Hypothesis test for normal mixture models: The EM approach

Normal mixture distributions are arguably the most important mixture models, and also the most technically challenging. The likelihood function of the normal mixture model is unbounded based on a set of random samples, unless an artificial bound is placed on its component variance parameter. Moreover, the model is not strongly identifiable so it is hard to differentiate between over dispersion caused by the presence of a mixture and that caused by a large variance, and it has infinite Fisher information with respect to mixing proportions. There has been extensive research on finite normal mixture models, but much of it addresses merely consistency of the point estimation or useful practical procedures, and many results require undesirable restrictions on the parameter space. We show that an EM-test for homogeneity is effective at overcoming many challenges in the context of finite normal mixtures. We find that the limiting distribution of the EM-test is a simple function of the $0.5\chi^2_0+0.5\chi^2_1$ and $\chi^2_1$ distributions when the mixing variances are equal but unknown and the $\chi^2_2$ when variances are unequal and unknown. Simulations show that the limiting distributions approximate the finite sample distribution satisfactorily. Two genetic examples are used to illustrate the application of the EM-test.Comment: Published in at http://dx.doi.org/10.1214/08-AOS651 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Hypersurfaces of Prescribed Curvature Measure

We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped $(n-k)$-convex bodies with prescribed $k$-th curvature measures ($k>0$) has been a longstanding problem. This is settled in this paper through the establishment of a crucial $C^2$ a priori estimate for the corresponding curvature equation on $\mathbb S^n$

Maximum Smoothed Likelihood Component Density Estimation in Mixture Models with Known Mixing Proportions

In this paper, we propose a maximum smoothed likelihood method to estimate the component density functions of mixture models, in which the mixing proportions are known and may differ among observations. The proposed estimates maximize a smoothed log likelihood function and inherit all the important properties of probability density functions. A majorization-minimization algorithm is suggested to compute the proposed estimates numerically. In theory, we show that starting from any initial value, this algorithm increases the smoothed likelihood function and further leads to estimates that maximize the smoothed likelihood function. This indicates the convergence of the algorithm. Furthermore, we theoretically establish the asymptotic convergence rate of our proposed estimators. An adaptive procedure is suggested to choose the bandwidths in our estimation procedure. Simulation studies show that the proposed method is more efficient than the existing method in terms of integrated squared errors. A real data example is further analyzed