Robustness of Bayes Prediction Under Error-in-Variables Superpopulation Model

In this article, we consider Bayes prediction in a finite population under the simple location error-in-variables superpopulation model. Bayes predictor of the finite population mean under Zellner's balanced loss function and the corresponding relative losses and relative savings loss are derived. The prior distribution of the unknown location parameter of the model is assumed to have a non-normal distribution belonging to the class of Edgeworth series distributions. Effects of non normality of the “true” prior distribution and that of a possible misspecification of the loss function on the Bayes predictor are illustrated for a hypothetical population.     

An Appreciation of Balanced Loss Functions Via Regret Loss

We examine balanced loss functions, which account for both estimation error and goodness of fit (or proximity to a “target” estimator), in terms of their regret losses, providing new insight and interpretations. This also shows a connection between quadratic balanced loss and usual quadratic loss, which easily converts frequentist and Bayesian results for quadratic loss to related results for quadratic balanced loss and vice versa. Some implications of these results for Stein-rule estimators under linear regression are discussed. We also examine regret losses corresponding to several non-quadratic balanced loss functions.     

Bayes Estimator of Inverse Gaussian Parameters Under General Entropy Loss Function Using Lindley's Approximation

In this article, we obtained Bayes estimators of parameters of Inverse Gaussian distributions under asymmetric loss function using Lindley's Approximation (L-Approximation). The proposed estimators have been compared with the corresponding estimators obtained under symmetric loss function and MLE for their risks. This comparison is illustrated using Monte-Carlo study of 2,000 simulated sample from the Inverse Gaussian distribution.     

A Loss Function Model for the Restoration of Grey Level Images

Common loss functions used for the restoration of grey scale images include the zero–one loss and the sum of squared errors. The corresponding estimators, the posterior mode and the posterior marginal mean, are optimal Bayes estimators with respect to their way of measuring the loss for different error configurations. However, both these loss functions have a fundamental weakness: the loss does not depend on the spatial structure of the errors. This is important because a systematic structure in the errors can lead to misinterpretation of the estimated image. We propose a new loss function that also penalizes strong local sample covariance in the error and we discuss how the optimal Bayes estimator can be estimated using a two-step Markov chain Monte Carlo and simulated annealing algorithm. We present simulation results for some artificial data which show improvement with respect to small structures in the image.     

Bayesian Estimation for Poisson-exponential Model under Progressive Type-II Censoring Data with Binomial Removal and Its Application to Ovarian Cancer Data

In this article, we propose Maximum likelihood estimators (MLEs) and Bayes estimators of parameters of Poisson-exponential distribution (PED) under General entropy loss function (GELF) and Squared error loss function (SELF) for Progressive type-II censored data with binomial removals (PT-II CBRs). The MLEs and corresponding Bayes estimators are compared in terms of their risks based on simulated samples from PED. The proposed methodology is illustrated on a real dataset of ovarian cancer.     

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.     

Bayes prediction of Poisson regression superpopulation mean with a non gamma prior

We consider Khamis' (1960) Laguerre expansion with gamma weight function as a class of “near-gamma” priors (K-prior) to obtain the Bayes predictor of a finite population mean under the Poisson regression superpopulation model using Zellner's balanced loss function (BLF). Kullback–Leibler (K-L) distance between gamma and some K-priors is tabulated to examine the quantitative prior robustness. Some numerical investigations are also conducted to illustrate the effects of a change in skewness and/or kurtosis on the Bayes predictor and the corresponding minimal Bayes predictive expected loss (MBPEL). Loss robustness with respect to the class of BLFs is also examined in terms of relative savings loss (RSL).     

A Unified Approach to Prior and Loss Robustness

ABSTRACT

We introduce a new statistical framework in order to study Bayesian loss robustness under classes of priors distributions, thus unifying both concepts of robustness. We propose measures that capture variation with respect to both prior selection and selection of loss function and explore general properties of these measures. We illustrate the approach for the continuous exponential family. Robustness in this context is studied first with respect to prior selection where we consider several classes of priors for the parameter of interest, including unimodal and symmetric and unimodal with positive support. After prior variation has been measured we investigate robustness to loss function, using Hellinger and Linex (Linear Exponential) classes of loss functions. The methods are applied to standard examples.     

Minimum Risk Estimation of Scalar Means under Convex Combination of Loss Functions

We derive the minimum risk estimates of the scalar means for Normal, Exponential, and Gamma distributions, under the convex combination of SEL and LINEX loss functions. The functional forms of the proposed estimates for the three examples are general in nature, and for the boundary conditions provide us with the corresponding estimates under SEL and LINEX loss, respectively. We authenticate our proposed models using different iterative as well as meta-heuristic techniques, and through extensive simulation as well as application of live data sets, validate the efficacy of our proposed results.     

Second-order asymptotic comparison of estimators under φ-divergence loss

Our main concern is on the second-order asymptotic optimality problem of estimators. The φ-divergence loss is used as a criterion for evaluating the performance of estimators. In the comparison problem of any two estimators, the condition that one estimator dominates another estimator under the φ-divergence risk is given by evaluating the second-order term in the difference between the risks. As a result, it is proved that the condition is characterized by a peculiar value of the φ-divergence loss, which is called the divergence-loss coefficient. Furthermore, it is shown that the comparison based on the φ-divergence loss does not correspond with that based on any standard loss functions including the mean squared error, the absolute loss and the 0-1 loss. In addition, a necessary and sufficient condition for an estimator to be second-order admissible is derived.     

Minimax estimation of a probability of success under LINEX loss

Minimax estimation of a binomial probability under LINEX loss function is considered. It is shown that no equalizer estimator is available in the statistical decision problem under consideration. It is pointed out that the problem can be solved by determining the Bayes estimator with respect to a least favorable distribution having finite support. In this situation, the optimal estimator and the least favorable distribution can be determined only by using numerical methods. Some properties of the minimax estimators and the corresponding least favorable prior distributions are provided depending on the parameters of the loss function. The properties presented are exploited in computing the minimax estimators and the least favorable distributions. The results obtained can be applied to determine minimax estimators of a cumulative distribution function and minimax estimators of a survival function.     

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.     

Logistic retainment interval dose exploration design for Phase I clinical trials of cytotoxic agents

Phase I studies of a cytotoxic agent often aim to identify the dose that provides an investigator specified target dose-limiting toxicity (DLT) probability. In practice, an initial cohort receives a dose with a putative low DLT probability, and subsequent dosing follows by consecutively deciding whether to retain the current dose, escalate to the adjacent higher dose, or de-escalate to the adjacent lower dose. This article proposes a Phase I design derived using a Bayesian decision-theoretic approach to this sequential decision-making process. The design consecutively chooses the action that minimizes posterior expected loss where the loss reflects the distance on the log-odds scale between the target and the DLT probability of the dose that would be given to the next cohort under the corresponding action. A logistic model is assumed for the log odds of a DLT at the current dose with a weakly informative t-distribution prior centered at the target. The key design parameters are the pre-specified odds ratios for the DLT probabilities at the adjacent higher and lower doses. Dosing rules may be pre-tabulated, as these only depend on the outcomes at the current dose, which greatly facilitates implementation. The recommended default version of the proposed design improves dose selection relative to many established designs across a variety of scenarios.     

Bayesian inference and prediction of the Pareto distribution based on ordered ranked set sampling

In this paper, order statistics from independent and non identically distributed random variables is used to obtain ordered ranked set sampling (ORSS). Bayesian inference of unknown parameters under a squared error loss function of the Pareto distribution is determined. We compute the minimum posterior expected loss (the posterior risk) of the derived estimates and compare them with those based on the corresponding simple random sample (SRS) to assess the efficiency of the obtained estimates. Two-sample Bayesian prediction for future observations is introduced by using SRS and ORSS for one- and m-cycle. A simulation study and real data are applied to show the proposed results.     

Oracle Inequalities for Convex Loss Functions with Nonlinear Targets

This article considers penalized empirical loss minimization of convex loss functions with unknown target functions. Using the elastic net penalty, of which the Least Absolute Shrinkage and Selection Operator (Lasso) is a special case, we establish a finite sample oracle inequality which bounds the loss of our estimator from above with high probability. If the unknown target is linear, this inequality also provides an upper bound of the estimation error of the estimated parameter vector. Next, we use the non-asymptotic results to show that the excess loss of our estimator is asymptotically of the same order as that of the oracle. If the target is linear, we give sufficient conditions for consistency of the estimated parameter vector. We briefly discuss how a thresholded version of our estimator can be used to perform consistent variable selection. We give two examples of loss functions covered by our framework.     

Testimator of the scale parameter of the exponential distribution using linex loss function

The present paper investigates the properties of a testimator of scale of an exponential distribution under Linex loss function. The risk function of testimator is derived and compared with that of an admissible estimator relative to Linex loss function. The shrinkage testimator is proposed which is the extension of testimator and its properties have been discussed. The level of significance of testimator is decided on the basis of Akaike information criterion following Hirano (1977, 1978). It is found that the testimator and shrinkage testimator dominates the admissible estimator in terms of risk in certain parametric space.     

Assessing point forecast accuracy by stochastic error distance

ABSTRACT

We propose point forecast accuracy measures based directly on distance of the forecast-error c.d.f. from the unit step function at 0 (“stochastic error distance,” or SED). We provide a precise characterization of the relationship between SED and standard predictive loss functions, and we show that all such loss functions can be written as weighted SEDs. The leading case is absolute error loss. Among other things, this suggests shifting attention away from conditional-mean forecasts and toward conditional-median forecasts.     

Minimax property of stein's estimator

Stein's estimator and some other estimators of the mean of a K-variate normal distribution are known to dominate the maximum likelihood estimator under quadratic loss for K > 3, and are therefore minimax. In this paper it is shown that the minimax property of Stein's rule is preserved with respect to a generalized loss function.     

On the Bayesian inference of Kumaraswamy distributions based on censored samples

In this article we discuss Bayesian estimation of Kumaraswamy distributions based on three different types of censored samples. We obtain Bayes estimates of the model parameters using two different types of loss functions (LINEX and Quadratic) under each censoring scheme (left censoring, singly type-II censoring, and doubly type-II censoring) using Monte Carlo simulation study with posterior risk plots for each different choices of the model parameters. Also, detailed discussion regarding elicitation of the hyperparameters under the dependent prior setup is discussed. If one of the shape parameters is known then closed form expressions of the Bayes estimates corresponding to posterior risk under both the loss functions are available. To provide the efficacy of the proposed method, a simulation study is conducted and the performance of the estimation is quite interesting. For illustrative purpose, real-life data are considered.