Statistical analysis for Kumaraswamy’s distribution based on record data

In this paper we review some results that have been derived on record values for some well known probability density functions and based on m records from Kumaraswamy’s distribution we obtain estimators for the two parameters and the future sth record value. These estimates are derived using the maximum likelihood and Bayesian approaches. In the Bayesian approach, the two parameters are assumed to be random variables and estimators for the parameters and for the future sth record value are obtained, when we have observed m past record values, using the well known squared error loss (SEL) function and a linear exponential (LINEX) loss function. The findings are illustrated with actual and computer generated data.     

The product <Emphasis Type="Italic">t</Emphasis> density distribution arising from the product of two Student’s <Emphasis Type="Italic">t</Emphasis> PDFs

The Student’s t distribution has become increasingly prominent and is considered as a competitor to the normal distribution. Motivated by real examples in Physics, decision sciences and Bayesian statistics, a new t distribution is introduced by taking the product of two Student’s t pdfs. Various structural properties of this distribution are derived, including its cdf, moments, mean deviation about the mean, mean deviation about the median, entropy, asymptotic distribution of the extreme order statistics, maximum likelihood estimates and the Fisher information matrix. Finally, an application to a Bayesian testing problem is illustrated.     

Bayesian estimation and prediction of the intensity of the power law process

In analyzing failure data pertaining to a repairable system, perhaps the most widely used parametric model is a nonhomogeneous Poisson process with Weibull intensity, more commonly referred to as the Power Law Process (PLP) model. Investigations relating to inference of parameters of the PLP under a frequentist framework abound in the literature. The focus of this article is to supplement those findings from a Bayesian perspective, which has thus far been explored to a limited extent in this context. Main emphasis is on the inference of the intensity function of the PLP. Both estimation and future prediction are considered under traditional as well as more complex censoring schemes. Modern computational tools such as Markov Chain Monte Carlo are exploited efficiently to facilitate the numerical evaluation process. Results from the Bayesian inference are contrasted with the corresponding findings from a frequentist analysis, both from a qualitative and a quantitative viewpoint. The developed methodology is implemented in analyzing interval-censored failure data of equipments in a fleet of marine vessels.     

Estimation and prediction of the Kumaraswamy distribution based on record values and inter-record times

ABSTRACT

The maximum likelihood and Bayesian approaches for estimating the parameters and the prediction of future record values for the Kumaraswamy distribution has been considered when the lower record values along with the number of observations following the record values (inter-record-times) have been observed. The Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. The Bayes and the maximum likelihood estimates are compared by using the estimated risk through Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record values arising from the Kumaraswamy distribution based on record values with their corresponding inter-record times and only record values. The comparison of the derived predictors are carried out by using Monte Carlo simulations. Real data are analysed for an illustration of the findings.     

Classical and Bayesian estimation of P(X<Y) using upper record values from Kumaraswamy’s distribution

In this paper, maximum likelihood and Bayesian approaches have been used to obtain the estimation of \(P(X<Y)\) based on a set of upper record values from Kumaraswamy distribution. The existence and uniqueness of the maximum likelihood estimates of the Kumaraswamy distribution parameters are obtained. Confidence intervals, exact and approximate, as well as Bayesian credible intervals are constructed. Bayes estimators have been developed under symmetric (squared error) and asymmetric (LINEX) loss functions using the conjugate and non informative prior distributions. The approximation forms of Lindley (Trabajos de Estadistica 3:281–288, 1980) and Tierney and Kadane (J Am Stat Assoc 81:82–86, 1986) are used for the Bayesian cases. Monte Carlo simulations are performed to compare the different proposed methods.     

A simple approach to the estimation of Tukey's gh distribution

ABSTRACT

The Tukey's gh distribution is widely used in situations where skewness and elongation are important features of the data. As the distribution is defined through a quantile transformation of the normal, the likelihood function cannot be written in closed form and exact maximum likelihood estimation is unfeasible. In this paper we exploit a novel approach based on a frequentist reinterpretation of Approximate Bayesian Computation for approximating the maximum likelihood estimates of the gh distribution. This method is appealing because it only requires the ability to sample the distribution. We discuss the choice of the input parameters by means of simulation experiments and provide evidence of superior performance in terms of Root-Mean-Square-Error with respect to the standard quantile estimator. Finally, we give an application to operational risk measurement.     

Bayes estimation of Gompertz distribution parameters and acceleration factor under partially accelerated life tests with type-I censoring

In this paper, the Bayesian approach is applied to the estimation problem in the case of step stress partially accelerated life tests with two stress levels and type-I censoring. Gompertz distribution is considered as a lifetime model. The posterior means and posterior variances are derived using the squared-error loss function. The Bayes estimates cannot be obtained in explicit forms. Approximate Bayes estimates are computed using the method of Lindley [D.V. Lindley, Approximate Bayesian methods, Trabajos Estadistica 31 (1980), pp. 223–237]. The advantage of this proposed method is shown. The approximate Bayes estimates obtained under the assumption of non-informative priors are compared with their maximum likelihood counterparts using Monte Carlo simulation.     

Linear discrimination for multi-level multivariate data with separable means and jointly equicorrelated covariance structure

In this article we study a linear discriminant function of multiple m-variate observations at u-sites and over v-time points under the assumption of multivariate normality. We assume that the m-variate observations have a separable mean vector structure and a “jointly equicorrelated covariance” structure. The new discriminant function is very effective in discriminating individuals in a small sample scenario. No closed-form expression exists for the maximum likelihood estimates of the unknown population parameters, and their direct computation is nontrivial. An iterative algorithm is proposed to calculate the maximum likelihood estimates of these unknown parameters. A discriminant function is also developed for unstructured mean vectors. The new discriminant functions are applied to simulated data sets as well as to a real data set. Results illustrating the benefits of the new classification methods over the traditional one are presented.     

Estimation with the generalized exponential distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches have been considered for the two-parameter generalized exponential distribution based on record values with the number of trials following the record values (inter-record times). The maximum likelihood estimates are obtained under the inverse sampling and the random sampling schemes. It is shown that the maximum likelihood estimator of the shape parameter converges in mean square to the true value when the scale parameter is known. The Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The confidence intervals for the parameters are constructed based on asymptotic and Bayesian methods. The Bayes and the maximum likelihood estimators are compared in terms of the estimated risk by the Monte Carlo simulations. The comparison of the estimators based on the record values and the record values with their corresponding inter-record times are performed by using Monte Carlo simulations.     

Posterior analysis of the compound Rayleigh distribution under balanced loss functions for censored data

This paper develops Bayesian analysis in the context of progressively Type II censored data from the compound Rayleigh distribution. The maximum likelihood and Bayes estimates along with associated posterior risks are derived for reliability performances under balanced loss functions by assuming continuous priors for parameters of the distribution. A practical example is used to illustrate the estimation methods. A simulation study has been carried out to compare the performance of estimates. The study indicates that Bayesian estimation should be preferred over maximum likelihood estimation. In Bayesian estimation, the balance general entropy loss function can be effectively employed for optimal decision-making.     

Some Recent Developments in Econometric Inference

Abstract

Recent results in information theory, see Soofi (1996; 2001) for a review, include derivations of optimal information processing rules, including Bayes' theorem, for learning from data based on minimizing a criterion functional, namely output information minus input information as shown in Zellner (1988; 1991; 1997; 2002). Herein, solution post data densities for parameters are obtained and studied for cases in which the input information is that in (1) a likelihood function and a prior density; (2) only a likelihood function; and (3) neither a prior nor a likelihood function but only input information in the form of post data moments of parameters, as in the Bayesian method of moments approach. Then it is shown how optimal output densities can be employed to obtain predictive densities and optimal, finite sample structural coefficient estimates using three alternative loss functions. Such optimal estimates are compared with usual estimates, e.g., maximum likelihood, two‐stage least squares, ordinary least squares, etc. Some Monte Carlo experimental results in the literature are discussed and implications for the future are provided.     

A Novel Bayesian Parameter Mapping Method for Estimating the Parameters of an Underlying Scientific Model

Population-parameter mapping (PPM) is a method for estimating the parameters of latent scientific models that describe the statistical likelihood function. The PPM method involves a Bayesian inference in terms of the statistical parameters and the mapping from the statistical parameter space to the parameter space of the latent scientific parameters, and obtains a model coherence estimate, P(coh). The P(coh) statistic can be valuable for designing experiments, comparing competing models, and can be helpful in redesigning flawed models. Examples are provided where greater estimation precision was found for small sample sizes for the PPM point estimates relative to the maximum likelihood estimator (MLE).     

Full likelihood inference in the Cox model with LTRC data when covariates are discrete

For the regression parameter β in the Cox model, there have been several estimates based on different types of approximated likelihood. For right-censored data, Ren and Zhou [Full likelihood inferences in the Cox model: an empirical approach. Ann Inst Statist Math. 2011;63:1005–1018] derive the full likelihood function for (β, F₀), where F₀ is the baseline distribution function in the Cox model. In this article, we extend their results to left-truncated and right-censored data with discrete covariates. Using the empirical likelihood parameterization, we obtain the full-profile likelihood function for β when covariates are discrete. Simulation results indicate that the maximum likelihood estimator outperforms Cox's partial likelihood estimator in finite samples.     

Estimation under modified Weibull distribution based on right censored generalized order statistics

In this paper, the maximum likelihood (ML) and Bayes, by using Markov chain Monte Carlo (MCMC), methods are considered to estimate the parameters of three-parameter modified Weibull distribution (MWD(β, τ, λ)) based on a right censored sample of generalized order statistics (gos). Simulation experiments are conducted to demonstrate the efficiency of the proposed methods. Some comparisons are carried out between the ML and Bayes methods by computing the mean squared errors (MSEs), Akaike's information criteria (AIC) and Bayesian information criteria (BIC) of the estimates to illustrate the paper. Three real data sets from Weibull(α, β) distribution are introduced and analyzed using the MWD(β, τ, λ) and also using the Weibull(α, β) distribution. A comparison is carried out between the mentioned models based on the corresponding Kolmogorov–Smirnov (K–S) test statistic, {AIC and BIC} to emphasize that the MWD(β, τ, λ) fits the data better than the other distribution. All parameters are estimated based on type-II censored sample, censored upper record values and progressively type-II censored sample which are generated from the real data sets.     

Power-linear models for incomplete contingency tables with nonignorable non-responses

For categorical data exhibiting nonignorable non-responses, it is well known that maximum likelihood (ML) estimates with a boundary solution are implausible and do not provide a perfect fit to the observed data even for saturated models. We provide the conditions under which ML estimates for the generalized linear model (GLM) with the usual log/logit link function have a boundary solution. These conditions introduce a new GLM with appropriately defined power link functions where its ML estimates resolve the problems arising from a boundary solution and offer useful statistics for identifying the non-response mechanism. This model is applied to a real dataset and compared with Bayesian models.     

Inference on progressive-stress model for the exponentiated exponential distribution under type-II progressive hybrid censoring

In this paper, progressive-stress accelerated life tests are applied when the lifetime of a product under design stress follows the exponentiated distribution [G(x)]^α. The baseline distribution, G(x), follows a general class of distributions which includes, among others, Weibull, compound Weibull, power function, Pareto, Gompertz, compound Gompertz, normal and logistic distributions. The scale parameter of G(x) satisfies the inverse power law and the cumulative exposure model holds for the effect of changing stress. A special case for an exponentiated exponential distribution has been discussed. Using type-II progressive hybrid censoring and MCMC algorithm, Bayes estimates of the unknown parameters based on symmetric and asymmetric loss functions are obtained and compared with the maximum likelihood estimates. Normal approximation and bootstrap confidence intervals for the unknown parameters are obtained and compared via a simulation study.     

Bias-reduced estimates for skewness, kurtosis, L-skewness and L-kurtosis

Estimates based on L-moments are less non-robust than estimates based on ordinary moments because the former are linear combinations of order statistics for all orders, whereas the later take increasing powers of deviations from the mean as the order increases. Estimates based on L-moments can also be more efficient than maximum likelihood estimates. Similarly, L-skewness and L-kurtosis are less non-robust and more informative than the traditional measures of skewness and kurtosis. Here, we give nonparametric bias-reduced estimates of both types of skewness and kurtosis. Their asymptotic computational efficiency is infinitely better than that of corresponding bootstrapped estimates.     

Estimation for the Parameters of Generalized Half-normal Distribution Based on Progressive Type-I Interval Censoring

Based on progressively Type-I interval censored sample, the problem of estimating unknown parameters of a two parameter generalized half-normal(GHN) distribution is considered. Different methods of estimation are discussed. They include the maximum likelihood estimation, midpoint approximation method, approximate maximum likelihood estimation, method of moments, and estimation based on probability plot. Several Bayesian estimates with respect to different symmetric and asymmetric loss functions such as squared error, LINEX, and general entropy is calculated. The Lindley’s approximation method is applied to determine Bayesian estimates. Monte Carlo simulations are performed to compare the performances of the different methods. Finally, analysis is also carried out for a real dataset.     

Bayesian Estimation of the Size of a Closed Population Using Photo-ID Data with Part of the Population Uncatchable

Abstract

We develop a Bayesian statistical model for estimating bowhead whale population size from photo-identification data when most of the population is uncatchable. The proposed conditional likelihood function is a product of Darroch's model, formulated as a function of the number of good photos, and a binomial distribution of captured whales given the total number of good photos at each occasion. The full Bayesian model is implemented via adaptive rejection sampling for log concave densities. We apply the model to data from 1985 and 1986 bowhead whale photographic studies and the results compare favorably with the ones obtained in the literature. Also, a comparison with the maximum likelihood procedure with bootstrap simulation is considered using different vague priors for the capture probabilities.     

Unimodality of the pareto distribution likelihood function for multicensored samples and implications for estimation

This paper discusses maximum likelihood parameter estimation in the Pareto distribution for multicensored samples. In particu-

lar, the modality of the associated conditional log-likelihood function is investigated in order to resolve questions concerninc

the existence and uniqurneas of the lnarimum likelihood estimates.For the cases with one parameter known, the maximum likelihood

estimates of the remaining unknown parameters are shown to exist and to be unique. When both parameters are unknown, the maximum likelihood estimates may or may not exist and be unique. That is, their existence and uniqueness would seem to depend solely upon the information inherent in the sample data. In viav of the possible nonexistence and/or non-uniqueness of the maximum likelihood estimates when both parameters are unknown, alternatives to standard iterative numerical methods are explored.