Nonparametric Prediction Intervals for Future Order Statistics in a Proportional Hazard Model

Consider k independent random samples with different sample sizes such that the ith sample comes from the cumulative distribution function (cdf) F _i  = 1 ? (1 ? F)^{α _i} , where α _i is a known positive constant and F is an absolutely continuous cdf. Also, suppose that we have observed the maximum and minimum of the first k samples. This article shows how one can construct the nonparametric prediction intervals for the order statistics of the future samples on the basis of these information. Three schemes are studied and in each case exact expressions for the prediction coefficients of prediction intervals are derived. Numerical computations are given for illustrating the results. Also, a comparison study is done while the complete samples are available.     

Two-Sample Bayesian Prediction Intervals of Generalized Order Statistics Based on Multiply Type II Censored Data

In this article, two-sample Bayesian prediction intervals of generalized order statistics (GOS) based on multiply Type II censored data are derived. To illustrate these results, the Pareto, Weibull, and Burr-Type XII distributions are used as examples. Finally, a numerical illustration of the sequential order statistics from the Pareto distribution is presented.     

Prediction Intervals for an Ordered Observation from Weibull Distribution Based on Censored Samples

In this article, we provide some suitable pivotal quantities for constructing prediction intervals for the jth future ordered observation from the two-parameter Weibull distribution based on censored samples. Our method is more general in the sense that it can be applied to any data scheme. We present a simulation of our method to analyze its performance. Two illustrative examples are also included. For further study, our method is easily applied to other location and scale family distributions.     

A Bayesian Approach for Nonlinear Regression Models with Continuous Errors

Abstract

In this paper we develop a Bayesian analysis for the nonlinear regression model with errors that follow a continuous autoregressive process. In this way, unequally spaced observations do not present a problem in the analysis. We employ the Gibbs sampler, (see Gelfand, A., Smith, A. (1990 Gelfand, A. and Smith, A. 1990. Sampling based approaches to calculating marginal densities. J. Amer. Statist. Assoc., 85: 398–409. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Sampling based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85:398–409.), as the foundation for making Bayesian inferences. We illustrate these Bayesian inferences with an analysis of a real data-set. Using these same data, we contrast the Bayesian approach with a generalized least squares technique.     

A Generalized Bayes Rule for Prediction

In the case of prior knowledge about the unknown parameter, the Bayesian predictive density coincides with the Bayes estimator for the true density in the sense of the Kullback-Leibler divergence, but this is no longer true if we consider another loss function. In this paper we present a generalized Bayes rule to obtain Bayes density estimators with respect to any α-divergence, including the Kullback-Leibler divergence and the Hellinger distance. For curved exponential models, we study the asymptotic behaviour of these predictive densities. We show that, whatever prior we use, the generalized Bayes rule improves (in a non-Bayesian sense) the estimative density corresponding to a bias modification of the maximum likelihood estimator. It gives rise to a correspondence between choosing a prior density for the generalized Bayes rule and fixing a bias for the maximum likelihood estimator in the classical setting. A criterion for comparing and selecting prior densities is also given.     

Estimation and prediction for an inverted exponentiated Rayleigh distribution under hybrid censoring

In this paper we consider estimation of unknown parameters of an inverted exponentiated Rayleigh distribution when it is known that data are hybrid Type I censored. The maximum likelihood and Bayes estimates are derived. In sequel interval estimates are also constructed. We further consider one- and two-sample prediction of future observations and also obtain prediction intervals. The performance of proposed methods of estimation and prediction is studied using simulations and an illustrative example is discussed in support of the suggested methods.     

Order Statistics from the Generalized Exponential Distribution and Applications

With linear dispersion effects, the standard factorial designs are not optimal estimation of a mean model. A sequential two-stage experimental design procedure has been proposed that first estimates the variance structure, and then uses the variance estimates and the variance optimality criterion to develop a second stage design that efficiency estimates the mean model. This procedure has been compared to an equal replicate design analyzed by ordinary least squares, and found to be a superior procedure in many situations.

However with small first stage sample sizes the variance estiamtes are not reliable, and hence an alternative procedure could be more beneficial. For this reason a Bayesian modification to the two-stage procedure is proposed which will combine the first stage variance estiamtes with some prior variance information that will produce a more efficient procedure. This Bayesian procedure will be compared to the non-Bayesian twostage procedure and to the two one-stage alternative procedures listed above. Finally, a recommendation will be made as to which procedure is preferred in certain situations.     

Statistical Inference Based on Progressively Type II Censored Data from Weibull Model

In this article, we consider the problem of estimating the shape and scale parameters and predicting the unobserved removed data based on a progressive type II censored sample from the Weibull distribution. Maximum likelihood and Bayesian approaches are used to estimate the scale and shape parameters. The sampling-based method is used to draw Monte Carlo (MC) samples and it has been used to estimate the model parameters and also to predict the removed units in multiple stages of the censored sample. Two real datasets are presented and analyzed for illustrative purposes and Monte carlo simulations are performed to study the behavior of the proposed methods.     

On a Characterization of Generalized Pareto Distribution Based on Generalized Order Statistics

In recent years, several attempts have been made to characterize the generalized Pareto distribution (GPD) based on the properties of order statistics and record values. In the present article, we give a characterization result on GPD based on the spacing of generalized order statistics.     

Prediction intervals for future Weibull residual data

In reliability theory, risk analysis, renewal processes and actuarial studies, the residual lifetimes data play an important essential role in studying the conditional tail of the lifetime data. In this paper, based on some observed ordered residual Weibull data, we introduce different prediction methods for obtaining prediction intervals (PIs) of future residual lifetimes including likelihood, Wald, moments, parametric bootstrap, and highest conditional methods. Monte Carlo simulations are performed to compare the performances of the so obtained PIs and one data analysis is performed for illustration purposes.     

Linear Estimators and Predictors Based on Generalized Order Statistics from Generalized Pareto Distributions

Linear estimation and prediction based on several samples of generalized order statistics from generalized Pareto distributions is considered. Representations of best linear unbiased estimators (BLUEs) and best linear equivariant estimators in location-scale families are derived, as well as corresponding optimal linear predictors. Moreover, we study positivity of the linear estimators of the scale parameter. An example illustrates that the BLUE may attain negative values with positive probability in certain situations.     

Sharp Bounds for Generalized Order Statistics via Logarithmic Moments

ABSTRACT

We present sharp bounds for expectations of generalized order statistics with random indices. The bounds are expressed in terms of logarithmic moments E X ^a (log max {1, X}) ^b of the underlying observation X. They are attainable and provide characterizations of some non trivial distributions. No restrictions are imposed on the parameters of the generalized order statistics model.     

Moments of Order Statistics from Weibull Distribution in the Presence of Multiple Outliers

In this article, we derive exact expressions for the single and product moments of order statistics from Weibull distribution under the contamination model. We assume that X₁, X₂, …, X_{n ? p} are independent with density function f(x) while the remaining, p observations (outliers) X_{n ? p + 1}, …, X_n are independent with density function arises from some modified version of f(x), which is called g(x), in which the location and/or scale parameters have been shifted in value. Next, we investigate the effect of the outliers on the BLUE of the scale parameter. Finally, we deduce some special cases.     

Bayesian adaptive Lasso for quantile regression models with nonignorably missing response data

Abstract

Handling data with the nonignorably missing mechanism is still a challenging problem in statistics. In this paper, we develop a fully Bayesian adaptive Lasso approach for quantile regression models with nonignorably missing response data, where the nonignorable missingness mechanism is specified by a logistic regression model. The proposed method extends the Bayesian Lasso by allowing different penalization parameters for different regression coefficients. Furthermore, a hybrid algorithm that combined the Gibbs sampler and Metropolis-Hastings algorithm is implemented to simulate the parameters from posterior distributions, mainly including regression coefficients, shrinkage coefficients, parameters in the non-ignorable missing models. Finally, some simulation studies and a real example are used to illustrate the proposed methodology.     

Joint Moment Generating Functions of Nonadjacent Dual Generalized Order Statistics from Reflected Generalized Pareto Distributions

In this article, a class of reflected generalized Pareto distributions (cf. Burkschat et al., 2003 Burkschat , M. , Cramer , E. , Kamps , U. ( 2003 ). Dual generalized order statistics . Metron LXI ( 1 ): 13 – 26 . [Google Scholar]) is considered. Recurrence relations for joint moment generating functions of higher non adjacent dual generalized order statistics based on a random sample drawn from the considered class are derived. Higher joint moments of non adjacent dual generalized order statistics (reversed ordered order statistics and lower k-records as special cases) are obtained. Recurrence relations for single and product moment generating functions and moments of higher non adjacent dual generalized order statistics are derived. Some results of higher moments of non adjacent generalized order statistics from generalized Pareto distributions (cf. Johnson et al., 1995 Johnson , N. L. , Kotz , S. , Balakrishnan , N. ( 1995 ). Continuous Univariate Distributions. , 2nd ed. Vol. 2. New York : Wiley & Sons . [Google Scholar]), are obtained by using a relation connecting higher moments of generalized order statistics and its dual.     

Inference Based on Order Statistics from the Generalized Pareto Distribution and Application

ABSTRACT

In this article, we derive exact explicit expressions for the single, double, triple, and quadruple moments of order statistics from the generalized Pareto distribution (GPD). Also, we obtain the best linear unbiased estimates of the location and scale parameters (BLUE's) of the GPD. We then use these results to determine the mean, variance, and coefficients of skewness and kurtosis of certain linear functions of order statistics. These are then utilized to develop approximate confidence intervals for the generalized Pareto parameters using Edgeworth approximation and compare them with those based on Monte Carlo simulations. To show the usefulness of our results, we also present a numerical example. Finally, we give an application to real data.     

A Computational Bayesian Method for Generalized Semiparametric Regression Models

To better understand the power shift and the U.S. role compared to China and others regional actors, the Chicago Council on Global Affairs and the East Asia Institute (EAI) surveyed people in six countries - China, Japan, South Korea, Vietnam, Indonesian, and the United States - in the first half of 2008 about regional security and economic integration in Asia and about how these nations perceive each other (Bouton et al., 2010 Bouton, M., Steven, K., Benjamin, P., and Gregory, H. (2010). Soft power in Asia survey, 2008. ICPSR25342-v1. Ann Arbor, MI: Inter-university Consortium for Political and Social Research [distributor], 2010-04-05. doi:10.3886/ICPSR25342.v1[Crossref] , [Google Scholar]). There exists latent variance that cannot be adequately explained by parametric models. This is, in large part, due to the hidden structures and latent stories that from in unexpected ways. Therefore, a new Gibbs sampler is developed here in order to reveal preciously unseen structures and latent variances found in the survey dataset of Bouton et al. This new sampler is based upon the semiparametric regression, a well-known tool frequently utilized in order to capture the functional dependence between variables with fixed effect parametric and nonlinear regression. This is then extended to a generalized semiparametric regression for binary responses with logit and probit link function. The new sampler is then developed for the generalized linear mixed model with a nonparametric random effect. It is expressed as nonparametric regression with the multinomial-Dirichlet distribution for the number and positions of knots.     

Prediction Intervals for Time Series: A Modified Sieve Bootstrap Approach

Traditional Box–Jenkins prediction intervals perform poorly when the innovations are not Gaussian. Nonparametric bootstrap procedures overcome this handicap, but most existing methods assume that the AR and MA orders of the process are known. The sieve bootstrap approach requires no such assumption but produces liberal coverage due to the use of residuals that underestimate the actual variance of the innovations and the failure of the methods to capture variations due to sampling error of the mean. A modified approach, that corrects these deficiencies, is implemented. Monte Carlo simulations results show that the modified version achieves nominal or near nominal coverage.     

Two-sample prediction for progressively Type-II censored Weibull lifetimes

Prediction on the basis of censored data has an important role in many fields. This article develops a non-Bayesian two-sample prediction based on a progressive Type-II right censoring scheme. We obtain the maximum likelihood (ML) prediction in a general form for lifetime models including the Weibull distribution. The Weibull distribution is considered to obtain the ML predictor (MLP), the ML prediction estimate (MLPE), the asymptotic ML prediction interval (AMLPI), and the asymptotic predictive ML intervals of the sth-order statistic in a future random sample (Y_s) drawn independently from the parent population, for an arbitrary progressive censoring scheme. To reach this aim, we present three ML prediction methods namely the numerical solution, the EM algorithm, and the approximate ML prediction. We compare the performances of the different methods of ML prediction under asymptotic normality and bootstrap methods by Monte Carlo simulation with respect to biases and mean square prediction errors (MSPEs) of the MLPs of Y_s as well as coverage probabilities (CP) and average lengths (AL) of the AMLPIs. Finally, we give a numerical example and a real data sample to assess the computational comparison of these methods of the ML prediction.