Sequential Minimax Research Articles

Sequential Minimax Search for Multi-Layer Gene Grouping

Many areas of exploratory data analysis need to deal with high-dimensional data sets. Some real life data like human gene have an inherent structure of hierarchy, which embeds multi-layer feature groups. In this paper, we propose an algorithm to search for the number of feature groups in high-dimensional data by sequential minimax method and detect the hierarchical structure of high-dimensional data. Several proper numbers of feature grouping can be discovered. The feature grouping and group weights are investigated for each group number. After the comparison of feature groupings, the multi-layer structure of feature groups is detected. The latent feature group learning (LFGL) algorithm is proposed to evaluate the effectiveness of the number of feature groups and provide a method of subspace clustering. In the experiments on several gene data sets, the proposed algorithm outstands several representative algorithms.

Minimax sequential estimation plans for exponential-type processes

The study of the theory of sequential estimation for continuous-time stochastic processes was initiated by Dvoretzky, Kiefer and Wolfowitz (1953). They proved that for the Poisson, negative-binomial, gamma and Wiener processes the minimax sequential plan reduces to a fixed-time plan if a weighted quadratic loss function is used. Their results were generalized by Magiera (1977) to an exponential class of processes with stationary independent increments and by Trybuła (1985) in case of the n-dimensional processes. It turns out, however, that for a certain class of processes there exist truly sequential minimax plans. It was showed by Trybuła (1985) that for the Poisson process an inverse plan is minimax. Wilczyński (1985) has obtained an analogous result for the multinomial process. In this paper we consider a class of exponential-type processes and for this class we prove that, with a properly weighted quadratic loss function, the inverse plans are minimax in the class of all sequential plans having finite risk. Examples illustrating the derived result are also presented.

Sequential estimation for the generalized exponential hyperbolic secant process

For the multidimensional GEHS process and a weighted quadratic loss function the sequential minimax plan is determined. For one-dimensional GEHS process a class of sequential minimax plans is obtained in the case wen the set of the prior distributions of parameter is restricted

On sequential minimax estimation for the exponential class of processes

On sequential minimax estimation for the exponential class of processes

Sequential Policies for Bank Money Management

Banks must hold an average deposit balance in the Federal Reserve Bank each week. Surplus or deficit balances may be traded on the Federal funds market. The rate on Fed funds behaves as a random variable over the week as does the size of the surplus or deficit to be offset. Incorporating both these sources of uncertainty, sequential minimax and expected utility maximizing strategies are derived for offsetting the surplus or deficit over the week.

Sufficiently informative functions and the minimax feedback control of uncertain dynamic systems

The problem of optimal feedback control of uncertain discrete-time dynamic systems is considered where the uncertain quantities do not have a stochastic description but instead are known to belong to given sets. The problem is converted to a sequential minimax problem and dynamic programming is suggested as a general method for its solution. The notion of a sufficiently informative function, which parallels the notion of a sufficient statistic of stochastic optimal control, is introduced, and conditions under which the optimal controller decomposes into an estimator and an actuator are identified. A limited class of problems for which this decomposition simplifies the computation and implementation of the optimal controller is delineated.

Sequential Minimax Search for a Maximum When Prior Information is Available

Previous article Next article Sequential Minimax Search for a Maximum When Prior Information is AvailableShmuel GalShmuel Galhttps://doi.org/10.1137/0121063PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. Kiefer, Sequential minimax search for a maximum, Proc. Amer. Math. Soc., 4 (1953), 502–506 MR0055639 0050.35702 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Techno-economic and life cycle analyses of battery-assisted hydrogen production systems from photovoltaic powerJournal of Cleaner Production, Vol. 298 Cross Ref Parallel Minimax Searching Algorithm for Extremum of Unimodal Unbounded FunctionInternational Journal of Communications, Network and System Sciences, Vol. 04, No. 09 Cross Ref Optimal sequential and parallel seach for finding a rootJournal of Combinatorial Theory, Series A, Vol. 23, No. 1 Cross Ref Literatur Cross Ref Volume 21, Issue 4| 1971SIAM Journal on Applied Mathematics History Submitted:21 July 1970Published online:12 July 2006 InformationCopyright © 1971 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0121063Article page range:pp. 590-595ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

Minimax Policies for Selling an Asset and Dollar Averaging

The results of this paper fall into two areas. First, it is shown that the conventional wisdom of dollar averaging is related to hedging against, large regrets rather than unfavorable outcomes. Suppose that a given sum of dollars is to be irreversibly converted into stock within a given number of periods. Suppose that the share price of dollars follows an arithmetic random walk. It is shown that dollar averaging is a nonsequential minimax strategy, if the largest possible price increase in each period is equal to the largest possible decrease. On the other hand, it is shown that dollar averaging cannot be a nonsequential expected utility maximizing strategy for any strictly concave utility function and any arithmetic random walk. Second, sequential minimax strategies are examined. Here the analysis may be viewed as an extension of the literature on the optimal time to sell an asset or the optimal stopping problem. The minimax sequential strategy is shown to be of the following form. At any time there exists a critical value which depends only on n, the difference between the maximum price since conversions began and the current price. Funds held in excess of this value are converted into stock, otherwise no funds are converted. The critical values are shown to be decreasing functions of n. When the share price of dollars rises (i.e., stock prices fall) few funds are converted while decreases produce large conversions. These results seem to correspond to the second thoughts of those, embarking on nonsequential policies of the dollar averaging type. Buying and selling minimax policies are shown to be symmetric. Sequential minimaxing will tend to have a reinforcing effect on price movements.

Optimum Sequential Search and Approximation Methods Under Minimum Regularity Assumptions

Previous article Next article Optimum Sequential Search and Approximation Methods Under Minimum Regularity AssumptionsJ. KieferJ. Kieferhttps://doi.org/10.1137/0105009PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. Kiefer, Sequential minimax search for a maximum, Proc. Amer. Math. Soc., 4 (1953), 502–506 MR0055639 0050.35702 CrossrefISIGoogle Scholar[2] Herbert Robbins and , Sutton Monro, A stochastic approximation method, Ann. Math. Statistics, 22 (1951), 400–407 MR0042668 0054.05901 CrossrefISIGoogle Scholar[3] J. Kiefer and , J. Wolfowitz, Stochastic estimation of the maximum of a regression function, Ann. Math. Statistics, 23 (1952), 462–466 MR0050243 0049.36601 CrossrefISIGoogle Scholar[4] K. L. Chung, On a stochastic approximation method, Ann. Math. Statistics, 25 (1954), 463–483 MR0064365 0059.13203 CrossrefISIGoogle Scholar[5] J. Sacks, Limit distributions of stochastic approximation methods, to be published Google Scholar[6] Colin R. Blyth, On minimax statistical decision procedures and their admissibility, Ann. Math. Statistics, 22 (1951), 22–42 MR0039966 0042.38303 CrossrefISIGoogle Scholar[7] A. Dvoretzky, , J. Kiefer and , J. Wolfowitz, Asymptotic minimax character of the sample distribution function, Ann. Math. Statist., 27 (1946), 642–669 CrossrefISIGoogle Scholar[8] J. L. Hodges, Jr. and , E. L. Lehmann, Some applications of the Cramér-Rao inequality, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, 1951, 13–22 MR0044795 0044.14404 Google Scholar[9] J. Kiefer, Invariance, minimax sequential estimation, and continuous time processes, Ann. Math. Statist., 28 (1957), 573–601 MR0092325 0080.13004 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Optimization by Direct Search: New Perspectives on Some Classical and Modern MethodsSIAM Review, Vol. 45, No. 3 | 4 August 2006AbstractPDF (1742 KB)Optimal Search for the Global Maximum of Functions with Bounded SeminormSIAM Journal on Numerical Analysis, Vol. 22, No. 5 | 14 July 2006AbstractPDF (1489 KB)Iterative Methods for the Localization of the Global MaximumSIAM Journal on Numerical Analysis, Vol. 19, No. 4 | 17 July 2006AbstractPDF (1200 KB)An Odd Order Search ProblemR. S. BoothSIAM Journal on Algebraic Discrete Methods, Vol. 3, No. 1 | 31 July 2006AbstractPDF (699 KB)On Finding the Maximal Range of Validity of a Constrained SystemSIAM Journal on Control and Optimization, Vol. 16, No. 3 | 18 July 2006AbstractPDF (2826 KB)A Sequential Method Seeking the Global Maximum of a FunctionSIAM Journal on Numerical Analysis, Vol. 9, No. 3 | 14 July 2006AbstractPDF (856 KB)Location of Zeros of DerivativesSIAM Journal on Applied Mathematics, Vol. 15, No. 6 | 17 February 2012AbstractPDF (540 KB)Minimization by Successive ApproximationSIAM Journal on Numerical Analysis, Vol. 3, No. 1 | 14 July 2006AbstractPDF (1065 KB)A Review of Minimization Techniques for Nonlinear FunctionsSIAM Review, Vol. 4, No. 4 | 18 July 2006AbstractPDF (2112 KB) Volume 5, Issue 3| 1957Journal of the Society for Industrial and Applied Mathematics89-169 History Submitted:19 February 1957Published online:10 July 2006 InformationCopyright © 1957 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0105009Article page range:pp. 105-136ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

Sequential Minimax Estimation for the Rectangular Distribution with Unknown Range

This paper is concerned with sequential minimax estimation of the parameter $\theta(0 < \theta < \infty)$ of the density function (3.1) when the observations are independently and identically distributed with this density, each observation costs the same amount $c > 0$, and the weight function is as given in Section 2. A procedure requiring a fixed sample size is shown to be a minimax solution for this problem.