently used in capture–recapture studies for estimating population size (Chao, 1987; Link, 2003; Böhning and Schön, 2005; Böhning et al., 2005; Böhning and Kuhnert, 2006). In this issue, Mao (2007) highlights a variety of sources of difficulties in statistical inference using mixture models and uses a binomial mixture model as an illustration. Random intercept models for binary data—as useful tools for addressing between-subject heterogeneity—are discussed by Caffo et al. (2007). The nonlinearity of link functions for binary data is blurred in probit models with a normally distributed random intercept because the resulting model implies a probit marginal link as well. Caffo et al. (2007) explore another family of random intercept models where the distribution associated with the marginal and conditional link function as well as the random effect distribution are all of the same family. Formann (2007) extends the latent class approach (as a specific discrete multivariate mixture model) for situations where the discrete outcome variables (such as longitudinal binary data) experience nonignorable associations and, in addition and most importantly, have missing entries as it is rather typical for repeated observations in longitudinal studies. The modelling also incorporates potential covariates. This is illustrated using data from the Muscatine Coronary Risk Factor Study. The contribution by Grün and Leisch (2007) introduces the R-package flexmixwhich provides flexible modelling of finite mixtures of regression models using the EM algorithm. Alfò et al. (2007) consider a semiparametric mixture model for detecting differentially expressed genes in microarray experiments.An important goal of microarray studies is the detection of genes that show significant changes in observed expressions when two or more classes of biological samples (e.g. treatment and control) are compared. With the c-fold rule a gene is declared to be differentially expressed if its average expression level varies by more than a constant (typically 2). Instead, Alfò et al. (2007) introduce a gene-specific random term to control for both dependence among genes and variability with respect to the probability of yielding a fold change over a threshold c. Likelihood based inference is accomplished with a two-level finite mixture model while nonparametric Bayesian estimation is performed through the counting distribution of exceedances. Mixtures-of-experts models (Jacobs et al., 1991) and their generalization, hierarchical mixtures-of-expert models (Jordan and Jacobs, 1994) have been introduced to account for nonlinearities and other complexities in the data.

Recent asymptotic results in testing for mixtures

Inferences concerning the number of components in a mixture distribution are often required. These can be performed under the framework of the generalized likelihood ratio test. The classical result giving a chi-squared asymptotic distribution in general does not apply, indeed the limiting distribution of the corresponding test statistic has long remained a mystery. The characterization of the asymptotic distribution in a general setting has been previously derived under a separation condition. The relaxation of the separation condition, the calculation of the percentile points and asymptotic power, both in the case of a bounded and an unbounded parameter set can be obtained. An overview of the very recent asymptotic results in the problem of testing homogeneity against a two-component mixture is provided. Illustrations of new and known results are presented.

Performance analysis of cyclic statistics for the estimation of harmonics in multiplicative and additive noise

The problem of estimating the frequencies of harmonics in multiplicative and additive noise is addressed. The cyclic mean (CM) can be used if the multiplicative noise has nonzero mean; the cyclic variance (CV) can be used whether or not the multiplicative noise has zero mean. This paper answers the following question: under what conditions should we use the CV instead of the CM? The criteria used are the ease of detection and the accuracy of estimation. The CV is preferable to the CM if the coherent to noncoherent harmonic power ratio is less than a threshold that depends on the first four cumulants; when the noises are colored, this threshold becomes frequency dependent. Third- and fourth-order cyclic statistics are also studied, and it is shown that they will always be outperformed either by the CM or CV when the multiplicative noise is symmetric.

Frequency polygons for weakly dependent processes

The purpose of this paper is to investigate the frequency polygon as a density estimator for stationary strong mixing processes. Optimal bin widths which asymptotically minimize integrated mean square errors (IMSE) are derived. Under weak conditions, frequency polygons achieve the same rate of convergence to zero of the IMSE as kernel estimators. They can also attain the optimal uniform rate of convergence ((n-1logn)1/3 under general conditions. Frequency polygons thus appear to be very good density estimators with respect to both criteria of IMSE and uniform convergence.

Revealing Some Unexpected Dependence Properties of Linear Combinations of Stable Random Variables Using Symmetric Covariation

Abstract The covariation is one of the possible dependence measures for variables where distribution is symmetric alpha-stable with parameter alpha between one and two. We introduce a symmetrized and normalized version of the covariation which enables us to reveal some unexpected dependence properties of stable variables.

Maximum likelihood estimation of amplitude-modulated time series

Abstract The concern of this paper is to study a class of nonstationary signals of the form x ( t ) c ( t ) where x ( t ) is a stationary Gaussian stochastic process and c ( t ) is a deterministic signal. The process x ( t ) is modeled by an autoregressive (AR) process. The deterministic signal c ( t ) is a known function of a finite-dimensional unknown vector. Closed-form expressions are derived for the finite-sample Cramer–Rao bound. Algorithms for the maximum likelihood estimation of c ( t ) and the spectral density of x ( t ) are developed. The proposed methods are applied to the problem of estimating abrupt change in multiplicative noise.

Optimal cyclic statistics for the estimation of harmonics in multiplicative and additive noise

The problem of detection and estimation of harmonics in multiplicative and additive noise is addressed. The problem may be solved using (i) the cyclic mean if the harmonic amplitude is not zero mean or (ii) the cyclic variance if the harmonic amplitude is zero mean. Solution (ii) may be used when the amplitude of the harmonic is not zero mean while solution (i) fails in the case of zero mean harmonic amplitudes. The paper answers the following questions: given a multiplicative and additive noisy environment, which solution is optimal? The paper determines thresholds on the coherent to non-coherent sine powers ratio which delimitate the regions of optimality of the two solutions. Comparison with higher-order cyclic statistics is presented. Gaussian as well as non-Gaussian noise sources are studied.

On AR modulated harmonics: CRB and parameter estimation

The problem of estimation of harmonics with randomly time-varying amplitude is addressed. We deduce closed forms of the Cramer-Rao bounds by modeling these fluctuations by a Gaussian or non-Gaussian AR process. For the frequency estimation, the paper compares a parametric approach based on the ARMA representation of the signal and a non-parametric approach based on filtered version of higher-order statistics. Convergence and asymptotic normality of the estimators are established.

Local Asymptotic Normality of Multivariate ARMA Processes with a Linear Trend

The local asymptotic normality (LAN) property is established for multivariate ARMA models with a linear trend or, equivalently, for multivariate general linear models with ARMA error term. In contrast with earlier univariate results, the central sequence here is correlogram-based, i.e. expressed in terms of a generalized concept of residual cross-covariance function.