Multivariate Gamma Distribution Research Articles

Transport of neutrals in turbulent Scrape-off-Layer plasmas

The effect of turbulence on the transport of neutral species (atom, molecules) in plasmas is investigated. A stochastic model relying on a multivariate gamma distribution is used to describe turbulent fluctuations. This model incorporates several important known features of Scrape-off-Layer turbulence, and has been implemented in the EIRENE Monte Carlo code. The effect of fluctuations on the neutral density, ionisation source and Balmer α radial profiles are investigated. Two important control parameters are identified, namely the ratio of the neutral mean free path to the size of turbulent structures, and the ratio of recycling time scales to the turbulent times.

On Bayesian inference for generalized multivariate gamma distribution

In this paper we define a generalized multivariate gamma (MG) distribution and develop various properties of this distribution. Then we consider a Bayesian decision theoretic approach to develop the inference technique for the related scale matrix Σ . We show that maximum posteriori (MAP) estimate is a Bayes estimator. We also develop the testing problem for Σ using a Bayes factor. This approach provides a mathematically closed form solution for Σ . The only other approach to Bayesian inference for the MG distribution is given in Tsionas (2004), which is based on Markov Chain Monte Carlo (MCMC) technique. The Tsionas (2004) technique involves a costly matrix inversion whose computational complexity increases in cubic order, hence making inference infeasible for Σ , for large dimensions. In this paper, we provide an elegant closed form Bayes factor for Σ .

Multivariate Tweedie distributions and some related capital-at-risk analyses

We study a multivariate extension of the univariate exponential dispersion Tweedie family of distributions. The class, referred to as the multivariate Tweedie family (MTwF), on the one hand includes multivariate Poisson, gamma, inverse Gaussian, stable and compound Poisson distributions and on the other hand introduces a high variety of new dependent probabilistic models unstudied so far. We investigate various properties of MTwF and discuss its possible applications to financial risk management.

Gamma Unobserved Heterogeneity and Duration Bias

Røed et al. (1999) demonstrate that the standard result of known negative duration bias does not necessarily hold in a two-state mixed proportional hazard (MPH) model. We show that the duration bias is still ambiguous in a MPH model with a multivariate gamma distribution. A discrete time two-state version of our MPH model is developed to analyze the duration of higher education. The estimation results show that we cannot reject the hypothesis that the two unobserved heterogeneity variables are uncorrelated. Accepting this hypothesis implies that the standard result holds in our analysis.

Estimating the polarization degree of polarimetric images in coherent illumination using maximum likelihood methods

We address the problem of estimating the polarization degree of polarimetric images in coherent illumination. It has been recently shown that the degree of polarization associated with polarimetric images can be estimated by the method of moments applied to two or four images assuming fully developed speckle. We show that the estimation can also be conducted by using maximum likelihood methods. The maximum likelihood estimators of the polarization degree are derived from the joint distribution of the image intensities. We show that the joint distribution of polarimetric images is a multivariate gamma distribution whose marginals are univariate, bivariate, or trivariate gamma distributions. This property is used to derive maximum likelihood estimators of the polarization degree using two, three, or four images. The proposed estimators provide better performance than the estimators of moments. These results are illustrated by estimations conducted on synthetic and real images.

Pairwise likelihood estimation for multivariate mixed Poisson models generated by Gamma intensities

Estimating the parameters of multivariate mixed Poisson models is an important problem in image processing applications, especially for active imaging or astronomy. The classical maximum likelihood approach cannot be used for these models since the corresponding masses cannot be expressed in a simple closed form. This paper studies a maximum pairwise likelihood approach to estimate the parameters of multivariate mixed Poisson models when the mixing distribution is a multivariate Gamma distribution. The consistency and asymptotic normality of this estimator are derived. Simulations conducted on synthetic data illustrate these results and show that the proposed estimator outperforms classical estimators based on the method of moments. An application to change detection in low-flux images is also investigated.

A new serially correlated gamma-frailty process for longitudinal count data

We describe a new multivariate gamma distribution and discuss its implication in a Poisson-correlated gamma-frailty model. This model is introduced to account for between-subjects correlation occurring in longitudinal count data. For likelihood-based inference involving distributions in which high-dimensional dependencies are present, it may be useful to approximate likelihoods based on the univariate or bivariate marginal distributions. The merit of composite likelihood is to reduce the computational complexity of the full likelihood. A 2-stage composite-likelihood procedure is developed for estimating the model parameters. The suggested method is applied to a meta-analysis study for survival curves.

Optimization in a multivariate generalized linear model situation

The purpose of this article is to find the settings of the factors which simultaneously optimize several mean responses in a multivariate generalized linear model (GLM) environment. The generalized distance approach, initially developed for the simultaneous optimization of several linear response surface models, is adapted to this multivariate GLM situation. An application of the proposed methodology is presented in the special case of a bivariate binary distribution resulting from a drug testing experiment concerning two responses, namely, the efficacy and toxicity of a particular drug combination. One of the objectives of this application is to find the dose levels of two drugs that simultaneously maximize their therapeutic effect and minimize any possible toxic effects. A second application is presented in the case of a multivariate gamma distribution.

Change Detection in Multisensor SAR Images Using Bivariate Gamma Distributions

This paper studies a family of distributions constructed from multivariate gamma distributions to model the statistical properties of multisensor synthetic aperture radar (SAR) images. These distributions referred to as multisensor multivariate gamma distributions (MuMGDs) are potentially interesting for detecting changes in SAR images acquired by different sensors having different numbers of looks. The first part of this paper compares different estimators for the parameters of MuMGDs. These estimators are based on the maximum likelihood principle, the method of inference function for margins, and the method of moments. The second part of the paper studies change detection algorithms based on the estimated correlation coefficient of MuMGDs. Simulation results conducted on synthetic and real data illustrate the performance of these change detectors.

Functionally Compatible Local Characteristics for the Local Specification of Priors in Graphical Models

Abstract. The local specification of priors in non‐decomposable graphical models does not necessarily yield a proper joint prior for all the parameters of the model. Using results concerning general exponential families with cuts, we derive specific results for the multivariate Gamma distribution (conjugate prior for Poisson counts) and the Wishart distribution (conjugate prior for Gaussian models). These results link the existence of a locally specified joint prior to the solvability of a related marginal problem over the cliques of the graph.

A Multivariate Gamma Distribution and its Characterizations

SYNOPTIC ABSTRACTIn this paper, we proffer a new multivariate gamma distribution with potential applications in survival and reliability modeling. The multivariate distribution is not necessarily restricted to those with gamma marginal distributions. We provide and characterize a generalized location scale family of multivariate gamma distributions. This family possesses three-parameter gamma marginals (in most cases) and it contains absolutely continuous classes, as well as, the Marshall Olkin type of distributions with a positive probability mass on a set of measure zero. Maximum likelihood estimators are developed in the bivariate case.

Step-Down Analysis for Changes in the Covariance Matrix and Other Parameters

Step-down diagnostic analysis to infer which means have changed (after detection of a shift in the mean vector) is familiar in multivariate statistical process control. We show how to include in the diagnostic analysis the elements of the covariance matrix and other higher order moments and give practical examples. We use the asymptotic normality of maximum-likelihood estimators to classify the parameters that describe the prior- and post-shift distributions into those for which there is significant evidence of a shift and those for which such evidence is lacking. This diagnostic information can be useful in guiding the search for the cause(s) of the shift. For practical applications, the choice of parameterization may affect the ease of interpretation of the diagnostic results. It may be more informative to use standard deviations and correlation coefficients instead of variances and covariances, and this approach is illustrated in our analysis and examples. We also consider application to distributions other than the multivariate normal and give an example with a multivariate gamma distribution. Although illustrated with independent and identically distributed observations, the proposed methodology is more general.

Bivariate Negative Binomial Generalized Linear Models for Environmental Count Data

We propose a new bivariate negative binomial model with constant correlation structure, which was derived from a contagious bivariate distribution of two independent Poisson mass functions, by mixing the proposed bivariate gamma type density with constantly correlated covariance structure (Iwasaki & Tsubaki, 2005), which satisfies the integrability condition of McCullagh & Nelder (1989, p. 334). The proposed bivariate gamma type density comes from a natural exponential family. Joe (1997) points out the necessity of a multivariate gamma distribution to derive a multivariate distribution with negative binomial margins, and the luck of a convenient form of multivariate gamma distribution to get a model with greater flexibility in a dependent structure with indices of dispersion. In this paper we first derive a new bivariate negative binomial distribution as well as the first two cumulants, and, secondly, formulate bivariate generalized linear models with a constantly correlated negative binomial covariance structure in addition to the moment estimator of the components of the matrix. We finally fit the bivariate negative binomial models to two correlated environmental data sets.

Integral Representations and Approximations for Multivariate Gamma Distributions

Let R be a p×p-correlation matrix with an “m-factorial” inverse R −1 = D − BB′ with diagonal D minimizing the rank m of B. A new $$\left(m+1 \atop 2\right)$$ -variate integral representation is given for p-variate gamma distributions belonging to R, which is based on the above decomposition of R −1 without the restriction D > 0 required in former formulas. This extends the applicability of formulas with small m. For example, every p-variate gamma cdf can be computed by an at most $$\left(p-1 \atop 2\right)$$ -variate integral if p = 3 or p = 4. Since computation is only feasible for small m, a given R is approximated by an m-factorial R 0. The cdf belonging to R is approximated by the cdf associated with R 0 and some additional correction terms with the deviations between R and R 0.

Closed-form Bayesian inferences for the logit model via polynomial expansions

Articles in Marketing and choice literatures have demonstrated the need for incorporating person-level heterogeneity into behavioral models (e.g., logit models for multiple binary outcomes as studied here). However, the logit likelihood extended with a population distribution of heterogeneity doesn’t yield closed-form inferences, and therefore numerical integration techniques are relied upon (e.g., MCMC methods). We present here an alternative, closed-form Bayesian inferences for the logit model, which we obtain by approximating the logit likelihood via a polynomial expansion, and then positing a distribution of heterogeneity from a flexible family that is now conjugate and integrable. For problems where the response coefficients are independent, choosing the Gamma distribution leads to rapidly convergent closed-form expansions; if there are correlations among the coefficients one can still obtain rapidly convergent closed-form expansions by positing a distribution of heterogeneity from a Multivariate Gamma distribution. The solution then comes from the moment generating function of the Multivariate Gamma distribution or in general from the multivariate heterogeneity distribution assumed. Closed-form Bayesian inferences, derivatives (useful for elasticity calculations), population distribution parameter estimates (useful for summarization) and starting values (useful for complicated algorithms) are hence directly available. Two simulation studies demonstrate the efficacy of our approach.

Multivariate Matsumoto–Yor property is rather restrictive

Matsumoto and Yor [2001. An analogue of Pitman's 2 M - X theorem for exponential Wiener functionals. Part II: the role of the GIG laws. Nagoya Math. J. 162, 65–86] discovered an interesting invariance property of a product of the generalized inverse Gaussian (GIG) and the gamma distributions. For univariate random variables or symmetric positive definite random matrices it is a characteristic property for this pair of distributions. It appears that for random vectors the Matsumoto–Yor property characterizes only very special families of multivariate GIG and gamma distributions: components of the respective random vectors are grouped into independent subvectors, each subvector having linearly dependent components. This complements the version of the multivariate Matsumoto–Yor property on trees and related characterization obtained in Massam and Wesołowski [2004. The Matsumoto–Yor property on trees. Bernoulli 10, 685–700].

A Probabilistic Model of Glenohumeral External Rotation Strength for Healthy Normals and Rotator Cuff Tear Cases

The reigning paradigm of musculoskeletal modeling is to construct deterministic models from parameters of an "average" subject and make predictions for muscle forces and joint torques with this model. This approach is limited because it does not perform well for outliers, and it does not model the effects of population parameter variability. The purpose of this study was to simulate variability in musculoskeletal parameters on glenohumeral external rotation strength in healthy normals, and in rotator cuff tear case using a Monte Carlo model. The goal was to determine if variability in musculoskeletal parameters could quantifiably explain variability in glenohumeral external rotation strength. Multivariate Gamma distributions for musculoskeletal architecture and moment arm were constructed from empirical data. Gamma distributions of measured joint strength were constructed. Parameters were sampled from the distributions and input to the model to predict muscle forces and joint torques. The model predicted measured joint torques for healthy normals, subjects with supraspinatus tears, and subjects with infraspinatus-supraspinatus tears with small error. Muscle forces for the three conditions were predicted and compared. Variability in measured torques can be explained by differences in parameter variability.

Counting and locating the solutions of polynomial systems of maximum likelihood equations, I

In statistical inference, mixture models consisting of several component subpopulations are used widely to model data drawn from heterogeneous sources. In this paper, we consider maximum likelihood estimation for mixture models in which the only unknown parameters are the component proportions. By applying the theory of multivariable polynomial equations, we derive bounds for the number of isolated roots of the corresponding system of likelihood equations. If the component densities belong to certain familiar continuous exponential families, including the multivariate normal or gamma distributions, then our upper bound is, almost surely, the exact number of solutions.

Risk capital decomposition for a multivariate dependent gamma portfolio

This paper examines the tail conditional expectation risk measure (TCE) in the case of a multivariate gamma portfolio of risks. Explicit formulas for both the TCE and the risk capital allocations based on it are provided in the context of the multivariate model possessing dependent gamma marginals. Some of our results exceed the frameworks of the multivariate gamma distributions and may be applied to other non-negative risks.

Bayesian inference for multivariate gamma distributions

The paper considers the multivariate gamma distribution for which the method of moments has been considered as the only method of estimation due to the complexity of the likelihood function. With a...