Bayesian analysis of covariance under inverse Gaussian model

This paper considers the problem of analysis of covariance (ANCOVA) under the assumption of inverse Gaussian distribution for response variable from the Bayesian point of view. We develop a fully Bayesian model for ANCOVA based on the conjugate prior distributions for parameters contained in the model. The Bayes estimator of parameters, ANCOVA model and adjusted effects for both treatments and covariates along with predictive distribution of future observations are developed. We also provide the essentials for comparing adjusted treatments effects and adjusted factor effects. A simulation study and a real world application are also performed to illustrate and evaluate the proposed Bayesian model.     

Inequalities on partial correlations in Gaussian graphical models containing star shapes

ABSTRACT

This short paper proves inequalities that restrict the magnitudes of the partial correlations in star-shaped structures in Gaussian graphical models. These inequalities have to be satisfied by distributions that are used for generating simulated data to test structure-learning algorithms, but methods that have been used to create such distributions do not always ensure that they are. The inequalities are also noteworthy because stars are common and meaningful in real-world networks.     

Transformation-Kernel Estimation of Copula Densities

ABSTRACT

The standard kernel estimator of copula densities suffers from boundary biases and inconsistency due to unbounded densities. Transforming the domain of estimation into an unbounded one remedies both problems, but also introduces an unbounded multiplier that may produce erratic boundary behaviors in the final density estimate. We propose an improved transformation-kernel estimator that employs a smooth tapering device to counter the undesirable influence of the multiplier. We establish the theoretical properties of the new estimator and its automatic higher-order improvement under Gaussian copulas. We present two practical methods of smoothing parameter selection. Extensive Monte Carlo simulations demonstrate the competence of the proposed estimator in terms of global and tail performance. Two real-world examples are provided. Supplementary materials for this article are available online.     

Estimating finite mixture of continuous trees using penalized mutual information

Abstract

In this paper we introduce continuous tree mixture model that is the mixture of undirected graphical models with tree structured graphs and is considered as multivariate analysis with a non parametric approach. We estimate its parameters, the component edge sets and mixture proportions through regularized maximum likalihood procedure. Our new algorithm, which uses expectation maximization algorithm and the modified version of Kruskal algorithm, simultaneosly estimates and prunes the mixture component trees. Simulation studies indicate this method performs better than the alternative Gaussian graphical mixture model. The proposed method is also applied to water-level data set and is compared with the results of Gaussian mixture model.     

Analysis of covariance under inverse Gaussian model

This paper considers the problem of analysis of covariance (ANCOVA) under the assumption of inverse Gaussian distribution for response variable. We develop the essential methodology for estimating the model parameters via maximum likelihood method. The general form of the maximum likelihood estimator is obtained in color closed form. Adjusted treatment effects and adjusted covariate effects are given, too. We also provide the asymptotic distribution of the proposed estimators. A simulation study and a real world application are also performed to illustrate and evaluate the proposed methodology.     

A Note on Item Response Theory Modeling for Online Customer Ratings

ABSTRACT

Online consumer product ratings data are increasing rapidly. While most of the current graphical displays mainly represent the average ratings, Ho and Quinn proposed an easily interpretable graphical display based on an ordinal item response theory (IRT) model, which successfully accounts for systematic interrater differences. Conventionally, the discrimination parameters in IRT models are constrained to be positive, particularly in the modeling of scored data from educational tests. In this article, we use real-world ratings data to demonstrate that such a constraint can have a great impact on the parameter estimation. This impact on estimation was explained through rater behavior. We also discuss correlation among raters and assess the prediction accuracy for both the constrained and the unconstrained models. The results show that the unconstrained model performs better when a larger fraction of rater pairs exhibit negative correlations in ratings.     

SOME REMARKS ON THE INFORMATION GEOMETRY OF THE GAMMA DISTRIBUTION

ABSTRACT

Bounds for the Fisher information metric associated with the Gamma statistical model are found in terms of Poincaré type metric. This results in the determination of bounds for the Rao distance, that is the Riemannian distance induced by the information metric, between Gamma distributions, and of bounds for the Gaussian curvature of the Gamma model. The bounds seem to be sharp, where the lower and upper Rao distance bounds are Poincaré distances with Gaussian curvatures ?1/4 and ?1/2, respectively. In addition, the sign of the Gaussian curvature of the Gamma model is shown to be negative which means, in particular, that the geometry of the model is hyperbolic.     

Joint models for a GLM-type longitudinal response and a time-to-event with smooth random effects

ABSTRACT

Longitudinal studies often entail non-Gaussian primary responses. When dropout occurs, potential non-ignorability of the missingness process may occur, and a joint model for the primary response and a time-to-event may represent an appealing tool to account for dependence between the two processes. As an extension to the GLMJM, recently proposed, and based on Gaussian latent effects, we assume that the random effects follow a smooth, P-spline based density. To estimate model parameters, we adopt a two-step conditional Newton–Raphson algorithm. Since the maximization of the penalized log-likelihood requires numerical integration over the random effect, which is often cumbersome, we opt for a pseudo-adaptive Gaussian quadrature rule to approximate the model likelihood. We discuss the proposed model by analyzing an original dataset on dilated cardiomyopathies and through a simulation study.     

The normal tempered stable regression model

Abstract

This paper deals with the statistical studies of the normal tempered stable model defined by Barndorff-Nielsen and Shephard. It represents the natural extension of the normal inverse Gaussian one introduced by Barndorff-Nielsen. We basically use the Monte-Carlo’s approximation in order to simulate this distribution. We introduce a linear regression model with normal tempered stable error. We apply this model for the analyzing of the daily logarithm returns data on CAC40 index. The parameters estimation results show that this model better deals with long tailed distribution which is the case for the CAC40 logarithm returns.     

A permutation-based Bayesian approach for inverse covariance estimation

Abstract

Covariance estimation and selection for multivariate datasets in a high-dimensional regime is a fundamental problem in modern statistics. Gaussian graphical models are a popular class of models used for this purpose. Current Bayesian methods for inverse covariance matrix estimation under Gaussian graphical models require the underlying graph and hence the ordering of variables to be known. However, in practice, such information on the true underlying model is often unavailable. We therefore propose a novel permutation-based Bayesian approach to tackle the unknown variable ordering issue. In particular, we utilize multiple maximum a posteriori estimates under the DAG-Wishart prior for each permutation, and subsequently construct the final estimate of the inverse covariance matrix. The proposed estimator has smaller variability and yields order-invariant property. We establish posterior convergence rates under mild assumptions and illustrate that our method outperforms existing approaches in estimating the inverse covariance matrices via simulation studies.     

Multiplicative Algorithms for Maximum Penalized Likelihood Inversion with Non Negative Constraints and Generalized Error Distributions

In many linear inverse problems the unknown function f (or its discrete approximation Θ _p×1), which needs to be reconstructed, is subject to the non negative constraint(s); we call these problems the non negative linear inverse problems (NNLIPs). This article considers NNLIPs. However, the error distribution is not confined to the traditional Gaussian or Poisson distributions. We adopt the exponential family of distributions where Gaussian and Poisson are special cases. We search for the non negative maximum penalized likelihood (NNMPL) estimate of Θ. The size of Θ often prohibits direct implementation of the traditional methods for constrained optimization. Given that the measurements and point-spread-function (PSF) values are all non negative, we propose a simple multiplicative iterative algorithm. We show that if there is no penalty, then this algorithm is almost sure to converge; otherwise a relaxation or line search is necessitated to assure its convergence.     

Shared frailty models with baseline generalized Pareto distribution

Abstract

In this article, we have considered three different shared frailty models under the assumption of generalized Pareto Distribution as baseline distribution. Frailty models have been used in the survival analysis to account for the unobserved heterogeneity in an individual risks to disease and death. These three frailty models are with gamma frailty, inverse Gaussian frailty and positive stable frailty. Then we introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters. We applied these three models to a kidney infection data and find the best fitted model for kidney infection data. We present a simulation study to compare true value of the parameters with the estimated values. Model comparison is made using Bayesian model selection criterion and a well-fitted model is suggested for the kidney infection data.     

Singular Gaussian graphical models: Structure learning

ABSTRACT

The goal of this article is to introduce singular Gaussian graphical models and their conditional independence properties. In fact, we extend the concept of Gaussian Markov Random Field to the case of a multivariate normally distributed vector with a singular covariance matrix. We construct, then, the associated graph’s structure from the covariance matrix’s pseudo-inverse on the basis of a characterization of the pairwise conditional independence. The proposed approach can also be used when the covariance matrix is ill-conditioned, through projecting data on a smaller subspace. In this case, our method ensures numerical stability and consistency of the constructed graph and significantly reduces the inference problem’s complexity. These aspects are illustrated using numerical experiments.     

Final outcome probabilities for SIR epidemic model

Abstract

We consider an SIR stochastic epidemic model in which new infections occur at rate f(x, y), where x and y are, respectively, the number of susceptibles and infectives at the time of infection and f is a positive sequence of real functions. A simple explicit formula for the final size distribution is obtained. Some efficient recursive methods are proved for the exact calculation of this distribution. In addition, we give a Gaussian approximation for the final distribution using a diffusion process approximation.     

THE LOG-EIG DISTRIBUTION: A NEW PROBABILITY MODEL FOR LIFETIME DATA

ABSTRACT

In this paper, we propose a new probability model called the log-EIG distribution for lifetime data analysis. Some important properties of the proposed model and maximum likelihood estimation of its parameters are discussed. Its relationship with the exponential inverse Gaussian distribution is similar to that of the lognormal and the normal distributions. Through applications to well-known datasets, we show that the log-EIG distribution competes well, and in some instances even provides a better fit than the commonly used lifetime models such as the gamma, lognormal, Weibull and inverse Gaussian distributions. It can accommodate situations where an increasing failure rate model is required as well as those with a decreasing failure rate at larger times.     

Estimating and Testing Nonlinear Local Dependence Between Two Time Series

ABSTRACT

The most common measure of dependence between two time series is the cross-correlation function. This measure gives a complete characterization of dependence for two linear and jointly Gaussian time series, but it often fails for nonlinear and non-Gaussian time series models, such as the ARCH-type models used in finance. The cross-correlation function is a global measure of dependence. In this article, we apply to bivariate time series the nonlinear local measure of dependence called local Gaussian correlation. It generally works well also for nonlinear models, and it can distinguish between positive and negative local dependence. We construct confidence intervals for the local Gaussian correlation and develop a test based on this measure of dependence. Asymptotic properties are derived for the parameter estimates, for the test functional and for a block bootstrap procedure. For both simulated and financial index data, we construct confidence intervals and we compare the proposed test with one based on the ordinary correlation and with one based on the Brownian distance correlation. Financial indexes are examined over a long time period and their local joint behavior, including tail behavior, is analyzed prior to, during and after the financial crisis. Supplementary material for this article is available online.     

A Multivariate Extension of Inverse Gaussian Distribution Derived from Inverse Relationship

Abstract

We propose a new multivariate extension of the inverse Gaussian distribution derived from a certain multivariate inverse relationship. First we define a multivariate extension of the inverse relationship between two sets of multivariate distributions, then define a reduced inverse relationship between two multivariate distributions. We derive the multivariate continuous distribution that has the reduced multivariate inverse relationship with a multivariate normal distribution and call it a multivariate inverse Gaussian distribution. This distribution is also characterized as the distribution of the location of a multivariate Brownian motion at some stopping time. The marginal distribution in one direction is the inverse Gaussian distribution, and the conditional distribution in the space perpendicular to this direction is a multivariate normal distribution. Mean, variance, and higher order cumulants are derived from the multivariate inverse relationship with a multivariate normal distribution. Other properties such as reproductivity and infinite divisibility are also given.     

Multiscale Gaussian convolution algorithm for estimate of Gaussian mixture model

Abstract

This paper introduces a multiscale Gaussian convolution model of Gaussian mixture (MGC-GMM) via the convolution of the GMM and a multiscale Gaussian window function. It is found that the MGC-GMM is still a Gaussian mixture model, and its parameters can be mapped back to the parameters of the GMM. Meanwhile, the multiscale probability density function (MPDF) of the MGC-GMM can be viewed as the mathematical expectation of a random process induced by the Gaussian window function and the GMM, which can be directly estimated by the use of sample data. Based on the estimated MPDF, a novel algorithm denoted by the MGC is proposed for the selection of model and the parameter estimates of the GMM, where the component number and the means of the GMM are respectively determined by the number and the locations of the maximum points of the MPDF, and the numerical algorithms for the weight and variance parameters of the GMM are derived. The MGC is suitable for the GMM with diagonal covariance matrices. A MGC-EM algorithm is also presented for the generalized GMM, where the GMM is estimated using the EM algorithm by taking the estimates from the MGC as initial parameters of the GMM model. The proposed algorithms are tested via a series of simulated sample sets from the given GMM models, and the results show that the proposed algorithms can effectively estimate the GMM model.     

Inequality Constrained State-Space Models

ABSTRACT

The standard Kalman filter cannot handle inequality constraints imposed on the state variables, as state truncation induces a nonlinear and non-Gaussian model. We propose a Rao-Blackwellized particle filter with the optimal importance function for forward filtering and the likelihood function evaluation. The particle filter effectively enforces the state constraints when the Kalman filter violates them. Monte Carlo experiments demonstrate excellent performance of the proposed particle filter with Rao-Blackwellization, in which the Gaussian linear sub-structure is exploited at both the cross-sectional and temporal levels.     

Mixed Marginal Copula Modeling

ABSTRACT

This article extends the literature on copulas with discrete or continuous marginals to the case where some of the marginals are a mixture of discrete and continuous components. We do so by carefully defining the likelihood as the density of the observations with respect to a mixed measure. The treatment is quite general, although we focus on mixtures of Gaussian and Archimedean copulas. The inference is Bayesian with the estimation carried out by Markov chain Monte Carlo. We illustrate the methodology and algorithms by applying them to estimate a multivariate income dynamics model. Supplementary materials for this article are available online.