Admissibility of linear estimators with respect to restricted parameter sets

In this paper the admissibility of a linear estimator for a linear regression parameter is characterized for such cases, where the considered parameter varies in an ellipsoid. We obtain a certain subset of the set of all linear estimators which are admissible with respect to the unrestricted parameter set. Furthermore, various linear estimators which have been proposed for improving the least squares estimator in cases of a restricted parameter set are investigated for admissibility. It turns out that only some shrunken estimators and some estimators of ridge type are admissible, whereas the KUKS-OLMAN estimator and all estimators of MARQUARDT type can be improved.     

SECOND ORDER ASYMPTOTIC COMPARISON OF ESTIMATORS UNDER UNIVERSAL DOMINATION CRITERION

The main purpose of this paper is to formulate theories of universal optimality, in the sense that some criteria for performances of estimators are considered over a class of loss functions. It is shown that the difference of the second order terms between two estimators in any risk functions is expressed as a form which is characterized by a peculiar value associated with the loss functions, which is referred to as the loss coefficient. This means that the second order optimal problem is completely characterized by the value of the loss coefficient. Furthermore, from the viewpoint of change of the loss coefficient, the relationship between two estimators is classified into six types. On the basis of this classification, the concept of universal second order admissibility is introduced. Some sufficient conditions are given to determine whether any estimators are universally admissible or not.     

Some remarks on an admissibility condition

Summary In this note we deal with some admissibility conditions proved by G. B. Tranquilli to be sufficient in the class of unbiased estimators of finite population parameters and with respect to (w.r.t.) a quadratic loss function. We show that the same conditions:i) are sufficient for the admissibility of an unbiased estimator with any loss function;ii) imply hyperadmissibility with reference to a particular (critical) population of the. From this fact we deduce that, for a fixed critical population, there is at most one estimator, in the class of all unbiased estimator of a finite population parameter, which satisfies Tranquilli condition. This research was partially supported by a M.U.R.S.T. grant ?Metodi inferenziali basati sul ricampionamento?.     

Characterization of Admissible Linear Estimators in Multivariate Linear Model with Respect to Inequality Constraints under Matrix Loss Function

In this article, we study the characterization of admissible linear estimators in a multivariate linear model with inequality constraint, under a matrix loss function. In the homogeneous class, we present several equivalent, necessary and sufficient conditions for a linear estimator of estimable functions to be admissible. In the inhomogeneous class, we find that the necessary and sufficient conditions depend on the rank of the matrix in the constraint. When the rank is greater than one, the necessary and sufficient conditions are obtained. When the rank is equal to one, we have necessary conditions and sufficient conditions separately. We also obtain the necessary and sufficient conditions for a linear estimator of inestimable function to be admissible in both classes.     

Linearly admissible estimators of stochastic regression coefficient under balanced loss function

The admissibility of linear estimators in a linear model with stochastic regression coefficient is investigated under a balanced loss function. The sufficient and necessary conditions for linear estimators to be admissible in classes of homogeneous and non-homogeneous linear estimators are obtained, respectively.     

Admissible estimation in an one parameter nonregular family of absolutely continuous distributions

Consider the problem of estimating under entropy loss an arbitrarily positive, strictly increasing or decreasing parametric function based on a sample of size n in an one parameter noregular family of absolutly continuous distributions with both endpoints of the support depending on a single parameter. We first provide sufficient conditions for the admissibility of generalized Bayes estimator with respect to some specific priors and then treat several examples which illustrate the admissibility of best invariant estimators is some location or scale parameter problems.     

A note on generalized ridge estimators

It is shown that a necessary and sufficient condition derived by Farebrother (1984)for a generalized ridge estimator to dominate the ordinary least-squares estimator with respect to the mean-square-error-matrix criterion in the linear regression model admits a similar interpretation as the well known criterion of Toro-Viz-carrondo and Wallace (1968)for the dominance of a restricted least-squares estimator over the ordinary least-squares estimator. Two other properties of the generalized ridge estimators, referring to the concept of admissibility, are also pointed out.     

Second order asymptotic and non-asymptotic optimality properties of combined tests

Let p independent test statistics be available to test a null hypothesis concerned with the same parameter. The p are assumed to be similar tests. Asymptotic and non-asymptotic optimality properties of combined tests are studied. The asymptotic study centers around two notions. The first is Bahadur efficiency. The second is based on a notion of second order comparisons. The non-asymptotic study is concerned with admissibility questions. Most of the popular combining methods are considered along with a method not studied in the past. Among the results are the following: Assume each of the p statistics has the same Bahadur slope. Then the combined test based on the sum of normal transforms, is asymptotically best among all tests studied, by virtue of second order considerations. Most of the popular combined tests are inadmissible for testing the noncentrality parameter of chi-square, t, and F distributions. For chi-square a combined test is offered which is admissible, asymptotically optimal (first order), asymptotically optimal (second order) among all tests studied, and for which critical values are obtainable in special cases. Extensions of the basic model are given.     

Estimated confidence under ancillary statistic everywhere-valid constraint

Consider the problem of estimating the coverage function of an usual confidence interval for a randomly chosen linear combination of the elements of the mean vector of a p-dimensional normal distribution. The usual constant coverage probability estimator is shown to be admissible under the ancillary statistic everywhere-valid constraint. Note that this estimator is not admissible under the usual sense if p⩾5. Since the criterion of admissibility under the ancillary statistic everywhere-valid constraint is a reasonable one, that the constant coverage probability estimator has been commonly accepted is justified.     

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.     

A Note on the Estimation of Binomial Probabilities

I am concerned with the admissibility under quadratic loss of certain estimators of binomial probabilities. The minimum variance unbiased estimator is shown to be admissible for Pr(X = 0) and Pr(X = n), but it is inadmissible for Pr(X = k), where 0 < k < n. An example is given of an admissible maximum likelihood estimator (MLE). It is conjectured that the MLE is always admissible.     

Dropping variables versus use of proxy variables in linear regression

In this paper we investigate under which conditions it is preferable to use proxies or to omit variables from the linear regression model with respect to the matrix mean square error criterion. Furthermore, some attention is paid to the admissibility of the proxies-based least squares estimator.     

On bagging and nonlinear estimation

We propose an elementary model for the way in which stochastic perturbations of a statistical objective function, such as a negative log-likelihood, produce excessive nonlinear variation of the resulting estimator. Theory for the model is transparently simple, and is used to provide new insight into the main factors that affect performance of bagging. In particular, it is shown that if the perturbations are sufficiently symmetric then bagging will not significantly increase bias; and if the perturbations also offer opportunities for cancellation then bagging will reduce variance. For the first property it is sufficient that the third derivative of a perturbation vanish locally, and for the second, that second and fourth derivatives have opposite signs. Functions that satisfy these conditions resemble sinusoids. Therefore, our results imply that bagging will reduce the nonlinear variation, as measured by either variance or mean-squared error, produced in an estimator by sinusoid-like, stochastic perturbations of the objective function. Analysis of our simple model also suggests relationships between the results obtained using different with-replacement and without-replacement bagging schemes. We simulate regression trees in settings that are far more complex than those explicitly addressed by the model, and find that these relationships are generally borne out.     

Estimating the exponential reliability function for a system in series with type II censored data

The problem of estimating the one parameter exponential reliability function for a system composed of l componentes in series is considered. Under the type II censoring scheme, the Bayes nature of the minimum variance unbiased estimator is demonstrated and the admissibility of related generalized Bayes estimators is established. For the one component case, the best unbiased estimator is admissible.     

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.     

On equality of ordinary least squares estimator,best linear unbiased estimator and best linear unbiased predictor in the general linear model

The equality of ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE) and best linear unbiased predictor (BLUP) in the general linear model with new observations is investigated through matrix rank method, some new necessary and sufficient conditions are given.     

A test of the mean square error criterion for linear admissible estimators

A test for choosing between a linear admissible estimator and the least squares estimator (LSE) is developed. A characterization of linear admissible estimators useful for comparing estimators is presented and necessary and sufficient conditions for superiority of a linear admissible estimator over the LS estimetor is derived for the test. The test is based on the MSE matrix superiority, but also new resl?!ts concerning covariance matrix comparisons of linear estimators are derived. Further,shown that the test of Toro - Vizcarrondo and Wailace applies iioi only the restricted least squares estimators but also to certain estimators outside this class.     

On Second Order Admissibilities in Two-Parameter Logistic Regression Model

In two-parameter family of distribution, conditions for a modified maximum likelihood estimator to be second-order admissible are given. Applying these results to two-parameter logistic regression model, it is shown that the maximum likelihood estimator is always second-order inadmissible and the Rao-Blackwellized minimum logit chi-squared estimator is second-order admissible if and only if the number of the doses is greater than or equal to 6.     

Local improvement of best linear unbiased estimation and admissibility under the weakly singular gauss-markov model

Under the weakly singular Gauss-Markov model, the class of linearly admissible estimators for the expectation of the observable random vector with respect to the mean square error criterion is considered. It is demonstrated that this class admits linearly admissible estimators for an arbitrary estimable parametric function, which locally improve the best linear estimator with respect to the mean square error matrix criterion.     

On the admissibility of the maximum-likelihood estimator of the binomial variance

This paper addresses the admissibility of the maximum-likelihood estimator (MLE) of the variance of a binomial distribution with parameters n and p under squared-error loss. We show that the MLE is admissible for n ≤ 5 and inadmissible for n≥ 6.