Bayes sequential estimation for a time-transformed exponential model

A problem of Bayesian sequential estimating an unknown parameter of a time-transformed exponential model is considered. It is supposed that the loss associated with the error of estimation is weighted squared or precautionary and the cost of observing the process is a function of time and the number of observations. Bayes sequential procedures for estimating the unknown parameter are presented.     

Unbiased estimation following selection for bernoulli populations

Although a large number of selection procedures have been published in the statistics literature, the selection approach has received only limited use in applications. One drawback to the use of such procedures has been the lack of parameter estimates, which prevents quantitative comparisons among the treatments. To partially address this criticism, we present a general method for constructing unbiased estimators of the success probabilities after the termination of a sequential experiment involving two or more Bernoulli populations. Some theoretical properties are presented, and examples are provided for several different selection procedures.     

On Spiring's normal loss function

Loss functions express the loss to society, incurred through the use of a product, in monetary units. Underlying this concept is the notion that any deviation from target of any product characteristic implies a degradation in the product performance and hence a loss. Spiring (1993), in response to criticisms of the quadratic loss function, developed the reflected normal loss function, which is based on the normal density function. We give some modifications of these loss functions to simplify their application and provide a framework for the reflected normal loss function that accomodates a broader class of symmetric loss situations. These modifications also facilitate the unification of both of these loss functions and their comparison through expected loss. Finally, we give a simple method for determing the parameters of the modified reflected normal loss function based on loss information for multiple values of the product characteristic, and an example to illustrate the flexibility of the proposed model and the determination of its parameters.     

A note on the location parameters of two exponential distributions under order restrictions

The problem of simultaneous estimation of location parameters of two independent exponential distributions is considered when location and/or scale parameters are ordered. We show that the standard estimators in the unrestricted case which use information only from the populations individually can be improved upon when various order restrictions are known to hold. The improved estimators are obtained under the quadratic loss function     

Some extensions of anova techniques to location-scale models

In the usual analysis of variance (ANOVA) framework, the different distributions being compared are assumed to differ only in location so that the various measures of comparison are based on these location parameters only. The shift functions, introduced by Doksum (1974, provide a natural basis for extending some of the ANOVA techniques to nonlinear model so We consider a location-scale model and discuss several measures for comparing the various populations. These measures have intuitive interpretations in "control-treatments" situations. We develop various estimation procedures and discuss their large sample properties. Asymptotically efficient multiple comparison procedures are also considered.     

Sequential Estimation of a Continuous Distribution Function from Delayed Observations

The problem of sequentially estimating a continuous distribution function is considered in the case when the observations become available at random times. A certain class of sequential estimation procedures which are composed of optimal stopping time and sequential minimum risk invariant estimator of a continuous distribution function is obtained under a nonparametric invariant loss function and with the observation cost determined by a convex function of the moment of stopping and the number of observations up to this moment.     

Balanced Bayesian LASSO for heavy tails

Regression procedures are not only hindered by large p and small n, but can also suffer in cases when outliers are present or the data generating mechanisms are heavy tailed. Since the penalized estimates like the least absolute shrinkage and selection operator (LASSO) are equipped to deal with the large p small n by encouraging sparsity, we combine a LASSO type penalty with the absolute deviation loss function, instead of the standard least squares loss, to handle the presence of outliers and heavy tails. The model is cast in a Bayesian setting and a Gibbs sampler is derived to efficiently sample from the posterior distribution. We compare our method to existing methods in a simulation study as well as on a prostate cancer data set and a base deficit data set from trauma patients.     

Further applications of bias to discriminant analysis

This article extends the biased minimum x² rule to the unequal covariance matrix case and to the case of several populations, The biased procedure is shown to improve the performance of the commonly used classification procedures. Results of sampling experiments over a broad range of conditions are provided to demonstrate this improvement.     

Use of asymmetric loss functions in sequential estimation problems for multiple linear regression

When estimating in a practical situation, asymmetric loss functions are preferred over squared error loss functions, as the former is more appropriate than the latter in many estimation problems. We consider here the problem of fixed precision point estimation of a linear parametric function in beta for the multiple linear regression model using asymmetric loss functions. Due to the presence of nuissance parameters, the sample size for the estimation problem is not known beforehand and hence we take the recourse of adaptive multistage sampling methodologies. We discuss here some multistage sampling techniques and compare the performances of these methodologies using simulation runs. The implementation of the codes for our proposed models is accomplished utilizing MATLAB 7.0.1 program run on a Pentium IV machine. Finally, we highlight the significance of such asymmetric loss functions with few practical examples.     

Asymptotics of cross-validated risk estimation in estimator selection and performance assessment

Risk estimation is an important statistical question for the purposes of selecting a good estimator (i.e., model selection) and assessing its performance (i.e., estimating generalization error). This article introduces a general framework for cross-validation and derives distributional properties of cross-validated risk estimators in the context of estimator selection and performance assessment. Arbitrary classes of estimators are considered, including density estimators and predictors for both continuous and polychotomous outcomes. Results are provided for general full data loss functions (e.g., absolute and squared error, indicator, negative log density). A broad definition of cross-validation is used in order to cover leave-one-out cross-validation, V-fold cross-validation, Monte Carlo cross-validation, and bootstrap procedures. For estimator selection, finite sample risk bounds are derived and applied to establish the asymptotic optimality of cross-validation, in the sense that a selector based on a cross-validated risk estimator performs asymptotically as well as an optimal oracle selector based on the risk under the true, unknown data generating distribution. The asymptotic results are derived under the assumption that the size of the validation sets converges to infinity and hence do not cover leave-one-out cross-validation. For performance assessment, cross-validated risk estimators are shown to be consistent and asymptotically linear for the risk under the true data generating distribution and confidence intervals are derived for this unknown risk. Unlike previously published results, the theorems derived in this and our related articles apply to general data generating distributions, loss functions (i.e., parameters), estimators, and cross-validation procedures.     

A note on the common location parameter of several exponential populations

The problem of estimation of an unknown common location parameter of several exponential populations with unknown and possibly unequal scale parameters is considered. A wide class of estimators, including both a modified maximum likelihood estimator (MLE), and the uniformly minimum variance unbiased estimator (Umvue) proposed by ghosh and razmpour(1984), is obtained under a class of convex loss functions.     

Predictive analysis for joint progressive censoring plans: a Bayesian approach

Comparative lifetime experiments are of particular importance in production processes when one wishes to determine the relative merits of several competing products with regard to their reliability. This paper confines itself to the data obtained by running a joint progressive Type-II censoring plan on samples in a combined manner. The problem of Bayesian predicting failure times of surviving units is discussed in details when parent populations are exponential. Two real data sets are analyzed in order to illustrate all the inferential procedures developed here. When destructive experiments under a censoring scheme finished, the researchers are usually interested to estimate remaining lifetimes of surviving units for sequel experiments. Findings of this paper are useful for these purposes specially when samples are non-homogeneous such as those taken from industrial storages.     

Comparison of linex and quadratic bayes estimators foe the rayleigh distribution

In this paper, the Bayes estimators for the parameter, the reliability function, and failure rate function of the Rayleigh distribution are obtained when based on complete or type II censored samples. Some types of the linex loss function are used. Comparieons in terms of risks of those under linex loss and squared error loss function with Bayes estimators relative to squared error loss function are made, Numerical example and simulation example are included.     

Preliminary test and Bayes Estimation of a Location Parameter Under Blinex Loss

In this article, the preliminary test estimator is considered under the BLINEX loss function. The problem under consideration is the estimation of the location parameter from a normal distribution. The risk under the null hypothesis for the preliminary test estimator, the exact risk function for restricted maximum likelihood and approximated risk function for the unrestricted maximum likelihood estimator, are derived under BLINEX loss and the different risk structures are compared to one another both analytically and computationally. As a motivation on the use of BLINEX rather than LINEX, the risk for the preliminary test estimator under BLINEX loss is compared to the risk of the preliminary test estimator under LINEX loss and it is shown that the LINEX expected loss is higher than BLINEX expected loss. Furthermore, two feasible Bayes estimators are derived under BLINEX loss, and a feasible Bayes preliminary test estimator is defined and compared to the classical preliminary test estimator.     

The reflected normal loss function

The use of loss functions in quality assurance has grown steadily with the introduction of Taguchi's philosophy. The quadratic loss function has been used by decision-theoretic statisticians and economists for many years. Taguchi uses a modified form of the quadratic loss function to demonstrate the need to consider proximity to the target while assessing quality. Several authors have suggested that the traditional quadratic loss function is inadequate for assessing quality and quality improvement. A new, easily understood loss function, based on a reflection of the normal density function, is presented, and some associated statistical properties discussed.     

Estimation of multiple gamma scale-parameters: bayes estimation subject to uniform domination

Simultaneous estimation of p gamma scale-parameters is considered under squared-error loss. The problem of minimizing, subject to uniform risk domination, the Bayes risk (or more generally the posterior expected loss) against certain conjugate or mixtures of conjugate priors is considered. Rather surprisingly, it is shown that the minimization can be done conditionally, thus avoiding variational arguments. Relative savings loss (and a posterior version thereof) are found, and it is found that in the most favorable situations, Bayesian robustness can be achieved without sacrificing substantial subjective Bayesian gains.     

Designing selection experiments with bernoulli populations

This paper is concerned with a fixed size subset selection problem for Bernoulli populations in the framework of the indifference zone approach. The goal is to select s populationswhich contain at least c of those with the t largest success probabilities. In order to control the probability of correct selection over the preference zone extensive tables of exact minimum sample sizes have been prepared to implement the single-stage procedure generalized from the well-known Sobel-Huyett procedure. It is shown how the tables can also be employed to design certain closedsequential procedures. These procedures curtail the sampling process of the single-stage procedureand may differ in their sampling rules. Two procedures working with play-the-winner rules are described in detail     

Posterior Regret Γ-Minimax Estimation and Prediction with Applications on k-Records Data Under Entropy Loss Function

Robust Bayesian analysis is connected with the effect of changing a prior within a class Γ instead of being specified exactly. The multiplicity of prior leads to a collection or a range of Bayes actions. It is interesting not only to investigate the range of estimators but also to recommend the optimal procedures. In this article, we deal with posterior regret Γ-minimax (PRGM) estimation and prediction of an unknown parameter θ and a value of a random variable Y under entropy loss function. Applications for k-records such as estimation and prediction problems are discussed.     

Bayes Predictor of One-Parameter Exponential Family Type Population Mean Under Balanced Loss Function

We obtain a Bayes predictor and a Bayes prediction risk of the mean of a finite population relative to the balanced loss function. The predictive expected losses associated with classical and standard Bayes predictors are derived and compared with that of a Bayes predictor under a balanced loss function. Specific expressions for a regular exponential family distributed superpopulation are presented and illustrated for some well-known superpopulations.     

Bayesian inference based on a jointly type-II censored sample from two exponential populations

In this paper, based on a jointly type-II censored sample from two exponential populations, the Bayesian inference for the two unknown parameters are developed with the use of squared-error, linear-exponential and general entropy loss functions. The problem of predicting the future failure times, both point and interval prediction, based on the observed joint type-II censored data, is also addressed from a Bayesian viewpoint. A Monte Carlo simulation study is conducted to compare the Bayesian estimators with the maximum likelihood estimator developed by Balakrishnan and Rasouli [Exact likelihood inference for two exponential populations under joint type-II censoring. Comput Stat Data Anal. 2008;52:2725–2738]. Finally, a numerical example is utilized for the purpose of illustration.