Characterization of an optimal matrix estimator under convex loss function

Characterization of an optimal vector estimator and an optimal matrix estimator are obtained. In each case appropriate convex loss functions are considered. The results are illustrated through the problems of simultaneous unbiased estimation, simultaneous equivariant estimation and simultaneous unbiased prediction. Further an optimality criterion is proposed for matrix unbiased estimation and it is shown that the matrix unbiased estimation of a matrix parametric function and the minimum variance unbiased estimation of its components are equivalent.     

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.     

Estimating the common hazard rate of several exponential distributions

Abstract

In the present communication, we consider the estimation of the common hazard rate of several exponential distributions with unknown and unequal location parameters with a common scale parameter under a general class of bowl-shaped scale invariant loss functions. We have shown that the best affine equivariant estimator (BAEE) is inadmissible by deriving a non smooth improved estimator. Further, we have obtained a smooth estimator which improves upon the BAEE. As an application, we have obtained explicit expressions of improved estimators for special loss functions. Finally, a simulation study is carried out for numerically comparing the risk performance of various estimators.     

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.     

On the improved estimation of a function of the scale parameter of an exponential distribution based on doubly censored sample

In the present article, we have studied the estimation of entropy, that is, a function of scale parameter ln⁡σ of an exponential distribution based on doubly censored sample when the location parameter is restricted to positive real line. The estimation problem is studied under a general class of bowl-shaped non monotone location invariant loss functions. It is established that the best affine equivariant estimator (BAEE) is inadmissible by deriving an improved estimator. This estimator is non-smooth. Further, we have obtained a smooth improved estimator. A class of estimators is considered and sufficient conditions are derived under which these estimators improve upon the BAEE. In particular, using these results we have obtained the improved estimators for the squared error and the linex loss functions. Finally, we have compared the risk performance of the proposed estimators numerically. One data analysis has been performed for illustrative purposes.     

An asymptotic expansion of Wishart distribution when the population eigenvalues are infinitely dispersed

Takemura and Sheena [A. Takemura, Y. Sheena, Distribution of eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues are infinitely dispersed and its application to minimax estimation of covariance matrix, J. Multivariate Anal. 94 (2005) 271–299] derived the asymptotic joint distribution of the eigenvalues and the eigenvectors of a Wishart matrix when the population eigenvalues become infinitely dispersed. They also showed necessary conditions for an estimator of the population covariance matrix to be tail minimax for typical loss functions by calculating the asymptotic risk of the estimator. In this paper, we further examine those distributions and risks by means of an asymptotic expansion. We obtain the asymptotic expansion of the distribution function of relevant elements of the sample eigenvalues and eigenvectors. We also derive the asymptotic expansion of the risk function of a scale and orthogonally equivariant estimator with respect to Stein’s loss. As an application, we prove non-minimaxity of Stein’s and Haff’s estimators, which has been an open problem for a long time.     

Sufficient admissibility conditions for scale parameter estimator

Although there are a number of results available for the admissibility of the best translation equivariant estimator of the parameter, there is hardly any stated explicitly for the best scale equivariant estimator of the scale parameter. In this paper, we derive sufficient conditions for the admissibility of the scale parameter estimators and compara them. The derivations use the well known results due to Brown [1], Farrell [2], and Portnoy [3]. The loss function has been taken to be quadratic.     

Reference prior bayes estimator for bivariate normal covariance matrix with risk comparison

The explicit form of the reference prior bayes estimator due to Yang and Ber-ger (1994) for bivariate normal covariance matrix under entropy loss is given in terms of Legendre polynomials when degrees of freedom is even and in terms of hypergeometric functions in general case. The finite series expression of the density function of the ratio of latent roots of bivariate Wishart matrix is obtained and the exact risk is compared with those of James-Stein minimax estimator and other orthogonally equivariant estimators. It is found numerically that the reference prior bayes estimator has the smallest risk among the class of equivariant estimators compared, when the ratio of the largest to the smallest population latent roots of covariance matrix lies in the middle of the interval [1, ∞]. It has larger risk than that of James-Stein minimax estimator when the ratio is large. Moreover it has larger risk than that of MLE when, for instance, degrees of freedom is 20 and the ratio lies between 4 and 8.     

Estimating the common hazard rate of two exponential distributions with ordered location parameters

This paper is concerned with estimating the common hazard rate of two exponential distributions with unknown and ordered location parameters under a general class of bowl-shaped scale invariant loss functions. The inadmissibility of the best affine equivariant estimator is established by deriving an improved estimator. Another estimator is obtained which improves upon the best affine equivariant estimator. A class of improving estimators is derived using the integral expression of risk difference approach of Kubokawa [A unified approach to improving equivariant estimators. Ann Statist. 1994;22(1):290–299]. These results are applied to specific loss functions. It is further shown that these estimators can be derived for four important sampling schemes: (i) complete and i.i.d. sample, (ii) record values, (iii) type-II censoring, and (iv) progressive Type-II censoring. A simulation study is carried out for numerically comparing the risk performance of these proposed estimators.     

Pseudo-bayes estimation of multinomial proportions

In the simultaneous estimation of multinomial proportions, two estimators are developed which incorporate prior means and a prior parameter which reflects the accuracy of the prior means. These estimators possess substantially smaller risk than the standard estimator in a region of the parameter space and are much more robust than the conjugate Bayes estimator with respect to parameter values far from the prior mean. When vague prior information is available, these estimators and confidence regions derived from them appear to be attractive alternatives to the procedures based on the standard estimator.     

Strawderman-type estimators for a scale parameter with application to the exponential distribution

It is shown that Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique for estimating the variance of a normal distribution can be extended to estimating a general scale parameter in the presence of a nuisance parameter. Employing standard monotone likelihood ratio-type conditions, a new class of improved estimators for this scale parameter is derived under quadratic loss. By imposing an additional condition, a broader class of improved estimators is obtained. The dominating procedures are in form analogous to those in Strawderman [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198]. Application of the general results to the exponential distribution yields new sufficient conditions, other than those of Brewster and Zidek [1974. Improving on equivariant estimators. Ann. Statist. 2, 21–38] and Kubokawa [1994. A unified approach to improving equivariant estimators. Ann. Statist. 22, 290–299], for improving the best affine equivariant estimator of the scale parameter. A class of estimators satisfying the new conditions is constructed. The results shed new light on Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique.     

On the Simultaneous Estimation of Scale Parameters

This note is an extension of Das Gupta's results (1986) on the estimation of multiparameter gamma distribution. Consider p (p ? 2) independent positive random variables with possibly different scale-parameter densities. For the estimation of the powers of the scale parameters it is shown that the “best multiple estimator” is inadmissible with respect to a large class of weighted quadratic loss functions.     

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.     

Comparison of estimators of a dispersion matrix under generalized pitman nearness criterion

Some equivariant estimators of the dispersion matrix of a multivariate normal population are compared using the generalized Pitman nearness criterion based on the entropy loss function. It is shown that, under the group of lower triangular transformations, a best equivariant estimator does not exist. Existence of best estimators in certain subclasses are discussed and the performances of two commonly used estimators are compared. Some properties of central chi-square distributions are obtained and used to derive the main results.     

Estimation of parameters of inverse Weibull distribution and application to multi-component stress-strength model

The problem of estimation of the parameters of two-parameter inverse Weibull distributions has been considered. We establish existence and uniqueness of the maximum likelihood estimators of the scale and shape parameters. We derive Bayes estimators of the parameters under the entropy loss function. Hierarchical Bayes estimator, equivariant estimator and a class of minimax estimators are derived when shape parameter is known. Ordered Bayes estimators using information about second population are also derived. We investigate the reliability of multi-component stress-strength model using classical and Bayesian approaches. Risk comparison of the classical and Bayes estimators is done using Monte Carlo simulations. Applications of the proposed estimators are shown using real data sets.     

Estimation of the parameter in the problem of the nile

Estimation of the parameter in the problem of the Nile is treated as a decision problem with squared error loss, It is shown that the minimum risk scale equivariant estimator dominates the incomplete sufficient unbiased estimators considered by Iwase and Seto, Sharper bounds for the equivariant estimator are derived which may be used to obtain the values of the same from the sample with sufficient accuracy.     

Estimation of a Gamma Scale Parameter Under Asymmetric Squared Log Error Loss

Abstract

Estimation of scale parameter under the squared log error loss function is considered with restriction to the principle of invariance and risk unbiasedness. An explicit form of minimum risk scale-equivariant estimator under this loss is obtained. The admissibility and inadmissibility of a class of linear estimators of the form (cT + d) are considered, where T follows a gamma distribution with an unknown scale parameter η and a known shape parameter ν. This includes the admissibility of the minimum risk equivariant estimator on η (MRE).     

Estimation of the MSE matrix of the stein estimator

Uniformly minimum-variance unbiased (UMVU) estimators of the total risk and the mean-squared-error (MSE) matrix of the Stein estimator for the multivariate normal mean with unknown covariance matrix are proposed. The estimated MSE matrix is helpful in identifying the components which contribute most to the total risk. It also contains information about the performance of the shrinkage estimator with respect to other quadratic loss functions.     

Shrinkage estimation of location parameters in a multivariate skew-normal distribution

Abstract

This paper studies decision theoretic properties of Stein type shrinkage estimators in simultaneous estimation of location parameters in a multivariate skew-normal distribution with known skewness parameters under a quadratic loss. The benchmark estimator is the best location equivariant estimator which is minimax. A class of shrinkage estimators improving on the best location equivariant estimator is constructed when the dimension of the location parameters is larger than or equal to four. An empirical Bayes estimator is also derived, and motivated from the Bayesian procedure, we suggest a simple skew-adjusted shrinkage estimator and show its dominance property. The performances of these estimators are investigated by simulation.     

Pitman closest equivariant estimators and predictors under location–scale models

For location, scale and location–scale models, which are common in practical applications, we derive optimum equivariant estimators and predictors using the Pitman closeness criterion. This approach is very robust with respect to the choice of the loss function as it only requires the loss function to be strictly monotone. We also prove that, in general, the Pitman closeness comparison of any two equivariant predictors depends on the unknown parameter only through a maximal invariant, and hence it is independent of the parameter when the parameter space is transitive. We present several examples illustrating applications of our theoretical results.