A Kalman filter as a minimax estimator

A minimax terminal state estimation problem is posed for a linear plant and a generalized quadratic loss function. Sufficient conditions are developed to insure that a Kalman filter will provide a minimax estimate for the terminal state of the plant. It is further shown that this Kalman filter will not generally be a minimax estimate for the terminal state if the observation interval is arbitrarily long. Consequently, a subminimax estimate is defined, subject to a particular existence condition. This subminimax estimate is related to the Kalman filter, and it may provide a useful estimate for the terminal state when the performance of the Kalman filter is no longer satisfactory.     

一类无约束离散Minimax问题的区间调节熵算法

In this paper,a class of unconstrained discrete minimax problems is described,in which the objective functions are in C^1. The paper deals with this problem by means of taking the place of maximum-entropy function with adjustable entropy function. By constructing an interval extension of adjustable entropy function and some region deletion test rules, a new interval algorithm is presented. The relevant properties are proven, The minimax value and the localization of the minimax points of the problem can be obtained by this method. This method can overcome the flow problem in the maximum-entropy algorithm. Both theoretical and numerical results show that the method is reliable and efficient.     

A multi-objective heuristic approach for the casualty collection points location problem

In this paper, we formulate the casualty collection points (CCPs) location problem as a multi-objective model. We propose a minimax regret multi-objective (MRMO) formulation that follows the idea of the minimax regret concept in decision analysis. The proposed multi-objective model is to minimize the maximum per cent deviation of individual objectives from their best possible objective function value. This new multi-objective formulation can be used in other multi-objective models as well. Our specific CCP model consists of five objectives. A descent heuristic and a tabu search procedure are proposed for its solution. The procedure is illustrated on Orange County, California.     

Linear Minimax Prediction in a Finite Population with Ellipsoidal Constraints

Under a matrix loss function, we investigate the prediction problem in a finite population with ellipsoidal restriction in this paper. Firstly, a class of homogeneous linear minimax predictors for finite population regression coefficient are obtained. Moreover, it is shown that the linear minimax predictors are admissible in the class of homogeneous linear predictors. Finally, a simulation study and a real data example are used to illustrate our results.     

Asymptotic minimax estimation in semiparametric models

Summary We give several conditions on the estimator of efficient score function for estimating the parametric component of semiparametric models. A semiparametric version of the one-step MLE using an estimator of efficient score function which fulfills the conditions is shown to converge to the normal distribution with minimum variance locally uniformly over a fairly large neighborhood around the assumed semiparametric model. Consequently, it is shown to be asymptotically minimax with bounded subconvex loss functions. A few examples are also considered.     

Minimax signal detection under weak noise assumptions

We consider minimax signal detection in the sequence model. Working with certain ellipsoids in the space of square-summable sequences of real numbers, with a ball of positive radius removed, we obtain upper and lower bounds for the minimax separation radius in the non-asymptotic framework, i.e., for a fixed value of the involved noise level. We use very weak assumptions on the noise (i.e., fourth moments are assumed to be uniformly bounded). In particular, we do not use any kind of Gaussian distribution or independence assumption on the noise. It is shown that the established minimax separation rates are not faster than the ones obtained in the classical sequence model (i.e., independent standard Gaussian noise) but, surprisingly, are of the same order as the minimax estimation rates in the classical setting. Under an additional condition on the noise, the classical minimax separation rates are also retrieved in benchmark well-posed and ill-posed inverse problems.     

A finite algorithm for monotone regression and the application of its principle to the aggregation of ranks and indifference curve fitting

Monotone regression makes optimal consistent adjusted value assignments to ordinal dependent data, or monotonically adjusts ratio-level-dependent variable data to achieve the best possible agreement with an explanatory linear (in the sense of parameters) model. Thus, for example, as a tool for building group or individual multi-attribute value (MAV) functions, it partially obviates the need to prespecify a particular MAV function class. The height of the unknown MAV function at each comparison bundle is determined along with the other model parameters.In this paper, the Kruskal stress criterion and iterative computational methodology are shown to be reducible to one of three simpler convex quadratic programming problems. The fundamental idea underlying the method is not restricted to least-squares objectives. An application to the problem of aggregation of ranks is shown. Here, the minimax data-fitting criterion is employed.     

On measurable minimax selectors

In this note we consider the upper value of a zero-sum game with payoff function depending on a state variable. We provide a new and much simpler proof of a measurable minimax selection theorem established 25 years ago by the author in Nowak (1985) [19]. A discussion of the basic assumptions and relations with the literature on stochastic games and (minimax) control models is also included.     

A polynomial time algorithm for a hemispherical minimax location problem

A polynomial time algorithm to obtain an exact solution for the equiweighted minimax location problem when the demand points are spread over a hemisphere is presented. It is shown that the solution of the minimax problem when the norm under consideration is geodesic is equivalent to solving a maximization problem using the Euclidean norm.     

A Lower Bound for the Penalty Parameter in the Exact Minimax Penalty Function Method for Solving Nondifferentiable Extremum Problems

In the paper, we consider the exact minimax penalty function method used for solving a general nondifferentiable extremum problem with both inequality and equality constraints. We analyze the relationship between an optimal solution in the given constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function under the assumption of convexity of the functions constituting the considered optimization problem (with the exception of those equality constraint functions for which the associated Lagrange multipliers are negative—these functions should be assumed to be concave). The lower bound of the penalty parameter is given such that, for every value of the penalty parameter above the threshold, the equivalence holds between the set of optimal solutions in the given extremum problem and the set of minimizers in its associated penalized optimization problem with the exact minimax penalty function.     

Multicriterion structure/control design for optimal maneuverability and fault tolerance of flexible spacecraft

A multicriterion design problem for optimal maneuverability and fault tolerance of flexible spacecraft is considered. The maneuverability index reflects the time required to perform rest-to-rest attitude maneuvers for a given set of angles, with the postmaneuver spillover within a specified bound. The performance degradation is defined to reflect the maximum possible attitude error after maneuver due to the effect of faults. The fault-tolerant design is to minimize the worst performance degradation from all admissible faults by adjusting the design of the spacecraft. It is assumed that admissible faults can be specified by a vector of real parameters. The multicriterion design for optimal maneuverability and fault tolerance is shown to be well defined, leading to a minimax problem. Analysis for this nonsmooth problem leads to closed-form expressions of the generalized gradient of the performance degradation function with respect to the fault parameters and structural design variables. Necessary and sufficient conditions for the optimum are derived, and the closed-form expressions of the generalized gradients are applied for their interpretation. The bundle method is applicable to this minimax problem. Approximate methods which efficiently solve this minimax problem with relatively little computational difficulties are presented. Numerical examples suggest that it is possible to improve the fault tolerance substantially with relatively little loss in maneuverability.     

Nonlinear programming using minimax techniques

A minimax approach to nonlinear programming is presented. The original nonlinear programming problem is formulated as an unconstrained minimax problem. Under reasonable restrictions, it is shown that a point satisfying the necessary conditions for a minimax optimum also satisfies the Kuhn-Tucker necessary conditions for the original problem. A leastpth type of objective function for minimization with extremely large values ofp is proposed to solve the problem. Several numerical examples compare the present approach with the well-known SUMT method of Fiacco and McCormick. In both cases, a recent minimization algorithm by Fletcher is used.This paper is based on work presented at the 5th Hawaii International Conference on System Sciences, Honolulu, Hawaii, 1972. The authors are greatly indebted to V. K. Jha for his programming assistance and J. H. K. Chen who obtained some of the numerical results. This work was supported in part by the National Research Council of Canada under Grant No. A7239, by a Frederick Gardner Cottrell Grant from the Research Corporation, and through facilities and support from the Communications Research Laboratory of McMaster University.     

Solving 0-1 Minimax Problems

It has been shown that it is possible to solve a minimax 0-1 programming problem by transforming the objective function coefficients and solving a minisum problem. This result is very useful for solving a ‘hard’ minimax problem when there exists a ‘relatively easy’ algorithm to solve the minisum problem. We discuss such problems and present a new transformation.     

Minimax confidence bound of the normal mean under an asymmetric loss function

This paper considers a minimax confidence bound of the normal mean under an asymmetric loss function. A minimax confidence bound is obtained for the case that the variance is known or unknown. The admissibility of the minimax confidence bound is also considered for the case of known variance.     

Adaptive estimation of linear functionals in functional linear models

We consider the estimation of the value of a linear functional of the slope parameter in functional linear regression, where scalar responses are modeled in dependence of randomfunctions. In Johannes and Schenk [2010] it has been shown that a plug-in estimator based on dimension reduction and additional thresholding can attain minimax optimal rates of convergence up to a constant. However, this estimation procedure requires an optimal choice of a tuning parameter with regard to certain characteristics of the slope function and the covariance operator associated with the functional regressor. As these are unknown in practice, we investigate a fully data-driven choice of the tuning parameter based on a combination of model selection and Lepski??s method, which is inspired by the recent work of Goldenshluger and Lepski [2011]. The tuning parameter is selected as theminimizer of a stochastic penalized contrast function imitating Lepski??smethod among a random collection of admissible values. We show that this adaptive procedure attains the lower bound for the minimax risk up to a logarithmic factor over a wide range of classes of slope functions and covariance operators. In particular, our theory covers pointwise estimation as well as the estimation of local averages of the slope parameter.     

Differential games with a given value function

The set of solutions of a differential game with a terminal payoff functional is investigated. A method is obtained that allows us to establish whether a given function is a value of some differential game with a terminal payoff functional. The condition obtained is in fact the condition for the given function to be a minimax (viscosity) solution of some Hamilton-Jacobi equation with Hamiltonian homogeneous in the third variable. We also obtain a sufficient condition for a function to belong to the set of values of differential games with a terminal payoff function.     

The local asymptotic minimax adaptive property of a recursive estimate

A locally asymptotically normal estiamtion problem generated by independent and identically distributed generalized random variables is considered. A recursive estimate based on a stochastic approximation method, a modification of an estimate proposed by Sakrison, is shown to be locally asymptotically minimax. For a nonparametric generalization of the estimation problem, a modification of the estimate is shown locally asymptotically minimax adaptive.     

Construction of a minimax solution for an eikonal-type equation

A formula for a minimax (generalized) solution of the Cauchy-Dirichlet problem for an eikonal-type equation is proved in the case of an isotropic medium providing that the edge set is closed; the boundary of the edge set can be nonsmooth. A technique of constructing a minimax solution is proposed that uses methods from the theory of singularities of differentiable mappings. The notion of a bisector, which is a representative of symmetry sets, is introduced. Special points of the set boundary—pseudovertices—are singled out and bisector branches corresponding to them are constructed; the solution suffers a “gradient catastrophe” on these branches. Having constructed the bisector, one can generate the evolution of wave fronts in smoothness domains of the generalized solution. The relation of the problem under consideration to one class of time-optimal dynamic control problems is shown. The efficiency of the developed approach is illustrated by examples of analytical and numerical construction of minimax solutions.     

Multi-period minimax hedging strategies

Option pricing and hedging under transaction costs are of major importance to marketmakers and investors. In this paper we present the basic minimax strategy which determines the optimum number of shares that minimizes the worst-case potential hedging error under transaction costs for the next period. We present two extensions of this strategy. The first extension is the two-period minimax where the worst case is defined over a two-period setting. The objective function of the basic minimax strategy is augmented to include the hedging error for the second period. The second extension is the variable minimax strategy where early rebalancing is triggered by the minimax hedging error. Simulation results suggest that the basic minimax strategy and its two extensions are superior in performance to delta hedging and that the variable minimax strategy is superior to both the basic and the two-period strategies. This result is due to the opportunity provided by the variable minimax strategy to rebalance early. The greatest amount of business for traded options is done for at-the-money options; in this paper, we have concluded that the minimax strategies are particularly suitable for managing the risk of such options. In the Appendix, we present the minimax algorithm used for the implementation of these strategies.     

A subclass of bayes linear estimators that are minimax

In the linear regression model with ellipsoidal parameter constraints, the problem of estimating the unknown parameter vector is studied. A well-described subclass of Bayes linear estimators is proposed in the paper. It is shown that for each member of this subclass, a generalized quadratic risk function exists so that the estimator is minimax. Moreover, some of the proposed Bayes linear estimators are admissible with respect to all possible generalized quadratic risks. Also, a necessary and sufficient condition is given to ensure that the considered Bayes linear estimator improves the least squares estimator over the whole ellipsoid whatever generalized risk function is chosen.