Exponential Power Family Research Articles

Component selection for exponential power mixture models

ABSTRACT Exponential Power (EP) family is a much flexible distribution family including Gaussian family as a sub-family. In this article, we study component selection and estimation for EP mixture models and regressions. The assumption on zero component mean in [X. Cao, Q. Zhao, D. Meng, Y. Chen, and Z. Xu, Robust low-rank matrix factorization under general mixture noise distributions, IEEE. Trans. Image. Process. 25 (2016), pp. 4677–4690.] is relaxed. To select components and estimate parameters simultaneously, we propose a penalized likelihood method, which can shrink mixing proportions to zero to achieve components selection. Modified EM algorithms are proposed, and the consistency of estimated component number is obtained. Simulation studies show the advantages of the proposed methods on accuracies of component number selection, parameter estimation, and density estimation. Analysis of value at risk of SHIBOR and a climate change data are given as illustration.

Sequentially Forecasting Economic Indices Using Mixture Linear Combinations of EP Distributions

This article displays an application of the statistical method motivated by Bruno de Finetti’s operational subjective theory of probability. We use exchangeable forecasting distributions based on mixtures of linear combinations of exponential power (EP) distributions to forecast the sequence of daily rates of return from the Dow-Jones index of stock prices over a 20 year period. The operational subjective statistical method for comparing distributions is quite different from that commonly used in data analysis, because it rejects the basic tenets underlying the practice of hypothesis testing. In its place, proper scoring rules for forecast distributions are used to assess the values of various forecasting strategies. Using a logarithmic scoring rule, we find that a mixture linear combination of EP distributions scores markedly better than does a simple mixture over the EP family, which scores much better than does a simple Normal mixture. Surprisingly, a mixture over a linear combination of three Normal distributions also makes a substantial improvement over a simple Normal mixture, although it does not quite match the performance of even the simple EP mixture. All substantive forecasting improvements become most marked after extreme tail phenomena were actually observed in the sequence, in particular after the abrupt drop in market prices in October, 1987. However, the improvements continue to be apparent over the long haul of 1985-2006 which has seen a number of extreme price changes. This result is supported by an analysis of the Negentropies embedded in the forecasting distributions, and a proper scoring analysis of these Negentropies as well.

Inferences from Asymmetric Multivariate Exponential Power Distribution

In this paper, we develop the asymptotic distribution of the covariance structure of an asymmetric multivariate exponential power distribution (AMEPD) by extending the existing Hotelling t2 distribution to its generalized form. The obtained results ‘generalised Hotelling t2 test statistic’, accommodates for the existing Mahalanobis distance between two multivariate data sets. The hazard rate and the characteristics function of the desired test statistics are established. The study further establishes the test of hypothesis and simultaneous confidence interval for the population mean vectors of p th attributes from AMEPD by using the Neymar-Pearson lemma and Pivotal quantity approaches. The obtained results, which are dependent on shape parameter flexibility and skewness parameter, generalize the test of hypothesis and simultaneous confidence intervals of various distributions which are members of the exponential power family. Simulated and real life data were used to demonstrate the usefulness of the obtained results.

Using Prior Expansions for Prior-Data Conflict Checking

Any Bayesian analysis involves combining information represented through different model components, and when different sources of information are in conflict it is important to detect this. Here we consider checking for prior-data conflict in Bayesian models by expanding the prior used for the analysis into a larger family of priors, and considering a marginal likelihood score statistic for the expansion parameter. Consideration of different expansions can be informative about the nature of any conflict, and an appropriate choice of expansion can provide more sensitive checks for conflicts of certain types. Extensions to hierarchically specified priors and connections with other approaches to prior-data conflict checking are considered, and implementation in complex situations is illustrated with two applications. The first concerns testing for the appropriateness of a LASSO penalty in shrinkage estimation of coefficients in linear regression. Our method is compared with a recent suggestion in the literature designed to be powerful against alternatives in the exponential power family, and we use this family as the prior expansion for constructing our check. A second application concerns a problem in quantum state estimation, where a multinomial model is considered with physical constraints on the model parameters. In this example, the usefulness of different prior expansions is demonstrated for obtaining checks which are sensitive to different aspects of the prior.

An alternative skew exponential power distribution formulation

In this note we propose a newly formulated skew exponential power distribution that behaves substantially better than previously defined versions. This new model performs very well in terms of the large sample behavior of the maximum likelihood estimation procedure when compared to the classically defined four parameter model defined by Azzalini. More recently, approaches to defining a skew exponential power distribution have used five or more parameters. Our approach improves upon previous attempts to extend the symmetric power exponential family to include skew alternatives by maintaining a minimum set of four parameters corresponding directly to location, scale, skewness and kurtosis. We illustrate the utility of our proposed model using translational and clinical data sets.

Bayesian analysis of some models that use the asymmetric exponential power distribution

The asymmetric exponential power (AEP) family includes the symmetric exponential power distribution as a particular case. It provides flexible distributions with lighter and heavier tails compared to the normal one. The distributions of this family can successfully handle both symmetry/asymmetry and light/heavy tails simultaneously. Even more, the distributions can fit each tail separately. This provides a great flexibility when fitting experimental data. The idea of using a scale mixture of uniform representation of the AEP distribution is exploited to derive efficient Gibbs sampling algorithms in three different Bayesian contexts. Firstly, a posterior exploration is performed, where the AEP distribution is considered for the likelihood model. Secondly, a linear regression model, that uses the AEP distribution for the error variable, is developed. And finally, a binary regression model is analyzed, by using the inverse of the AEP cumulative distribution function as the link function. These three models have been built in such a way that they share some full conditional distributions to sample from their respective posterior distributions. The theoretical results are illustrated by comparing with other competing models using some previously published datasets.

Assessing the effect of kurtosis deviations from Gaussianity on conditional distributions

The multivariate exponential power family is considered for n-dimensional random variables, Z, with a known partition Z≡(Y,X) of dimensions p and n-p, respectively, with interest focusing on the conditional distribution Y|X. An infinitesimal variation of any parameter of the joint distribution produces perturbations in both the conditional and marginal distributions. The aim of the study was to determine the local effect of kurtosis deviations using the Kullback–Leibler divergence measure between probability distributions. The additive decomposition of this measure in terms of the conditional and marginal distributions, Y|X and X, is used to define a relative sensitivity measure of the conditional distribution family {Y|X=x}. Finally, simulated results suggest that for large dimensions, the measure is approximately equal to the ratio p/n, and then the effect of non-normality with respect to kurtosis depends only on the relative size of the variables considered in the partition of the random vector.

Bayesian binary regression with exponential power link

A flexible Bayesian approach to a generalized linear model is proposed to describe the dependence of binary data on explanatory variables. The inverse of the exponential power cumulative distribution function is used as the link to the binary regression model. The exponential power family provides distributions with both lighter and heavier tails compared to the normal distribution, and includes the normal and an approximation to the logistic distribution as particular cases. The idea of using a data augmentation framework and a mixture representation of the exponential power distribution is exploited to derive efficient Gibbs sampling algorithms for both informative and noninformative settings. Some examples are given to illustrate the performance of the proposed approach when compared with other competing models.

A study of the effect of kurtosis on discriminant analysis under elliptical populations

This paper is concerned with the role some parameters indexing four important families within the multivariate elliptically contoured distributions play as indicators of multivariate kurtosis. The problem is addressed for the exponential power family, for a subclass of the Kotz family and for the Pearson type II and type VII distributions. Once such a problem is analyzed, we study the effect these parameters have, as kurtosis indicators, on binary discriminant analysis by exploring their relationship with the error rate of the Bayes discriminant rule. The effect is analyzed under mild conditions on the kernel function generating the elliptical density. Some numerical examples are given in order to illustrate our theoretical insights and findings.

Conditional Specification with Exponential Power Distributions

The problem of modeling Bayesian networks with continuous nodes deals with discrete approximations and conditional linear Gaussian models. In this article we have considered the possibility of using the exponential power family as conditional probability densities. It will be shown that for some platikurtic conditional distributions\\ in this family, conditional regression functions are constant. These results give conditions to avoid compatibility problems when distributions with lighter tails than the normal are used in the description of conditional densities to specify joint densities, like in Bayesian networks.

Relative Sensitivity of Conditional Distributions to Kurtosis Deviations in the Joint Model

The Multivariate Exponential Power family is considered for n-dimensional random variables, Z, with a known partition Z (Y,X) of dimensions p and n-p, respectively. An infinitesimal variation of any parameter produces both conditional and marginal distributions perturbations. The aim of our study is to determine the local effect of kurtosis deviations by means of the Kullback-Leibler divergence measure between probability distributions. The additive decomposition of this measure in terms of the conditional and marginal distributions, YǀX and X, has been used to define the relative sensitivity of the conditional distributions family {YǀX=x}. The obtained results show that, for large dimensions, it is nearly p/n.

Estimating a bounded parameter for symmetric distributions

For the problem of estimating under squared error loss the parameter of a symmetric distribution which is subject to an interval constraint, we develop general theory which provides improvements on various types of inadmissible procedures, such as maximum likelihood procedures. The applications and further developments given include: (i) symmetric location families such as the exponential power family including double-exponential and normal, Student and Cauchy, a Logistic type family, and scale mixture of normals in cases where the variance is lower bounded; (ii) symmetric exponential families such as those related to a Binomial(n,p) model with bounded |p−1/2| and to a Beta(α + θ, α −θ) model; and (iii) symmetric location distributions truncated to an interval (−c,c). Finally, several of the dominance results are studied with respect to model departures yielding robustness results, and specific findings are given for scale mixture of normals and truncated distributions.

The extended exponential power distribution and Bayesian robustness

In this paper, it is demonstrated that an extension to the exponential power family allows for robustness characteristics for the normal location parameter problem, previously thought to be restricted to the Student- t and a subclass of the positive stable families.

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.

Comparing the Sample Mean and the Sample Median: An Exploration in the Exponential Power Family

We compare the (asymptotic) efficiencies of the sample mean and the sample median in the exponential power family. This single family of densities is an ideal one to illustrate the effect of “heavy tails” on the two estimators. We find the density for which the mean and median are equally efficient, and observe what happens for limiting cases within the family. Only basic tools are needed in the analysis and statistical development.

Comparing the Sample Mean and the Sample Median: An Exploration in the Exponential Power Family

Comparing the Sample Mean and the Sample Median: An Exploration in the Exponential Power Family

Near ignorance classes of log-concave priors for the location model

Near ignorance classes (NIC) for the normal location model are defined in Pericchi and Walley (1991) as a proper alternative to the improper uniform prior. In this paper we study the feasibility of such classes using log-concave distributions such as the logistic and the Box and Tiao exponential power family. We use Meeden and Isaacson’s theory of posterior moments for exponential family likelihoods, as well as results for posterior distributions under log-concave priors. The main conclusion that emerges is that exponential tails are the lightest permitted in order to produce NICs with non-vacuous posterior inference, for the normal location model.

Properties of modelling the error distribution with an extra shape parameter

A robust extension of normal theory regression is to add an extra parameter to model the kurtosis of the error distribution, for example by using the T-family or the power-exponential family of distributions. The statistical properties of maximum likelihood estimation schemes for both families of models are considered. This article extends the work of Lange et al. (1989) in which the usefulness of the multivariate T-family for modelling data was demonstrated. The cost of adding the extra kurtosis parameter to the standard normal model is considered. This cost is measured by comparing the variance of the quantity of interest when the estimation of the extra parameter is taken into account and the variance when the estimated value of the extra parameter is treated as if it were known. In Lange et al. (1989), it is shown in a general setting that asymptotically there is no cost due to the estimation of the extra parameter, where the quantity of interest is the location or regression parameters. Whether this finding remains valid in small samples for the T and power-exponential families is investigated in this paper using Monte Carlo simulations and a real dataset. The efficiency of parameter estimates and coverage rates are also considered under three different scenarios: when the extra kurtosis parameter is estimated from the data, when the extra parameter is fixed at the true value, and when it is fixed at a wrong value. The expected information matrix is used to estimate the confidence intervals, and the comparisons are based on asymptotic calculations and Monte Carlo simulations. It is found that for the T-family there is very little cost due to estimation of the extra parameter, except for small sample sizes. The inflation in variance due to the estimation of the extra parameter increases as the sample size decreases. In the Monte Carlo simulations of simple regression settings the inflation in variance is found to be at most 14% for the T-family and at most 62% for the power-exponential family. For the T-family, there is a considerable loss of efficiency in fitting a normal model when the true degrees of freedom is small, but only a small loss of efficiency if a model with a low number of degrees of freedom is fit to normal observations. In contrast, the coverage rates of confidence intervals are close to the nominal level if a normal model is fit to the data whatever the true degrees of freedom, but the coverage rates can be too low if a model with a low number of degrees of freedom is fit to normal data. The coverage rates of confidence intervals when the degrees of freedom is estimated from the data are satisfactory, except at small sample size. Similar results are obtained for the power-exponential family. In addition to the larger inflation in variance due to estimation of the extra parameter, the power-exponential family is less satisfactory then the T-family because the extra kurtosis parameter is frequently estimated to be at the boundary of its range in small samples. The findings in this article support the notion that the extra kurtosis

On scale mixtures of normal distributions

SUMMARY The exponential power family of distributions of Box & Tiao (1973) is shown to be a subset of the class of scale mixtures of normals. The corresponding mixing distributions are explicitly obtained, identifying a close relationship between the exponential power family and a further class of normal scale mixtures, namely the stable distributions.

A Diagnostic Test for Normality Within the Power Exponential Family

This article develops the locally uniformly most powerful unbiased Lagrange multiplier test of normality of regression disturbances within the family of power exponential distributions. The small sample power properties of the test are compared in a Monte Carlo study with 6 well-known tests across 12 alternative nonnormal distributions. In addition, the finite sample power properties for nonnormal alternatives within the power exponential family are summarized by estimating response surfaces. The results suggest that the proposed text is computationally convenient and possesses relatively attractive power properties even against alternatives outside the power exponential family.