Consistent least squares nonparametric regression (Q578796)

The regression model is studied. Let \(\{\epsilon_ i\}\) be an independent sequence of identically distributed random variables with mean equal to zero. Let f be a function `almost' in a known finite dimensional collection of functions, F. The data \(\{(z_ i,t_ i)\}\) are modeled as \(z_ i=f(t_ i)+\epsilon_ i.\) The least squares estimate examined is the minimizer over \(h\in S+F\) of \(\sum^{n}_{i=1}(z_ i-h(t_ i))^ 2\). S is a compact (in sup-norm) set of functions. Given a weak condition on the covariates \(\{t_ i\}\), the least squares estimate, \(f^*_ n\), converges in sup-norm to f almost surely. Under further conditions on S, derivatives of \(f^*_ n\) converge almost surely to the derivative of f.