Nonparametric Extension of Regression Functions Outside Domain

The article refers to the problem of regression functions estimation in the points situated near the edges but outside of function domain. We investigate the model

$y_i = R\left( {x_i } \right) + \epsilon _i ,\,i = 1,2, \ldots n$

, where

x

i

is assumed to be the set of deterministic inputs,

x

i

 ∈ 

D

,

y

i

is the set of probabilistic outputs, and

ε

i

is a measurement noise with zero mean and bounded variance.

R

(.) is a completely unknown function. In the literature the possible ways of finding unknown function are based on the algorithms derived from the Parzen kernel. These algorithms were also applied to estimation of the derivatives of unknown functions. The commonly known disadvantage of the kernel algorithms is that the error of estimation dramatically increases if the point of estimation

x

is approaching to the left or right bound of interval

D

. Algorithms on predicting values in the boundary region outside the function domain

D

are unknown for the author, so far.

The main result of this paper is a new algorithm based on integral version of Parzen methods for local prediction of values of the function

R

near boundaries in the region outside domain. The results of numerical experiments are presented.