Interval estimation for the generalized inverted exponential distribution under progressive first failure censoring

In this paper, we consider the inferential procedures for the generalized inverted exponential distribution under progressive first failure censoring. The exact confidence interval for the scale parameter is derived. The generalized confidence intervals (GCIs) for the shape parameter and some commonly used reliability metrics such as the quantile and the reliability function are explored. Then the proposed procedure is extended to the prediction interval for the future measurement. The GCIs for the reliability of the stress-strength model are discussed under both equal scale and unequal scale scenarios. Extensive simulations are used to demonstrate the performance of the proposed GCIs and prediction interval. Finally, an example is used to illustrate the proposed methods.     

Comparison of Interval Estimators of Pr(X < Y) in the Two-parameter Exponential Distribution

We consider confidence intervals for the stress–strength reliability Pr(X< Y) in the two-parameter exponential distribution. We have derived the Bayesian highest posterior density interval using non-informative prior distributions. We have compared its performance with the intervals based on the generalized pivot variable intervals in terms of their coverage probabilities and expected lengths. Our simulation study shows that the Bayesian interval performs better according to the criteria used, especially when the sample sizes are very small. An example is given.     

Statistical inference for two exponential populations under joint progressive type-I censored scheme

In this article, we introduce a new scheme called joint progressive type-I (JPC-I) censored and as a special case, joint type-I censored scheme. Bayesian and non Bayesian estimators have been obtained for two exponential populations under both JPC-I censored scheme and joint type-I censored. The maximum likelihood estimators of the parameters, the asymptotic variance covariance matrix, have been obtained. Bayes estimators have been developed under squared error loss function using independent gamma prior distributions. Moreover, approximate confidence region based on the asymptotic normality of the maximum likelihood estimators and credible confidence region from a Bayesian viewpoint are also discussed and compared with two Bootstrap confidence regions. A numerical illustration for these new results is given.     

Bayesian prediction of future number of failures based on finite mixture of general class of distributions

This paper is concerned with the Bayesian prediction problem of the number of components which will fail in a future time interval. The failure times are distributed according to a finite mixture of a general class of distributions. Type-I censored sample from this nonhomogeneous population and a general class of prior density functions are used. A one-sample scheme is used to predict the number of failures in a future time interval. An example of a finite mixture of k exponential components is given to illustrate our results.     

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.     

Curtailed Bayesian sampling plans for exponential distributions based on Type-II censored samples

This paper studies the problem of designing a curtailed Bayesian sampling plan (CBSP) with Type-II censored data. We first derive the Bayesian sampling plan (BSP) for exponential distributions based on Type-II censored samples in a general loss function. For the conjugate prior with quadratic loss function, an explicit expression for the Bayes decision function is derived. Using the property of monotonicity of the Bayes decision function, a new Bayesian sampling plan modified by the curtailment procedure, called a CBSP, is proposed. It is shown that the risk of CBSP is less than or equal to that of BSP. Comparisons among some existing BSPs and the proposed CBSP are given. Monte Carlo simulations are conducted, and numerical results indicate that the CBSP outperforms those early existing sampling plans if the time loss is considered in the loss function.     

Bayesian confidence intervals for means and variances of lognormal and bivariate lognormal distributions

The lognormal distribution is currently used extensively to describe the distribution of positive random variables. This is especially the case with data pertaining to occupational health and other biological data. One particular application of the data is statistical inference with regards to the mean of the data. Other authors, namely Zou et al. (2009), have proposed procedures involving the so-called “method of variance estimates recovery” (MOVER), while an alternative approach based on simulation is the so-called generalized confidence interval, discussed by Krishnamoorthy and Mathew (2003). In this paper we compare the performance of the MOVER-based confidence interval estimates and the generalized confidence interval procedure to coverage of credibility intervals obtained using Bayesian methodology using a variety of different prior distributions to estimate the appropriateness of each. An extensive simulation study is conducted to evaluate the coverage accuracy and interval width of the proposed methods. For the Bayesian approach both the equal-tail and highest posterior density (HPD) credibility intervals are presented. Various prior distributions (Independence Jeffreys' prior, Jeffreys'-Rule prior, namely, the square root of the determinant of the Fisher Information matrix, reference and probability-matching priors) are evaluated and compared to determine which give the best coverage with the most efficient interval width. The simulation studies show that the constructed Bayesian confidence intervals have satisfying coverage probabilities and in some cases outperform the MOVER and generalized confidence interval results. The Bayesian inference procedures (hypothesis tests and confidence intervals) are also extended to the difference between two lognormal means as well as to the case of zero-valued observations and confidence intervals for the lognormal variance. In the last section of this paper the bivariate lognormal distribution is discussed and Bayesian confidence intervals are obtained for the difference between two correlated lognormal means as well as for the ratio of lognormal variances, using nine different priors.     

Bayesian confidence interval for difference of the proportions in a 2×2 table with structural zero

This article studies the construction of a Bayesian confidence interval for risk difference in a 2×2 table with structural zero. The exact posterior distribution of the risk difference is derived under the Dirichlet prior distribution, and a tail-based interval is used to construct the Bayesian confidence interval. The frequentist performance of the tail-based interval is investigated and compared with the score-based interval by simulation. Our results show that the tail-based interval at Jeffreys prior performs as well as or better than the score-based confidence interval.     

Generalized fiducial inference for generalized exponential distribution

This article mainly considers interval estimation of the scale and shape parameters of the generalized exponential (GE) distribution. We adopt the generalized fiducial method to construct a kind of new confidence intervals for the parameters of interest and compare them with the frequentist and Bayesian methods. In addition, we give the comparison of the point estimation based on the frequentist, generalized fiducial and Bayesian methods. Simulation results show that a new procedure based on generalized fiducial inference is more applicable than the non-fiducial methods for the point and interval estimation of the GE distribution. Finally, two lifetime data sets are used to illustrate the application of our new procedure.     

Bayesian confidence interval for the risk ratio in a correlated 2 × 2 table with structural zero

This paper studies the construction of a Bayesian confidence interval for the risk ratio (RR) in a 2 × 2 table with structural zero. Under a Dirichlet prior distribution, the exact posterior distribution of the RR is derived, and tail-based interval is suggested for constructing Bayesian confidence interval. The frequentist performance of this confidence interval is investigated by simulation and compared with the score-based interval in terms of the mean coverage probability and mean expected width of the interval. An advantage of the Bayesian confidence interval is that it is well defined for all data structure and has shorter expected width. Our simulation shows that the Bayesian tail-based interval under Jeffreys’ prior performs as well as or better than the score-based confidence interval.     

Bayes estimation in exponential censored samples with incomplete information

A Bayesian procedure is proposed to estimate the exponential mean lifetime and the reliability function in a time censored sampling with incomplete information. On the basis of a Monte Carlo study, the Bayes point and interval estimators are compared to the maximum likelihood ones, taking into account several factors, such as prior information, sample size, and censoring time. It is found that only a vague (from an engineering viewpoint) prior knowledge on the mean lifetime is required to make attractive the Bayesian procedure.     

SHRINKAGE PREDICTION IN THE EXPONENTIAL DISTRIBUTION WITH A PRIOR INTERVAL FOR THE SCALE PARAMETER

The author proposes the best shrinkage predictor of a preassigned dominance level for a future order statistic of an exponential distribution, assuming a prior estimate of the scale parameter is distributed over an interval according to an arbitrary distribution with known mean. Based on a Type II censored sample from this distribution, we predict the future order statistic in another independent sample from the same distribution. The predictor is constructed by incorporating a preliminary confidence interval for the scale parameter and a class of shrinkage predictors constructed here. It improves considerably classical predictors for all values of the scale parameter within its dominance interval containing the confidence interval of a preassigned level.     

The inverted Nadarajah–Haghighi distribution: estimation methods and applications

The 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.     

Bayesian Confidence Interval for the Ratio of Marginal Probabilities in the Matched-Pair Design

This article studies the construction of a Bayesian confidence interval for the ratio of marginal probabilities in matched-pair designs. Under a Dirichlet prior distribution, the exact posterior distribution of the ratio is derived. The tail confidence interval and the highest posterior density (HPD) interval are studied, and their frequentist performances are investigated by simulation in terms of mean coverage probability and mean expected length of the interval. An advantage of Bayesian confidence interval is that it is always well defined for any data structure and has shorter mean expected width. We also find that the Bayesian tail interval at Jeffreys prior performs as well as or better than the frequentist confidence intervals.     

Approximate Bayesianity of Frequentist Confidence Intervals for a Binomial Proportion

The well-known Wilson and Agresti–Coull confidence intervals for a binomial proportion p are centered around a Bayesian estimator. Using this as a starting point, similarities between frequentist confidence intervals for proportions and Bayesian credible intervals based on low-informative priors are studied using asymptotic expansions. A Bayesian motivation for a large class of frequentist confidence intervals is provided. It is shown that the likelihood ratio interval for p approximates a Bayesian credible interval based on Kerman’s neutral noninformative conjugate prior up to O(n^{? 1}) in the confidence bounds. For the significance level α ? 0.317, the Bayesian interval based on the Jeffreys’ prior is then shown to be a compromise between the likelihood ratio and Wilson intervals. Supplementary materials for this article are available online.     

Confidence Intervals for Quantiles of a Two-parameter Exponential Distribution under Progressive Type-II Censoring

Confidence intervals for the pth-quantile Q of a two-parameter exponential distribution provide useful information on the plausible range of Q, and only inefficient equal-tail confidence intervals have been discussed in the statistical literature so far. In this article, the construction of the shortest possible confidence interval within a family of two-sided confidence intervals is addressed. This shortest confidence interval is always shorter, and can be substantially shorter, than the corresponding equal-tail confidence interval. Furthermore, the computational intensity of both methodologies is similar, and therefore it is advantageous to use the shortest confidence interval. It is shown how the results provided in this paper can apply to data obtained from progressive Type II censoring, with standard Type II censoring as a special case. The applications of more complex confidence interval constructions through acceptance set inversions that can employ prior information are also discussed.     

Multiple comparisons of several populations with more than one control with respectto scale parameter

In this paper, one-sided and two-sided test procedures for comparing several treatments with more than one control with respect to scale parameter are proposed. The proposed test procedures are inverted to obtain the associated simultaneous confidence intervals. The multiple comparisons of test treatments with the best control are also developed. The computation of the critical points, required to implement the proposed procedures, is discussed by taking the normal probability model. Applications of the proposed test procedures to two-parameter exponential probability model are also demonstrated.     

Bayesian inference for quantum state tomography

We present a Bayesian approach to the problem of estimating density matrices in quantum state tomography. A general framework is presented based on a suitable mathematical formulation, where a study of the convergence of the Monte Carlo Markov Chain algorithm is given, including a comparison with other estimation methods, such as maximum likelihood estimation and linear inversion. This analysis indicates that our approach not only recovers the underlying parameters quite properly, but also produces physically acceptable punctual and interval estimates. A prior sensitive study was conducted indicating that when useful prior information is available and incorporated, more accurate results are obtained. This general framework, which is based on a reparameterization of the model, allows an easier choice of the prior and proposal distributions for the Metropolis–Hastings algorithm.     

Confidence and Likelihood*

Confidence intervals for a single parameter are spanned by quantiles of a confidence distribution, and one‐sided p‐values are cumulative confidences. Confidence distributions are thus a unifying format for representing frequentist inference for a single parameter. The confidence distribution, which depends on data, is exact (unbiased) when its cumulative distribution function evaluated at the true parameter is uniformly distributed over the unit interval. A new version of the Neyman–Pearson lemma is given, showing that the confidence distribution based on the natural statistic in exponential models with continuous data is less dispersed than all other confidence distributions, regardless of how dispersion is measured. Approximations are necessary for discrete data, and also in many models with nuisance parameters. Approximate pivots might then be useful. A pivot based on a scalar statistic determines a likelihood in the parameter of interest along with a confidence distribution. This proper likelihood is reduced of all nuisance parameters, and is appropriate for meta‐analysis and updating of information. The reduced likelihood is generally different from the confidence density. Confidence distributions and reduced likelihoods are rooted in Fisher–Neyman statistics. This frequentist methodology has many of the Bayesian attractions, and the two approaches are briefly compared. Concepts, methods and techniques of this brand of Fisher–Neyman statistics are presented. Asymptotics and bootstrapping are used to find pivots and their distributions, and hence reduced likelihoods and confidence distributions. A simple form of inverting bootstrap distributions to approximate pivots of the abc type is proposed. Our material is illustrated in a number of examples and in an application to multiple capture data for bowhead whales.     

Efficient Bayesian sampling plans for exponential distributions with random censoring

This paper considers Bayesian sampling plans for exponential distribution with random censoring. The efficient Bayesian sampling plan for a general loss function is derived. This sampling plan possesses the property that it may make decisions prior to the end of the life test experiment, and its decision function is the same as the Bayes decision function which makes decisions based on data collected at the end of the life test experiment. Compared with the optimal Bayesian sampling plan of Chen et al. (2004), the efficient Bayesian sampling plan has the smaller Bayes risk due to the less duration time of life test experiment. Computations of the efficient Bayes risks for the conjugate prior are given. Numerical comparisons between the proposed efficient Bayesian sampling plan and the optimal Bayesian sampling plan of Chen et al. (2004) under two special decision losses, including the quadratic decision loss, are provided. Numerical results also demonstrate that the performance of the proposed efficient sampling plan is superior to that of the optimal sampling plan by Chen et al. (2004).