Bayesian estimation of P (Y<X) from Burr-type X model containing spurious observations

We consider the problem of estimating R=P(Y<X) when X and Y are independent Burr-type X random variables. We assume that the sample from each population contains one spurious observation. Bayes estimates are derived for exchangeable and identifiable cases. Monte Carlo simulation is carried out to compare the bias and the expected loss of R.     

Classical and Bayesian estimation of P(Y < X) for Kumaraswamy's distribution

The aim of this paper is to study the estimation of the reliability R=P(Y<X) when X and Y are independent random variables that follow Kumaraswamy's distribution with different parameters. If we assume that the first shape parameter is common and known, the maximum-likelihood estimator (MLE), the exact confidence interval and the uniformly minimum variance unbiased estimator of R are obtained. Moreover, when the first parameter is common but unknown, MLEs, Bayes estimators, asymptotic distributions and confidence intervals for R are derived. Furthermore, Bayes and empirical Bayes estimators for R are obtained when the first parameter is common and known. Finally, when all four parameters are different and unknown, the MLE of R is obtained. Monte Carlo simulations are performed to compare the different proposed methods and conclusions on the findings are given.     

Bayesian inference on P(X > Y) in bivariate Rayleigh model

In the literature, assuming independence of random variables X and Y, statistical estimation of the stress–strength parameter R = P(X > Y) is intensively investigated. However, in some real applications, the strength variable X could be highly dependent on the stress variable Y. In this paper, unlike the common practice in the literature, we discuss on estimation of the parameter R where more realistically X and Y are dependent random variables distributed as bivariate Rayleigh model. We derive the Bayes estimates and highest posterior density credible intervals of the parameters using suitable priors on the parameters. Because there are not closed forms for the Bayes estimates, we will use an approximation based on Laplace method and a Markov Chain Monte Carlo technique to obtain the Bayes estimate of R and unknown parameters. Finally, simulation studies are conducted in order to evaluate the performances of the proposed estimators and analysis of two data sets are provided.     

Empirical Bayes estimation for archaeological stratigraphy

Models and algorithms are presented for the stratigraphic analysis of earth core samples collected at archaeological sites. The aim is to separate the occupation of the site into distinct periods, by dividing the earth core into well-defined blocks of uniform magnetic susceptibility. The models describe the response of detector equipment by using both a spread function and an error process, and they incorporate prior beliefs regarding the nature of the true susceptibility values. The prior parameters are estimated by using pseudolikelihood and the susceptibilities by maximum a posteriori methods via the one-step-late algorithm. These procedures are illustrated with data from synthetic and real core specimens. The new procedures prove to be far superior to other approaches, producing reconstructions which clearly show distinct periods of uniform magnetic susceptibility separated by sharp discontinuities.     

Shrinkage estimation of P(Y < X) in the exponential distribution mixing with exponential distribution

ABSTRACT

The problem of estimation of R = P(Y < X) have been used in the paper. Let X has exponential distribution mixing with exponential distribution with parameters β and θ and Y independently of X has exponential distribution with parameter λ. By using a prior guess or estimate R₀, different shrinkage estimators of R are derived. Then the performance of the estimators are discussed. Finally, we compare these results with Baklizei and Dayyeh (2003) approaches.     

Irregularities in X(Y) from Y(X) in linear calibration

Let X be an input measurement and Y the output reading of a calibrated instrument, with Y(X) as the calibration curve. Solving X(Y) projects an instrumental reading back onto the scale of measurements as an object of pivotal interest. The arrays of instrumental readings are projected in this manner in practice, yielding arrays of calibrated measurements, typically subject to errors of calibration. The effects of calibration errors on the properties of calibrated measurements are examined here under linear calibration. Irregularities arise as induced dependencies, inflated variances, non-standard distributions, inconsistent sample means, the underestimation of measurement variance, and other unintended consequences. On the other hand, conventional properties are seen to remain largely in place in the use of selected regression diagnostics and in one-way comparative experiments using calibrated data.     

Weighted likelihood based inference for P(X < Y)

This contribution deals with the statistical problem of evaluating the stress–strength reliability parameter R = P(X < Y), when both stress and strength data are prone to contamination. Standard likelihood inference can be badly affected by mild data inadequacies, that often occur in the form of several outliers. Then, robust tools are recommended. Here, inference relies on the weighted likelihood methodology. This approach has the advantage to lead to robust estimators, tests, and confidence intervals that share the main asymptotic properties of their classical counterparts. The accuracy of the proposed methodology is illustrated both by numerical studies and real-data applications.     

Empirical Bayes estimation of undercount in the decennial census

Empirical Bayes methods are used to estimate the extent of the undercount at the local level in the 1980 U.S. census. "Grouping of like subareas from areas such as states, counties, and so on into strata is a useful way of reducing the variance of undercount estimators. By modeling the subareas within a stratum to have a common mean and variances inversely proportional to their census counts, and by taking into account sampling of the areas (e.g., by dual-system estimation), empirical Bayes estimators that compromise between the (weighted) stratum average and the sample value can be constructed. The amount of compromise is shown to depend on the relative importance of stratum variance to sampling variance. These estimators are evaluated at the state level (51 states, including Washington, D.C.) and stratified on race/ethnicity (3 strata) using data from the 1980 postenumeration survey (PEP 3-8, for the noninstitutional population)."     

Empirical Bayes sequential estimation of a finite population mean

In this article, we consider the Bayes and empirical Bayes problem of the current population mean of a finite population when the sample data is available from other similar (m-1) finite populations. We investigate a general class of linear estimators and obtain the optimal linear Bayes estimator of the finite population mean under a squared error loss function that considered the cost of sampling. The optimal linear Bayes estimator and the sample size are obtained as a function of the parameters of the prior distribution. The corresponding empirical Bayes estimates are obtained by replacing the unknown hyperparameters with their respective consistent estimates. A Monte Carlo study is conducted to evaluate the performance of the proposed empirical Bayes procedure.     

Toward the evaluation of P(X(t) > Y(t)) when both X(t) and Y(t) are inactivity times of two systems

The inactivity time, also known as reversed residual life, has been a topic of increasing interest in the literature. In this investigation, based on the comparison of inactivity times of two devices, we introduce and study a new estimate of the probability of the inactivity time of one device exceeding that of another device. The problem studied in this paper is important for engineers and system designers. It would enable them to compare the inactivity times of the products and, hence to design better products. Several properties of this probability are established. Connections between the target probability and the reversed hazard rates of the two devices are established. In addition, some of the reliability properties of the new concept are investigated extending the well-known probability ordering. Finally, to illustrate the introduced concepts, many examples and applications in the context of reliability theory are included.     

Inference on P(Y < X) in Bivariate Rayleigh Distribution

This paper deals with the estimation of reliability R = P(Y < X) when X is a random strength of a component subjected to a random stress Y, and (X, Y) follows a bivariate Rayleigh distribution. The maximum likelihood estimator of R and its asymptotic distribution are obtained. An asymptotic confidence interval of R is constructed using the asymptotic distribution. Also, two confidence intervals are proposed based on Bootstrap method and a computational approach. Testing of the reliability based on asymptotic distribution of R is discussed. Simulation study to investigate performance of the confidence intervals and tests has been carried out. Also, a numerical example is given to illustrate the proposed approaches.     

Estimation of P(Y < X) for Weibull Distribution Under Progressive Type-II Censoring

Based on progressively Type II censored samples, we consider the estimation of R = P(Y < X) when X and Y are two independent Weibull distributions with different shape parameters, but having the same scale parameter. The maximum likelihood estimator, approximate maximum likelihood estimator, and Bayes estimator of R are obtained. Based on the asymptotic distribution of R, the confidence interval of R are obtained. Two bootstrap confidence intervals are also proposed. Analysis of a real data set is given for illustrative purposes. Monte Carlo simulations are also performed to compare the different proposed methods.     

Empirical best linear unbiased and empirical Bayes prediction in multivariate small area estimation

Small area estimation plays a prominent role in survey sampling due to a growing demand for reliable small area estimates from both public and private sectors. Popularity of model-based inference is increasing in survey sampling, particularly, in small area estimation. The estimates of the small area parameters can profitably ‘borrow strength’ from data on related multiple characteristics and/or auxiliary variables from other neighboring areas through appropriate models. Fay (1987, Small Area Statistics, Wiley, New York, pp. 91–102) proposed multivariate regression for small area estimation of multiple characteristics. The success of this modeling rests essentially on the strength of correlation of these dependent variables. To estimate small area mean vectors of multiple characteristics, multivariate modeling has been proposed in the literature via a multivariate variance components model. We use this approach to empirical best linear unbiased and empirical Bayes prediction of small area mean vectors. We use data from Battese et al. (1988, J. Amer. Statist. Assoc. 83, 28 –36) to conduct a simulation which shows that the multivariate approach may achieve substantial improvement over the usual univariate approach.     

Bayes and empirical Bayes shrinkage estimation of regression coefficients

For a moderate or large number of regression coefficients, shrinkage estimates towards an overall mean are obtained by Bayes and empirical Bayes methods. For a special case, the Bayes and empirical Bayes shrinking weights are shown to be asymptotically equivalent as the amount of shrinkage goes to zero. Based on comparisons between Bayes and empirical Bayes solutions, a modification of the empirical Bayes shrinking weights designed to guard against unreasonable overshrinking is suggested. A numerical example is given.     

Constrained Bayes and empirical Bayes estimation under random effects normal ANOVA model with balanced loss function

The paper develops constrained Bayes and empirical Bayes estimators in the random effects ANOVA model under balanced loss functions. In the balanced normal–normal model, estimators of the Bayes risks of the constrained Bayes and constrained empirical Bayes estimators are provided which are correct asymptotically up to O(m^-1)$\mathrm{O}(m^{-1})$, that is the remainder term is o(m^-1)$\mathrm{o}(m^{-1})$, m$m$ denoting the number of strata.     

Empirical Bayes estimation of parameters in Markov transition probability matrix with computational methods

Empirical Bayes estimator for the transition probability matrix is worked out in the cases where we have belief regarding the parameters, For example, where the states seem to be equal or not. In both cases, priors are in accordance with our beliefs. Using EM algorithm, computational methods for different hyperparameters of the empirical Bayes are described. Also, robustness of empirical Bayes procedure is investigated.     

The Sampling Variance of r(X,Y) for Unrelated X and Y: A Simple Algebraic Demonstration

Abstract

A demonstration of the variance of Pearson's correlation coefficient r for samples of unrelated X, Y variables is laid out in elementary algebra. An approximation of var(r) for samples of ρ-related binormal variables is also provided.