Parameter and reliability estimation for an exponentiated half-logistic distribution under progressive type II censoring

In this article, we deal with a two-parameter exponentiated half-logistic distribution. We consider the estimation of unknown parameters, the associated reliability function and the hazard rate function under progressive Type II censoring. Maximum likelihood estimates (M LEs) are proposed for unknown quantities. Bayes estimates are derived with respect to squared error, linex and entropy loss functions. Approximate explicit expressions for all Bayes estimates are obtained using the Lindley method. We also use importance sampling scheme to compute the Bayes estimates. Markov Chain Monte Carlo samples are further used to produce credible intervals for the unknown parameters. Asymptotic confidence intervals are constructed using the normality property of the MLEs. For comparison purposes, bootstrap-p and bootstrap-t confidence intervals are also constructed. A comprehensive numerical study is performed to compare the proposed estimates. Finally, a real-life data set is analysed to illustrate the proposed methods of estimation.     

Estimation and prediction for the generalized inverted exponential distribution based on progressively first-failure-censored data with application

In this paper, the estimation of parameters for a generalized inverted exponential distribution based on the progressively first-failure type-II right-censored sample is studied. An expectation–maximization (EM) algorithm is developed to obtain maximum likelihood estimates of unknown parameters as well as reliability and hazard functions. Using the missing value principle, the Fisher information matrix has been obtained for constructing asymptotic confidence intervals. An exact interval and an exact confidence region for the parameters are also constructed. Bayesian procedures based on Markov Chain Monte Carlo methods have been developed to approximate the posterior distribution of the parameters of interest and in addition to deduce the corresponding credible intervals. The performances of the maximum likelihood and Bayes estimators are compared in terms of their mean-squared errors through the simulation study. Furthermore, Bayes two-sample point and interval predictors are obtained when the future sample is ordinary order statistics. The squared error, linear-exponential and general entropy loss functions have been considered for obtaining the Bayes estimators and predictors. To illustrate the discussed procedures, a set of real data is analyzed.     

Spatially-adaptive Penalties for Spline Fitting

The paper studies spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. The estimates are p th degree piecewise polynomials with p − 1 continuous derivatives. A large and fixed number of knots is used and smoothing is achieved by putting a quadratic penalty on the jumps of the p th derivative at the knots. To be spatially adaptive, the logarithm of the penalty is itself a linear spline but with relatively few knots and with values at the knots chosen to minimize the generalized cross validation (GCV) criterion. This locally-adaptive spline estimator is compared with other spline estimators in the literature such as cubic smoothing splines and knot-selection techniques for least squares regression. Our estimator can be interpreted as an empirical Bayes estimate for a prior allowing spatial heterogeneity. In cases of spatially heterogeneous regression functions, empirical Bayes confidence intervals using this prior achieve better pointwise coverage probabilities than confidence intervals based on a global-penalty parameter. The method is developed first for univariate models and then extended to additive models.     

Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring

In this paper, the estimation of parameters, reliability and hazard functions of a inverted exponentiated half logistic distribution (IEHLD) from progressive Type II censored data has been considered. The Bayes estimates for progressive Type II censored IEHLD under asymmetric and symmetric loss functions such as squared error, general entropy and linex loss function are provided. The Bayes estimates for progressive Type II censored IEHLD parameters, reliability and hazard functions are also obtained under the balanced loss functions. However, the Bayes estimates cannot be obtained explicitly, Lindley approximation method and importance sampling procedure are considered to obtain the Bayes estimates. Furthermore, the asymptotic normality of the maximum likelihood estimates is used to obtain the approximate confidence intervals. The highest posterior density credible intervals of the parameters based on importance sampling procedure are computed. Simulations are performed to see the performance of the proposed estimates. For illustrative purposes, two data sets have been analyzed.     

Estimation of parameters of Kumaraswamy-Exponential distribution under progressive type-II censoring

In this paper, the problem of estimating unknown parameters of a two-parameter Kumaraswamy-Exponential (Kw-E) distribution is considered based on progressively type-II censored sample. The maximum likelihood (ML) estimators of the parameters are obtained. Bayes estimates are also obtained using different loss functions such as squared error, LINEX and general entropy. Lindley's approximation method is used to evaluate these Bayes estimates. Monte Carlo simulation is used for numerical comparison between various estimates developed in this paper.     

On the Bayesian Estimation for the Uniform Scale Parameter via a Functional Equation

Bayes uniform model under the squared error loss function is shown to be completely identifiable by the form of the Bayes estimates of the scale parameter. This results in solving a specific functional equation. A complete characterization of differentiable Bayes estimators (BE) and generalized Bayes estimators (GBE) is given as well as relations between degrees of smoothness of the estimators and the priors. Characterizations of strong (generalized Bayes) Bayes sequence (SBS or SGBS) are also investigated. A SBS is a sequence of estimators (one for each sample size) where all its components are BE generated by the same prior measure. A complete solution is given for polynomial Bayesian estimation.     

Bayesian estimation and prediction based on lognormal record values

In this paper we consider the problems of estimation and prediction when observed data from a lognormal distribution are based on lower record values and lower record values with inter-record times. We compute maximum likelihood estimates and asymptotic confidence intervals for model parameters. We also obtain Bayes estimates and the highest posterior density (HPD) intervals using noninformative and informative priors under square error and LINEX loss functions. Furthermore, for the problem of Bayesian prediction under one-sample and two-sample framework, we obtain predictive estimates and the associated predictive equal-tail and HPD intervals. Finally for illustration purpose a real data set is analyzed and simulation study is conducted to compare the methods of estimation and prediction.     

Nonparametric estimation of varying-coefficient single-index models

The varying-coefficient single-index model has two distinguishing features: partially linear varying-coefficient functions and a single-index structure. This paper proposes a nonparametric method based on smoothing splines for estimating varying-coefficient functions and an unknown link function. Moreover, the average derivative estimation method is applied to obtain the single-index parameter estimates. For interval inference, Bayesian confidence intervals were obtained based on Bayes models for varying-coefficient functions and the link function. The performance of the proposed method is examined both through simulations and by applying it to Boston housing data.     

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.     

Estimation and prediction of the Burr type XII distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the 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. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.     

Estimation under inverse Weibull distribution based on Balakrishnan’s unified hybrid censored scheme

In this article, point and interval estimations of the parameters α and β of the inverse Weibull distribution (IWD) have been studied based on Balakrishnan’s unified hybrid censoring scheme (UHCS), see Balakrishnan et al. In point estimation, the maximum likelihood (ML) and Bayes (B) methods have been used. The Bayes estimates have been computed based on squared error loss (SEL) function and Linex loss function and using Markov Chain Monte Carlo (MCMC) algorithm. In interval estimation, a (1 ? τ) × 100% approximate, bootstrap-p, credible and highest posterior density (HPD) confidence intervals (CI′s) for the parameters α and β have been introduced. Based on Monte Carlo simulation, Bayes estimates have been compared with their corresponding maximum likelihood estimates by computing the mean squared errors (MSE′s) of all estimators. Finally, point and interval estimations of all parameters have been studied based on a real data set as an illustrative example.     

On the Bayesian analysis of the mixture of power function distribution using the complete and the censored sample

The power function distribution is often used to study the electrical component reliability. In this paper, we model a heterogeneous population using the two-component mixture of the power function distribution. A comprehensive simulation scheme including a large number of parameter points is followed to highlight the properties and behavior of the estimates in terms of sample size, censoring rate, parameters size and the proportion of the components of the mixture. The parameters of the power function mixture are estimated and compared using the Bayes estimates. A simulated mixture data with censored observations is generated by probabilistic mixing for the computational purposes. Elegant closed form expressions for the Bayes estimators and their variances are derived for the censored sample as well as for the complete sample. Some interesting comparison and properties of the estimates are observed and presented. The system of three non-linear equations, required to be solved iteratively for the computations of maximum likelihood (ML) estimates, is derived. The complete sample expressions for the ML estimates and for their variances are also given. The components of the information matrix are constructed as well. Uninformative as well as informative priors are assumed for the derivation of the Bayes estimators. A real-life mixture data example has also been discussed. The posterior predictive distribution with the informative Gamma prior is derived, and the equations required to find the lower and upper limits of the predictive intervals are constructed. The Bayes estimates are evaluated under the squared error loss function.     

On the Bayesian inference of Kumaraswamy distributions based on censored samples

In this article we discuss Bayesian estimation of Kumaraswamy distributions based on three different types of censored samples. We obtain Bayes estimates of the model parameters using two different types of loss functions (LINEX and Quadratic) under each censoring scheme (left censoring, singly type-II censoring, and doubly type-II censoring) using Monte Carlo simulation study with posterior risk plots for each different choices of the model parameters. Also, detailed discussion regarding elicitation of the hyperparameters under the dependent prior setup is discussed. If one of the shape parameters is known then closed form expressions of the Bayes estimates corresponding to posterior risk under both the loss functions are available. To provide the efficacy of the proposed method, a simulation study is conducted and the performance of the estimation is quite interesting. For illustrative purpose, real-life data are considered.     

Estimation of P(Y < X) for progressively first-failure-censored generalized inverted exponential distribution

In this article, we consider the problem of estimation of the stress–strength parameter δ?=?P(Y?<?X) based on progressively first-failure-censored samples, when X and Y both follow two-parameter generalized inverted exponential distribution with different and unknown shape and scale parameters. The maximum likelihood estimator of δ and its asymptotic confidence interval based on observed Fisher information are constructed. Two parametric bootstrap boot-p and boot-t confidence intervals are proposed. We also apply Markov Chain Monte Carlo techniques to carry out Bayes estimation procedures. Bayes estimate under squared error loss function and the HPD credible interval of δ are obtained using informative and non-informative priors. A Monte Carlo simulation study is carried out for comparing the proposed methods of estimation. Finally, the methods developed are illustrated with a couple of real data examples.     

Inferences on stress–strength parameter based on GLD5 distribution

Let X and Y be independent random variables distributed as generalized Lindley distribution type 5 (GLD5). This article deals with the estimation of the stress–strength parameter R = P(Y < X), which plays an important role in reliability analysis. For this purpose, the maximum likelihood and the uniformly minimum variance unbiased estimators are presented in the explicit form. Moreover, considering Arnold and Strauss’ bivariate Gamma distribution as an informative prior and Jeffreys’ as noninformative prior, the Bayes estimators are derived. Various bootstrap confidence intervals are also proposed and, finally, the presented methods are compared using a simulation study.     

Inferential statistics on the dynamic system model with time-dependent failure-rate

This study focuses on the classical and Bayesian analysis of a k-components load-sharing parallel system in which components have time-dependent failure rates. In the classical set up, the maximum likelihood estimates of the load-share parameters with their standard errors (SEs) are obtained. (1?γ) 100% simultaneous and two bootstrap confidence intervals for the parameters and system reliability and hazard functions have been constructed. Further, on recognizing the fact that life-testing experiments are very time consuming, the parameters involved in the failure time distribution of the system are expected to follow some random variations. Therefore, Bayes estimates along with their posterior SEs of the parameters and system reliability and hazard functions are obtained by assuming gamma and Jeffrey's priors of the unknown parameters. Markov chain Monte Carlo technique such as Gibbs sampler has been used to obtain Bayes estimates and highest posterior density credible intervals.     

Inference for Weibull distribution under adaptive Type-I progressive hybrid censored competing risks data

In this paper, a competing risks model is considered under adaptive type-I progressive hybrid censoring scheme (AT-I PHCS). The lifetimes of the latent failure times have Weibull distributions with the same shape parameter. We investigate the maximum likelihood estimation of the parameters. Bayes estimates of the parameters are obtained based on squared error and LINEX loss functions under the assumption of independent gamma priors. We propose to apply Markov Chain Monte Carlo (MCMC) techniques to carry out a Bayesian estimation procedure and in turn calculate the credible intervals. To evaluate the performance of the estimators, a simulation study is carried out.     

Bayesian inference of the inverse Weibull mixture distribution using type-I censoring

A large number of models have been derived from the two-parameter Weibull distribution including the inverse Weibull (IW) model which is found suitable for modeling the complex failure data set. In this paper, we present the Bayesian inference for the mixture of two IW models. For this purpose, the Bayes estimates of the parameters of the mixture model along with their posterior risks using informative as well as the non-informative prior are obtained. These estimates have been attained considering two cases: (a) when the shape parameter is known and (b) when all parameters are unknown. For the former case, Bayes estimates are obtained under three loss functions while for the latter case only the squared error loss function is used. Simulation study is carried out in order to explore numerical aspects of the proposed Bayes estimators. A real-life data set is also presented for both cases, and parameters obtained under case when shape parameter is known are tested through testing of hypothesis procedure.     

Profile likelihood-based confidence interval for the dispersion parameter in count data

The importance of the dispersion parameter in counts occurring in toxicology, biology, clinical medicine, epidemiology, and other similar studies is well known. A couple of procedures for the construction of confidence intervals (CIs) of the dispersion parameter have been investigated, but little attention has been paid to the accuracy of its CIs. In this paper, we introduce the profile likelihood (PL) approach and the hybrid profile variance (HPV) approach for constructing the CIs of the dispersion parameter for counts based on the negative binomial model. The non-parametric bootstrap (NPB) approach based on the maximum likelihood (ML) estimates of the dispersion parameter is also considered. We then compare our proposed approaches with an asymptotic approach based on the ML and the restricted ML (REML) estimates of the dispersion parameter as well as the parametric bootstrap (PB) approach based on the ML estimates of the dispersion parameter. As assessed by Monte Carlo simulations, the PL approach has the best small-sample performance, followed by the REML, HPV, NPB, and PB approaches. Three examples to biological count data are presented.     

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.