K. M. Wong and S. Chen [9] analyzed the Shannon entropy of a sequence of random variables under order restrictions. Using $(r,s)$-entropies, I. J. Taneja [8], these results are generalized. Upper and lower bounds to the entropy reduction when the sequence is ordered and conditions under which they are achieved are derived. Theorems are presented showing the difference between the average entropy of the individual order statistics and the entropy of a member of the original independent identically distributed...

The problem of nonparametric function fitting using the complete orthogonal system of Whittaker cardinal functions $s_k$, k = 0,±1,..., for the observation model $y_j=f\left(u_j\right)+{\eta }_j$, j = 1,...,n, is considered, where f ∈ L²(ℝ) ∩ BL(Ω) for Ω > 0 is a band-limited function, $u_j$ are independent random variables uniformly distributed in the observation interval [-T,T], ${\eta }_j$ are uncorrelated or correlated random variables with zero mean value and finite variance, independent of the observation points. Conditions for convergence...

This paper is concerned with general conditions for convergence rates of nonparametric orthogonal series estimators of the regression function. The estimators are obtained by the least squares method on the basis of an observation sample $Y_i=f\left(X_i\right)+{\eta }_i$, i=1,...,n, where $X_i\in A\subset {ℝ}^d$ are independently chosen from a distribution with density ϱ ∈ L¹(A) and ${\eta }_i$ are zero mean stationary errors with long-range dependence. Convergence rates of the error $n^{-1}{\sum }_{i=1}^n(f\left(X_i\right)-f{̂}_N\left(X_i\right))²$ for the estimator $f{̂}_N\left(x\right)={\sum }_{k=1}^{N}c{̂}_k{e}_k\left(x\right)$, constructed using an orthonormal system $e_k$, k=1,2,...,...

Nonparametric orthogonal series regression function estimation is investigated in the case of a fixed point design where the observation points are irregularly spaced in a finite interval [a,b]i ⊂ ℝ. Convergence rates for the integrated mean-square error and pointwise mean-square error are obtained in the case of estimators constructed using the Legendre polynomials and Haar functions for regression functions satisfying the Lipschitz condition.