Mixtures of regression models with incomplete and noisy data

The estimation of the mixtures of regression models is usually based on the normal assumption of components and maximum likelihood estimation of the normal components is sensitive to noise, outliers, or high-leverage points. Missing values are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this article, we propose the mixtures of regression models for contaminated incomplete heterogeneous data. The proposed models provide robust estimates of regression coefficients varying across latent subgroups even under the presence of missing values. The methodology is illustrated through simulation studies and a real data analysis.     

Asymptotic inference for mixture models by using data-dependent priors

For certain mixture models, improper priors are undesirable because they yield improper posteriors. However, proper priors may be undesirable because they require subjective input. We propose the use of specially chosen data-dependent priors. We show that, in some cases, data-dependent priors are the only priors that produce intervals with second-order correct frequentist coverage. The resulting posterior also has another interpretation: it is the product of a fixed prior and a pseudolikelihood.     

Exact likelihood inference for Laplace distribution based on Type-II censored samples

We develop exact inference for the location and scale parameters of the Laplace (double exponential) distribution based on their maximum likelihood estimators from a Type-II censored sample. Based on some pivotal quantities, exact confidence intervals and tests of hypotheses are constructed. Upon conditioning first on the number of observations that are below the population median, exact distributions of the pivotal quantities are expressed as mixtures of linear combinations and of ratios of linear combinations of standard exponential random variables, which facilitates the computation of quantiles of these pivotal quantities. Tables of quantiles are presented for the complete sample case.     

A comparison of various methods for estimating the parameters in mixtures of von mises distributions

In this paper we compare five methods for estimating the unknown parameters in a mixture of two von Mises distributions. We propose a new method based on the characteristic function and compare it with the classical methods based on maximum likelihood and moments. Thus far these methods have been successfully applied only to linear data. Here we show that the application to circular data is reasonably straightforward and that convergence to the final estimates is fairly rapid. For various simulated known mixtures the results obtained are satisfactory. Finally, we introduce a modification of the method of moments which is considerably faster in CPU time than any of the other methods used and gives good results.     

Mixtures of Mixtures Based on Multiresolution Analysis Theory

In this article, the approaches for exploiting mixtures of mixtures are expanded by using the Multiresolution family of probability density functions (MR pdf). The flexibility and the properties of local analysis of the MR pdf facilitate the location of subpopulations into a given population. In order to do this, two algorithms are provided.

The MR model is more flexible in adapting to the different subpopulations than the traditional mixtures. In addition, the problems of identification of mixtures distributions and the label-switching do not appear in the MR pdf context.     

On the mixtures of Weibull and Pareto (IV) distribution: An alternative to Pareto distribution

Finite mixture models have provided a reasonable tool to model various types of observed phenomena, specially those which are random in nature. In this article, a finite mixture of Weibull and Pareto (IV) distribution is considered and studied. Some structural properties of the resulting model are discussed including estimation of the model parameters via expectation maximization (EM) algorithm. A real-life data application exhibits the fact that in certain situations, this mixture model might be a better alternative than the rival popular models.     

A Family of Near-Exact Distributions Based on Truncations of the Exact Distribution for the Generalized Wilks Lambda Statistic

For the case where at least two sets have an odd number of variables we do not have the exact distribution of the generalized Wilks Lambda statistic in a manageable form, adequate for manipulation. In this article, we develop a family of very accurate near-exact distributions for this statistic for the case where two or three sets have an odd number of variables. We first express the exact characteristic function of the logarithm of the statistic in the form of the characteristic function of an infinite mixture of Generalized Integer Gamma distributions. Then, based on truncations of this exact characteristic function, we obtain a family of near-exact distributions, which, by construction, match the first two exact moments. These near-exact distributions display an asymptotic behaviour for increasing number of variables involved. The corresponding cumulative distribution functions are obtained in a concise and manageable form, relatively easy to implement computationally, allowing for the computation of virtually exact quantiles. We undertake a comparative study for small sample sizes, using two proximity measures based on the Berry-Esseen bounds, to assess the performance of the near-exact distributions for different numbers of sets of variables and different numbers of variables in each set.     

Development and comparative study of two near-exact approximations to the distribution of the product of an odd number of independent Beta random variables

Using the concept of near-exact approximation to a distribution we developed two different near-exact approximations to the distribution of the product of an odd number of particular independent Beta random variables (r.v.'s). One of them is a particular generalized near-integer Gamma (GNIG) distribution and the other is a mixture of two GNIG distributions. These near-exact distributions are mostly adequate to be used as a basis for approximations of distributions of several statistics used in multivariate analysis. By factoring the characteristic function (c.f.) of the logarithm of the product of the Beta r.v.'s, and then replacing a suitably chosen factor of that c.f. by an adequate asymptotic result it is possible to obtain what we call a near-exact c.f., which gives rise to the near-exact approximation to the exact distribution. Depending on the asymptotic result used to replace the chosen parts of the c.f., one may obtain different near-exact approximations. Moments from the two near-exact approximations developed are compared with the exact ones. The two approximations are also compared with each other, namely in terms of moments and quantiles.     

Bayesian Clustering of Many Garch Models

We consider the estimation of a large number of GARCH models, of the order of several hundreds. Our interest lies in the identification of common structures in the volatility dynamics of the univariate time series. To do so, we classify the series in an unknown number of clusters. Within a cluster, the series share the same model and the same parameters. Each cluster contains therefore similar series. We do not know a priori which series belongs to which cluster. The model is a finite mixture of distributions, where the component weights are unknown parameters and each component distribution has its own conditional mean and variance. Inference is done by the Bayesian approach, using data augmentation techniques. Simulations and an illustration using data on U.S. stocks are provided.     

Some properties of discrete bathtub-shaped distributions

The existence and the usefulness of discrete bathtub-shaped and upside down bathtub-shaped distributions have been demonstrated in some papers of recent origin. However, the general properties of these two classes of distributions do not seem to have been discussed. This article proposes to study some reliability properties of such distributions. We investigate the closure properties with reference to convolution, mixing, series and parallel systems, etc. and existence of bounds on reliability functions, moment properties, and convergence.