On Bayes Linear Unbiased Estimator Under the Balanced Loss Function

In this paper, the Bayes linear unbiased estimator (Bayes LUE) is derived under the balanced loss function. Moreover, the superiority of Bayes LUE over ordinary least square estimator is studied under the mean square error matrix criterion and Pitman closeness criterion. Furthermore, we compare Bayes LUE under the balanced loss function with Bayes LUE under the quadratic loss function.     

Estimation of an exponential quantile under pitman's measure of closeness

We consider the problem of estimating a quantile of an exponential distribution with unknown location and scale parameters under Pitman's measure of closeness (PMC). The loss function is required to satisfy some mild conditions but is otherwise arbitrary. An optimal estimator is obtained in the class of location-scale-equivariant estimators, and its admissibility in the sense of PMC is investigated.     

Robust importance sampling for some typical types of utility-based shortfall risk measures using exponential twisting and kernel density techniques

A robust algorithm for utility-based shortfall risk (UBSR) measures is developed by combining the kernel density estimation with importance sampling (IS) using exponential twisting techniques. The optimal bandwidth of the kernel density is obtained by minimizing the mean square error of the estimators. Variance is reduced by IS where exponential twisting is applied to determine the optimal IS distribution. Conditions for the best distribution parameters are derived based on the piecewise polynomial loss function and the exponential loss function. The proposed method not only solves the problem of sampling from the kernel density but also reduces the variance of the UBSR estimator.     

Estimation for a scale parameter with known coefficient of variation

A loss function proposed by Wasan (1970) is well-fitted for a measure of inaccuracy for an estimator of a scale parameter of a distribution defined onR ⁺=(0, ∞). We refer to this loss function as the K-loss function. A relationship between the K-loss and squared error loss functions is discussed. And an optimal estimator for a scale parameter with known coefficient of variation under the K-loss function is presented.     

The performance of the adaptive optimal estimator under the extended balanced loss function

The adaptive optimal estimator of Farebrother (1975) is discussed by many authors, but the goodness of fitted model criterion that is used to investigate the performance of estimators is quite often ignored. Shalabh, Toutenburg, and Heumann (2009) proposed the extended balanced loss function in which the mean squared error and the Zellner's balanced loss function are just special cases of it. In this paper, we discuss the performance of the adaptive optimal estimator of Farebrother (1975) under the extended balanced loss function. Moreover, a Monte Carlo simulation experiment is conducted to examine the performance of the estimator in finite samples.     

Some Pitman closeness properties pertinent to symmetric populations

In this paper, we focus on Pitman closeness probabilities when the estimators are symmetrically distributed about the unknown parameter θ. We first consider two symmetric estimators θ?₁ and θ?₂ and obtain necessary and sufficient conditions for θ?₁ to be Pitman closer to the common median θ than θ?₂. We then establish some properties in the context of estimation under the Pitman closeness criterion. We define Pitman closeness probability which measures the frequency with which an individual order statistic is Pitman closer to θ than some symmetric estimator. We show that, for symmetric populations, the sample median is Pitman closer to the population median than any other independent and symmetrically distributed estimator of θ. Finally, we discuss the use of Pitman closeness probabilities in the determination of an optimal ranked set sampling scheme (denoted by RSS) for the estimation of the population median when the underlying distribution is symmetric. We show that the best RSS scheme from symmetric populations in the sense of Pitman closeness is the median and randomized median RSS for the cases of odd and even sample sizes, respectively.     

Kernel Density Estimation with Generalized Binning

We propose kernel density estimators based on prebinned data. We use generalized binning schemes based on the quantiles points of a certain auxiliary distribution function. Therein the uniform distribution corresponds to usual binning. The statistical accuracy of the resulting kernel estimators is studied, i.e. we derive mean squared error results for the closeness of these estimators to both the true function and the kernel estimator based on the original data set. Our results show the influence of the choice of the auxiliary density on the binned kernel estimators and they reveal that non-uniform binning can be worthwhile.     

Robust joint estimation of location and scale parameters in ranked set samples

A robust estimator is developed for the location and scale parameters of a location-scale family. The estimator is defined as the minimizer of a minimum distance function that measures the distance between the ranked set sample empirical cumulative distribution function and a possibly misspecified target model. We show that the estimator is asymptotically normal, robust, and has high efficiency with respect to its competitors in literature. It is also shown that the location estimator is consistent within the class of all symmetric distributions whereas the scale estimator is Fisher consistent at the true target model. The paper also considers an optimal allocation procedure that does not introduce any bias due to judgment error classification. It is shown that this allocation procedure is equivalent to Neyman allocation. A numerical efficiency comparison is provided.     

Bayesian inference using product of spacings function for Progressive hybrid Type-I censoring scheme

This article is devoted to the development of product of spacings estimator for a Progressive hybrid Type-I censoring scheme with binomial removals. The experimental units are assumed to follow inverse Lindley distribution. We propose a Bayes estimator of associated scale parameter based on the product of spacings function and simultaneously compare it with that obtained under a usual Bayesian estimation procedure. The estimators are obtained under the squared error loss function along with corresponding HP intervals evaluated by using the Markov chain Monte-Carlo technique. The classical product of spacings estimator has also been derived and compared with the maximum likelihood estimator in addition to 95% average asymptotic confidence intervals. The applicability of the proposed methods is demonstrated by analysing a real data of guinea pigs affected with tuberculosis for the considered censoring scheme.     

On risk comparisons of some estimators in a linear regression model under inagaki’S loss function and nonnormal error terms

In this paper we consider the risk performances of some estimators for both location and scale parameters in a linear regression model under Inagaki’s loss function We prove that the pre-test estimator for location parameter is dominated by the Stein-rule estimator under Inagaki’s loss function when the distribution of error terms is expressed by the scale mixture of normal distribution and the variance of error terms is unknown.. It is an extension of the results in Nagata (1983) to our situation Also we perform numerical calculations to draw the shapes of the risks.     

A Note on Estimation of a Distribution Function in a Nonparametric Set-up Using Stein’s Shrinkage Estimation Technique

Based on Stein’s famous shrinkage estimation of a multivariate normal distribution, we propose a new type of estimators of the distribution function of a random variable in a nonparametric setup. The proposed estimators are then compared with the empirical distribution function, which is the best equivariant estimator under a well-known loss function. Our extensive simulation study shows that our proposed estimators can perform better for moderate to large sample sizes.     

The concept of risk unbiasedness in statistical prediction

This paper extends the concept of risk unbiasedness for applying to statistical prediction and nonstandard inference problems, by formalizing the idea that a risk unbiased predictor should be at least as close to the “true” predictant as to any “wrong” predictant, on the average. A novel aspect of our approach is measuring closeness between a predicted value and the predictant by a regret function, derived suitably from the given loss function. The general concept is more relevant than mean unbiasedness, especially for asymmetric loss functions. For squared error loss, we present a method for deriving best (minimum risk) risk unbiased predictors when the regression function is linear in a function of the parameters. We derive a Rao–Blackwell type result for a class of loss functions that includes squared error and LINEX losses as special cases. For location-scale families, we prove that if a unique best risk unbiased predictor exists, then it is equivariant. The concepts and results are illustrated with several examples. One interesting finding is that in some problems a best unbiased predictor does not exist, but a best risk unbiased predictor can be obtained. Thus, risk unbiasedness can be a useful tool for selecting a predictor.     

A note on new feasible generalized ridge regression estimator under the linex loss function

In this paper, using the asymmetric LINEX loss function we derive and numerically evaluate the risk function of the new feasible ridge regression estimator.We also examine the risk performance of this estimator when the LINEX loss function is used.     

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.     

Risk performance of a pre-test estimator for normal variance with the Stein-variance estimator under the LINEX loss function

In this paper, we derive the exact formula of the risk function of a pre-test estimator for normal variance with the Stein-variance (PTSV) estimator when the asymmetric LINEX loss function is used. Fixing the critical value of the pre-test to unity which is a suggested critical value in some sense, we examine numerically the risk performance of the PTSV estimator based on the risk function derived. Our numerical results show that although the PTSV estimator does not dominate the usual variance estimator when under-estimation is more severe than over-estimation, the PTSV estimator dominates the usual variance estimator when over-estimation is more severe. It is also shown that the dominance of the PTSV estimator over the original Stein-variance estimator is robust to the extension from the quadratic loss function to the LINEX loss function.     

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.     

A semiparametric estimator of the distribution function of a variable measured with error

The estimation of the distribution functon of a random variable X measured with error is studied. Let the i-th observation on X be denoted by Y_iX_i+ε_i where ε_i is the measuremen error. Let {Y_i} (i=1,2,…,n) be a sample of independent observations. It is assumed that {X_i} and {∈_i} are mutually independent and each is identically distributed. As is standard in the literature for this problem, the distribution of e is assumed known in the development of the methodology. In practice, the measurement error distribution is estimated from replicate observations.

The proposed semiparametric estimator is derived by estimating the quantises of X on a set of n transformed V-values and smoothing the estimated quantiles using a spline function. The number of parameters of the spline function is determined by the data with a simple criterion, such as AIC. In a simulation study, the semiparametric estimator dominates an optimal kernel estimator and a normal mixture estimator for a wide class of densities.

The proposed estimator is applied to estimate the distribution function of the mean pH value in a field plot. The density function of the measurement error is estimated from repeated measurements of the pH values in a plot, and is treated as known for the estimation of the distribution function of the mean pH value.     

Bayesian Estimation of Regression Coefficients Under Extended Balanced Loss Function

Appreciating the desirability of simultaneously using both the criteria of goodness of fitted model and clustering of estimates around true parameter values, an extended version of the balanced loss function is presented and the Bayesian estimation of regression coefficients is discussed. The thus obtained optimal estimator is then compared with the least squares estimator and posterior mean vector with respect to the criteria like posterior expected loss, Bayes risk, bias vector, mean squared error matrix and risk function.     

The Bayes rule of the variance parameter of the hierarchical normal and inverse gamma model under Stein's loss

For the variance parameter of the hierarchical normal and inverse gamma model, we analytically calculate the Bayes rule (estimator) with respect to a prior distribution IG (alpha, beta) under Stein's loss function. This estimator minimizes the posterior expected Stein's loss (PESL). We also analytically calculate the Bayes rule and the PESL under the squared error loss. Finally, the numerical simulations exemplify that the PESLs depend only on alpha and the number of observations. The Bayes rules and PESLs under Stein's loss are unanimously smaller than those under the squared error loss.     

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.