Analysis to Neyman-Pearson classification with convex loss function

Neyman-Pearson classification has been studied in several articles before.But they all proceeded in the classes of indicator functions with indicator function as the loss function,which make the calculation to be difficult.This paper investigates NeymanPearson classification with convex loss function in the arbitrary class of real measurable functions.A general condition is given under which Neyman-Pearson classification with convex loss function has the same classifier as that with indicator loss function.We give analysis to NP-ERM with convex loss function and prove it's performance guarantees.An example of complexity penalty pair about convex loss function risk in terms of Rademacher averages is studied,which produces a tight PAC bound of the NP-ERM with convex loss function.     

Testing a sub-hypothesis in linear regression models with long memory covariates and errors

This paper considers the problem of testing a sub-hypothesis in homoscedastic linear regression models when the covariate and error processes form independent long memory moving averages. The asymptotic null distribution of the likelihood ratio type test based on Whittle quadratic forms is shown to be a chi-square distribution. Additionally, the estimators of the slope parameters obtained by minimizing the Whittle dispersion is seen to be n ^1/2-consistent for all values of the long memory parameters of the design and error processes. Research of the first author was partly supported by the NSF DMS Grant 0701430. Research of the second author was partly supported by the bilateral France-Lithuania scientific project Gilibert and the Lithuanian State Science and Studies Foundation grant T-15/07.     

Controlled Markov processes on the infinite planning horizon: Weighted and overtaking cost criteria

Stochastic control problems for controlled Markov processes models with an infinite planning horizon are considered, under some non-standard cost criteria. The classical discounted and average cost criteria can be viewed as complementary, in the sense that the former captures the short-time and the latter the long-time performance of the system. Thus, we study a cost criterion obtained as weighted combinations of these criteria, extending to a general state and control space framework several recent results by Feinberg and Shwartz, and by Krass et al. In addition, a functional characterization is given for overtaking optimal policies, for problems with countable state spaces and compact control spaces; our approach is based on qualitative properties of the optimality equation for problems with an average cost criterion.Research partially supported by the Engineering Foundation under grant RI-A-93-10, in part by the National Science Foundation under grant NSF-INT 9201430, and in part by a grant from the AT&T Foundation.Research partially supported by the Air Force Office of Scientific Research under Grant F49620-92-J-0045, and in part by the National Science Foundation under Grant CDR-8803012.     

Optimal control of linear stochastic systems described by functional differential equations

The optimal control is determined for a class of systems by assuming the configuration of the feedback loop. The feedback loop consists of an unbiased estimator and controller. The gain matrices of the estimator and the controller are so determined that the mean-squared estimation error and the average value of a quadratic cost functional, respectively, are minimized. This is accomplished by the application of the matrix maximum principle to a distributed parameter system. The results indicate that the optimal estimation and the optimal control can be computed independently (separation principle).This work was supported in part by the Air Force Office of Scientific Research, Grant No. 69-1776.     

Inductive theories and their forcing companions

A decomposition theorem for ideals of a distributive lattice is related to a classification of the generic models of an arbitrary inductive theory, generalizing, for example, the classification of algebraically closed fields according to their characteristics. Research supported in part by the National Science Foundation Grant No. GP-29218.     

Deflation in Krylov subspace methods and distance to uncontrollability

The task of extracting from a Krylov decomposition the approximation to an eigenpair that yields the smallest backward error can be phrased as finding the smallest perturbation which makes an associated matrix pair uncontrollable. Exploiting this relationship, we propose a new deflation criterion, which potentially admits earlier deflations than standard deflation criteria. Along these lines, a new deflation procedure for shift-and-invert Krylov methods is developed. Numerical experiments demonstrate the merits and limitations of this approach. This author has been supported by a DFG Emmy Noether fellowship and in part by the Swedish Foundation for Strategic Research under the Frame Programme Grant A3 02:128.     

The facets of the polyhedral set determined by the Gale—Hoffman inequalities

The Gale—Hoffman inequalities characterize feasible external flow in a (capacitated) network. Among these inequalities, those that are redundant can be identified through a simple arc-connectedness criterion.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.Research supported in part by NATO Collaborative Research Grant 0785/87.Supported in part by a grant of the Air Force Office of Scientific Research.     

Global Error Bound for the Generalized Linear Complementarity Problem over a Polyhedral Cone

In this paper, the global error bound estimation for the generalized linear complementarity problem over a polyhedral cone (GLCP) is considered. To obtain a global error bound for the GLCP, we first develop some equivalent reformulations of the problem under milder conditions and then characterize the solution set of the GLCP. Based on this, an easily computable global error bound for the GLCP is established. The results obtained in this paper can be taken as an extension of the existing global error bound for the classical linear complementarity problems. This work was supported by the Research Grant Council of Hong Kong, a Chair Professor Fund of The Hong Kong Polytechnic University, the Natural Science Foundation of China (Grant No. 10771120) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.     

The subgaussian constant and concentration inequalities

We study concentration inequalities for Lipschitz functions on graphs by estimating the optimal constant in exponential moments of subgaussian type. This is illustrated on various graphs and related to various graph constants. We also settle, in the affirmative, a question of Talagrand on a deviation inequality for the discrete cube. Research supported in part by NSF Grant No. DMS-0405587 and by EPSRC Visiting Fellowship. Research supported in part by NSF Grant No. DMS-9803239, DMS-0100289. Research supported in part by NSF Grant No. DMS-0401239.     

L-functions of second-order cusp forms

We discuss equivalent definitions of holomorphic second-order cusp forms and prove bounds on their Fourier coefficients. We also introduce their associated L-functions, prove functional equations for twisted versions of these L-functions and establish a criterion for a Dirichlet series to originate from a second order form. In the last section we investigate the effect of adding an assumption of periodicity to this criterion. 2000 Mathematics Subject Classification Primary—11F12, 11F66 G. Mason: Research supported in part by NSF Grant DMS 0245225. C. O’Sullivan: Research supported in part by PSC CUNY Research Award No. 65453-00 34.     

Uniform consistency of automatic and location-adaptive delta-sequence estimators

Summary The class of delta-sequence estimators for a probability density includes the kernel, histogram and orthogonal series types, because each can be characterized as a collection of averages of some function that is indexed by a smoothing parameter. There are two important extensions of this class. The first allows a random smoothing parameter, for example that specified by a cross-validation method. The second allows the smoothing parameter to be a function of location, for example an estimator based on nearest-neighbor distance. In this paper a general method is presented which establishes uniform consistency for all of these estimators.Research partially supported by AFOSR Grant No. S-49620-82-C-0144, and by NSF Grant DMS-850-3347Research supported by NSF Grant DMS-8400602     

Bilinear programming and structured stochastic games

One-step algorithms are presented for two classes of structured stochastic games, namely, those with additive rewards and transitions and those which have switching controllers. Solutions to such classes of games under the average reward criterion can be derived from optimal solutions to appropriate bilinear programs. The validity of using bilinear programming as a solution method follows from two preliminary theorems, the first of which is a complete classification of undiscounted stochastic games with optimal stationary strategies. The second of these preliminary theorems relaxes the conditions of the classification theorem for certain classes of stochastic games and provides the basis for the bilinear programming results. Analogous results hold for the discounted stochastic games with the above special structures.This research was supported in part by the Air Force Office of Scientific Research and by the National Science Foundation under Grant No. ECS-850-3440.     

A minimal cutset of the boolean lattice with almost all members

Two almost explicit constructions are given satisfying the title.This research was done while the authors visited the Institute for Mathematics and its Applications at the University of Minnesota, Minneapolis, MN 55455.Research supported in part by Hungarian National Foundation for Scientific Research, grant no. 1812.Research supported in part by NSF Grant MDS 87-01475.Research supported in part by NSF grant DMS 86-06225 and Airforce Grant OSR-86-0076.     

Limit theorems for sums of dependent random variables occurring in statistical mechanics

Summary By the use of conditioning, we extend previously obtained results on the asymptotic behavior of partial sums for certain triangular arrays of dependent random variables, known as Curie-Weiss models. These models arise naturally in statistical mechanics. The relation of these results to multiple phases, metastable states, and other physical phenomena is explained.Alfred P. Sloan Research Fellow. Research supported in part by National Science Foundation Grant MPS 76-06644A01Alfred P. Sloan Research Fellow. Research supported in part by National Science Foundation Grant MCS 77-20683 and by U.S.-Israel Binational Science FoundationResearch supported in part by National Science Foundation Grant PHY77-02172     

Truncation error analysis by means of approximant systems and inclusion regions

Summary A general approach to truncation error analysis is described, in which bounds for the truncation error are determined by means of inclusion regions, and the notion of bestness is meaningfully formulated. A new mathematical structure (approximant system) is introduced and developed. It consists of a family of infinite processes having a natural structure for truncation error analysis. Applications of the methods are included for infinite series, Cesaro sums, approximate integration, an iterative method for solving equations, Padé approximants and continued fractions.Research supported in part by the National Science Foundation under Grant No. MPS 74-22111 and by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-70-1888. The United States Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon     

On simulating arbitrary events

In simulation studies, the distributional properties of random variables associated witharbitrary events in point processes, queues, and other stochastic models are to be understood as appropriate averages over long simulation runs. We caution against trying to generate arbitrary events explicitly by some randomized selection. Because of the likelihood of hidden selection biases, that easily results in significant errors. The point is illustrated by an example for which explicit formulas yield computational results that allow comparisons with the simulation estimates. This work was supported by the Australian Research Council and was done at the ARC Special Research Centre for Ultra-Broadband Information Networks (CUBIN), Department of Electrical and Electronic Engineering, The University of Melbourne, Victoria 3010, Australia. The research of M.F. Neuts was supported in part by NSF Grant Nr. DMI-9988749.     

Adaptive control of discounted Markov decision chains

In this paper, we consider discounted-reward finite-state Markov decision processes which depend on unknown parameters. An adaptive policy inspired by the nonstationary value iteration scheme of Federgruen and Schweitzer (Ref. 1) is proposed. This policy is briefly compared with the principle of estimation and control recently obtained by Schäl (Ref. 4).This research was supported in part by the Consejo Nacional de Ciencia y Tecnología under Grant No. PCCBBNA-005008, in part by a grant from the IBM Corporation, in part by the Air Force Office of Scientific Research under Grant No. AFOSR-79-0025, in part by the National Science Foundation under Grant No. ECS-0822033, and in part by the Joint Services Electronics Program under Contract No. F49620-77-C-0101.     

Simultaneous estimation of parameters in exponential families

Summary The paper considers estimation of the natural parameter vector or the mean vector from independent distributions each belonging to the one-parameter discrete or absolutely continuous exponential family. The usual estimators (maximum likelihood, minimum variance unbiased or best invariant) are improved simultaneously under various weighted squared error losses. Research supported by the NSF Grant Number MCS-8202116.     

Locally adaptive hazard smoothing

Summary We consider nonparametric estimation of hazard functions and their derivatives under random censorship, based on kernel smoothing of the Nelson (1972) estimator. One critically important ingredient for smoothing methods is the choice of an appropriate bandwidth. Since local variance of these estimates depends on the point where the hazard function is estimated and the bandwidth determines the trade-off between local variance and local bias, data-based local bandwidth choice is proposed. A general principle for obtaining asymptotically efficient data-based local bandwiths, is obtained by means of weak convergence of a local bandwidth process to a Gaussian limit process. Several specific asymptotically efficient bandwidth estimators are discussed. We propose in particular an, asymptotically efficient method derived from direct pilot estimators of the hazard function and of the local mean squared error. This bandwidth choice method has practical advantages and is also of interest in the uncensored case as well as for density estimation.Research supported by UC Davis Faculty Research Grant and by Air Force grant AFOSR-89-0386Research supported by Air Force grant AFOSR-89-0386     

Optimal designs with respect to Elfving's partial minimax criterion in polynomial regression

For the polynomial regression model on the interval [a, b] the optimal design problem with respect to Elfving's minimax criterion is considered. It is shown that the minimax problem is related to the problem of determining optimal designs for the estimation of the individual parameters. Sufficient conditions are given guaranteeing that an optimal design for an individual parameter in the polynomial regression is also minimax optimal for a subset of the parameters. The results are applied to polynomial regression on symmetric intervals [–b, b] (b1) and on nonnegative or nonpositive intervals where the conditions reduce to very simple inequalities, involving the degree of the underlying regression and the index of the maximum of the absolute coefficients of the Chebyshev polynomial of the first kind on the given interval. In the most cases the minimax optimal design can be found explicitly.Research supported in part by the Deutsche Forschungsgemeinschaft.Research supported in part by NSF Grant DMS 9101730.