Exact inference and prediction for k-sample exponential case under type-ii censoring

The exact inference and prediction intervals for the K-sample exponential scale parameter under doubly Type-II censored samples are derived using an algorithm of Huffer and Lin [Huffer, F.W. and Lin, C.T., 2001, Computing the joint distribution of general linear combinations of spacings or exponen-tial variates. Statistica Sinica, 11, 1141–1157.]. This approach provides a simple way to determine the exact percentage points of the pivotal quantity based on the best linear unbiased estimator in order to develop exact inference for the scale parameter as well as to construct exact prediction intervals for failure times unobserved in the ith sample. Similarly, exact prediction intervals for failure times of units from a future sample can also be easily obtained.     

Bayesian prediction of the total time on test using doubly censored rayleigh data

In a Bayesian setting, and on the basis of a doubly censored random sample of failure times drawn from a Rayleigh distribution, Fernandez (2000, Statist. Probab. Lett. , 48 , 393-399) considered the problem of predicting an independent future sample from the same distribution. In this article, we extend his work to include the estimation of the predictive distribution of the total time on test up to a certain failure in a future sample, as well as that of the remaining testing time time until all the items in the original sample have failed. Two examples are used to illustrate the prediction procedure.     

Outer and inner prediction intervals for order statistics intervals based on current records

This paper considers the largest and smallest observations at the times when a new record of either kind (upper or lower) occurs. These are called the upper and lower current records and are denoted by ${R^l_m}$ and ${R^s_m}$ , respectively. The interval ${(R^s_m,R^l_m)}$ is then referred to as the record coverage. The prediction problem in the two-sample case is then discussed and, specifically, the exact outer and inner prediction intervals are derived for order statistics intervals from an independent future Y-sample based on the m-th record coverage from the X-sequence when the underlying distribution of the two samples are the same. The coverage probabilities of these intervals are exact and do not depend on the underlying distribution. Distribution-free prediction intervals as well as upper and lower prediction limits for spacings from a future Y-sample are obtained in terms of the record range from the X-sequence.     

ON THE PREDICTION OF FUTURE FAILURES FOR A REPAIRABLE EQUIPMENT SUBJECT TO OVERHAULS

This paper deals with the prediction, from a Bayes viewpoint, of future failures for a repairable equipment subjected both to minimal repairs and periodic overhauls. The effect of major overhauls on the reliability of the equipment is modeled by a proportional age reduction model, while the failure process between two successive overhaul epochs is modeled by the power law process. Prediction both of the future failure times and of the number of failures in a future time interval are provided on the basis of the observed data and of a number of suitable prior densities, which reflect different degrees of belief on the failure mechanism and overhaul effectiveness. Finally, a numerical application illustrates the proposed prediction procedures and their use in assessing the adequacy of the model to describe the observed data set.     

Bayes inference for the modulated power law process

The Modulated Power Law process has been recently proposed as a suitable model for describing the failure pattern of repairable systems when both renewal-type behaviour and time trend are present. Unfortunately, the maximum likelihood method provides neither accurate confidence intervals on the model parameters for small or moderate sample sizes nor predictive intervals on future observations.

This paper proposes a Bayes approach, based on both non-informative and vague prior, as an alternative to the classical method. Point and interval estimation of the parameters, as well as point and interval prediction of future failure times, are given. Monte Carlo simulation studies show that the Bayes estimation and prediction possess good statistical properties in a frequentist context and, thus, are a valid alternative to the maximum likelihood approach.

Numerical examples illustrate the estimation and prediction procedures.     

Bayesian prediction based on Pareto doubly censored data

Based on Doubly type II censored data, this paper present Bayesian prediction intervals for future ordered failure times of components whose failure times have the classical Pareto distribution. Two different sampling schemes have been considered. Conjugate priors for either the one or the two-parameter cases are outlined. Illustrative examples and a simulation study are included.     

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.     

Inference for the two-parameter half-logistic distribution using pivotal quantities under progressively Type-II censoring schemes

This article addresses estimation and prediction problems for the two-parameter half-logistic distribution based on pivotal quantities when a sample is available from the progressively Type-II censoring scheme. An unbiased estimator of the location parameter based on a pivotal quantity is derived. To estimate the scale parameter, a new method based on a pivotal quantity is proposed. The proposed method provides a simpler estimation equation than the maximum likelihood equation. In addition, confidence intervals for the location and scale parameters are derived from these pivotal quantities. In the prediction of censored failure times, the shortest-length predictive intervals for the censored failure times are derived using a pivotal quantity. Finally, the validity of the proposed method is assessed through Monte Carlo simulations and a real data set is presented for illustration purposes.     

NONPARAMETRIC META‐ANALYSIS FOR MINIMAL‐REPAIR SYSTEMS

We consider the situation that repair times of several identically structured technical systems are observed. As an example of such data we discuss the Boeing air conditioner data, consisting of successive failures of the air conditioning system of each member of a fleet of Boeing jet airplanes. The repairing process is assumed to be performed according to a minimal‐repair strategy. This reflects the idea that only those operations are accomplished that are absolutely necessary to restart the system after a failure. The ‘after‐repair‐state’ of the system is the same as it was shortly before the failure. Clearly, the observed repair times contain valuable information about the repair times of an identically structured system put into operation in the future. Thus, for statistical analysis and prediction, it is certainly favourable to take into account all repair times from each system. The resulting pooled sample is used to construct nonparametric prediction intervals for repair times of a future minimal‐repair system. To illustrate our results we apply them to the above‐mentioned data set. As expected, the maximum coverage probabilities of prediction intervals based on two samples exceed those based on one sample. We show that the relative gain for a two‐sample prediction over a one‐sample prediction can be substantial. One of the advantages of the present approach is that it allows nonparametric prediction intervals to be constructed directly. This provides a beneficial alternative to existing nonparametric methods for minimal‐repair systems that construct prediction intervals via the asymptotic distribution of quantile estimators. Moreover, the prediction intervals presented here are exact regardless of the sample size.     

Improved Prediction Intervals and Distribution Functions

Abstract.  The plug-in solution is usually not entirely adequate for computing prediction intervals, as their coverage probability may differ substantially from the nominal value. Prediction intervals with improved coverage probability can be defined by adjusting the plug-in ones, using rather complicated asymptotic procedures or suitable simulation techniques. Other approaches are based on the concept of predictive likelihood for a future random variable. The contribution of this paper is the definition of a relatively simple predictive distribution function giving improved prediction intervals. This distribution function is specified as a first-order unbiased modification of the plug-in predictive distribution function based on the constrained maximum likelihood estimator. Applications of the results to the Gaussian and the generalized extreme-value distributions are presented.     

A note about calibrated prediction regions and distributions

In this work we address the problem of the construction of prediction regions and distribution functions, with particular regard to the multidimensional setting. Firstly, we define a simple procedure for calculating the predictive distribution function which gives improved prediction limits. Secondly, with a multivariate generalization of a result presented in Ueki and Fueda (2007), we propose a method for correcting estimative prediction regions, to reduce their coverage error to the third-order accuracy. The improved prediction regions and the associated distribution functions are easy to calculate using a suitable bootstrap procedure. Examples of application are included, showing the good performance of the proposed method, even if we consider an approximated model for prediction purposes.     

Calibrating predictive distributions

This paper concerns prediction from the frequentist point of view. The aim is to define a well-calibrated predictive distribution giving prediction intervals, and in particular prediction limits, with coverage probability equal or close to the target nominal value. This predictive distribution can be considered in a number of situations, including discrete data and non-regular cases, and it is founded on the idea of calibrating prediction limits to control the associated coverage probability. Whenever the computation of the proposed distribution is not feasible, this can be approximated using a suitable bootstrap simulation procedure or by considering high-order asymptotic expansions, giving predictive distributions already known in the literature. Examples and applications of the results to different contexts show the wide applicability and the very good performance of the proposed predictive distribution.     

Bayesian estimation and prediction with multiply Type-II censored samples of sequential order statistics from one- and two-parameter exponential distributions

Based on multiply Type-II censored samples of sequential order statistics, Bayesian estimators are derived for the parameters of one- and two-parameter exponential distributions. In the one-parameter set-up, the posterior density is obtained under the assumption that the prior distribution is given by an inverse Gamma distribution, and the Bayes estimator with respect to squared error loss is calculated. Its performance is illustrated by a numerical example and compared with two non-Bayesian estimators, namely the BLUE and the approximate maximum likelihood estimator (AMLE). Moreover, prediction of future failure times is considered. Minimum risk equivariant estimators and predictors are deduced from the given results. Finally, similar results are presented for the two-parameter situation.     

Forecasting future values of changing sequences

Sequences of independent random variables are observed and on the basis of these observations future values of the process are forecast. The Bayesian predictive density of k future observations for normal, exponential, and binomial sequences which change exactly once are analyzed for several cases. It is seen that the Bayesian predictive densities are mixtures of standard probability distributions. For example, with normal sequences the Bayesian predictive density is a mixture of either normal or t-distributions, depending on whether or not the common variance is known. The mixing probabilities are the same as those occurring in the corresponding posterior distribution of the mean(s) of the sequence. The predictive mass function of the number of future successes that will occur in a changing Bernoulli sequence is computed and point and interval predictors are illustrated.     

Prediction in trend-renewal processes for repairable systems

Some problems of point and interval prediction in a trend-renewal process (TRP) are considered. TRP’s, whose realizations depend on a renewal distribution as well as on a trend function, comprise the non-homogeneous Poisson and renewal processes and serve as useful reliability models for repairable systems. For these processes, some possible ideas and methods for constructing the predicted next failure time and the prediction interval for the next failure time are presented. A method of constructing the predictors is also presented in the case when the renewal distribution of a TRP is unknown (and consequently, the likelihood function of this process is unknown). Using the prediction methods proposed, simulations are conducted to compare the predicted times and prediction intervals for a TRP with completely unknown renewal distribution with the corresponding results for the TRP with a Weibull renewal distribution and power law type trend function. The prediction methods are also applied to some real data.     

Bayesian inference for contingency tables with given marginals

Summary In this paper we introduce a class of prior distributions for contingency tables with given marginals. We are interested in the structrre of concordance/discordance of such tables. There is actually a minor limitation in that the marginals are required to assume only rational values. We do argue, though, that this is not a serious drawback for all applicatory purposes. The posterior and predictive distributions given anM-sample are computed. Examples of Bayesian estimates of some classical indices of concordance are also given. Moreover, we show how to use simulation in order to overcome some difficulties which arise in the computation of the posterior distribution.     

Bayesian inference and prediction of order statistics for a Type-II censored Weibull distribution

This paper describes the Bayesian inference and prediction of the two-parameter Weibull distribution when the data are Type-II censored data. The aim of this paper is twofold. First we consider the Bayesian inference of the unknown parameters under different loss functions. The Bayes estimates cannot be obtained in closed form. We use Gibbs sampling procedure to draw Markov Chain Monte Carlo (MCMC) samples and it has been used to compute the Bayes estimates and also to construct symmetric credible intervals. Further we consider the Bayes prediction of the future order statistics based on the observed sample. We consider the posterior predictive density of the future observations and also construct a predictive interval with a given coverage probability. Monte Carlo simulations are performed to compare different methods and one data analysis is performed for illustration purposes.     

Prediction of future failures times based on Type-I hybrid censored samples of random sample sizes

In this article, we consider the prediction of future failure times based on Type-I hybrid censored samples. Point predictors and prediction intervals using different procedures are discussed for a general model. The exponential and Rayleigh distributions are used as illustrative examples to show the most simplified forms of the so obtained predictors as well as prediction intervals. Intensive simulation study and a real life dataset are presented to illustrate our findings and results.     

Bayes prediction based on right censored data

This paper is concerned with a Bayes prediction problem in the exponential distribution under random censorship. Using censored samples, we work out a prediction interval for a sum of interest which consists of some future samples. Differing from the general Bayes approach, we do not specify the prior distribution of the parameter, and only a first moment condition on the prior is assumed. Simulation studies are conducted to exhibit the coverage probabilities of the prediction interval. Financial support from the IAP research network (#P5/24) of the Belgian Government (Belgian Science Policy) is gratefully acknowledged.     

A mutual information-based k-sample test for discrete distributions

The two-sample problem and its extension to the k-sample problem are well known in the statistical literature. However, the discrete version of the k-sample problem is relatively less explored. Here in this work we suggest a k-sample non-parametric test procedure for discrete distributions based on mutual information. A detailed power study with comparison with other alternatives is provided. Finally, a comparison of some English soccer league teams based on their goal-scoring pattern is discussed.