Improvement of mixed predictors in linear mixed models

In this paper, we introduce stochastic-restricted Liu predictors which will be defined by combining in a special way the two approaches followed in obtaining the mixed predictors and the Liu predictors in the linear mixed models. Superiorities of the linear combination of the new predictor to the Liu and mixed predictors are done in the sense of mean square error matrix criterion. Finally, numerical examples and a simulation study are done to illustrate the findings. In numerical examples, we took some arbitrary observations from the data as the prior information since we did not have historical data or additional information about the data sets. The results show that this case does the new estimator gain efficiency over the constituent estimators and provide accurate estimation and prediction of the data.     

Monitoring processes with data censored owing to competing risks by using exponentially weighted moving average control charts

In industry, process monitoring is widely employed to detect process changes rapidly. However, in some industrial applications observations are censored. For example, when testing breaking strengths and failure times often a limited stress test is performed. With censored observations, a direct application of traditional monitoring procedures is not appropriate. When the censoring occurs due to competing risks, we propose a control chart based on conditional expected values to detect changes in the mean strength. To protect against possible confounding caused by changes in the mean of the censoring mechanism we also suggest a similar chart to detect changes in the mean censoring level. We provide an example of monitoring bond strength to illustrate the application of this methodology.     

Approximate maximum likelihood predictors of future failure times of shifted exponential distributions under multiple type II censoring

Suppose we consider a general multiple type II censored sample (some middle observations being censored) from a shifted exponential distribution. The maximum likelihood prediction method does not admit explicit solutions. We introduce a simple approximation to one of prediction likelihood equations and derive approximate predictors of missing failure times. We compute their mean square prediction errors by simulation and compare them with the best linear predictors. Further, we present two real examples to illustrate this method of prediction.AMS Subject Classification (2000): 62G30, 62M20, 62F99     

Efficiency estimation of type-I censored sample from the Weibull distribution based on sup-entropy

On the basis of Awad sup-entropy, the efficiency function for type-I censored sample from the Weibull distribution is numerically introduced. The properties of the derived efficiency are discussed. Furthermore, for a given efficiency, the termination time of the experiment, and the maximum likelihood estimates for the Weibull parameters, are proposed. Simulation results are tabulated and discussed. Censored and complete samples are compared for a wide range of the efficiency. The comparisons show the quality of the developed algorithms and the effectiveness of using censoring in estimating with the Weibull distribution.     

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.     

On Mean Squared Prediction Error Estimation in Small Area Estimation Problems

For a general linear mixed normal model, a new linearized weighted jackknife method is proposed to estimate the mean squared prediction error (MSPE) of an empirical best linear unbiased predictor (EBLUP) of a general mixed effect. Different MSPE estimators are compared using a Monte Carlo simulation study.     

Prediction for future failures in Weibull distribution under hybrid censoring

In this paper, we consider the prediction of a future observation based on a type-I hybrid censored sample when the lifetime distribution of experimental units is assumed to be a Weibull random variable. Different classical and Bayesian point predictors are obtained. Bayesian predictors are obtained using squared error and linear-exponential loss functions. We also provide a simulation consistent method for computing Bayesian prediction intervals. Monte Carlo simulations are performed to compare the performances of the different methods, and one data analysis has been presented for illustrative purposes.     

Two-sample Pitman closeness comparison under progressive Type-II censoring

In this paper, we consider two problems concerning two independent progressively Type-II censored samples. We first consider the Pitman closeness (PC) of order statistics from two independent progressively censored samples to a specific population quantile. We then consider the point prediction of a future progressively censored order statistic and discuss the determination of the closest progressively censored order statistic from the current sample according to the simultaneous closeness probabilities. For both these problems, explicit expressions are derived for the pertinent PC probabilities, and then special cases are given as examples. For various censoring schemes, we also present numerical results for the standard uniform, standard exponential, and standard normal distributions. Finally, a distribution-free result for the median is obtained.     

Pitman comparisons of predictors of censored observations from progressively censored samples for exponential distribution

On the basis of a progressively censored sample, Basak et al. [On some predictors of times to failure of censored items in progressively censored samples. Comput Statist Data Anal. 2006;50:1313 –1337] considered the problem of predicting the unobserved censored units at various stages of progressive censoring. They then discussed several different point predictors of these censored units and compared them with respect to mean square prediction error. In this work, we use the Pitman closeness (PC) criterion to compare the maximum likelihood, best linear unbiased, best linear equivariant, and conditional median predictors (CMPs) of these progressively censored units. Next, we compare all these with respect to the median unbiased predictor in terms of PC. Numerical computations are then performed to compare all these predictors. By comparing our results to those of Basak et al. (2006), we note that our findings in the sense of PC are similar to theirs in which the CMP competes well when compared to all other predictors.     

Development of life-test sampling plans for exponential distributions based on accelerated life testing

This paper presents a procedure for developing life-test sampling plans for exponential distributions based upon accelerated life testing(ALT). Type II censoring is assumed at each overstress level. The derived test statistic is shown to be a quotient of two independent random variables, each of which is a rational power of a Chi-square random variable. The distribution of the test statistic is characterized by the H-function, which can be numerically evaluated to obtain desired sampling plans.     

Assessing different uncertainty measures of EBLUP: a resampling-based approach

The empirical best linear unbiased prediction approach is a popular method for the estimation of small area parameters. However, the estimation of reliable mean squared prediction error (MSPE) of the estimated best linear unbiased predictors (EBLUP) is a complicated process. In this paper we study the use of resampling methods for MSPE estimation of the EBLUP. A cross-sectional and time-series stationary small area model is used to provide estimates in small areas. Under this model, a parametric bootstrap procedure and a weighted jackknife method are introduced. A Monte Carlo simulation study is conducted in order to compare the performance of different resampling-based measures of uncertainty of the EBLUP with the analytical approximation. Our empirical results show that the proposed resampling-based approaches performed better than the analytical approximation in several situations, although in some cases they tend to underestimate the true MSPE of the EBLUP in a higher number of small areas.     

Gap-ratio goodness of fit tests for weibull or extreme value distribution assumptions with left or right censored data

With data collection in environmental science and bioassay, left censoring because of nondetects is a problem. Similarly in reliability and life data analysis right censoring frequently occurs. There is a need for goodness of fit tests that can adapt to left or right censored data and be used to check important distributional assumptions without becoming too difficult to regularly implement in practice. A new test statistic is derived from a plot of the standardized spacings between the order statistics versus their ranks. Any linear or curvilinear pattern is evidence against the null distribution. When testing the Weibull or extreme value null hypothesis this statistic has a null distribution that is approximately F for most combinations of sample size and censoring of practical interest. Our statistic is compared to the Mann-Scheuer-Fertig statistic which also uses the standardized spacings between the order statistics. The results of a simulation study show the two tests are competitive in terms of power. Although the Mann-Scheuer-Fertig statistic is somewhat easier to compute, our test enjoys advantages in the accuracy of the F approximation and the availability of a graphical diagnostic.     

An estimator for the scale parameter of the two parameter weibull distribution for type i singly right-censored data

For type I censoring, in addition to the failure times, the number failures is also observed as part of the data. Using this feature of type I singly right-censored data a simple estimator is obtained for the scale parameter of the two parameter Weibull distribution. The exact mean and variance of the estimator are derived and computed for finite sample sizes. Its limiting properties such as asymptotic normality and asymptotic relative efficiency are obtained. The estimator has high efficiency for moderate and heavy censoring. Its use is illustrated by means of an example.     

Interval estimation for the Pareto distribution based on the progressive Type II censored sample

For the complete sample and the right Type II censored sample, Chen [Joint confidence region for the parameters of Pareto distribution. Metrika 44 (1996), pp. 191–197] proposed the interval estimation of the parameter θ and the joint confidence region of the two parameters of Pareto distribution. This paper proposed two methods to construct the confidence region of the two parameters of the Pareto distribution for the progressive Type II censored sample. A simulation study comparing the performance of the two methods is done and concludes that Method 1 is superior to Method 2 by obtaining a smaller confidence area. The interval estimation of parameter ν is also given under progressive Type II censoring. In addition, the predictive intervals of the future observation and the ratio of the two future consecutive failure times based on the progressive Type II censored sample are also proposed. Finally, one example is given to illustrate all interval estimations in this paper.     

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.     

Variable screening for survival data in the presence of heterogeneous censoring

Variable screening for censored survival data is most challenging when both survival and censoring times are correlated with an ultrahigh-dimensional vector of covariates. Existing approaches to handling censoring often make use of inverse probability weighting by assuming independent censoring with both survival time and covariates. This is a convenient but rather restrictive assumption which may be unmet in real applications, especially when the censoring mechanism is complex and the number of covariates is large. To accommodate heterogeneous (covariate-dependent) censoring that is often present in high-dimensional survival data, we propose a Gehan-type rank screening method to select features that are relevant to the survival time. The method is invariant to monotone transformations of the response and of the predictors, and works robustly for a general class of survival models. We establish the sure screening property of the proposed methodology. Simulation studies and a lymphoma data analysis demonstrate its favorable performance and practical utility.     

Adjusting win statistics for dependent censoring

For composite outcomes whose components can be prioritized on clinical importance, the win ratio, the net benefit and the win odds apply that order in comparing patients pairwise to produce wins and subsequently win proportions. Because these three statistics are derived using the same win proportions and they test the same hypothesis of equal win probabilities in the two treatment groups, we refer to them as win statistics. These methods, particularly the win ratio and the net benefit, have received increasing attention in methodological research and in design and analysis of clinical trials. For time-to-event outcomes, however, censoring may introduce bias. Previous work has shown that inverse-probability-of-censoring weighting (IPCW) can correct the win ratio for bias from independent censoring. The present article uses the IPCW approach to adjust win statistics for dependent censoring that can be predicted by baseline covariates and/or time-dependent covariates (producing the CovIPCW-adjusted win statistics). Theoretically and with examples and simulations, we show that the CovIPCW-adjusted win statistics are unbiased estimators of treatment effect in the presence of dependent censoring.     

Bayesian estimation and prediction for the inverse weibull distribution under general progressive censoring

Abstract

This paper deals with Bayesian estimation and prediction for the inverse Weibull distribution with shape parameter α and scale parameter λ under general progressive censoring. We prove that the posterior conditional density functions of α and λ are both log-concave based on the assumption that λ has a gamma prior distribution and α follows a prior distribution with log-concave density. Then, we present the Gibbs sampling strategy to estimate under squared-error loss any function of the unknown parameter vector (α, λ) and find credible intervals, as well as to obtain prediction intervals for future order statistics. Monte Carlo simulations are given to compare the performance of Bayesian estimators derived via Gibbs sampling with the corresponding maximum likelihood estimators, and a real data analysis is discussed in order to illustrate the proposed procedure. Finally, we extend the developed methodology to other two-parameter distributions, including the Weibull, Burr type XII, and flexible Weibull distributions, and also to general progressive hybrid censoring.     

Approximate MLEs of the parameters of location-scale models under type II censoring

Log-location-scale distributions are widely used parametric models that have fundamental importance in both parametric and semiparametric frameworks. The likelihood equations based on a Type II censored sample from location-scale distributions do not provide explicit solutions for the para-meters. Statistical software is widely available and is based on iterative methods (such as, Newton Raphson Algorithm, EM algorithm etc.), which require starting values near the global maximum. There are also many situations that the specialized software does not handle. This paper provides a method for determining explicit estimators for the location and scale parameters by approximating the likelihood function, where the method does not require any starting values. The performance of the proposed approximate method for the Weibull distribution and Log-Logistic distributions is compared with those based on iterative methods through the use of simulation studies for a wide range of sample size and Type II censoring schemes. Here we also examine the probability coverages of the pivotal quantities based on asymptotic normality. In addition, two examples are given.     

Exact likelihood inference based on an unified hybrid censored sample from the exponential distribution

Recently, exact confidence bounds and exact likelihood inference have been developed based on hybrid censored samples by Chen and Bhattacharyya [Chen, S. and Bhattacharyya, G.K. (1998). Exact confidence bounds for an exponential parameter under hybrid censoring. Communications in Statistics—Theory and Methods, 17, 1857–1870.], Childs et al. [Childs, A., Chandrasekar, B., Balakrishnan, N. and Kundu, D. (2003). Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution. Annals of the Institute of Statistical Mathematics, 55, 319–330.], and Chandrasekar et al. [Chandrasekar, B., Childs, A. and Balakrishnan, N. (2004). Exact likelihood inference for the exponential distribution under generalized Type-I and Type-II hybrid censoring. Naval Research Logistics, 51, 994–1004.] for the case of the exponential distribution. In this article, we propose an unified hybrid censoring scheme (HCS) which includes many cases considered earlier as special cases. We then derive the exact distribution of the maximum likelihood estimator as well as exact confidence intervals for the mean of the exponential distribution under this general unified HCS. Finally, we present some examples to illustrate all the methods of inference developed here.