Nonparametric Estimation in a Model with a Trend

Let Yt be a stochastic process taking values in R and defined by the model Yt=azt+(Xt)+ t where {zt} is a deterministic sequence, {Xt} is strictly stationary and strongly mixing, and {t} is i.i.d. We study asymptotic properties of nonparametric estimators of density and regression with rates of convergence, and their behavior on estimation when () is polynomial. It is shown that the estimator of the coefficients of () constructed from the nonparametric estimators of regression is consistent when the deterministic {zt} converges in Cesàro mean, and may be inconsistent when {zt} is periodic under some ordinary conditions.     

Determining the limiting solutions of nonstationary axisymmetric Hele-Shaw problems

Numerical solution of the Hele-Shaw problem reduces to solution of three boundary-value problems of determining analytic functions of a complex variable in each time step: conformal mapping of the range of the parametric variable to the physical plane, the Dirichlet problems for determining the electric-field strength, and the Riemann-Hilbert problem for calculating partial time derivatives of the coordinates of points of the interelectrode space (the images of the points on the boundary of the parametric plane are fixed). Unlike in the two-dimensional problem, the electric-field strength is determined using integral transformations of an analytic function. Approximation by spline function is performed, and more accurate and steady (than the well-known ones) general solution algorithms for the nonstationary axisymmetric problems are described. Results of a numerical study of the formation of stationary and self-similar configurations are presented. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 87–99, July–August, 2009.     

Nonstationarity in statistical process control — issues,cases, ideas

Statistical process control (SPC) is a powerful framework that is used in many industries to decrease process variability and to pinpoint special cause variation. Although a broad range of techniques have been developed to do so, often the real‐life situation does not fully comply with the basic assumptions that are made in SPC resulting in poor results. One of the main violations against the assumptions is the fact that industrial processes rarely behave in a stationary manner — this is evidently the case for biological processes but is also an important issue when monitoring industrial processes. Besides, the ever increasing amount of data, with a clear shift towards multivariate and even multiway quality control, makes the classical univariate approach not feasible anymore. These two observations pose important challenges to statisticians to develop novel SPC algorithms that are broadly applicable in modern industries. In this contribution we discuss both issues and use two very different case studies to show the reader recent directions and developments in the SPC landscape. Copyright © 2011 John Wiley & Sons, Ltd.