# INFERENCE FOR NORMAL MIXTURES IN MEAN AND VARIANCE

@inproceedings{Chen2008INFERENCEFN, title={INFERENCE FOR NORMAL MIXTURES IN MEAN AND VARIANCE}, author={Jiahua Chen and Xianming Tan and Runchu Zhang}, year={2008} }

A finite mixture of normal distributions, in both mean and variance parameters, is a typical finite mixture in the location and scale families. Because the likelihood function is unbounded for any sample size, the ordinary maximum likelihood estimator is not consistent. Applying a penalty to the likelihood function to control the estimated component variances is thought to restore the optimal properties of the likelihood approach. Yet this proposal lacks practical guidelines, has not been…

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## References

SHOWING 1-10 OF 25 REFERENCES

### Estimating the components of a mixture of normal distributions

- Mathematics
- 1969

SUMMARY The problem of estimating the components of a mixture of two normal distributions, multivariate or otherwise, with common but unknown covariance matrices is examined. The maximum likelihood…

### TESTS FOR HOMOGENEITY IN NORMAL MIXTURES IN THE PRESENCE OF A STRUCTURAL PARAMETER

- Mathematics
- 2000

Often a question arises as to whether observed data are a sample from a homogeneous population or from a heterogeneous population. If in particular, one wants to test for a single normal distribution…

### A Graphical Technique for Determining the Number of Components in a Mixture of Normals

- Mathematics
- 1994

Abstract When a population is assumed to be composed of a finite number of subpopulations, a natural model to choose is the finite mixture model. It will often be the case, however, that the number…

### Penalized maximum likelihood estimator for normal mixtures

- Mathematics, Computer Science
- 2003

The estimation of the parameters of a mixture of Gaussian densities is considered, within the framework of maximum likelihood, and a solution to likelihood function degeneracy which consists in penalizing the likelihood function is adopted.

### A Constrained Formulation of Maximum-Likelihood Estimation for Normal Mixture Distributions

- Mathematics
- 1985

In this context, the normal densities are sometimes referred to as component densities. The log-likelihood function L(zy), corresponding to a random sample {xl, *, xnj, is defined by L(Ty) = Ek=1…

### Penalized maximum likelihood estimation for univariate normal mixture distributions

- Mathematics
- 1999

Due to singularities of the likelihood function, the maximum likelihood approach for the estimation of the parameters of normal mixture models is an acknowledged ill posed optimization problem. Ill…

### Mixture densities, maximum likelihood, and the EM algorithm

- Computer Science
- 1984

This work discusses the formulation and theoretical and practical properties of the EM algorithm, a specialization to the mixture density context of a general algorithm used to approximate maximum-likelihood estimates for incomplete data problems.

### CONSISTENCY OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN THE PRESENCE OF INFINITELY MANY INCIDENTAL PARAMETERS

- Mathematics
- 1956

0 and ai. The parameter 0, upon which all the distributions depend, is called "structural"; the parameters {aiI} are called "incidental". Throughout this paper we shall assume that the Xi, are…

### Optimal Rate of Convergence for Finite Mixture Models

- Mathematics
- 1995

In finite mixture models, we establish the best possible rate of convergence for estimating the mixing distribution. We find that the key for estimating the mixing distribution is the knowledge of…