A minimax property of Lahiri-Midzuno-Sen’s sampling scheme

We consider parameter spaces which are generalizations of spaces discussed so far in connection with minimax strategies. We give a lower bound for the minimax value and derive, under weak assumptions, minimax strategies consisting of the expansion estimator and an appropriate design. This design is of the Lahiri-Midzuno-Sen type for an important subclass of parameter spaces.     

A conditional minimax approach in survey sampling

Summary Applying the usual minimax criterion in finite sampling theory yields complicated solutions except the parameter space has certain invariance properties. A conditional minimax criterion is suggested. After a sample is selected it is reasonable to seek an estimator that has good properties (e.g. minimaxity) for that sample. Explicit solutions are given in the case where the parameter space is described by quadratic forms.     

Optimal rules and robust Bayes estimation of a Gamma scale parameter

For estimating an unknown scale parameter of Gamma distribution, we introduce the use of an asymmetric scale invariant loss function reflecting precision of estimation. This loss belongs to the class of precautionary loss functions. The problem of estimation of scale parameter of a Gamma distribution arises in several theoretical and applied problems. Explicit form of risk-unbiased, minimum risk scale-invariant, Bayes, generalized Bayes and minimax estimators are derived. We characterized the admissibility and inadmissibility of a class of linear estimators of the form $cX\,{+}\,d$ , when $X\sim \varGamma (\alpha ,\eta )$ . In the context of Bayesian statistical inference any statistical problem should be treated under a given loss function by specifying a prior distribution over the parameter space. Hence, arbitrariness of a unique prior distribution is a critical and permanent question. To overcome with this issue, we consider robust Bayesian analysis and deal with Gamma minimax, conditional Gamma minimax, the stable and characterize posterior regret Gamma minimax estimation of the unknown scale parameter under the asymmetric scale invariant loss function in detail.     

On least favourable two point priors and minimax estimators under absolute error loss

Summary In case of absolute error loss we investigate for an arbitrary class of probability distributions, if or if not a two point prior can be least favourable and a corresponding Bayes estimator can be minimax when the parameter is restricted to a closed and bounded interval of ℝ. The general results are applied to several examples, for instance location and scale parameter families are considered. We give examples for which, independent of the length of the parameter interval, no two point priors exist. On the other hand examples are given having a least favourable two point prior when the parameter interval is sufficiently small.     

Admissible and minimax estimators of 363-1363-1363-1in the gamma distribution with truncated parameter space

In this paper we consider asmissible and minimax estimation of the parameter in the gamma distribution with truncated parameter space. We give a necessary and sufficient condition for minimaxity (Theorem 1) and obtain the classes of new minimax and asmissible estimators. The results of the paper can be applied to estimation of parameters in the normal, lognormal, Pareto, generalized gamma, generalized Laplace and other distributions.     

An unexpected property of minimax estimation in the relative squared error approach to linear regression analysis

An unexpected property of the relative squared error approach to linear regression analysis is derived: It is shown that an estimator being minimax among all linear affine estimators is also minimax in the set of all estimators. Two illustrative special cases are mentioned, where a generalized least squares estimator and a general ridge or Kuks-Olman estimator turn out to be minimax.     

Gamma-Minimax Prediction for the Multinomial Distribution

Characterizations of gamma-minimax predictors for the linear combinations of the unknown parameter and the random variable having the multinomial distribution under arbitrary squared error loss are established in two situations – when the sample size is fixed and when the sample size is a realization of a random variable. It is always assumed that the available vague prior information about the unknown parameter can be described by a class of priors whose vector of first moments belongs to a suitable convex and compact set. Several known gamma-minimax and minimax results can be obtained from the characterizations derived in the present paper.     

Minimax‐Optimal Strength of Statistical Evidence for a Composite Alternative Hypothesis

While the likelihood ratio measures statistical support for an alternative hypothesis about a single parameter value, it is undefined for an alternative hypothesis that is composite in the sense that it corresponds to multiple parameter values. Regarding the parameter of interest as a random variable enables measuring support for a composite alternative hypothesis without requiring the elicitation or estimation of a prior distribution, as described below. In this setting, in which parameter randomness represents variability rather than uncertainty, the ideal measure of the support for one hypothesis over another is the difference in the posterior and prior log‐odds. That ideal support may be replaced by any measure of support that, on a per‐observation basis, is asymptotically unbiased as a predictor of the ideal support. Such measures of support are easily interpreted and, if desired, can be combined with any specified or estimated prior probability of the null hypothesis. Two qualifying measures of support are minimax‐optimal. An application to proteomics data indicates that a modification of optimal support computed from data for a single protein can closely approximate the estimated difference in posterior and prior odds that would be available with the data for 20 proteins.     

Local minimax pointwise estimation of a multivariate density

We consider the problem of the nonparametric minimax estimation of a multivariate density at a given point. A concept of smoothness classes in nonparametric minimax estimation problems is proposed. The smoothness of a function is characterized by the approximability of the function at a point by an integral of the product of this function with an approximate identity. We propose a singular integral estimator, an integral of this approximate identity with respect to the empirical distribution function. Under some assumptions on the approximate identity, the bias of the estimator is shown to be of smaller order asymptotically than the variance, and the estimator itself is shown to be asymptotically locally minimax with respect to the quadratic risk in a proper topology.     

Limited deviation estimators of the Poisson parameter

Estimators for the Poisson parameter are proposed which perform well with respect to both a weighted, i.e. Bayes, and unweighted risk criterion. The estimators follow the Bayes rule (with respect to a conjugate gamma prior) as closely as possible subject to a restraint imposed on the allowable deviation from the minimax estimate. The resulting class of rules maintains good performance with respect to the Bayes criterion while at the same time possessing bounded risk functions. The excess Bayes risk incurred is compared to a lower bound on the optimal restricted Bayes risk.     

Procedures to determine optimum two-stage sampling plans by attributes

Summary In order to compare two sampling plans we use the minimax regret principle, i.e. the minimax principle applied to regret functions. It is shown that among all two-stage sampling plans there exists an optimum sampling plan which can be computed with the aid of a procedure presented in this paper; furthermore another procedure is described how to obtain an approximately optimum two-stage sampling plan in a more direct way. Finally only those two-stage sampling plans are regarded which satisfy an additional condition; among these sampling plans an optimum one exists and is to be determined, too.     

Minimax estimation of constrained parametric functions for discrete families of distributions

For a vast class of discrete model families where the natural parameter is constrained to an interval, we give conditions for which the Bayes estimator with respect to a boundary supported prior is minimax under squared error loss type functions. Building on a general development of éric Marchand and Ahmad Parsian, applicable to squared error loss, we obtain extensions to various parametric functions and squared error loss type functions. We provide illustrations for various distributions and parametric functions, and these include examples for many common discrete distributions, as well as when the parametric function is a zero-count probability, an odds-ratio, a Binomial variance, and a Negative Binomial variance, among others. The Research of M. Jafari Jozani is supported by a grant of the Institute for Research and Planning in Higher Education, Ministry of Science, Research and Technology, Iran. The Research of é. Marchand is supported by NSERC of Canada.     

极大极小投资组合模型

本文讨论了摩擦市场中,可以卖空的基础上,构造最优投资组合选择的极大极小模型。应用经典的Ky Fan极大极小不等式定理,将其转化为两个二次规划问题,并给出最优投资组合的表达形式。     

Decision theory for continuous observations: Minimax solutions

This paper is concerned with the existence of minimax solutions in a decision theoretic model for continuous observations as described inIrle, Schmitz, [1914]. There is given a result on the existence of minimax solutions with constant time observation, followed by a discussion of the conditions which are used to prove the above statement.     

«Una condizione necessaria e sufficiente di minimax per il problema dei dadi truccati»

Nell'articolo si fornisce una condizione necessaria e sufficiente di minimax locale per il problema dei dadi truccati». Tale condizione integra una condizione necessaria di minimax proposta dall'Autore in un precedente lavoro. La verifica della condizione non presenta difficoltà in relazione alla regolarità dei vincoli. Una applicazione significativa à esposta relativamente al caso di dimensione 5.

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Summary A necessary and sufficient condition for the minimax loaded dice problem is proposed. An application of this condition for the casen=5 is given.

     

Approximately optimum two-stage sampling plans

Summary In order to compare two-stage sampling plans we use the minimax regret principle i.e. the minimax principle applied to the corresponding regret functions. In this paper approximation formulas for optimum two-stage sampling plans are derived in the case of sampling by attributes as well as in the case of sampling by variables; furthermore a method is presented how to improve the approximate solution in the first case.     

Minimax regret treatment choice with covariates or with limited validity of experiments

This paper continues the investigation of minimax regret treatment choice initiated by Manski (2004). Consider a decision maker who must assign treatment to future subjects after observing outcomes experienced in a sample. A certain scoring rule is known to achieve minimax regret in simple versions of this decision problem. I investigate its sensitivity to perturbations of the decision environment in realistic directions. They are as follows. (i) Treatment outcomes may be influenced by a covariate whose effect on outcome distributions is bounded (in one of numerous probability metrics). This is interesting because introduction of a covariate with unrestricted effects leads to a pathological result. (ii) The experiment may have limited validity because of selective noncompliance or because the sampling universe is a potentially selective subset of the treatment population. Thus, even large samples may generate misleading signals. These problems are formalized via a “bounds” approach that turns the problem into one of partial identification.In both scenarios, small but positive perturbations leave the minimax regret decision rule unchanged. Thus, minimax regret analysis is not knife-edge-dependent on ignoring certain aspects of realistic decision problems. Indeed, it recommends to entirely disregard covariates whose effect is believed to be positive but small, as well as small enough amounts of missing data or selective attrition. All findings are finite sample results derived by game theoretic analysis.     

Minimax optimality of CUSUM for an autoregressive model

Different change point models for AR(1) processes are reviewed. For some models, the change is in the distribution conditional on earlier observations. For others, the change is in the unconditional distribution. Some models include an observation before the first possible change time – others not. Earlier and new CUSUM type methods are given, and minimax optimality is examined. For the conditional model with an observation before the possible change, there are sharp results of optimality in the literature. The unconditional model with possible change at (or before) the first observation is of interest for applications. We examined this case and derived new variants of four earlier suggestions. By numerical methods and Monte Carlo simulations, it was demonstrated that the new variants dominate the original ones. However, none of the methods is uniformly minimax optimal.     

Ungünstige Verteilungen bei einparametrigen Parameterschätzproblemen

Summary One of the commonly used methods for determining minimax point estimators is based on least favorable distributions, because Bayes estimators with respect to a least favorable distribution are frequently minimax point estimators. Therefore it is worthwhile to investigate the structure of least favorable distributions. In the present paper it will be proved that, under certain conditions, each least favorable distribution is finite discrete. We give sufficient conditions for point estimation problems with the property that each least favorable distribution is finite discrete. Some examples are presented.     

Model reference adaptive systems applied to regression analyses

The adaptive estimation procedure of model reference adaptive systems is modified and applied to linear models. In general the principle can be used for almost any time series model. Because of the recursive nature of the resulting estimator, it is computationally appealing, especially when a time series is considered as a flow of data. In addition, the estimator turns out to have certain statistical optimality properties.
In the linear regression setting, Ridge estimators turn out to constitute a subclass of the adaptive estimators considered, whereas for unknown measurement variance, the resulting estimators are related to J ames -S tkin type estimators, and have better properties than the latter. The estimator is shown to be strongly consistent and to converge in law to a normal variate under the standard assumptions of linear models. Further it is shown to be admissible and minimax in restricted parameter spaces. The connection between K alman filters and the classical least-squares estimator is also pointed out.