Simultaneous equivariant estimation of the parameters of linear models

Consider a family of distributions which is invariant under a group of transformations. In this paper, we define an optimality criterion with respect to an arbitrary convex loss function and we prove a characterization theorem for an equivariant estimator to be optimal. Then we consider a linear model Y=Xβ+ε, in which ε has a multivariate distribution with mean vector zero and has a density belonging to a scale family with scale parameter σ. Also we assume that the underlying family of distributions is invariant with respect to a certain group of transformations. First, we find the class of all equivariant estimators of regression parameters and the powers of σ. By using the characterization theorem we discuss the simultaneous equivariant estimation of the parameters of the linear model.     

A robust support vector machine for labeling errors

Support vector machine (SVM) is sparse in that its classifier is expressed as a linear combination of only a few support vectors (SVs). Whenever an outlier is included as an SV in the classifier, the outlier may have serious impact on the estimated decision function. In this article, we propose a robust loss function that is convex. Our learning algorithm is more robust to outliers than SVM. Also the convexity of our loss function permits an efficient solution path algorithm. Through simulated and real data analysis, we illustrate that our method can be useful in the presence of labeling errors.     

Pareto-optimal reinsurance for both the insurer and the reinsurer with general premium principles

Abstract

In this paper, we study Pareto-optimal reinsurance policies from the perspectives of an insurer and a reinsurer, assuming reinsurance premium principles satisfy risk loading and stop-loss ordering preserving. By geometric approach, we determine the forms of the optimal policies among two classes of ceded loss functions, the class of increasing convex ceded loss functions and the class that the constraints on both ceded and retained loss functions are relaxed to increasing functions. Then we demonstrate the applicability of our results by giving the parameters of the optimal ceded loss functions under Dutch premium principle and Wang’s premium principle.     

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.     

Conditional ¡-minimax actions under convex losses

The problem of making decisions about an unknown parameter is examined under a convex loss function, when its prior distribution may not be uniquely specified on the basis of the available information. Following the conditional ¡-minimax approach, an action is chosen such that it minimises the maximum posterior expected loss. The characterising properties of such an action, called a conditional ¡-minimax action, are found and illustrated in three examples.     

Minimax rules under zero–one loss for a restricted location parameter

In this paper, we obtain minimax and near-minimax nonrandomized decision rules under zero–one loss for a restricted location parameter of an absolutely continuous distribution. Two types of rules are addressed: monotone and nonmonotone. A complete-class theorem is proved for the monotone case. This theorem extends the previous work of Zeytinoglu and Mintz (1984) to the case of 2e-MLR sampling distributions. A class of continuous monotone nondecreasing rules is defined. This class contains the monotone minimax rules developed in this paper. It is shown that each rule in this class is Bayes with respect to nondenumerably many priors. A procedure for generating these priors is presented. Nonmonotone near-minimax almost-equalizer rules are derived for problems characterized by non-2e-MLR distributions. The derivation is based on the evaluation of a distribution-dependent function Q_c. The methodological importance of this function is that it is used to unify the discrete- and continuous-parameter problems, and to obtain a lower bound on the minimax risk for the non-2e-MLR case.     

Construction of optimal designs in random coefficient regression models

We consider the problem of optimal experimental design in random coefficient regression models with respect to a quadratic loss function. By application of WHITTLE'S general equivalence theorem we obtain the structure of optimal designs. An alogrithm is given which allows, under certain assumptions, the construction of the information matrix of an optimal design. Moreover, we give conditions on the equivalence of optimal designs with respect to optimality criteria which are analogous to usual A-D- and _E/-optimality.     

Quadratic reflected BSDEs and related obstacle problems for PDEs

Abstract

In this paper, we study a kind of reflected backward stochastic differential equations (BSDEs) whose generators are of quadratic growth in z and linear growth in y. We first give an estimate of solutions to such reflected BSDEs. Then under the condition that the generators are convex with respect to z, we can obtain a comparison theorem, which implies the uniqueness of solutions for this kind of reflected BSDEs. Besides, the assumption of convexity also leads to a stability property in the spirit of above estimate. We further establish the nonlinear Feynman-Kac formula of the related obstacle problems for partial differential equations (PDEs) in our framework. At last, a numerical example is given to illustrate the applications of our theoretical results, as well as its connection with an optimal stopping time problem.     

Optimal Partitioning of Probability Distributions under General Convex Loss Functions in Selective Assembly

Selective assembly is an effective approach for improving the quality of a product which is composed of two mating components. This article studies optimal partitioning of the dimensional distributions of the components in selective assembly. It extends previous results for squared error loss function to cover general convex loss functions, including asymmetric convex loss functions. Equations for the optimal partition are derived. Assuming that the density function of the dimensional distribution is log-concave, uniqueness of solutions is established and some properties of the optimal partition are shown. Some numerical results compare the optimal partition with some heuristic partitioning schemes.     

A robust principal component analysis

This work is concerned with robustness in Principal Component Analysis (PCA). The approach, which we adopt here, is to replace the criterion of least squares by another criterion based on a convex and sufficiently differentiable loss function ρ. Using this criterion we propose a robust estimate of the location vector and introduce an orthogonality with respect to (w.r.t.) ρ in order to define the different steps of a PCA. The influence functions of a vector mean and principal vectors are developed in order to provide method for obtaining a robust PCA. The practical procedure is based on an alternative-steps algorithm.     

Tail Exactness of Multivariate Saddlepoint Approximations

We consider a log-concave density f in R ^m satisfying certain weak conditions, particularly on the Hessian matrix of log f . For such a density, we prove tail exactness of the multivariate saddlepoint approximation. The proof is based on a local limit theorem for the exponential family generated by f . However, the result refers not to asymptotic behaviour under repeated sampling, but to a limiting property at the boundary of the domain of f . Our approach does not apply any complex analysis, but relies totally on convex analysis and exponential models arguments.     

Wahrscheinlichkeitstransformationen in Entscheidungsmodellen bei linearer partieller Information

In decision models with linear partial information the incomplete knowledge of probabilities is expressed by means of convex polyhedra. The transformation of incompletely specified probabilities by Bayes' theorem, the law of total probabilities, and the multiplication law for independent events is analysed with reference to recent results on this topic. As a new result the incorrectness of a central theorem in Kofler/Menges (1976) is shown. But the theory is partly reestablished by an equivalence between two optimization problems, which is derived from bilinearity.     

A necessary and sufficient condition for an estimator to be optimal

A necessary condition for an estimator δ₀ to be optimal in a class Δ of estimators is derived. Thic condition becomes also sufficient when the loss function is convex.     

Risk of a homoscedasticity pre-test estimator of the regression scale under LINEX loss

In this paper we consider the risk of an estimator of the error variance after a pre-test for homoscedasticity of the variances in the two-sample heteroscedastic linear regression model. This particular pre-test problem has been well investigated but always under the restrictive assumption of a squared error loss function. We consider an asymmetric loss function — the LINEX loss function — and derive the exact risks of various estimators of the error variance.     

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.     

Robust Bayesian prediction under a general linear-exponential posterior risk function and its application in finite population

We consider robust Bayesian prediction of a function of unobserved data based on observed data under an asymmetric loss function. Under a general linear-exponential posterior risk function, the posterior regret gamma-minimax (PRGM), conditional gamma-minimax (CGM), and most stable (MS) predictors are obtained when the prior distribution belongs to a general class of prior distributions. We use this general form to find the PRGM, CGM, and MS predictors of a general linear combination of the finite population values under LINEX loss function on the basis of two classes of priors in a normal model. Also, under the general ε-contamination class of prior distributions, the PRGM predictor of a general linear combination of the finite population values is obtained. Finally, we provide a real-life example to predict a finite population mean and compare the estimated risk and risk bias of the obtained predictors under the LINEX loss function by a simulation study.     

Static Hedging of Geometric Average Asian Options with Standard Options

In this article, we first establish a theorem that represents the price of an Asian option in terms of standard European options with a shorter term and different strikes. Then using Gauss–Hermite numerical integration, we discretize our theorem so as to use Monte Carlo simulation to examine the error of the static hedging under the Black–Scholes model and the Merton jump-diffusion model. For ease of comparison, we also provide the error of the dynamic hedging. The numerical results show that the static hedging strategy performs better than the dynamic one under both models.     

Decision-theoretic estimation of generalized variance and generalized precision

Consider a random data matrix X=(X₁,...,X_k):pXk with independent columns [sathik] and an independent p X p Wishart matrix [sathik]. Estimators dominating the best affine equivariant estimators of [sathik] are obtained under four types of loss functions. Improved estimators (Testimators) of generalized variance and generalized precision are also considered under convex entropy loss (CEL).     

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.     

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.