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 closeness for hierarchical bayes predictors in mixed linear models

The present article considers the Pitman Closeness (PC) criterion of certain hierarchical Bayes (HB) predictors derived under a normal mixed linear models for known ratios of variance components using a uniform prior for the vector of fixed effects and some proper or improper prior on the error variance. For a generalized Euclidean error, simultaneous HB predictors of several linear combinations of vector of effects are shown to be the Pitman-closest in the frequentist sense in the class of equivariant predictors for location group of transformations. The normality assumption can be relaxed to show that these HB predictors are the Pitman-closest for location-scale group of transformations for a wider family of elliptically symmetric distributions. Also for this family, the HB predictors turn out to be Pitman-closest in the class of all linear unbiased predictors (LUPs). All these results are extended for the HB predictor of finite population mean vector in the context of finite population sampling.     

Pitman closeness comparison of linear estimators:a canonical form

A general form is presented for the comparison of two linear estimators of a common parameter by means of the Pitman measure of closeness. Several asymptotic results are given. The case in which the estimators are linear combinations of the order statistics is discussed. The asymptotic comparison of the sample mean versus the sample median is derived for the Laplace distribution, and two other examples are given.     

Order statistics from extreme value distribution,ii: best linear unbiased estimates and some other uses

Let Y_r+1:n ≤ _Y:r+2:n ≤≤… <Y_n?6:n-<: TYPE-II censored sample from an extreme value population with µ and α as the location and scale parameters, respectively. Tables of coefficients for the best linear unbiased estimators (BLUEs) of µ and α are presented for various choices of censoring and sample sizes n = 2(1)15(5)30; variances and covariance of these estimators are also presented. The computational formulae and procedure used and some checks employed are explained. We finally illustrate some uses of the tables by taking examples.     

Comparison of preliminary test estimators based on generalized order statistics from proportional hazard family using Pitman measure of closeness

In this article, based on generalized order statistics from a family of proportional hazard rate model, we use a statistical test to generate a class of preliminary test estimators and shrinkage preliminary test estimators for the proportionality parameter. These estimators are compared under Pitman measure of closeness (PMC) as well as MSE criteria. Although the PMC suffers from non transitivity, in the first class of estimators, it has the transitivity property and we obtain the Pitman-closest estimator. Analytical and graphical methods are used to show the range of parameter in which preliminary test and shrinkage preliminary test estimators perform better than their competitor estimators. Results reveal that when the prior information is not too far from its real value, the proposed estimators are superior based on both mentioned criteria.     

Improved best linear unbiased estimators for the simple linear regression model using double ranked set sampling schemes

ABSTRACT

In this paper, we consider the best linear unbiased estimators (BLUEs) based on double ranked set sampling (DRSS) and ordered DRSS (ODRSS) schemes for the simple linear regression model with replicated observations. We assume three symmetric distributions for the random error term, i.e., normal, Laplace and some scale contaminated normal distributions. The proposed BLUEs under DRSS (BLUEs-DRSS) and ODRSS (BLUEs-ODRSS) are compared with the BLUEs based on ordered simple random sampling (OSRS), ranked set sampling (RSS), and ordered RSS (ORSS) schemes. These estimators are compared in terms of relative efficiency (RE), RE of determinant (RED), and RE of trace (RET). It is found that the BLUEs-ODRSS are uniformly better than the BLUEs based on OSRS, RSS, ORSS, and DRSS schemes. We also compare the estimators based on imperfect RSS (IRSS) schemes. It is worth mentioning here that the BLUEs under ordered imperfect DRSS (OIDRSS) are better than their counterparts based on IRSS, ordered IRSS (OIRSS), and imperfect DRSS (IDRSS) methods. Moreover, for sensitivity analysis of the BLUEs, we calculate REs and REDs of the BLUEs under the assumption of normality when in fact the parent distribution follows a non normal symmetric distribution. It turns out that even under violation of normality assumptions, BLUEs of the intercept and the slope parameters are found to be unbiased with equal REs under each sampling scheme. It is also observed that the BLUEs under ODRSS are more efficient than the existing BLUEs.     

Criteria for the equality between ordinary least squares and best linear unbiased estimators under certain linear models

A new necessary and sufficient condition is derived for the equality between the ordinary least-squares estimator and the best linear unbiased estimator of the expectation vector in linear models with certain specific design matrices. This condition is then applied to special cases involving one-way and two-way classification models.     

Estimation and prediction in the presence of an outlier under Type-II censoring

A single-outlier data set containing some independent random variables is considered such that all of observations expect one have the same distribution. To describe the model of interested, a location-scale family of distributions is used and the estimation problem of the parameters is studied when the data are collected under Type-II censoring scheme. Moreover, three different predictors are presented to predict the censored order statistics. They are also compared regarding both of mean squared prediction error and Pitman's measure of closeness criteria. The role of outlier parameter as well as censorship rate is studied on performance of proposed estimator and predictors. The results of the paper are illustrated via a real data set. Finally, some conclusions are stated.     

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.     

Improved estimators for the location of double exponential distribution

In this paper, attention is focused on estimation of the location parameter in the double exponential case using a weighted linear combination of the sample median and pairs of order statistics, with symmetric distance to both sides from the sample median. Minimizing with respect to weights and distances we get smaller asymptotic variance in the second order. If the number of pairs is taken as infinite and the distances as null we attain the least asymptotic variance in this class of estimators. The Pitman estimator is also noted. Similarly improved estimators are scanned over their probability of concentration to investigate its bound. Numerical comparison of the estimators is shown.     

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.     

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     

Best linear unbiased estimation of location and scale parameters of the log-logistic distribution

If an assumption, such as homoscedasticity, or some other aspect of an inference problem, such as the number of cases, is altered, our conclusions may change and different parts of the conclusions can be affected in different ways. Most diagnostic procedures measure the influence on one particular aspect of the conclusion - such as model fit or change in parameter estimates. The effect on all aspects of the conclusions can be described by the difference in two log likelihood functions and when the log likelihood functions come from an exponential family or are quasi-likelihoods, this difference can be factored into three terms: one depending only on the alteration, another depending only on the aspects of the conclusions to be considered, and a third term depending on both. The third term is interesting because it shows which aspects of the conclusions are relatively insensitive even to large alterations.     

Comparison of two sampling schemes for generating record-breaking data from the proportional hazard rate models

ABSTRACT

In this article, we consider a sampling scheme in record-breaking data set-up, as record ranked set sampling. We compare the proposed sampling with the well-known sampling scheme in record values known as inverse sampling scheme when the underlying distribution follows the proportional hazard rate model. Various point estimators are obtained in each sampling schemes and compared with respect to mean squared error and Pitman measure of closeness criteria. It is observed in most of the situations that the new sampling scheme provides more efficient estimators than their counterparts. Finally, one data set has been analyzed for illustrative purposes.     

Optimum linear unbiased estimation of the scale parameter by absolute values of order statistics in the double exponential and the double weibull distributions

The first two moments and product moments of absolute values of order statistics are obtained for the double exponential and the double Weibull distributions. In both of the distributions an optimum linear unbiased estimator of the scale parameter, by absolute values of the order statistics, is obtained from complete and censored samples of size n=3(1)10. It is found that the new estimator is generally more efficient than the best linear unbiased estimator (BLUE) of the scale parameter by order statistcs in both of the distributions.     

Parameter estimation and prediction of order statistics for the Burr Type XII distribution with Type II censoring

This article deals with the statistical inference and prediction on Burr Type XII parameters based on Type II censored sample. It is observed that the maximum likelihood estimators (MLEs) cannot be obtained in closed form. We use the expectation-maximization algorithm to compute the MLEs. We also obtain the Bayes estimators under symmetric and asymmetric loss functions such as squared error and Linex By applying Lindley's approximation and Markov chain Monte Carlo (MCMC) technique. Further, MCMC samples are used to calculate the highest posterior density credible intervals. Monte Carlo simulation study and two real-life data-sets are presented to illustrate all of the methods developed here. Furthermore, we obtain a prediction of future order statistics based on the observed ordered because of its important application in different fields such as medical and engineering sciences. A numerical example carried out to illustrate the procedures obtained for prediction of future order statistics.     

Computational testing algorithmic procedure of assessment for lifetime performance index of products with two-parameter exponential distribution based on the multiply type II censored sample

Process capability indices have been widely used to evaluate the process performance to the continuous improvement of quality and productivity. Since the lifetime of products may possess a two-parameter exponential distribution, 14 estimators are used to estimate the lifetime performance index based on the multiply type II censored sample. These estimators are utilized to develop the new hypothesis testing algorithmic procedure in the condition of known L. Finally, two practical examples are illustrated to employ the testing algorithmic procedure to determine whether the process is capable.     

Approximate maximum likelihood estimation of the mean and standard deviation of the normal distribution based on type ii censored samples

It is known that the maximum likelihood methods does not provide explicit estimators for the mean and standard deviation of the normal distribution based on Type II censored samples. In this paper we present a simple method of deriving explicit estimators by approximating the likelihood equations appropriately. We obtain the variances and covariance of these estimators. We also show that these estimators are almost as eficient as the maximum likelihood (ML) estimators and just as eficient as the best linear unbiased (BLU), and the modified maximum likelihood (MML) estimators. Finally, we illustrate this method of estimation by applying it to Gupta's and Darwin's data.     

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.     

Estimating E-bayesian and hierarchical bayesian of scalar parameter of gompertz distribution under type II censoring schemes based on fuzzy data

In this study, the E-Bayesian and hierarchical Bayesian of the scalar parameter of a Gompertz distribution under Type II censoring schemes were estimated based on fuzzy data under the squared error (SE) loss function and the efficiency of the proposed methods was compared with each other and with the Bayesian estimator using Monte Carlo simulation.