Consideramos una clase de problemas de optimización que surgen en estimaciones de la densidad de datos en dimensión elevada a partir de proyecciones en subespacios de dimensión más baja. Los criterios que se usan para la selección óptima del modelo son máxima entropía y máxima verosimilitud. En cada caso nuestro planteamiento requiere estimadores de la densidad univariados y a este respecto exploramos el uso de modelos mezcla de densidades gaussianas y de estimadores de Parzen para los datos proyectados....

We formulate the notion of Q-independence which generalizes the classical independence of random variables and free independence introduced by Voiculescu. Here Q stands for a family of polynomials indexed by tiny partitions of finite sets. The analogs of the central limit theorem and Poisson limit theorem are proved. Moreover, it is shown that in some special cases this kind of independence leads to the q-probability theory of Bożejko and Speicher.

This paper is a survey of recent results on some problems of supervised learning in the setting formulated by Cucker and Smale. Supervised learning, or learning-from-examples, refers to a process that builds on the base of available data of inputs $x_i$ and outputs $y_i$, i = 1,...,m, a function that best represents the relation between the inputs x ∈ X and the corresponding outputs y ∈ Y. The goal is to find an estimator $f_z$ on the base of given data $z:=((x₁,y₁),...,(x_m,y_m))$ that approximates well the regression function $f_{\rho }$ of...

We present optimal upper bounds for expectations of order statistics from i.i.d. samples with a common distribution function belonging to the restricted family of probability measures that either precede or follow a given one in the star ordering. The bounds for families with monotone failure density and rate on the average are specified. The results are obtained by projecting functions onto convex cones of Hilbert spaces.

In the general geometric asset price model, the asset price P(t) at time t satisfies the relation $P\left(t\right)=P₀·e^{\alpha ·f\left(t\right)+\sigma ·F\left(t\right)}$, t ∈ [0,T], where f is a deterministic trend function, the stochastic process F describes the random fluctuations of the market, α is the trend coefficient, and σ denotes the volatility. The paper examines the problem of optimal trend estimation by utilizing the concept of kernel reproducing Hilbert spaces. It characterizes the class of trend functions with the property that the trend coefficient...