Bayesian analysis of the generalized gamma distribution using non-informative priors

The Generalized gamma (GG) distribution plays an important role in statistical analysis. For this distribution, we derive non-informative priors using formal rules, such as Jeffreys prior, maximal data information prior and reference priors. We have shown that these most popular formal rules with natural ordering of parameters, lead to priors with improper posteriors. This problem is overcome by considering a prior averaging approach discussed in Berger et al. [Overall objective priors. Bayesian Analysis. 2015;10(1):189–221]. The obtained hybrid Jeffreys-reference prior is invariant under one-to-one transformations and yields a proper posterior distribution. We obtained good frequentist properties of the proposed prior using a detailed simulation study. Finally, an analysis of the maximum annual discharge of the river Rhine at Lobith is presented.     

Estimation for the parameters of the generalized half-normal distribution

As an applicable and flexible lifetime model, the two-parameter generalized half-normal (GHN) distribution has been received wide attention in the field of reliability analysis and lifetime study. In this paper maximum likelihood estimates of the model parameters are discussed and we also proposed corresponding bias-corrected estimates. Unweighted and weighted least squares estimates for the parameters of the GHN distribution are also presented for comparison purpose. Moreover, the likelihood ratio test is provided as complementary. Simulation study and illustrative examples are provided to compare the performance of the proposed methods.     

A new weighted distribution as an extension of the generalized half-normal distribution with applications

In this study, a new extension of generalized half-normal (GHN) distribution is introduced. Since this new distribution can be viewed as weighted version of GHN distribution, it is called as weighted generalized half-normal (WGHN) distribution. It is shown that WGHN distribution can be observed as a single constrained and hidden truncation model. Therefore, the new distribution is more flexible than the GHN distribution. Some statistical properties of the WGHN distribution are studied, i.e. moments, cumulative distribution function, hazard rate function are derived. Furthermore, maximum likelihood estimation of the parameters is considered. Some real-life data sets taken from the literature are modelled using the WGHN distribution. It is seen that for these data sets the WGHN distribution provides better fitting than the GHN and slashed generalized half-normal (SGHN) distributions.     

The odd log-logistic generalized half-normal lifetime distribution: Properties and applications

Cooray and Ananda (2008 Cooray, K., Ananda, M.M.A. (2008). A Generalization of the half-normal distribution with applications to lifetime data. Commun. Stat. - Theory Methods 37:1323–1337.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) pioneered a lifetime model commonly used in reliability studies. Based on this distribution, we propose a new model called the odd log-logistic generalized half-normal distribution for describing fatigue lifetime data. Various of its structural properties are derived. We discuss the method of maximum likelihood to fit the model parameters. For different parameter settings and sample sizes, some simulation studies compare the performance of the new lifetime model. It can be very useful, and its superiority is illustrated by means of a real dataset.     

Estimation for the generalized half-normal distribution based on progressive type-II censoring

Based on progressively type-II censored data, the problem of estimating unknown parameters and reliability function of a two-parameter generalized half-normal distribution is considered. Maximum likelihood estimates are obtained by applying expectation-maximization algorithm. Since they do not have closed forms, approximate maximum likelihood estimators are proposed. Several Bayesian estimates with respect to different symmetric and asymmetric loss functions such as squared error, LINEX and general entropy are calculated. The Lindley approximation method is applied to determine Bayesian estimates. Monte Carlo simulations are performed to compare the performances of the different methods. Finally, one real data set is analysed.     

Noninformative priors for linear combinations of normal means with unequal variances

For normal populations with unequal variances, we develop matching priors and reference priors for a linear combination of the means. Here, we find three second-order matching priors: a highest posterior density (HPD) matching prior, a cumulative distribution function (CDF) matching prior, and a likelihood ratio (LR) matching prior. Furthermore, we show that the reference priors are all first-order matching priors, but that they do not satisfy the second-order matching criterion that establishes the symmetry and the unimodality of the posterior under the developed priors. The results of a simulation indicate that the second-order matching prior outperforms the reference priors in terms of matching the target coverage probabilities, in a frequentist sense. Finally, we compare the Bayesian credible intervals based on the developed priors with the confidence intervals derived from real data.     

Noninformative priors for the normal variance ratio

In this paper, we consider noninformative priors for the ratio of variances in two normal populations. We develop first and second order matching priors. We find that the second order matching prior matches alternative coverage probabilities up to the second order and is also a HPD matching prior. It turns out that among the reference priors, only one-at-a-time reference prior satisfies a second order matching criterion. Our simulation study indicates that the one-at-a-time reference prior performs better than other reference priors in terms of matching the target coverage probabilities in a frequentist sense. This work is supported by Korea Research Foundation Grant (KRF-2004-002-C00041).     

General properties for the beta extended half-normal model

We formulate and study a four-parameter lifetime model called the beta extended half-normal distribution. This model includes as sub-models the exponential, extended half-normal and half-normal distributions. We derive expansions for the new density function which do not depend on complicated functions. We obtain explicit expressions for the moments and incomplete moments, generating function, mean deviations, Bonferroni and Lorenz curves and Rényi entropy. In addition, the model parameters are estimated by maximum likelihood. We provide the observed information matrix. The new model is modified to cope with possible long-term survivors in the data. The usefulness of the new distribution is shown by means of two real data sets.     

Noninformative priors for linear combinations of the normal means

In this paper, we develop noninformative priors for linear combinations of the means under the normal populations. It turns out that among the reference priors the one-at-a-time reference prior satisfies a second order probability matching criterion. Moreover, the second order probability matching priors match alternative coverage probabilities up to the second order and are also HPD matching priors. Our simulation study indicates that the one-at-a-time reference prior performs better than the other reference priors in terms of matching the target coverage probabilities in a frequentist sense.     

Inference and optimal design of multiple constant-stress testing for generalized half-normal distribution under type-II progressive censoring

The generalized half-normal (GHN) distribution and progressive type-II censoring are considered in this article for studying some statistical inferences of constant-stress accelerated life testing. The EM algorithm is considered to calculate the maximum likelihood estimates. Fisher information matrix is formed depending on the missing information law and it is utilized for structuring the asymptomatic confidence intervals. Further, interval estimation is discussed through bootstrap intervals. The Tierney and Kadane method, importance sampling procedure and Metropolis-Hastings algorithm are utilized to compute Bayesian estimates. Furthermore, predictive estimates for censored data and the related prediction intervals are obtained. We consider three optimality criteria to find out the optimal stress level. A real data set is used to illustrate the importance of GHN distribution as an alternative lifetime model for well-known distributions. Finally, a simulation study is provided with discussion.     

Generalized variable method inference for the location parameter of the general half-normal distribution

This paper considers the development of inferential techniques based on the generalized variable method (GV-Method) for the location parameter of the general half-normal distribution. We are interested in hypothesis testing of, and interval estimation for, the location parameter. Body fat data, urinary excretion rate data, and simulated data are used to illustrate the application and to demonstrate the advantages of the proposed GV-Method over the large-sample method and the Bayesian method.     

The use of Jeffreys priors for the Student-t distribution

The Jeffreys-rule prior and the marginal independence Jeffreys prior are recently proposed in Fonseca et al. [Objective Bayesian analysis for the Student-t regression model, Biometrika 95 (2008), pp. 325–333] as objective priors for the Student-t regression model. The authors showed that the priors provide proper posterior distributions and perform favourably in parameter estimation. Motivated by a practical financial risk management application, we compare the performance of the two Jeffreys priors with other priors proposed in the literature in a problem of estimating high quantiles for the Student-t model with unknown degrees of freedom. Through an asymptotic analysis and a simulation study, we show that both Jeffreys priors perform better in using a specific quantile of the Bayesian predictive distribution to approximate the true quantile.     

Non-informative priors for the inverse Weibull distribution

In this paper, we develop the non-informative priors for the inverse Weibull model when the parameters of interest are the scale and the shape parameters. We develop the first-order and the second-order matching priors for both parameters. For the scale parameter, we reveal that the second-order matching prior is not a highest posterior density (HPD) matching prior, does not match the alternative coverage probabilities up to the second order and is not a cumulative distribution function (CDF) matching prior. Also for the shape parameter, we reveal that the second-order matching prior is an HPD matching prior and a CDF matching prior and also matches the alternative coverage probabilities up to the second order. For both parameters, we reveal that the one-at-a-time reference prior is the second-order matching prior, but Jeffreys’ prior is not the first-order and the second-order matching prior. A simulation study is performed to compare the target coverage probabilities and a real example is given.     

A new destructive Poisson odd log-logistic generalized half-normal cure rate model

We propose a new cure rate survival model by assuming that the initial number of competing causes of the event of interest follows a Poisson distribution and the time to event has the odd log-logistic generalized half-normal distribution. This survival model describes a realistic interpretation for the biological mechanism of the event of interest. We estimate the model parameters using maximum likelihood. For different sample sizes, various simulation scenarios are performed. We propose the diagnostics and residual analysis to verify the model assumptions. The potentiality of the new cure rate model is illustrated by means of a real data.     

Noninformative priors for the nested design

This paper considers noninformative priors for three-stage nested designs. It turns out that the noninformative prior given by Li and Stern (1997) is the one-at-a-time reference prior satisfying a second-order matching criterion when either the variance ratio or linear combinations of the means is of interest. Moreover, it is a joint probability matching prior when both the variance ratio and linear combinations of the means are of interest. These priors are compared with Jeffreys' prior in light of how accurately the coverage probabilities of Bayesian credible intervals match the corresponding frequentist coverage probabilities.     

Noninformative priors for the between-group variance in the unbalanced one-way random effects model with heterogeneous error variances

For the unbalanced one-way random effects model with heterogeneous error variances, we propose the non-informative priors for the between-group variance and develop the first- and second-order matching priors. It turns out that the second-order matching priors do not exist and the reference prior and Jeffreys prior do not satisfy a first-order matching criterion. We also show that the first-order matching prior meets the frequentist target coverage probabilities much better than the Jeffreys prior and reference prior through simulation study, and the Bayesian credible intervals based on the matching prior and reference prior give shorter intervals than the existing confidence intervals by examples.     

A matching prior for extreme quantile estimation of the generalized Pareto distribution

Extreme quantile estimation plays an important role in risk management and environmental statistics among other applications. A popular method is the peaks-over-threshold (POT) model that approximate the distribution of excesses over a high threshold through generalized Pareto distribution (GPD). Motivated by a practical financial risk management problem, we look for an appropriate prior choice for Bayesian estimation of the GPD parameters that results in better quantile estimation. Specifically, we propose a noninformative matching prior for the parameters of a GPD so that a specific quantile of the Bayesian predictive distribution matches the true quantile in the sense of Datta et al. (2000).     

Noninformative priors for the common mean in the bivariate normal distribution

In this paper, we consider some noninformative priors for the common mean in a bivariate normal population. We develop the first-order and second-order matching priors and reference priors. We find that the second-order matching prior is also an HPD matching prior, and matches the alternative coverage probabilities up to the second order. It turns out that derived reference priors do not satisfy a second-order matching criterion. Our simulation study indicates that the second-order matching prior performs better than the reference priors in terms of matching the target coverage probabilities in a frequentist sense. We also illustrate our results using real data.     

On priors that match posterior and frequentist distribution functions

In the multiparameter case, this paper characterizes priors so as to match, up to o(n^-1/2), the posterior joint cumulative distribution function (c.d.f.) of a posterior standardized version of the parametric vector with the corresponding frequentist c.d.f.     

Parameter estimation for generalized Pareto distribution by generalized probability weighted moment-equations

The generalized Pareto distribution (GPD) has been widely used to model exceedances over a threshold. This article generalizes the method of generalized probability weighted moments, and applies this method to estimate the parameters of GPD. The estimator is computationally easy. Some asymptotic results of this method are provided. Two simulations are carried out to investigate the behavior of this method and to compare them with other methods suggested in the literature. The simulation results show that the performance of the proposed method is better than some other methods. Finally, this method is applied to analyze a real-life data.