We study the problem of nonparametric classification with repeated observations. Let${\mathbf {X}}$be the$d$dimensional feature vector and let$Y$denote the label taking values in$\lbrace 1,\ldots, M\rbrace$. In contrast to usual setup with large sample size$n$and relatively low dimension$d$, this letter deals with the situation, when instead of observing a single feature vector${\mathbf {X}}$we are given$t$repeated feature vectors${\mathbf {V}}_{1},\ldots, {\mathbf {V}}_{t}$. Some simple classification rules are presented such that the conditional error probabilities have exponential rate of convergence as$t\to \infty$.

Cover's algorithm yields an iterative portfolio choice for maximizing expected log investment return where the distribution function of the stock market vector is known. In the case that the stock market vectors form a stationary ergodic sequence with unknown distribution, by stochastic approximation and nonparametric regression estimation the algorithm is modified for iterative estimation of a log-optimal portfolio selection function of the last$d$observed vectors (fixed$d\in {\bf N}$) on the basis of an observed training sequence of vectors. Under a boundedness and a mild$\alpha $-mixing condition, a strong consistency result is established.

Discrete time growth optimal investment in stock markets with proportional transactions costs is considered. The market process is modeled by a first-order Markov process. Not assuming that the distribution of the market process is known, we show empirical investment strategies such that, in the long run, the growth rate on trajectories achieves the maximum with probability 1.

Static forecasting of stationary and ergodic time series is considered, i.e., inference of the conditional expectation of the response variable at time zero given the infinite past. It is shown that the mean squared error of a combination of suitably defined localized least squares estimates converges to zero for all distributions where the response variable is square integrable.

We consider American options also called Bermudan options in discrete time.We use the dual approach to derive upper bounds on the price of such options using only a reduced number of nested Monte Carlo steps. The key idea is to use nonparametric regression to estimate continuation values and all other required conditional expectations and to combine the resulting estimate with another estimate computed by using only a reduced number of nested Monte Carlo steps. The mean value of the resulting estimate is an upper bound on the option price. One can show that the estimates of the option prices are universally consistent, i.e., they converge to the true price regardless of the structure of the continuation values. The finite sample behavior is validated by experiments on simulated data.

This paper presents relative stability properties of various nonparametric density estimators (histogram, kernel estimates) and of regression estimators (partitioning, kernel, and nearest neighbor estimates). In density estimation, let En denote the L/sub 1/ error of an estimate calculated from n data, whereas in regression estimation, the L/sub 2/ error of the estimate is used. Sufficient conditions for E/sub n//E{E/sub n/}/spl rarr/1 in probability are provided. If this limit holds, the asymptotic behavior of the random error E/sub n/ can be characterized by its expectation E{E/sub n/},, and one may apply, for example, the established rate-of-convergence results for E{En}.

Let stock market vectors form a stationary ergodic sequence. For fixed d /spl isin/ N, a log-optimal portfolio selection function of the past d observed vectors is iteratively estimated on the basis of a training sequence by use of gradients and nonparametric regression. Strong consistency is obtained under a boundedness and /spl alpha/-mixing condition without further assumptions on the distribution.