A method to estimate power parameter in exponential power distribution via polynomial regression

The exponential power distribution (EPD), also known as generalized error distribution, is a flexible symmetrical unimodal family that belongs to the exponential one. The EPD becomes the density function of a range of symmetric distributions with different values of its power parameter β. A closed-form estimator for β does not exist, so the power parameter is usually estimated numerically. Unfortunately, the optimization algorithms do not always converge, especially when the true value of β is close to its parametric space frontier. In this paper, we present an alternative method to estimate β. Our proposal is based on the normal standardized Q–Q plot, and it exploits the relationship between β and the kurtosis. Furthermore, it is a direct method which does not require computational efforts nor the use of optimization algorithms.     

Approximating probability density functions in hybrid Bayesian networks with mixtures of truncated exponentials

Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization and Monte Carlo methods for solving hybrid Bayesian networks. Any probability density function (PDF) can be approximated by an MTE potential, which can always be marginalized in closed form. This allows propagation to be done exactly using the Shenoy-Shafer architecture for computing marginals, with no restrictions on the construction of a join tree. This paper presents MTE potentials that approximate standard PDF’s and applications of these potentials for solving inference problems in hybrid Bayesian networks. These approximations will extend the types of inference problems that can be modelled with Bayesian networks, as demonstrated using three examples.     

Dimensionally reduced mixtures of regression models

A mixture of regression models for multivariate observed variables which contextually involves a dimension reduction step through a linear factor model is proposed. The model estimation is performed via the EM-algorithm and a procedure to compute asymptotic standard errors for the parameter estimates is developed. The proposed approach is applied to the study of students satisfaction towards different aspects of their school as a function of various covariates.     

Hierarchical Gaussian process mixtures for regression

As a result of their good performance in practice and their desirable analytical properties, Gaussian process regression models are becoming increasingly of interest in statistics, engineering and other fields. However, two major problems arise when the model is applied to a large data-set with repeated measurements. One stems from the systematic heterogeneity among the different replications, and the other is the requirement to invert a covariance matrix which is involved in the implementation of the model. The dimension of this matrix equals the sample size of the training data-set. In this paper, a Gaussian process mixture model for regression is proposed for dealing with the above two problems, and a hybrid Markov chain Monte Carlo (MCMC) algorithm is used for its implementation. Application to a real data-set is reported.     

Finite mixtures approach to ecological regression

During past few years great attention has been devoted to the analysis of disease incidence and mortality rates, with an explicit focus on modelling geographical variation of rates observed in spatially adjacent regions. The general aim of these contributes has been both to highlight clusters of regions with homogeneous relative risk and to determine the effects of observed and unobserved risk factors related to the analyzed disease. Most of the proposed modelling approaches can be derived as alternative specifications of the components of a general convolution model (Molliè, 1996). In this paper, we consider the semiparametric approach discussed by Schlattmann and Böhning (1993); in particular, we focus on models with an explicit spatially structured component (see Biggeri et al., 2000), and propose alternative choices for the structure of the spatial component.     

Finite mixtures of quantile and M-quantile regression models

In this paper we define a finite mixture of quantile and M-quantile regression models for heterogeneous and /or for dependent/clustered data. Components of the finite mixture represent clusters of individuals with homogeneous values of model parameters. For its flexibility and ease of estimation, the proposed approaches can be extended to random coefficients with a higher dimension than the simple random intercept case. Estimation of model parameters is obtained through maximum likelihood, by implementing an EM-type algorithm. The standard error estimates for model parameters are obtained using the inverse of the observed information matrix, derived through the Oakes (J R Stat Soc Ser B 61:479–482, 1999) formula in the M-quantile setting, and through nonparametric bootstrap in the quantile case. We present a large scale simulation study to analyse the practical behaviour of the proposed model and to evaluate the empirical performance of the proposed standard error estimates for model parameters. We considered a variety of empirical settings in both the random intercept and the random coefficient case. The proposed modelling approaches are also applied to two well-known datasets which give further insights on their empirical behaviour.     

Random effects regression mixtures for analyzing infant habituation

Random effects regression mixture models are a way to classify longitudinal data (or trajectories) having possibly varying lengths. The mixture structure of the traditional random effects regression mixture model arises through the distribution of the random regression coefficients, which is assumed to be a mixture of multivariate normals. An extension of this standard model is presented that accounts for various levels of heterogeneity among the trajectories, depending on their assumed error structure. A standard likelihood ratio test is presented for testing this error structure assumption. Full details of an expectation-conditional maximization algorithm for maximum likelihood estimation are also presented. This model is used to analyze data from an infant habituation experiment, where it is desirable to assess whether infants comprise different populations in terms of their habituation time.     

Linear regression through the origin with constant coefficient of variation for the inverse gaussian distribution

A simple linear regression model with no intercept term for the situation where the response variable obeys an inverse Gaussian distribution and the coefficient of variation is an unknown constant is discussed. Maximum likelihood estimators and the confidence limits of the regression parameter are obtained. Finally uniformly minimum variance unbiased estimators of parameters are given.     

Asymptotic distribution of regression correlation coefficient for Poisson regression model

This study treats an asymptotic distribution for measures of predictive power for generalized linear models (GLMs). We focus on the regression correlation coefficient (RCC) that is one of the measures of predictive power. The RCC, proposed by Zheng and Agresti is a population value and a generalization of the population value for the coefficient of determination. Therefore, the RCC is easy to interpret and familiar. Recently, Takahashi and Kurosawa provided an explicit form of the RCC and proposed a new RCC estimator for a Poisson regression model. They also showed the validity of the new estimator compared with other estimators. This study discusses the new statistical properties of the RCC for the Poisson regression model. Furthermore, we show an asymptotic normality of the RCC estimator.     

On mitigating collinearity through mixtures

In linear models having near collinear columns of X, ridge and surrogate estimators often are used to mitigate collinearity. A new class of estimators is based on mixtures, either of X and a design minimal in an ordered class or of the Fisher information and a scalar matrix. Comparisons are drawn among choices for the mixing parameter, and the estimators are found to be admissible relative to ordinary least squares. Case studies demonstrate that selected mixture designs are perturbed from the original design to a lesser extent than are those of the surrogate method, while retaining reasonable efficiency characteristics.     

Robust regression through robust covariances

This paper discusses the estimation of regression parameters after summarizing the data by a covariance matrix of the concatenated vector of explanatory variables and response variable. A robust estimate of the covariance matrix leads to a robust regression estimator. An M-estimator at the covariance estimation step is studied in the paper, and the resulting regression estimator is compared to a few previously proposed robust regression estimators.     

Median estimation through a regression transformation

The author considers the problem of constructing confidence intervals for the median of a future observation at certain values of exogenous variables, following a normalizing transformation. He shows that when this transformation is estimated, the usual interval obtained through an inverse transformation needs to be corrected, even when the sample size is large. He then gives a simple analytical solution to this problem and provides simulation results confirming the good small‐sample properties of the corrected interval. He also presents two concrete illustrations.     

Multiple-output quantile regression through optimal quantization

A new nonparametric quantile regression method based on the concept of optimal quantization was developed recently and was showed to provide estimators that often dominate their classical, kernel-type, competitors. In the present work, we extend this method to multiple-output regression problems. We show how quantization allows approximating population multiple-output regression quantiles based on halfspace depth. We prove that this approximation becomes arbitrarily accurate as the size of the quantization grid goes to infinity. We also derive a weak consistency result for a sample version of the proposed regression quantiles. Through simulations, we compare the performances of our estimators with (local constant and local bilinear) kernel competitors. The results reveal that the proposed quantization-based estimators, which are local constant in nature, outperform their kernel counterparts and even often dominate their local bilinear kernel competitors. The various approaches are also compared on artificial and real data.     

Linear censored regression models with skew scale mixtures of normal distributions

A special source of difficulty in the statistical analysis is the possibility that some subjects may not have a complete observation of the response variable. Such incomplete observation of the response variable is called censoring. Censorship can occur for a variety of reasons, including limitations of measurement equipment, design of the experiment, and non-occurrence of the event of interest until the end of the study. In the presence of censoring, the dependence of the response variable on the explanatory variables can be explored through regression analysis. In this paper, we propose to examine the censorship problem in context of the class of asymmetric, i.e., we have proposed a linear regression model with censored responses based on skew scale mixtures of normal distributions. We develop a Monte Carlo EM (MCEM) algorithm to perform maximum likelihood inference of the parameters in the proposed linear censored regression models with skew scale mixtures of normal distributions. The MCEM algorithm has been discussed with an emphasis on the skew-normal, skew Student-t-normal, skew-slash and skew-contaminated normal distributions. To examine the performance of the proposed method, we present some simulation studies and analyze a real dataset.     

Inference and diagnostics in skew scale mixtures of normal regression models

Skew scale mixtures of normal distributions are often used for statistical procedures involving asymmetric data and heavy-tailed. The main virtue of the members of this family of distributions is that they are easy to simulate from and they also supply genuine expectation-maximization (EM) algorithms for maximum likelihood estimation. In this paper, we extend the EM algorithm for linear regression models and we develop diagnostics analyses via local influence and generalized leverage, following Zhu and Lee's approach. This is because Cook's well-known approach cannot be used to obtain measures of local influence. The EM-type algorithm has been discussed with an emphasis on the skew Student-t-normal, skew slash, skew-contaminated normal and skew power-exponential distributions. Finally, results obtained for a real data set are reported, illustrating the usefulness of the proposed method.     

Heteroscedastic and heavy-tailed regression with mixtures of skew Laplace normal distributions

Joint modelling skewness and heterogeneity is challenging in data analysis, particularly in regression analysis which allows a random probability distribution to change flexibly with covariates. This paper, based on a skew Laplace normal (SLN) mixture of location, scale, and skewness, introduces a new regression model which provides a flexible modelling of location, scale and skewness parameters simultaneously. The maximum likelihood (ML) estimators of all parameters of the proposed model via the expectation-maximization (EM) algorithm as well as their asymptotic properties are derived. Numerical analyses via a simulation study and a real data example are used to illustrate the performance of the proposed model.     

The uniform power distribution

This paper introduces a generalization of the normal distribution: the uniform power distribution. It is a symmetric and unimodal family of distributions, defined on the real line, and is closely related to the exponential power family. The exponential power family was introduced to allow the modelling of kurtosis. The uniform power family matches the exponential power family with respect to the range of kurtosis. However, whereas the exponential is somewhat difficult to work with, the contrary is true for the uniform power family.     

Robust symmetric distribution location estimation and regression

A procedure for estimating the location parameter of an unknown symmetric distribution is developed for application to samples from very light-tailed through very heavy-tailed distributions. This procedure has an easy extension to a technique for estimating the coefficients in a linear regression model whose error distribution is symmetric with arbitrary tail weights. The regression procedure is, in turn, extended to make it applicable to situations where the error distribution is either symmetric or skewed. The potentials of the procedures for robust location parameter and regression coefficient estimation are demonstrated by simulation studies.