Model robust extrapolation designs
A regression model is considered where for each x in an interval X an experiment can be performed whose outcome Y(x) is a random variable with mean f(x) and variance \(\sigma^ 2\). The regression function f is assumed to belong to the class \[ F_ 0=\{f | \quad | f^{(h+1)}(x)| \leq \epsilon,\quad x_ 0<x<\infty \}, \] where \(\epsilon >0\) and \(0>x_ 0\in X^ c\) are given numbers. An n point experimental design \(\xi\) with support \(x_ 1,x_ 2,...,x_ r\in X\) is defined to be a probability measure on X putting mass \(m_ i>0\) at \(x_ i\). The design is exact if \(n_ i=n m_ i\) are integers and approximate otherwise. An estimator of \(f(x_ 0)\) of the form \(\hat f= \sum^{r}_{i=1}a_ i\bar Y(x_ i)\), based on \(\xi\), where \(\bar Y(x_ i)\) is the mean of the \(n_ i\) observations at \(x_ i\), is to be used to extrapolate the value of f at \(x_ 0\). Given \(x_ 0\) and \(\epsilon\), the optimal design problem is to determine an approximate design \(\xi\) on a minimal number \((h+1)\) of design points and constants \(\{a_ i\}\) to minimize the quantity \(\sup_{f\in F_ 0}E(\hat f- f(x_ 0))^ 2.\) Two cases are considered: (1) \(X=[0,\infty)\) and (2) \(X=[-1,1]\). Examples are given to illustrate the allocation of observations in the two cases.
scientific article; zbMATH DE number 4060568