Inverted Gamma Research Articles

Bayesian estimation and prediction for Burr‐Rayleigh mixture model using censored data

SummaryIn this study, Burr‐XII and Rayleigh distributions are combined to form a new mixture model that is considered to model heterogeneous data. Our objective is to estimate parameters of the proposed mixture model using Bayesian technique under type‐I censoring. Bayesian parameter estimation for the said mixture model is conducted by using informative priors, ie, gamma and squared root inverted gamma (SRIG) as well as noninformative prior, ie, Jeffrey's prior. Squared error loss function (SELF) and quadratic loss function (QLF) are employed to obtain and Bayes estimators. Properties of the proposed Bayes estimators are highlighted through a simulation study. When prior distributions and loss functions utilized in the study are compared in terms of posterior risks, informative prior found to be more suitable and decision turns out to be in favor of QLF. Prediction limits for the single sample case and two sample case are obtained to provide an insight into future sample data. Application of the proposed model is also elaborated using a real‐life example.

Bayesian inference from the mixture of half-normal distributions under censoring

This study considers the Bayesian inference for the mixture of two components of half-normal distribution using non-informative and informative prior. Several of its structural properties were derived, including explicit expression for mean, median, mode, reliability and hazard rate functions. Due to cost and time constraints, in most lifetime testing experiments censoring is an obligatory feature of lifetime datasets. We investigated Bayesian estimation of the parameters using various loss functions. The prior belief of the mixture model is represented by the uniform and square-root inverted gamma priors. Some properties of the model with graphs of the mixture density and hazard function are also discussed. The efficiencies of the proposed set of estimates of the mixture model parameters were studied through simulation and a real life dataset. Posterior risks of the Bayes estimators are evaluated and compared to explore the effect of prior belief and loss functions.

The inverted Nadarajah–Haghighi distribution: estimation methods and applications

ABSTRACTThe inverted (or inverse) distributions are sometimes very useful to explore additional properties of the phenomenons which non-inverted distributions cannot. We introduce a new inverted model called the inverted Nadarajah–Haghighi distribution which exhibits decreasing and unimodal (right-skewed) density while the hazard rate shapes are decreasing and upside-down bathtub. Our main focus is the estimation (from both frequentist and Bayesian points of view) of the unknown parameters along with some mathematical properties of the new model. The Bayes estimators and the associated credible intervals are obtained using Markov Chain Monte Carlo techniques under squared error loss function. The gamma priors are adopted for both scale and shape parameters. The potentiality of the distribution is analysed by means of two real data sets. In fact, it is found to be superior in its ability to sufficiently model the data as compared to the inverted Weibull, inverted Rayleigh, inverted exponential, inverted gamma, inverted Lindley and inverted power Lindley models.

Convergence Analysis of MCMC Algorithms for Bayesian Multivariate Linear Regression with Non‐Gaussian Errors

AbstractWhen Gaussian errors are inappropriate in a multivariate linear regression setting, it is often assumed that the errors are iid from a distribution that is a scale mixture of multivariate normals. Combining this robust regression model with a default prior on the unknown parameters results in a highly intractable posterior density. Fortunately, there is a simple data augmentation (DA) algorithm and a corresponding Haar PX‐DA algorithm that can be used to explore this posterior. This paper provides conditions (on the mixing density) for geometric ergodicity of the Markov chains underlying these Markov chain Monte Carlo algorithms. Letting d denote the dimension of the response, the main result shows that the DA and Haar PX‐DA Markov chains are geometrically ergodic whenever the mixing density is generalized inverse Gaussian, log‐normal, inverted Gamma (with shape parameter larger than d/2) or Fréchet (with shape parameter larger than d/2). The results also apply to certain subsets of the Gamma, F and Weibull families.

On length and area-biased Maxwell distributions

ABSTRACTThis article addresses the various properties and different methods of estimation of the unknown parameter of length and area-biased Maxwell distributions. Although, our main focus is on estimation from both frequentist and Bayesian point of view, yet, various mathematical and statistical properties of length and area-biased Maxwell distributions (such as moments, moment-generating function (mgf), hazard rate function, mean residual lifetime function, residual lifetime function, reversed residual life function, conditional moments and conditional mgf, stochastic ordering, and measures of uncertainty) are derived. We briefly describe different frequentist approaches, namely, maximum likelihood estimator, moments estimator, least-square and weighted least-square estimators, maximum product of spacings estimator and compare them using extensive numerical simulations. Next we consider Bayes estimation under different types of loss function (symmetric and asymmetric loss functions) using inverted gamma prior for the scale parameter. Furthermore, Bayes estimators and their respective posterior risks are computed and compared using Markov chain Monte Carlo (MCMC) algorithm. Also, bootstrap confidence intervals using frequentist approaches are provided to compare with Bayes credible intervals. Finally, a real dataset has been analyzed for illustrative purposes.

An Efficient Single Stage Shrinkage Estimator for the Scale parameter of Inverted Gamma Distribution

The present paper agrees with estimation of scale parameter θ of the Inverted Gamma (IG) Distribution when the shape parameter α is known (α=1), bypreliminarytestsinglestage shrinkage estimators using suitable shrinkage weight factor and region. The expressions for the Bias, Mean Squared Error [MSE] for the proposed estimators are derived. Comparisons between the considered estimator with the usual estimator (MLE) and with the existing estimator are performed .The results are presented in attached tables.

On the Ratio of Inverted Gamma Variates

In this paper the distribution and moments of the ratio of independent inverted gamma variates have been considered. Unbiased estimators of the parameter involved in the distribution have been proposed. As a particular case, the ratio of independent Levy variates have been studied.

Estimation procedures for Maxwell distribution under type-I progressive hybrid censoring scheme

The Maxwell (or Maxwell–Boltzmann) distribution was invented to solve the problems relating to physics and chemistry. It has also proved its strength of analysing the lifetime data. For this distribution, we consider point and interval estimation procedures in the presence of type-I progressively hybrid censored data. We obtain maximum likelihood estimator of the parameter and provide asymptotic and bootstrap confidence intervals of it. The Bayes estimates and Bayesian credible and highest posterior density intervals are obtained using inverted gamma prior. The expression of the expected number of failures in life testing experiment is also derived. The results are illustrated through the simulation study and analysis of a real data set is presented.

Inference on Reliability in the Gamma and Inverted Gamma Distributions

We derive distributions of ratio for two independent gamma variables and two independent inverted gamma variables and then we observe the skewness of two ratio densities. We then consider inference on reliability in two independent gamma random variables and two independent inverted gamma random variables each having known shape parameters.

Log-normal, log-gamma and log-negative inverted gamma scenarios in multifractal products of stochastic processes

This paper presents the log-normal, log-gamma and log-negative inverted gamma scenarios in limit theorems for multifractal products of stochastic processes with long-range dependence.

Analysis of Hybrid Life-tests in Presence of Competing Risks

The mixture of type-I and type-II censoring schemes, called the hybrid censoring scheme is quite common in life-testing or reliability experiments. In this paper, we consider the competing risks model in presence of hybrid censored data. Under this set up, it is assumed that the item may fail due to various causes and the corresponding lifetime distributions are independent and exponentially distributed with different scale parameters. We obtain the maximum likelihood estimators of the mean life of the different causes and derive their exact distributions. Using the exact distributions, all the moments can be obtained. Asymptotic confidence intervals and two bootstrap confidence intervals are also proposed. Bayes estimates and credible intervals of the unknown parameters are obtained under the assumptions of independent inverted gamma priors of the mean life of the different causes. Different methods have been compared using Monte Carlo simulations. Onereal data set has been analyzed for illustrative purposes.

Joined-Up Pensions Policy in the UK: An Asset-Liability Model for Simultaneously Determining the Asset Allocation and Contribution Rate

The trustees of funded defined benefit pension schemes must make two vital and inter-related decisions - setting the asset allocation and the contribution rate. While these decisions are usually taken separately, it is argued that they are intimately related and should be taken jointly. The objective of funded pension schemes is taken to be the minimization of both the mean and the variance of the contribution rate, where the asset allocation decision is designed to achieve this objective. This is done by splitting the problem into two main steps. First, the Markowitz mean-variance model is generalized to include three types of pension scheme liabilities (actives, deferreds and pensioners), and this model is used to generate the efficient set of asset allocations. Second, for each point on the risk-return efficient set of the asset-liability portfolio model, the mathematical model of Haberman (1992) is used to compute the corresponding mean and variance of the contribution rate and funding ratio. Since the Haberman model assumes that the discount rate for computing the present value of liabilities equals the investment return, it is generalized to avoid this restriction. This generalization removes the trade-off between contribution rate risk and funding ratio risk for a fixed spread period. Pension schemes need to choose a spread period, and it is shown how this can be set to minimize the variance of the contribution rate. Finally, using the result that the funding ratio follows an inverted gamma distribution, shortfall risk and expected tail loss are computed for funding below the minimum funding requirement, and funding above the taxation limit. This model is then applied to one of the largest UK pension schemes - the Universities Superannuation Scheme.

Financial Data Analysis with Two Symmetric Distributions

AbstractThe normal inverted gamma mixture or generalized Student t and the symmetric double Weibull, as well as their logarithmic counterparts, are proposed for modeling some loss distributions in non-life insurance and daily index return distributions in financial markets. For three specific data sets, the overall goodness-offit from these models, as measured simultaneously by the negative log-likelihood, chi-square and minimum distance statistics, is found to be superior to that of various “good” competitive models including the log-normal, the Burr, and the symmetric α-stable distribution. Furthermore, the study justifies on a statistical basis different important models of financial returns like the model of Black-Scholes (1973), the log-Laplace model of Hürlimann (1995), the normal mixture by Praetz (1972), the symmetric α-stable model by Mandelbrot (1963) and Fama (1965), and the recent double Weibull as limiting geometric-multiplication stable scheme in Mittnik and Rachev (1993). As an application, the prediction of one-year index returns from daily index returns is discussed.

Bayesian reliability modeling with a new loss function and numerical simulation

The aim the present study is to develop a Bayesian reliability model using a new loss function that has been introduced by Higgins and Tsokos. The underlyng failure model is the exponential and four priors are being used: the inverted gamma , the general uniform, the exponential and the lognormal. The concept of efficienncy along with the notion of integrated mean square are being used as a criterion to analytically and numerically compare our findings with respect to the natural conjugate prior. It is shown that the Bayesian reliability estimates are in general sensitive to the choice of the prior probability distribution.

Shortest bayes credibility intervals for the lognormal failure model

The lognormal distribution with density function f(x|λ,δ 2)= (2π) 1 2 δx −1 exp − ( In x−λ) 2 2δ 2 , forx>0,δ>0,−∞<λ<∞) is considered as a failure model from the Bayesian point of view. The shortest Bayesian confidence intervals for the parameters and reliability function are obtained for two cases. First, it is assumed that σ 2 is known and that γ has a normal prior distribution. Then, the case that γ is known and σ 2 has an inverted gamma prior distribution is considered. Some computer simulations are given to demonstrate the advantage of the shortest Bayesian confidence intervals over the ordinary Bayesian confidence intervals in terms of their length ratios.

A multiplicative model for volume and volatility

We first present prima facie evidence for the predictions generated by the mixture of distributions hypothesis, using daily German stock returns and their corresponding daily trading volumes and number of trades. These last two variables are used as proxies for the stochastic rate of information arrival when one wishes to explain GARCH effects by adhering to the mixture of distributions hypothesis. We show that there is no need for these proxies when the stochastic rate of information arrival follows an inverted gamma distribution. Daily trading volume and the daily number of trades, however, empirically provide an explanation for the occurrence of conditional heteroskedasticity of the GARCH form. We estimate several specifications where daily trading volume is included in the conditional variance equation additively and multiplicatively. The new multiplicative specification clearly outperforms the additive specification.

Bayes shrinkage estimation of reliability and the parameters of a finite range failure time model

A finite range failure time distribution has been proposed and studied. For estimating the two parameters of this distribution, this paper considers a prior assumption that (1 − b) is the probability that the scale parameter θ and shape parameter p have the values θ 0 and p 0, respectively, and that the rest of the probability mass b(0 ≤ b ≤ 1) is distributed as h( p, θ) = h 1( p) h 2( θ). The value h 1( p) is a uniform density for p and h 2( θ) is an inverted gamma density for θ. With this prior density, Bayes estimators are first obtained and then Bayesian shrinkage estimators are defined. Bayesian shrinkage estimators are compared with maximum likelihood estimators (m.l.e.) and it was found that the proposed estimators are better than m.l.e. for quite a wide range of parameters.

Bayesian inference in truncated and censored exponential distribution and reliability estimation

Point and confidence intervals for the scale parameter θ and the reliability function R(to) of the doubly truncated and censored exponential distribution are considered using inverted gamma as a prior distribution for θ. The Bayes risk of Bayesian and non-Bayesian estimators are obtained for comparison. Illustrative examples, using real data are included.

Bayesian reliability analysis when multiple early failures may be present

This paper discusses the Bayesian reliability analysis for an exponential failure mode on the basis of some ordered observations when the first p observations may represent early failures or outliers. The Bayes estimators of the mean life and reliability are obtained for the underlying parametric model referred to as the SB(p) model under the assumption of the squared error loss function, the inverted gamma prior for scale parameter and a generalized uniform prior for the nuisance parameter

Robust bayes analysis in normal linear regression with an improper mixture prior

This paper examines the linear regression model when prior information of the regression parameterβ ∊ Rk and the error varianceσ2 > 0 is represented by the improper mixture pdf , where π0 is the usual normal inverted gamma base priorq is a specific noninformative prior and (1-∊),0 ≤ ∊ ≤ 1, is the degree of confidence about π0. This prior is specially useful in multiparameter situations since no new hyperparameters are required in the specification of q. The prior to posterior analysis is shown to produce a mixture pdf which may be multimodal. Robustness is obtained in the following sense:if π0 is notcompatible withthe data,its influence inthe posterior is discounted, and more weight is automatically placed on the posterior that emerges with the noninformative prior. A numerical example is presented that illustrates the ideas developed in the paper.