Noncentral Gamma Research Articles

Reliability of repairable systems with Non-Central Gamma frailty

Maintenance actions on industrial equipment are essential to reduce expenses associated with equipment failures. Based on a well-fitted model, it is possible, through the estimated parameters, to predict several functions of interest, such as the cumulative average and reliability functions. In this paper, a new frailty model is proposed to analyze failure times of repairable systems subject to unobserved heterogeneity actions. The Non-Central Gamma distribution is assumed to the frailty random variable effect. The class of minimal repair models for repairable systems is explored considering an approach that includes the frailty term to estimate the unobserved heterogeneity over the systems’ failure process. Classical inferential methods were used to parameter estimation and define the reliability prediction functions. A simulation study was conducted to confirm the properties expected in the estimators. Two real-world data known in literature were used to illustrate the estimation procedures and validate the proposed model as a viable alternative to those already established in the literature. The results obtained highlight the potential of our proposed approach, particularly for industries dealing with such systems, where unquantifiable factors may impact equipment failure times.

Generalized Dirichlet Distribution Based on Confluent Hypergeometric Series

Dirichlet distribution is a kind of high-dimensional continuous probability distribution, which has important applications in the fields of statistics, machine learning and bioinformatics. In this paper, based on gamma distribution we study two two-dimensional random variables. Then we derive the properties of these two two-dimensional random variables by using the properties of non-central gamma distribution and confluent hypergeometric series. From these properties, we find the two random variables follow generalized Dirichlet distributions. Applying hypergeometric series to Dirichlet distribution broadens the research of Dirichlet distribution.

Graduation of term assurance data using frailty approach

Frailty models have been used in literature to account for heterogeneity among insureds in-terms of mortality. In this article, we compare the gamma and the non-central gamma as frailty distributions with the exponentiated exponential and exponentiated Weibull as baseline hazards. We adopt a fully Bayesian approach to calibrate the baselines based on crude mortality rates from a major Kenyan insurer. Comparing the gamma-exponentiated Weibull with the non-central gamma-exponentiated Weibull models shows that the non-central gamma provides a good fit to the real life data-set and is therefore recommended for valuation.

Graduation of term assurance data using frailty approach

Frailty models have been used in literature to account for heterogeneity among insureds in-terms of mortality. In this article, we compare the gamma and the non-central gamma as frailty distributions with the exponentiated exponential and exponentiated Weibull as baseline hazards. We adopt a fully Bayesian approach to calibrate the baselines based on crude mortality rates from a major Kenyan insurer. Comparing the gamma-exponentiated Weibull with the non-central gamma-exponentiated Weibull models shows that the non-central gamma provides a good fit to the real life data-set and is therefore recommended for valuation.

GammaCHI: A package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral). New version announcement

This is a revised and updated version of the package GammaCHI. The package includes routines for computing and inverting the gamma and chi-square cumulative distribution functions (central and noncentral). Additionally, the package also provides routines for computing the gamma function, the error function and other functions related to the gamma function (the logarithm of the gamma function, the regulated gamma function and the ratio of two gamma functions). In this new version, the range of computation of the inversion routine of the central gamma distribution function invcdfgamC(ichi,a,p,q,x,ierr) is extended; p or q input values close to the underflow limit are now allowed. New version program summaryProgram Title: GammaCHICPC Library link to program files:https://doi.org/10.17632/d2kwwvsyny.1Licensing provisions: GPLv2Programming language: Fortran 90Journal reference of previous version: Comput. Phys. Commun. 191 (2015) 132–139.Does the new version supersede the previous version?: YesReasons for the new version: With minor modifications of few of the functions included in the module GammaCHI, it is possible to enlarge the range of computation of the routine for the inversion of the central gamma distribution.Summary of revisions: The basic algorithms are unchanged. The range of computation of the routine for the inversion of the central gamma distribution function is extended in this new version. Specifically in the routine invcdfgamC(ichi,a,p,q,x,ierr) values of p or q close to the underflow limit are now allowed.Nature of problem: The computation and inversion of gamma and chi-square cumulative distribution functions (central and noncentral) as well as the computation of the error and gamma functions is needed in many problems of applied and mathematical physics. For example, central and noncentral gamma distributions are used in the analysis of signal detection in different physical scenarios such as optics or quantum detection.Solution method: Different methods of computation are used depending on the range of parameters: asymptotic expansions, quadrature methods, etc.

Doubly Non-Central Generalized Beta Distributions

The doubly non-central generalized beta type 1 and type 2 distributions have been derived by using two independent non-central gamma variables with different scale parameters. These distributions generalize several well known central and non-central beta distributions. The doubly non-central generalized beta densities are much more flexible than many exiting beta models and can assume a large variety of shapes. Several properties of these distributions have been studied.

Predicting the Estrogen Receptor Activity of Environmental Chemicals by Single-Cell Image Analysis and Data-driven Modeling

A comprehensive evaluation of toxic chemicals and understanding their potential harm to human physiology is vital in mitigating their adverse effects following exposure from environmental emergencies. In this work, we develop data-driven classification models to facilitate rapid decision making in such catastrophic events and predict the estrogenic activity of environmental toxicants as estrogen receptor-α (ERα) agonists or antagonists. By combining high-content analysis, big-data analytics, and machine learning algorithms, we demonstrate that highly accurate classifiers can be constructed for evaluating the estrogenic potential of many chemicals. We follow a rigorous, high throughput microscopy-based high-content analysis pipeline to measure the single cell-level response of benchmark compounds with known in vivo effects on the ERα pathway. The resulting high-dimensional dataset is then pre-processed by fitting a non-central gamma probability distribution function to each feature, compound, and concentration. The characteristic parameters of the distribution, which represent the mean and the shape of the distribution, are used as features for the classification analysis via Random Forest (RF) and Support Vector Machine (SVM) algorithms. The results show that the SVM classifier can predict the estrogenic potential of benchmark chemicals with higher accuracy than the RF algorithm, which misclassifies two antagonist compounds.

Statistical characterization of the linear attenuation coefficient in polychromatic CT scans.

To provide a unifying statistical model that characterizes the integrated x-ray intensity at the detector after logarithmic transformation and can be extended to the characterization of computed tomography (CT) numbers in the reconstructed image. We study the statistical characteristics of polyenergetic x-ray beams in the detector. Firstly, we consider the characterization of the integrated x-ray intensity at the detector through a probabilistic model (compound Poisson) that describes its statistics. We analyze its properties and derive the probabilistic distribution after the logarithmic transformation analytically. Finally, we propose a more tractable probabilistic distribution with the same features observed in the characterization, the noncentral Gamma (nc-Gamma). This distribution exhibits desirable properties for the statistical characterization across the reconstruction process. We assess the assumptions adopted in the derivation of the statistical models throughout Monte Carlo simulations and validate them with a water phantom and a lung phantom acquired in a Siemens clinical CT scan. We evaluate the statistical similarities between the theoretical distribution and the nc-Gamma using a power analysis with a Kolmogorov-Smirnov test for a 95% confidence level. The Kolmogorov-Smirnov goodness-of-fit test obtained for the Monte Carlo simulation shows an extremely high agreement between the empirical distribution of the post-logarithmic-integrated x-ray intensity and the nc-Gamma. The experimental validation performed with both phantoms confirmed the excellent match between the theoretical distribution, the proposed nc-Gamma, and sample distributions in all situations. We derive an analytical model describing the post-log distribution of the linear attenuation coefficient in the sensor for polychromatic CT scans. We also demonstrate that the nc-Gamma distribution approximates well the theoretical distribution. This distribution also approximates well the CT numbers after reconstruction since it naturally extends across linear operations involved in filtered back projection reconstructions. This probabilistic model may provide the analytical foundation to define new likelihood-based reconstruction methodologies for polychromatic scans.

A gamma-mixture class of distributions with Bayesian application

ABSTRACTIn this article, a subjective Bayesian approach is followed to derive estimators for the parameters of the normal model by assuming a gamma-mixture class of prior distributions, which includes the gamma and the noncentral gamma as special cases. An innovative approach is proposed to find the analytical expression of the posterior density function when a complicated prior structure is ensued. The simulation studies and a real dataset illustrate the modeling advantages of this proposed prior and support some of the findings.

Performance prediction of noncoherent detector for independent and nonidentically distributed NCG fluctuating targets

We address the performance prediction of the noncoherent detector (i.e., squared-law plus integrator) for independent but possibly nonidentically distributed (non-id) noncentral Gamma (NCG) fluctuating target model, which is more general and can describe more complex targets. Initially, we obtain the probability density function (pdf) of the sum of independent and non-id NCG random variables in terms of an infinite sum of Gamma pdfs. Furthermore, we develop an analytic expression of the detection probability in terms of converging series based on the generalized Marcum Q function. Finally, at the analysis stage, we first study the truncation error of the derived expression and then evaluate the impacts on the detection performance of the target fluctuating parameters via numerical simulations.

NON-CENTRAL MULTIVARIATE CHI-SQUARE AND GAMMA DISTRIBUTIONS

A (p-1)-variate integral representation is given for the cumulative distribution function of the general p-variate non-central gamma distribution with a non-centrality matrix of any admissible rank. The real part of products of well known analytical functions is integrated over arguments from (-pi,pi). To facilitate the computation, these formulas are given more detailed for p=2 and p=3. These (p-1)-variate integrals are also derived for the diagonal of a non-central complex Wishart Matrix. Furthermore, some alternative formulas are given for the cases with an associated one-factorial (pxp)-correlation matrix R, i.e. R differs from a suitable diagonal matrix only by a matrix of rank 1, which holds in particular for all (3x3)-R with no vanishing correlation.

DERIVATION OF A TEST STATISTIC FOR EMPHYSEMA QUANTIFICATION.

Density masking is the de-facto quantitative imaging phenotype for emphysema that is widely used by the clinical community. Density masking defines the burden of emphysema by a fixed threshold, usually between -910 HU and -950 HU, that has been experimentally validated with histology. In this work, we formalized emphysema quantification by means of statistical inference. We show that a non-central Gamma is a good approximation for the local distribution of image intensities for normal and emphysema tissue. We then propose a test statistic in terms of the sample mean of a truncated non-central Gamma random variable. Our results show that this approach is well-suited for the detection of emphysema and superior to standard density masking. The statistical method was tested in a dataset of 1337 samples obtained from 9 different scanner models in subjects with COPD. Results showed an increase of 17% when compared to the density masking approach, and an overall accuracy of 94.09%.

Sharp bounds for cumulative distribution functions

Ratios of integrals can be bounded in terms of ratios of integrands under certain monotonicity conditions. This result, related to L'Hôpital's monotone rule, can be used to obtain sharp bounds for cumulative distribution functions. We consider the case of noncentral cumulative gamma and beta distributions. Three different types of sharp bounds for the noncentral gamma distributions (also called Marcum functions) are obtained in terms of modified Bessel functions and one additional type of function: a second modified Bessel function, two error functions or one incomplete gamma function. For the noncentral beta case the bounds are expressed in terms of Kummer functions and one additional Kummer function or an incomplete beta function. These bounds improve previous results with respect to their range of application and/or its sharpness.

Simulations of full multivariate Tweedie with flexible dependence structure

We employ a variables-in-common method for constructing multivariate Tweedie distributions, based on linear combinations of independent univariate Tweedie variables. The method lies on the convolution and scaling properties of the Tweedie laws, using the cumulant generating function for characterization of the distributions and correlation structure. The routine allows the equivalence between independence and zero correlation and gives a parametrization through given values of the mean vector and dispersion matrix, similarly to the Gaussian vector. Our approach leads to a matrix representation of multivariate Tweedie models, which permits the simulations of many known distributions, including Gaussian, Poisson, non-central gamma, gamma, and inverse Gaussian, both positively or negatively correlated.

GammaCHI: A package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral)

A Fortran 90 module GammaCHI for computing and inverting the gamma and chi-square cumulative distribution functions (central and noncentral) is presented. The main novelty of this package is the reliable and accurate inversion routines for the noncentral cumulative distribution functions. Additionally, the package also provides routines for computing the gamma function, the error function and other functions related to the gamma function. The module includes the routines cdfgamC, invcdfgamC, cdfgamNC, invcdfgamNC, errorfunction, inverfc, gamma, loggam, gamstar and quotgamm for the computation of the central gamma distribution function (and its complementary function), the inversion of the central gamma distribution function, the computation of the noncentral gamma distribution function (and its complementary function), the inversion of the noncentral gamma distribution function, the computation of the error function and its complementary function, the inversion of the complementary error function, the computation of: the gamma function, the logarithm of the gamma function, the regulated gamma function and the ratio of two gamma functions, respectively. Program summaryProgram title: Module GammaCHICatalogue identifier: AEVM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEVM_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 7653No. of bytes in distributed program, including test data, etc.: 85283Distribution format: tar.gzProgramming language: Fortran 90.Computer: Any supporting a FORTRAN compiler.Operating system: Any supporting a FORTRAN compiler.RAM: a few MBClassification: 4.7.Nature of problem: The computation and inversion of gamma and chi-square cumulative distribution functions (central and noncentral) as well as the computation of the error and gamma functions is needed in many problems of applied and mathematical physics.Solution method: The algorithms use different methods of computation depending on the range of parameters: asymptotic expansions, quadrature methods, etc.Restrictions: In the inversion of the central gamma/chi-square distribution functions, very small input function values Pμ(x,y), Qμ(x,y) (lower than 10−150) are not admissible.The admissible input parameter ranges for computing the noncentral cumulative gamma distribution functions Pμ(x,y), Qμ(x,y) in standard IEEE double precision arithmetic are 0≤x≤10,000, 0≤y≤10,000,0.5≤μ≤ 10,000 and the related parameter ranges for the noncentral chi-square cumulative distribution function. In the inversion of the noncentral gamma/chi-square distribution functions, very small input function values Pμ(x,y), Qμ(x,y) (lower than 10−25 and 10−35, respectively) are not admissible.Running time: It varies depending on the function and the parameter range. The test provided takes milliseconds.

The Asymptotic and Numerical Inversion of the Marcum Q‐Function

The generalized Marcum functions appear in problems of technical and scientific areas such as, for example, radar detection and communications. In mathematical statistics and probability theory these functions are called the noncentral gamma or the noncentral chi‐squared cumulative distribution functions. In this paper, we describe a new asymptotic method for inverting the generalized Marcum Q‐function and for the complementary Marcum P‐function. Also, we show how monotonicity and convexity properties of these functions can be used to find initial values for reliable Newton or secant methods to invert the function. We present details of numerical computations that show the reliability of the asymptotic approximations.

Algorithm 939

Methods and an algorithm for computing the generalized MarcumQ–function (Qμ(x,y)) and the complementary function (Pμ(x,y)) are described. These functions appear in problems of different technical and scientific areas such as, for example, radar detection and communications, statistics, and probability theory, where they are called the noncentral chi-square or the noncentral gamma cumulative distribution functions.The algorithm for computing the Marcum functions combines different methods of evaluation in different regions: series expansions, integral representations, asymptotic expansions, and use of three-term homogeneous recurrence relations. A relative accuracy close to 10−12can be obtained in the parameter region (x,y,μ) ∈ [0,A] ×[0,A] × [1,A],A= 200, while for larger parameters the accuracy decreases (close to 10−11forA= 1000 and close to 5 × 10−11forA= 10000).

Distributions of Sum, Difference, Product and Quotient of Independent Non-central Beta Type 3 Variables

Let X and Y be independent random variables, X having a gamma distribution with shape parameter a and Y having a non-central gamma distribution with shape and non-centrality parameters b and δ, respectively. Define Z = X/(X + 2Y ). Then, the random variable Z has a non-central beta type 3 distribution, Z ∼ NCB3(a, b; δ). In this article we derive density functions of sum, difference, product and quotient of two independent random variables each having noncentral beta type 3 distribution. These density functions are expressed in series involving first hypergeometric function of Appell.

Non-Central Beta Type 3 Distribution

Let $X$ and $Y$ be independent random variables, $X$ having a gamma distribution with shape parameter $a$ and $Y$ having a non-central gamma distribution with shape and non-centrality parameters $b$ and $\delta$, respectively. Define $ W ={X}/(X + 2 Y)$. Then, the random variable $W$ has a non-central beta type 3 distribution, $W\sim \textnormal{NCB3} (a,b;\delta)$. In this article we study several of its properties. We also give a multivariate generalization of the non-central beta type 3 distribution and derive its properties.

Computing the noncentral gamma distribution, its inverse and the noncentrality parameter

The noncentral gamma distribution can be viewed as a generalization of the noncentral chi-squared distribution and it can be expressed as a mixture of a Poisson density function with a incomplete g...