Notes on minimax approaches in nonparametric regression
Consider the nonparametric regression model \[ Y_ i = g(t_ i) + \varepsilon_ i, \quad i = 1,\dots,n, \] where observations are taken at design points \(t_ i\) for \(i=1, \ldots,n\), and the errors \(\varepsilon_ i\) are independent and identically distributed as normal distribution with mean zero and variance \(\sigma^ 2\). In one section the normality assumption will be unnecessary. The response function \(g\) is assumed to belong to a space \(W = \{g:g\) and \(g'\) are absolutely continuous, and \(\int^ 1_ 0 | g''(t) |^ 2dt<\infty\}\). We deal with minimax estimators of \(g\) and \(\sigma^ 2\) in some sense, based on a restricted class of the response function \(W_ C=\{g \in W: \int^ 1_ 0 | g''(t) |^ 2dt<C\}\). To simplify the minimax problem, we shall use a natural coordinate system.
scientific article; zbMATH DE number 597295