On the \(L^p\)-consistency of wavelet estimators (Q327270)

The size-biased data is the collected data if the probability of inclusion depends on a certain known function (weight function) of an observation. This paper provides some \(L^{p}\)-consistency results of wavelet estimators under size-biased samples. The authors show the \(L^{p}\)-consistency (\(1\leq p<\infty\)) for both independent and identically distributed random vectors in \(\mathbb R^{d}\) and negatively associated samples in \(\mathbb R\). They also discuss the case \(p=\infty\) and obtain similar results under additional conditions on the weight function \(g\) and the unknown density function \(f_{X}\).