Stochastic Programming and Approximation

We consider a regression function having a unique mode on a compact interval and a curve estimator which is given as a weighted mean of the observations (including most of the common kernel estimators, such as those of Gasser-Müller, Priestley-Chao and Nadaraya-Watson type). As it is done in practice, we determine the empirical mode by maximization of the curve estimator not over the whole interval, but only over a discrete lattice.

We investigate now how the lattice width has to be chosen (asymptotically) in order to optimize the rate of consistency of the empirical mode.

We prove the results for (alpha-)mixing observations, and a potentially non-smooth regression function that is only assumed to be Hölder-continuous.

Our method consists in a rather direct evaluation of the mean squared error of the empirical mode. A recent moment inequality of Rosenthal type due to Yang (2007) for mixing random variables will be used. In the non-smooth case, the resulting rate of consistency of the empirical mode turns out to be even slightly better than any results for any estimator that are known to us so far in this case in the previous literature.

The ultimate goal of formulating any mathematical programming problem is to obtain a solution, usually by sending the problem to a linear or nonlinear solver. This talk explores some of the formats available to represent instances of stochastic programs. Three formats are studied in depth: the algebraic modelling language AMPL, the so-called AMPS format, as well as the XML-based schema OSIL.

The problem of prediction of stationary and ergodic time series with side information is considered. For squared loss, asymptotically optimal prediction is presented, where the basic principles of universally consistent regression estimation (local averaging) and of machine learning (aggregation of estimates) are applied. This prediction rule implies a classification rule for time series, too.