Nonquadratic estimators of a quadratic functional

Estimation of a quadratic functional over parameter spaces that are not quadratically convex is considered. It is shown, in contrast to the theory for quadratically convex parameter spaces, that optimal quadratic rules are often rate suboptimal. In such cases minimax rate optimal procedures are constructed based on local thresholding. These nonquadratic procedures are sometimes fully efficient even when optimal quadratic rules have slow rates of convergence. Moreover, it is shown that when estimating a quadratic functional nonquadratic procedures may exhibit different elbow phenomena than quadratic procedures.Comment: Published at http://dx.doi.org/10.1214/009053605000000147 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Phase-space Formulation of the Belavkin-Kushner-Stratonovich Filtering Equation for Nonlinear Quantum Stochastic Systems

This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic systems with multichannel nondemolition measurements. The system-observation dynamics are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation driven by quantum Wiener processes of bosonic fields in vacuum state. The Hamiltonian and system-field coupling operators, as functions of the system variables, are represented in a Weyl quantization form. Using the Wigner-Moyal phase-space framework, we obtain a stochastic integro-differential equation for the posterior quasi-characteristic function (QCF) of the system conditioned on the measurements. This equation is a spatial Fourier domain representation of the Belavkin-Kushner-Stratonovich stochastic master equation driven by the innovation process associated with the measurements. We also discuss a more specific form of the posterior QCF dynamics in the case of linear system-field coupling and outline a Gaussian approximation of the posterior quantum state.Comment: 12 pages, a brief version of this paper to be submitted to the IEEE 2016 Conference on Norbert Wiener in the 21st Century, 13-15 July, Melbourne, Australi

Robust Stability Analysis of Nonlinear Hybrid Systems

We present a methodology for robust stability analysis of nonlinear hybrid systems, through the algorithmic construction of polynomial and piecewise polynomial Lyapunov-like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials. Several improvements compared to previous approaches are discussed, such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems

The Davidon-Fletcher-Powell penalty function method: A generalized iterative technique for solving parameter optimization problems

The Fletcher-Powell version of the Davidon variable metric unconstrained minimization technique is described. Equations that have been used successfully with the Davidon-Fletcher-Powell penalty function technique for solving constrained minimization problems and the advantages and disadvantages of using them are discussed. The experience gained in the behavior of the method while iterating is also related

US Monetary Policy Rules: the Case for Asymmetric Preferences

This paper investigates the empirical relevance of a new framework for monetary policy analysis in which decision makers are allowed to weight differently positive and negative deviations of inflation and output from the target values. The specification of the central bank objective is general enough to nest the symmetric quadratic form as a special case, thereby making the derived policy rule potentially nonlinear. This forms the basis of our identification strategy which is used to evelop a formal hypothesis testing for the presence of asymmetric preferences. Reduced-form estimates of postwar US policy rules indicate that the preferences of the Fed have been highly asymmetric with respect to both inflation and output gaps, with the latter being the dominant source of nonlinearity after 1983.nonlinear optimal monetary policy rules, asymmetric loss function, linearized central bank Euler equation

NONQUADRATIC COST AND NONLINEAR FEEDBACK CONTROL

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/57832/1/BernsteinNonquadraticCost1993.pd