Gamma Mixture Research Articles

A novel claim size distribution based on a Birnbaum–Saunders and gamma mixture capturing extreme values in insurance: estimation, regression, and applications

A novel claim size distribution based on a Birnbaum–Saunders and gamma mixture capturing extreme values in insurance: estimation, regression, and applications

Bayesian estimation of electricity price risk with a multi-factor mixture of densities

The risks in daily electricity prices are becoming substantial and it is clear that improvements in price density forecasting can translate into improved risk management. However, the specification of the most appropriate price density function is challenging as the best functional forms differ by time of day evolve over time, dynamically respond to fluctuating exogenous factors such as wind speed and solar irradiance. This research develops and tests a new flexible, functional form based upon the Gamma Mixture of Uniform (GMU) densities which effectively avoids the choice of a particular density function and has conditional moments specified as a function of the dynamic exogenous drivers. Empirical testing shows that it outperforms the multi-factor skewed student-t family of densities, previously advocated in this context. Additionally, using Bayesian estimation the new methodology provides a complete description of the uncertainty in the estimation of the coefficients for those exogenous factors. Empirical testing on day-ahead hourly electricity prices in the German market from 2012 to 2016, where renewable energy sources, such as wind and solar, play a critical role in the formation of electricity price risk, validates the extra accuracy of this formulation.

Bayesian sample size determination using commensurate priors to leverage preexperimental data.

This paper develops Bayesian sample size formulae for experiments comparing two groups, where relevant preexperimental information from multiple sources can be incorporated in a robust prior to support both the design and analysis. We use commensurate predictive priors for borrowing of information and further place Gamma mixture priors on the precisions to account for preliminary belief about the pairwise (in)commensurability between parameters that underpin the historical and new experiments. Averaged over the probability space of the new experimental data, appropriate sample sizes are found according to criteria that control certain aspects of the posterior distribution, such as the coverage probability or length of a defined density region. Our Bayesian methodology can be applied to circumstances that compare two normal means, proportions, or event times. When nuisance parameters (such as variance) in the new experiment are unknown, a prior distribution can further be specified based on preexperimental data. Exact solutions are available based on most of the criteria considered for Bayesian sample size determination, while a search procedure is described in cases for which there are no closed-form expressions. We illustrate the application of our sample size formulae in the design of clinical trials, where pretrial information is available to be leveraged. Hypothetical data examples, motivated by a rare-disease trial with an elicited expert prior opinion, and a comprehensive performance evaluation of the proposed methodology are presented.

Bayesian sample size determination using commensurate priors to leverage pre-experimental data.

This paper develops Bayesian sample size formulae for experiments comparing two groups, where relevant pre-experimental information from multiple sources can be incorporated in a robust prior to support both the design and analysis. We use commensurate predictive priors for borrowing of information, and further place Gamma mixture priors on the precisions to account for preliminary belief about the pairwise (in)commensurability between parameters that underpin the historical and new experiments. Averaged over the probability space of the new experimental data, appropriate sample sizes are found according to criteria that control certain aspects of the posterior distribution, such as the coverage probability or length of a defined density region. Our Bayesian methodology can be applied to circumstances that compare two normal means, proportions or event times. When nuisance parameters (such as variance) in the new experiment are unknown, a prior distribution can further be specified based on pre-experimental data. Exact solutions are available based on most of the criteria considered for Bayesian sample size determination, while a search procedure is described in cases for which there are no closed-form expressions. We illustrate the application of our sample size formulae in the design of clinical trials, where pre-trial information is available to be leveraged. Hypothetical data examples, motivated by a rare-disease trial with elicited expert prior opinion, and a comprehensive performance evaluation of the proposed methodology are presented.

Valid properties of truncated Student-t regression model with applications in analysis of censored data

Kim (J. Korean Stat. Soc. 37 (2008) 81–87) introduced an incorrect stochastic representation (SR) for the truncated Student-t (Tt) random variable. By pointing out that the gamma mixture based on a truncated normal distribution actually cannot result in a true Tt distribution, in this paper, we first propose three correct SRs and then recalculate the corresponding moments of the Tt distribution. Different from those derived by following the invalid SR of Kim (J. Korean Stat. Soc. 37 (2008) 81–87), the correct moments of the Tt distribution play a crucial role in parameter estimations. Based on the third SR proposed and the correct expressions of truncated moments, expectation–maximization (EM) algorithms are developed for calculating the maximum likelihood estimates of parameters in the Tt distribution. Extensions to a Tt regression model and a t interval-censored regression model are provided as well. Simulated experiments are conducted to evaluate the performance of the proposed methods. Finally, two real data analyses corroborate the theoretical results.

Consistency of the MLE under a two-parameter Gamma mixture model with a structural shape parameter

Finite Gamma mixture models are often used to describe randomness in income data, insurance data, and data in applications where the response values are intrinsically positive. The popular likelihood approach for model fitting, however, does not work for this model because its likelihood function is unbounded. Because of this, the maximum likelihood estimator is not well-defined. Other approaches have been developed to achieve consistent estimation of the mixing distribution, such as placing an upper bound on the shape parameter or adding a penalty to the log-likelihood function. In this paper, we show that if the shape parameter in the finite Gamma mixture model is structural, then the direct maximum likelihood estimator of the mixing distribution is well-defined and strongly consistent. We also present simulation results demonstrating the consistency of the estimator. We illustrate the application of the model with a structural shape parameter to household income data. The fitted mixture distribution leads to several possible subpopulation structures with regard to the level of disposable income.

Daily Mortality/Morbidity and Air Quality: Using Multivariate Time Series with Seasonally Varying Covariances

Abstract We study the associations between daily mortality and short-term variations in the ambient concentrations of fine particulate matter (PM2.5), nitrogen dioxide (NO2) and ozone (O3) in four cities in Canada. First, a novel multivariate time series model within Bayesian framework is proposed for exposure assessment, where the response is a mixture of Gamma and Half-Cauchy distributions and the correlations between pollutants vary seasonally. A case-crossover design and conditional logistic regression model is used to relate exposure to disease data for each city, which then are combined to obtain a global estimate of exposure health effects allowing exposure uncertainty. The results suggest that every 10 ppb increase in O3 is associated with a 3.88% (95% credible interval [CI], 2.5%, 5.18%) increase in all-cause mortality, a 5.04% (2.84%, 7.43%) increase in circulatory mortality, a 7.87% (2.4%, 12.9%) increase in respiratory mortality, a 0.76% (0.19%, 1.35%) increase in all-cause morbidity and a 6.6% (0.58%, 12.7%) increase in respiratory morbidity. Similarly, every 10 ppb increase in NO2 is associated with a 2.13% (0.42%, 3.87%) increase in circulatory morbidity. The health impacts of PM2.5 are not found to be present once other pollutants are accounted for.

Infinite Switching Dynamic Probabilistic Network With Bayesian Nonparametric Learning

To model sequentially observed multivariate nonstationary count data, we propose a switching Poisson-gamma dynamical systems (SPGDS), a dynamic probabilistic network with switching mechanism. Different from previous models, SPGDS assigns its latent variables into mixture of gamma distributed parameters to model complex sequences and describe the nonlinear dynamics, meanwhile, capture various temporal dependencies. Moreover, SPGDS can model all discrete and nonnegative real data by linking them to latent counts. To take advantage of Bayesian nonparametrics in handling the unknown number of mixture components, we integrate Dirichlet process (DP) mixture into SPGDS and develop an infinite switching Poisson-gamma dynamical systems (iSPGDS). For efficient and nonparametric inference, we develop a infinite switching recurrent variational inference network, combined with a scalable hybrid stochastic gradient-MCMC and variational inference method, which is scalable to large scale sequences and fast in out-of-sample prediction. Besides, to handle the time-series categorization task, we further propose an supervised attention iSPGDS (attn-iSPGDS), which combines the representation power of iSPGDS, discriminative power of deep neural networks, and selection power of the attention mechanism under a principled probabilistic framework. Experiments on both unsupervised and supervised tasks demonstrate that the proposed model not only has excellent fitting and prediction performance on complex sequences, but also separates different dynamical patterns within them.

Hierarchical Dirichlet Process Based Gamma Mixture Modeling for Terahertz Band Wireless Communication Channels

Due to the unique channel characteristics of Terahertz (THz), comprehensive propagation channel modeling is essential to understand the spectrum and develop reliable communication systems in these bands. In this work, we propose the utilization of the hierarchical Dirichlet Process Gamma Mixture Model (DPGMM) to characterize THz channels statistically in the absence of any prior knowledge. DPGMM provides mixture component parameters and the required number of components. A revised expectation-maximization (EM) algorithm is also proposed as a pre-step for DPGMM. Kullback-Leibler Divergence (KL-divergence) is utilized as an error metric to examine the amount of inaccuracy of the EM algorithm and DPGMM when modeling the experimental probability density functions (PDFs). DPGMM and EM algorithm are implemented over the measurements taken at frequencies between 240 GHz and 300 GHz. By comparing the results of the DPGMM and EM algorithms for the measurement datasets, we demonstrate how well the DPGMM fits the target distribution. It is shown that the proposed DPGMM can accurately describe the various THz channels as well as the EM algorithm, and its flexibility allows it to represent more complex distributions better than the EM algorithm. We also demonstrated that DPGMM can be used to model any wireless channel due to its versatility.

A Hierarchical Gamma Mixture Model Toward Hidden Markov Random Field for High-Resolution SAR Image Segmentation

Accurately modeling the distributions of spectral intensities is an effective way to obtain accurate segmentation results. Existing mixture models mostly fail to establish accurate statistical models of high-resolution SAR images due to the complex statistical features of their spectral intensities. We propose a hierarchical gamma mixture model (HGaMM)-based high-resolution SAR images segmentation algorithm. In the proposed algorithm, a statistical model for SAR images is built using HGaMM, which can model heavy-tailed, asymmetric, multimodal, or flat distributions with its hierarchical structure. The algorithm allows for the effective utilization of spectral intensity information required for image segmentation. The HGaMM consists of several components that serve as the first layer, and several elements under each component serve as the second layer. The component weight is constructed using posterior probabilities of local pixels aimed at reducing noise effects. The segmentation model can then be established by the Bayesian theorem. A new expectation maximization/adding or deleting Markov chain Monte Carlo (EM/ADMCMC) is incorporated to implement parameter estimation and determine the optimum number of components. Several experiments were implemented on simulated and real high-resolution SAR images to evaluate the applicability and efficiency of the proposed approach. The experimental results show that the proposed HGaMM algorithm outperforms traditional algorithms and can obtain accurate results.

Effective Frameworks Based on Infinite Mixture Model for Real-World Applications

Interest in automated data classification and identification systems has increased over the past years in conjunction with the high demand for artificial intelligence and security applications. In particular, recognizing human activities with accurate results have become a topic of high interest. Although the current tools have reached remarkable successes, it is still a challenging problem due to various uncontrolled environments and conditions. In this paper two statistical frameworks based on nonparametric hierarchical Bayesian models and Gamma distribution are proposed to solve some real-world applications. In particular, two nonparametric hierarchical Bayesian models based on Dirichlet process and Pitman-Yor process are developed. These models are then applied to address the problem of modelling grouped data where observations are organized into groups and these groups are statistically linked by sharing mixture components. The choice of the Gamma mixtures is motivated by its flexibility for modelling heavy-tailed distributions. In addition, deploying the Dirichlet process prior is justified by its advantage of automatically finding the right number of components and providing nice properties. Moreover, a learning step via variational Bayesian setting is presented in a flexible way. The priors over the parameters are selected appropriately and the posteriors are approximated effectively in a closed form. Experimental results based on a real-life applications that concerns texture classification and human actions recognition show the capabilities and effectiveness of the proposed framework.

Entropy-Based Variational Scheme with Component Splitting for the Efficient Learning of Gamma Mixtures.

Finite Gamma mixture models have proved to be flexible and can take prior information into account to improve generalization capability, which make them interesting for several machine learning and data mining applications. In this study, an efficient Gamma mixture model-based approach for proportional vector clustering is proposed. In particular, a sophisticated entropy-based variational algorithm is developed to learn the model and optimize its complexity simultaneously. Moreover, a component-splitting principle is investigated, here, to handle the problem of model selection and to prevent over-fitting, which is an added advantage, as it is done within the variational framework. The performance and merits of the proposed framework are evaluated on multiple, real-challenging applications including dynamic textures clustering, objects categorization and human gesture recognition.

Application of Modified Generalized–Gamma Mixture Cure Model in the Analysis of Ovarian Cancer Data

The modeling and analysis of lifetime for terminal diseases such as cancer is a significant aspect of statistical work. This study considered data from thirty-seven women diagnosed with Ovarian Cancer and hospitalized for care at theDepartment of Obstetrics and Gynecology, University of Ibadan, Nigeria. Focus was on the application of a parametric mixture cure model that can handle skewness associated with survival data – a modified generalized-gamma mixture cure model (MGGMCM). The effectiveness of MGGMCM was compared with existing parametric mixture cure models using Akaike Information Criterion, median time-to-cure and variance of the cure rate. It was observed that the MGGMCM is an improved parametric model for the mixture cure model.

Modeling in Forestry Using Mixture Models Fitted to Grouped and Ungrouped Data

The creation and maintenance of complex forest structures has become an important forestry objective. Complex forest structures, often expressed in multimodal shapes of tree size/diameter (DBH) distributions, are challenging to model. Mixture probability density functions of two- or three-component gamma, log-normal, and Weibull mixture models offer a solution and can additionally provide insights into forest dynamics. Model parameters can be efficiently estimated with the maximum likelihood (ML) approach using iterative methods such as the Newton-Raphson (NR) algorithm. However, the NR algorithm is sensitive to the choice of initial values and does not always converge. As an alternative, we explored the use of the iterative expectation-maximization (EM) algorithm for estimating parameters of the aforementioned mixture models because it always converges to ML estimators. Since forestry data frequently occur both in grouped (classified) and ungrouped (raw) forms, the EM algorithm was applied to explore the goodness-of-fit of the gamma, log-normal, and Weibull mixture distributions in three sample plots that exhibited irregular, multimodal, highly skewed, and heavy-tailed DBH distributions where some size classes were empty. The EM-based goodness-of-fit was further compared against a nonparametric kernel-based density estimation (NK) model and the recently popularized gamma-shaped mixture (GSM) models using the ungrouped data. In this example application, the EM algorithm provided well-fitting two- or three-component mixture models for all three model families. The number of components of the best-fitting models differed among the three sample plots (but not among model families) and the mixture models of the log-normal and gamma families provided a better fit than the Weibull distribution for grouped and ungrouped data. For ungrouped data, both log-normal and gamma mixture distributions outperformed the GSM model and, with the exception of the multimodal diameter distribution, also the NK model. The EM algorithm appears to be a promising tool for modeling complex forest structures.

Gamma Mixture Density Networks and their application to modelling insurance claim amounts

We discuss how mixtures of Gamma distributions with mixing probabilities, shape and rate parameters depending on features can be fitted with neural networks. We develop two versions of the EM algorithm for fitting so-called Gamma Mixture Density Networks, which we call the EM network boosting algorithm and the EM forward network algorithm, and we test their implementation together with the choices of hyperparameters. A simulation study shows that our algorithms perform very well on synthetic data sets. We further illustrate the application of the Gamma Mixture Density Network on a real data set of motor insurance claim amounts and conclude that Gamma Mixture Density Networks can improve the fit of the regression model and the predictions of the claim severities used for rate-making compared to classical actuarial techniques.

Oil Spill Detection in SAR Images Using Online Extended Variational Learning of Dirichlet Process Mixtures of Gamma Distributions

In this paper, we propose a Dirichlet process (DP) mixture model of Gamma distributions, which is an extension of the finite Gamma mixture model to the infinite case. In particular, we propose a novel online nonparametric Bayesian analysis method based on the infinite Gamma mixture model where the determination of the number of clusters is bypassed via an infinite number of mixture components. The proposed model is learned via an online extended variational Bayesian inference approach in a flexible way where the priors of model’s parameters are selected appropriately and the posteriors are approximated effectively in a closed form. The online setting has the advantage to allow data instances to be treated in a sequential manner, which is more attractive than batch learning especially when dealing with massive and streaming data. We demonstrated the performance and merits of the proposed statistical framework with a challenging real-world application namely oil spill detection in synthetic aperture radar (SAR) images.

Samade Probability Distribution: Its Properties and Application to Real Lifetime Data

A new two-parameter lifetime distribution has been proposed in this study. The distribution is called Samade distribution. The model is motivated by the wide use of the lifetime models derived from the mixture of gamma and exponential distributions. Its mathematical properties which include the first four moments, variance as well as coefficient of variation, reliability function, hazard function, survival function, Renyi entropy measure and distribution of order statistics have been successfully derived. The maximum likelihood estimation of its parameters and application to real life data have been discussed. Application of this model to three real datasets shown that the proposed model yields a satisfactorily better fit than other existing lifetime distributions. The comparism of goodness-of-fits were established using -2Loglikelihood, AIC and BIC.

Bayesian credibility under a bivariate prior on the frequency and the severity of claims

In this paper, as opposed to the usual assumption of independence, we propose a credibility model in which the (unobservable) risk profiles of the claim frequency and the claim severity are dependent. Given the risk profiles, the (conditional) marginal distributions of frequency and severity are assumed to belong to the exponential family. A bivariate conjugate prior is proposed for the risk profiles, where the dependency is incorporated via a factorization structure of the joint density. The bivariate posterior is derived, and in turn the Bayesian premium for the aggregate claim is given along with some results on the predictive joint and marginal distributions involving the claim number and the aggregate claim in the next period. To demonstrate the generality of our proposed model, we provide four different examples of bivariate conjugate priors in relation to mixed Erlang, gamma mixture, Farlie-Gumbel-Morgenstern (FGM) copula, and bivariate beta, where each choice has different merits. In these examples, more explicit results can be obtained, and in particular the predictive variance and Value-at-Risk (VaR) of the aggregate claim certainly provide more information on the inherent risk than the Bayesian premium which is merely the predictive mean. Finally, numerical examples will be given to illustrate the effect of dependence on the results, including the use of a real data set that further takes observable risk factors into consideration under a regression setting.

Verifying Clinically Derived Reference Intervals for Daily Excretion Rates of Fractionated Metanephrines Using Modern Indirect Reference Interval Models.

Biochemical testing of urinary metanephrines is useful in the diagnosis and monitoring of pheochromocytoma and paragangliomas. We investigated the feasibility of mixture decomposition (ie, indirect) methods in verifying clinically derived reference intervals for urinary deconjugated metanephrine metabolites. Urinary 24-hour metanephrine and normetanephrine excretion results were extracted from our data warehouse and intervals were estimated by the modern variant of the Hoffmann method, maximum likelihood estimation (MLE), and gamma mixture model using R software. Hoffmann, MLE, and gamma mixture models provided metanephrine and normetanephrine intervals that closely matched those derived from clinical studies. However, three-component MLE and gamma models were required for normetanephrine in adult women because the Hoffmann method was not suitable. Some data transformations caused blending of the mixed distributions and subsequent widening of the reference interval estimation, emphasizing the importance of careful data transformation for Hoffmann and MLE analyses. Gamma mixture models gave overall good agreement without the need for data transformation. Indirect methods have utility in verifying reference intervals in 24-hour urine specimens collected by patients. We emphasize the benefits of applying multiple decomposition methods to corroborate findings and careful application of data transformation when using Gaussian-based models.

Characterization of generalized Gamma-Lindley distribution using truncated moments of order statistics

AbstractHaving only two parameters, the Gamma-Lindley distribution does not provide enough flexibility for analyzing different types of lifetime data. From this perspective, in order to further enhance its flexibility, we set forward in this paper a new class of distributions named Generalized Gamma-Lindley distribution with four parameters. Its construction is based on certain mixtures of Gamma and Lindley distributions. The truncated moment, as a characterization method, has drawn a little attention in the statistical literature over the great popularity of the classical methods. We attempt to prove that the Generalized Gamma-Lindley distribution is characterized by its truncated moment of the first order statistics. This method rests upon finding a survival function of a distribution, that is a solution of a first order differential equation. This characterization includes as special cases: Gamma, Lindley, Exponential, Gamma-Lindley and Weighted Lindley distributions. Finally, a simulation study is performed to help the reader check whether the available data follow the underlying distribution.