Trimmed minimax estimator of a covariance matrix

Summary In the problem of estimating the covariance matrix of a multivariate normal population, James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob.,1, 361–380, Univ. of California Press) obtained a minimax estimator under a scale invariant loss. In this paper we propose an orthogonally invariant trimmed estimator by solving certain differential inequality involving the eigenvalues of the sample covariance matrix. The estimator obtained, truncates the extreme eigenvalues first and then shrinks the larger and expands the smaller sample eigenvalues. Adaptive version of the trimmed estimator is also discussed. Finally some numerical studies are performed using Monte Carlo simulation method and it is observed that the trimmed estimate shows a substantial improvement over the minimax estimator. The second author's research was supported by NSF Grant Number MCS 82-12968.     

A minimax regret estimator of a normal mean after preliminary test

Summary This paper considers the problem of estimating a normal mean from the point of view of the estimation after preliminary test of significance. But our point of view is different from the usual one. The difference is interpretation about a null hypothesis. Let denote the sample mean based on a sample of sizen from a normal population with unknown mean μ and known varianceσ ². We consider the estimator that assumes the value when and the value when where ω is a real number such that 0≤ω≤1 andC is some positive constant. We prove the existence of ω, satisfying the minimax regret criterion and make a numerical comparison among estimators by using the mean square error as a criterion of goodness of estimators.     

A skewed Kalman filter

The popularity of state-space models comes from their flexibilities and the large variety of applications they have been applied to. For multivariate cases, the assumption of normality is very prevalent in the research on Kalman filters. To increase the applicability of the Kalman filter to a wider range of distributions, we propose a new way to introduce skewness to state-space models without losing the computational advantages of the Kalman filter operations. The skewness comes from the extension of the multivariate normal distribution to the closed skew-normal distribution. To illustrate the applicability of such an extension, we present two specific state-space models for which the Kalman filtering operations are carefully described.     

求解极小极大问题的非单调过滤算法

提出一种改进的求解极小极大问题的信赖域滤子方法,利用SQP子问题来求一个试探步,尾服用滤子来衡量是否接受试探步,避免了罚函数的使用;并且借用已有文献的思想, 使用了Lagrange函数作为效益函数和非单调技术,在适当的条件下,分析了算法的全局和局部收敛性,并进行了数值实验.     

Fuzzy variant of a statistical test point Kalman filter

In this paper, we propose the conceptual use of fuzzy clustering techniques as iterative spatial methods to estimate a posteriori statistics in place of the weighted averaging scheme of the Unscented Kalman filter. Specifically, instead of a linearization methodology involving the statistical linear regression of the process and measurement functions through some deterministically chosen set of test points (sigma points) contained within the “uncertainty region” around the state estimate, we present a variant of the Unscented transformation involving fuzzy clustering techniques which will be applied to the test points yielding “degrees of membership” in which Gaussian shapes can be “fit” using a least squares scheme. Implementation into the Kalman methodology will be shown along with simple state and parameter estimation examples.     

Credibility theory and the Kalman filter

Following Mehra (1975) we indicate how some of the well known credibility models may be formulated as Kalman filters. The formulation yields recursive premium forecasts including recursive predictions errors which are of importance to practitioners.     

An improved tracking Kalman filter using a multilayered neural network

This paper presents a method for improving the estimation accuracy of a tracking Kalman filter (TKF) by using a multilayered neural network (MNN). Estimation accuracy of the TKF is degraded due to the uncertainties which cannot be expressed by the linear state-space model given a priori. The MNN capable of learning an arbitrary nonlinear mapping is thus added to the TKF to compensate the uncertainties. The MNN is trained so that it realizes a mapping from, the measurements to the corrections of estimations of the TKF. Simulation results show that the estimation accuracy is much improved by using the MNN.     

An admissible minimax estimator of a bounded scale-parameter in a subclass of the exponential family under scale-invariant squared-error loss

A subclass of the scale-parameter exponential family is considered and for the rth power of the scale parameter, which is lower bounded, an admissible minimax estimator under scale-invariant squared-error loss is presented. Also, an admissible minimax estimator of a lower-bounded parameter in the family of transformed chi-square distributions is given. These estimators are the pointwise limits of a sequence of Bayes estimators. Some examples are given.     

A minimax theorem

A minimax theorem is proved, under a continuity condition complementing the celebrated von Neumann–Sion's minimax theorem. The result is based on an intersection theorem which may be of interest on its own right.     

A general minimax theorem

This paper is concerned with minimax theorems for two-person zero-sum games (X, Y, f) with payofff and as main result the minimax equality inf supf (x, y)=sup inff (x, y) is obtained under a new condition onf. This condition is based on the concept of averaging functions, i.e. real-valued functions defined on some subset of the plane with min {x, y}< (x, y)x, y} forx y and (x, x)=x. After establishing some simple facts on averaging functions, we prove a minimax theorem for payoffsf with the following property: Forf there exist averaging functions and such that for any x₁, x₂ X, > 0 there exists x₀ X withf (x₀, y) > f (x₁,y),f (x₂,y))– for ally Y, and for any y₁, y₂ Y, > 0 there exists y₀ Y withf (x, y₀) (f (x, y₁),f (x, y₂))+. This result contains as a special case the Fan-König result for concave-convex-like payoffs in a general version, when we take linear averaging with (x, y)=x+(1–)y, (x, y)=x+(1–)y, 0 <, < 1.Then a class of hide-and-seek games is introduced, and we derive conditions for applying the minimax result of this paper.

---

Zusammenfassung In dieser Arbeit werden Minimaxsätze für Zwei-Personen-Nullsummenspiele (X, Y,f) mit Auszahlungsfunktionf behandelt, und als Hauptresultat wird die Gültigkeit der Minimaxgleichung inf supf (x, y)=sup inff (x, y) unter einer neuen Bedingung an f nachgewiesen. Diese Bedingung basiert auf dem Konzept mittelnder Funktionen, d.h. reellwertiger Funktionen, welche auf einer Teilmenge der Ebene definiert sind und dort der Eigenschaft min {x, y} < < (x, y)x, y} fürx y, (x, x)=x, genügen. Nach der Herleitung einiger einfacher Aussagen über mittelnde Funktionen beweisen wir einen Minimaxsatz für Auszahlungsfunktionenf mit folgender Eigenschaft: Zuf existieren mittelnde Funktionen und, so daß zu beliebigen x₁, x₂ X, > 0 mindestens ein x₀ X existiert mitf (x₀,y) (f (x ₁,y),f (x₂,y)) – für alley Y und zu beliebigen y₁, y₂ Y, > 0 mindestens ein y₀ Y existiert mitf (x, y₀) (f (x, y₁),f (x, y ₂))+ für allex X. Dieses Resultat enthält als Spezialfall den Fan-König'schen Minimaxsatz für konkav-konvev-ähnliche Auszahlungsfunktionen in einer allgemeinen Version, wenn wir lineare Mittelung mit (x, y)=x+(1–)y, (x, y)= x+(1–)y, 0 <, < 1, betrachten.Es wird eine Klasse von Suchspielen eingeführt, welche mit dem vorstehenden Resultat behandelt werden können.

     

On the convergence of the ensemble Kalman filter

Convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman filter is proved. In each step of the filter, convergence of the ensemble sample covariance follows from a weak law of large numbers for exchangeable random variables, the continuous mapping theorem gives convergence in probability of the ensemble members, and L ^p bounds on the ensemble then give L ^p convergence.     

An approach to construction of modified Kalman filter

We propose a modified dynamic filtering algorithm for estimating perturbations of programmed values of state parameters. The implementation of the proposed Kaiman filtering approach for estimating the state of an aircraft under given modeling assumptions has reduced the multidimensional filtering problem to several one-dimensional problems.     

On a minimax theorem

The aim of this note is to prove the following statement. It gives an extension of the theorem proved by O. Y. Borenshtein and V. S. Shul'man in [1].