A theorem on the convergence of algorithms of static stochastic optimization

.The problem of this paper is to minimize a function f, which is scalar-valued and defined on a finite dimensional vector space. An iterative algorithm is of the form X(n+1)=A(n)(X(n)) and can take the usual form X(n+1)=X(n)−a(n)Y(n), where Y(n) can be as in the Kiefer-Wolfowitz procedure, but an is random. Making use of the theorem on convergence of supermartingales the author gives several theorems on the convergence of the procedure to the minimal point of f

Testing for a difference between conditional variance functions of nonlinear time series

In this report, the problem of testing for a difference between conditional variance fnuctions (or volatilites) of two independent nonlinear time series is investigated by means of an extensive simulation study. Empirical results on the properties of the test proposed confirm the test's validity, at least for some types of heteroscedasticity as contrasted with homnoscedastic erroos as well as for some types of differences in heteooscedasticity. Moreover, interesting properties of several estimators of conditional mean, variance and fourth moment functions are empirically found too

J. Koronacki

Let X (1) ; : : : ; X (m) be m independent, real-valued random variables with unknown densities and let Y = \Psi(X (1) ; : : : ; X (m) ) be their known functional. The aim is to estimate the density of the random variable Y , given n independent realizations of the vector (X (1) ; : : : ; X (m) ). In the paper, three natural ways of solving the estimation problem, all based on the Rosenblatt-Parzen estimator, are proposed and compared by means of a simulation study. Applicability of each of the three methods is thus assessed. Simulations suggest that the "direct" method based on the observations of Y only, is "almost always" outperformed by the other two methods. Recommendations for choosing between the methods are given. Key words: density estimation, kernel estimators, estimating density of a functional. Subject classification: Primary 62G05. ) The paper was completed while on leave of this author to the Department of Statistics, Rice University, Houston, Texas. Partia..

The Sarmatian Review, Vol. 12, No. 1

Jacek Koronacki, "Liberty and the Polish Intelligentsia"; Witold J. Lukaszewski, "Security in the post-Soviet Era", James R. Thompson, "A Glimpse at the Dark Side of Socialist Industrial Planning"; Adam A. Hetnal, "Soviet Women Walking the Tightrope"; Polish Women Artists' Exhibit in Washington; Independence for Croatia and Slovenia; From the Editor; The Sarmatian Review Recommends; BOOKS; LETTERS; PIASA MEETIN