Independent Gamma Distributions Research Articles

Different Statistical Inference Algorithms for the New Pareto Distribution Based on Type-II Progressively Censored Competing Risk Data with Applications

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.

A Bayesian approach to estimate the capacity factor of offshore wind farms on the Brazilian coast

This work presents a Bayesian approach for inferring the capacity factor of an offshore wind farm. The calculation of the capacity factor depends on the power curve model and the probabilistic wind speed model, which in this work follows a Weibull distribution, whose scale and shape parameters are estimated using a Bayesian approach where the priors are independent gamma distributions, the likelihood is the Weibull distribution, and the posteriors are obtained by Hamiltonian Monte Carlo. The application of the described methodology is illustrated through a case study of an offshore wind power project located on the Northern coast of the Rio de Janeiro State, Brazil.

On the gamma difference distribution

The gamma difference distribution is defined as the difference of two independent gamma distributions, with in general different shape and rate parameters. Starting with knowledge of the corresponding characteristic function, a second order linear differential equation specification of the probability density function is given. This is used to derive a Stein-type differential identity relating to the expectation with respect to the gamma difference distribution of a general twice differentiable function g(x). Choosing g(x)=xk gives a second order recurrence for the positive integer moments, which are also shown to permit evaluations in terms of 2F1 hypergeometric polynomials. A hypergeometric function evaluation is given for the absolute continuous moments. Specialising the gamma difference distribution gives the variance gamma distribution. Results of the type obtained herein have previously been obtained for this distribution, allowing for comparisons to be made.

Estimation for two Gompertz populations under a balanced joint progressive Type-II censoring scheme

Comparative lifetime experiments are remarkable when the study is to ascertain the relative merits of two competing products regarding the duration of their service life. This paper considers the comparative lifetime experiments of two Gompertz populations under a balanced joint progressive Type-II censoring scheme. The lifetime distributions of the units are assumed to follow the Gompertz distribution with a common shape but different scale parameters. The maximum likelihood estimates of the unknown parameters are derived. The existence of the maximum likelihood estimates is proved. Expectation-maximization and stochastic expectation-maximization algorithms are provided to calculate the estimates. The bootstrap-p, bootstrap-t, and approximate confidence intervals are established. To obtain the Bayesian estimates, it is assumed that the prior of scale parameters is a Beta-Gamma distribution and the prior of the common shape parameter is an independent Gamma distribution. Under squared error loss and LINEX loss functions, the Metropolis-Hastings algorithm is provided to compute the Bayes estimates and the credible intervals. Further, the statistical inferences with order restriction are studied when it is known a priori that the expectation of the lifespan of one population is shorter than that of the other population. A wide range of simulation experiments is conducted to evaluate the performance of the proposed methods. Finally, the lifetimes of white organic light-emitting diodes and the breaking strengths of jute fiber of gauge lengths are analyzed to illustrate the practical application of the proposed model and methods.

Inference of a competing risks model with partially observed failure causes under improved adaptive type-II progressive censoring

In this paper, a competing risks model is analyzed based on improved adaptive type-II progressive censored sample (IAT-II PCS). Two independent competing causes of failures are considered. It is assumed that lifetimes of the competing causes of failure follow exponential distributions with different means. Maximum likelihood estimators (MLEs) for the unknown model parameters are obtained. Using asymptotic normality property of MLE, the asymptotic confidence intervals are constructed. Existence and uniqueness properties of the MLEs are studied. Further, bootstrap confidence intervals are computed. The Bayes estimators are obtained under symmetric and asymmetric loss functions with non-informative and informative priors. For informative priors, independent gamma distributions are considered. Highest posterior density (HPD) credible intervals are obtained. A Monte Carlo simulation study is carried out to compare performance of the established estimates. Furthermore, three different optimality criteria are proposed to obtain the optimal censoring plan. Finally, a real-life data set is considered for illustrative purposes.

Simplified population data analysis using gamma distribution for nutritional requirements and its application to the estimation of iron requirements for women of child-bearing age

BackgroundEstimation of nutritional requirements is a difficult task, especially for nutrients whose requirements have skewed distribution like iron in menstruating women. ObjectiveTo mathematically develop a simplified population data analysis using gamma distribution for the estimation of nutritional requirements, not depending on extensive numerical calculation. MethodsThe required equations for the estimation of requirements were devised and solved. Existing data of iron intakes and iron losses in literature were fitted to cumulative distribution curves including gamma distribution. The proposed method was applied to the estimation of iron requirements in women using the National Health and Nutrition Survey data from 2003 to 2007 in Japan. ResultsThe type 2 equation of nutritional state is first introduced: prevalence of inadequate nutritional state = Σ(individuals with intake/requirement <1)/(total number of individuals). The prevalence of inadequate nutritional status is determined by F-distribution with positive real number parameters, if the intake and the requirement have independent gamma distributions. The sum of basal and menstrual iron losses is well approximated by gamma distribution, if the basal and menstrual iron losses have independent gamma distributions. Using these relationships, an approximate estimate of iron requirements was determined. Iron intakes and losses were found to be well approximated by gamma distribution. The median of the coefficient of variation (CV) of basal iron loss was 34 %. A new correction method for intra-individual variation in gamma distribution under constant CV is presented also. The median of the estimated median iron requirement was 8.12 mg, 8.15 mg and 8.18 mg for women aged 18−29 years, respectively, as the exact numerical estimate, the approximate estimate using BETA.INV in Excel and the approximate estimate using F.INV in Excel. For women aged 30−49 years, it was 9.15 mg, 9.17 mg and 9.14 mg. The intake covering the needs of 97.7 % women was 14.55 mg, 14.44 mg and 14.49 mg for women aged 18−29 years, and 15.70 mg, 15.77 mg and 15.60 mg for women aged 30−49 years. ConclusionThe approximate estimates of iron requirements agreed well with the exact numerical estimate. The proposed method is useful for the estimation of iron requirements in menstruating women that usually requires extensive numerical calculations.

Irregular Tessellation and Statistical Modeling Based Regionalized Segmentation for SAR Intensity Image

This paper presents a regionalized segmentation method for synthetic aperture radar (SAR) intensity images based on tessellation with irregular polygons. In the proposed method, the image domain is partitioned into a collection of irregular polygons, which are constructed using sets of nodes and are used to fit homogeneous regions with arbitrary shapes. Each partitioned polygon is taken as the basic processing unit. Assuming the intensities of the pixels in the polygon follow an independent and identical gamma distribution, the likelihood of the image intensities is modeled. After defining the prior distributions of the tessellation and the parameters for the likelihood model, a posterior probability model can be built based on the Bayes theorem as a segmentation model. To obtain optimal segmentation, a reversible-jump Markov chain Monte Carlo (RJMCMC) algorithm is designed to simulate from the segmentation model, where the move operations include updating the gamma distribution parameter, updating labels, moving nodes, merging polygons, splitting polygons, adding nodes, and deleting nodes. Experiments were carried out on synthetic and real SAR intensity images using the proposed method while the regular and Voronoi tessellation-based methods were also preformed for comparison. Our results show the proposed method overcomes some intrinsic limitations of current segmentation methods and is able to generate good results for homogeneous regions with different shapes.

A multivariate projected Gamma model for directional data

Analysis of some real phenomena involve directional variables that by their nature are defined only in certain subsets of the k-dimensional unit sphere, For example, when working with axial data, the support of the associated directional variables turns out to be the interval Thus, from a methodological point of view it is important to have probability distributions defined in bounded subsets of Specifically, in order to describe directional variables restricted to the first orthant, in this paper we introduce the Multivariate Projected Gamma model (MPG). This model is flexible enough and treats observations as projections onto the unit sphere of unobserved responses from a multivariate distribution which is generated as a product of k independent univariate Gamma distributions. Inference about the parameters of the model is based on samples from the corresponding joint posterior density, which is obtained using a Gibbs sampling after the introduction of suitable latent variables. The proposed methodology is illustrated using simulated data sets as well as a real data set.

Understanding narwhal diving behaviour using Hidden Markov Models with dependent state distributions and long range dependence.

Diving behaviour of narwhals is still largely unknown. We use Hidden Markov models (HMMs) to describe the diving behaviour of a narwhal and fit the models to a three-dimensional response vector of maximum dive depth, duration of dives and post-dive surface time of 8,609 dives measured in East Greenland over 83 days, an extraordinarily long and rich data set. Narwhal diving patterns have not been analysed like this before, but in studies of other whale species, response variables have been assumed independent. We extend the existing models to allow for dependence between state distributions, and show that the dependence has an impact on the conclusions drawn about the diving behaviour. We try several HMMs with 2, 3 or 4 states, and with independent and dependent log-normal and gamma distributions, respectively, and different covariates to characterize dive patterns. In particular, diurnal patterns in diving behaviour is inferred, by using periodic B-splines with boundary knots in 0 and 24 hours.

Detecting dark spots from SAR intensity images by a point process with irregular geometry marks

ABSTRACTAiming at the difficulties to determine the geometric shape of dark spots, this paper proposes an Irregular Geometry Marked Point Process (IGMPP) to detect dark spots from Synthetic Aperture Radar (SAR) images. Firstly, a series of random points used to define the locations of dark spots in the SAR image domain, and an irregular polygon mark associated with each random point indicates the shape of each dark spot. Subsequently, the pixels intensities in and out of the polygons are characterized with independent and identical Gamma distributions, respectively. By Bayesian paradigm, the number, sites, and geometry parameters of polygons are modelled with a posterior distribution. In order to simulate the posterior, a Reversible Jump Markov Chain Monte Carlo (RJMCMC) strategy is developed. Then, the optimal parameters concerning the dark spots can be obtained by a Maximum A Posteriori (MAP) scheme. Experiments are performed by simulated SAR images. The experimental results show that the proposed algorithm is effectively and efficiently applied to detect the dark spots.

Improved estimation of the smallest scale parameter of gamma distributions

In this work improved point and interval estimation of the smallest scale parameter of independent gamma distributions with known shape parameters are studied in an integrated fashion. The approach followed is based on formulating the model in such a way that enables us to treat the estimation of the smallest scale parameter as a problem of estimating an unrestricted scale parameter in the presence of a nuisance parameter. The class of improved point and interval estimators is enriched. Within this class, a subclass of generalized Bayes estimators of a simple form is identified.

Objective and subjective prior distributions for the Gompertz distribution.

This paper takes into account the estimation for the unknown parameters of the Gompertz distribution from the frequentist and Bayesian view points by using both objective and subjective prior distributions. We first derive non-informative priors using formal rules, such as Jefreys prior and maximal data information prior (MDIP), based on Fisher information and entropy, respectively. We also propose a prior distribution that incorporate the expert's knowledge about the issue under study. In this regard, we assume two independent gamma distributions for the parameters of the Gompertz distribution and it is employed for an elicitation process based on the predictive prior distribution by using Laplace approximation for integrals. We suppose that an expert can summarize his/her knowledge about the reliability of an item through statements of percentiles. We also present a set of priors proposed by Singpurwala assuming a truncated normal prior distribution for the median of distribution and a gamma prior for the scale parameter. Next, we investigate the effects of these priors in the posterior estimates of the parameters of the Gompertz distribution. The Bayes estimates are computed using Markov Chain Monte Carlo (MCMC) algorithm. An extensive numerical simulation is carried out to evaluate the performance of the maximum likelihood estimates and Bayes estimates based on bias, mean-squared error and coverage probabilities. Finally, a real data set have been analyzed for illustrative purposes.

A Bayesian Motivated Laplace Inversion for Multivariate Probability Distributions

The paper introduces a recursive procedure to invert the multivariate Laplace transform of probability distributions. The procedure involves taking independent samples from the Laplace transform; these samples are then used to update recursively an initial starting distribution. The update is Bayesian driven. The final estimate can be written as a mixture of independent gamma distributions, making it the only methodology which guarantees to numerically recover a probability distribution with positive support. Proof of convergence is given by a fixed point argument. The estimator is fast, accurate and can be run in parallel since the target distribution is evaluated on a grid of points. The method is illustrated on several examples and compared to the bivariate Gaver–Stehfest method.

COUPLING REGULAR TESSELLATION WITH RJMCMC ALGORITHM TO SEGMENT SAR IMAGE WITH UNKNOWN NUMBER OF CLASSES

Abstract. This paper presents a Synthetic Aperture Radar (SAR) image segmentation approach with unknown number of classes, which is based on regular tessellation and Reversible Jump Markov Chain Monte Carlo (RJMCMC') algorithm. First of all, an image domain is portioned into a set of blocks by regular tessellation. The image is modeled on the assumption that intensities of its pixels in each homogeneous region satisfy an identical and independent Gamma distribution. By Bayesian paradigm, the posterior distribution is obtained to build the region-based image segmentation model. Then, a RJMCMC algorithm is designed to simulate from the segmentation model to determine the number of homogeneous regions and segment the image. In order to further improve the segmentation accuracy, a refined operation is performed. To illustrate the feasibility and effectiveness of the proposed approach, two real SAR image is tested.

COUPLING REGULAR TESSELLATION WITH RJMCMC ALGORITHM TO SEGMENT SAR IMAGE WITH UNKNOWN NUMBER OF CLASSES

This paper presents a Synthetic Aperture Radar (SAR) image segmentation approach with unknown number of classes, which is based on regular tessellation and Reversible Jump Markov Chain Monte Carlo (RJMCMC') algorithm. First of all, an image domain is portioned into a set of blocks by regular tessellation. The image is modeled on the assumption that intensities of its pixels in each homogeneous region satisfy an identical and independent Gamma distribution. By Bayesian paradigm, the posterior distribution is obtained to build the region-based image segmentation model. Then, a RJMCMC algorithm is designed to simulate from the segmentation model to determine the number of homogeneous regions and segment the image. In order to further improve the segmentation accuracy, a refined operation is performed. To illustrate the feasibility and effectiveness of the proposed approach, two real SAR image is tested.

Likelihood ratio tests for comparing several gamma distributions

Likelihood ratio tests (LRTs) for comparing several independent gamma distributions with respect to shape parameters, scale parameters, and means are derived. The LRT for testing homogeneity of several gamma distributions is also derived. Extensive simulation studies indicate that the null distributions of the LRT statistics for all four problems depend on the parameters only weakly, and for practical purposes, they are independent of the parameters. Furthermore, our simulation studies suggest that the null distribution of the LRT statistic for testing the equality of shape parameters and that of the LRT statistic for testing the equality of scale parameters are essentially the same. Numerical algorithms to compute the maximum likelihood estimates and p‐values are given. Percentiles of the null distributions are tabulated for some selected sample sizes to compare three and five gamma distributions. The methods are illustrated using two practical examples. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Segmentation of high-resolution SAR image with unknown number of classes based on regular tessellation and RJMCMC algorithm

This article presents a statistics- and region-based approach to segmentation of synthetic aperture radar (SAR) images. The proposed approach can automatically determine the number of classes and segment the image simultaneously. First of all, an image domain is partitioned into a set of blocks by regular tessellation and the image is modelled on the assumption that intensities of its pixels in each homogeneous region satisfy an identical and independent gamma distribution. The Bayesian paradigm is followed to build an image segmentation model. Then, a Reversible Jump Markov Chain Monte Carlo scheme is designed to simulate the segmentation model, which determines the number of classes and segments the image roughly. Furthermore, in order to improve the accuracy of the segmentation results, refined operation is performed. The results obtained from both real and simulated SAR images show that the proposed approach works well and efficient.

On the Exact and Near-Exact Distributions of the Product of Generalized Gamma Random Variables and the Generalized Variance

In this article, the authors first obtain the exact distribution of the logarithm of the product of independent generalized Gamma r.v.’s (random variables) in the form of a Generalized Integer Gamma distribution of infinite depth, where all the rate and shape parameters are well identified. Then, by a routine transformation, simple and manageable expressions for the exact distribution of the product of independent generalized Gamma r.v.’s are derived. The method used also enables us to obtain quite easily very accurate, manageable and simple near-exact distributions in the form of Generalized Near-Integer Gamma distributions. Numerical studies are carried out to assess the precision of different approximations to the exact distribution and they show the high accuracy of the approximations provided by the near-exact distributions. As particular cases of the exact distributions obtained we have the distribution of the product of independent Gamma, Weibull, Frechet, Maxwell-Boltzman, Half-Normal, Rayleigh, and Exponential distributions, as well as the exact distribution of the generalized variance, the exact distribution of discriminants or Vandermonde determinants and the exact distribution of any linear combination of generalized Gumbel distributions, as well as yet the distribution of the product of any power of the absolute value of independent Normal r.v.’s.

Bayesian estimation of Dirichlet mixture model with variational inference

In statistical modeling, parameter estimation is an essential and challengeable task. Estimation of the parameters in the Dirichlet mixture model (DMM) is analytically intractable, due to the integral expressions of the gamma function and its corresponding derivatives. We introduce a Bayesian estimation strategy to estimate the posterior distribution of the parameters in DMM. By assuming the gamma distribution as the prior to each parameter, we approximate both the prior and the posterior distribution of the parameters with a product of several mutually independent gamma distributions. The extended factorized approximation method is applied to introduce a single lower-bound to the variational objective function and an analytically tractable estimation solution is derived. Moreover, there is only one function that is maximized during iterations and, therefore, the convergence of the proposed algorithm is theoretically guaranteed. With synthesized data, the proposed method shows the advantages over the EM-based method and the previously proposed Bayesian estimation method. With two important multimedia signal processing applications, the good performance of the proposed Bayesian estimation method is demonstrated.

Reliability Estimators for the Components of Series and Parallel Systems: The Weibull Model

This paper presents a hierarchical Bayesian approach to the estimation of components' reliability (survival) using a Weibull model for each of them. The proposed method can be used to estimation with general survival censored data, because the estimation of a component's reliability in a series (parallel) system is equivalent to the estimation of its survival function with right- (left-) censored data. Besides the Weibull parametric model for reliability data, independent gamma distributions are considered at the first hierarchical level for the Weibull parameters and independent uniform distributions over the real line as priors for the parameters of the gammas. In order to evaluate the model, an example and a simulation study are discussed.