Large deviations behavior for the quadratic error of density estimate (Q1969266)

Let \(f_{nM}\) be the projection estimator of an unknown density \(f\) based on the first \(n\) observations and some orthonormal system \(\{e_j \}^M_{j=-M}\), where \(M=M(n) \to\infty\). Then \(L_{nM}=\int (f_{nM}-f)^2\) is the integrated square error of \(f_{nM}\). Under suitable conditions it is shown that \[ P\bigl(nb^{-1} (L_{nM} -EL_{nM})> r\bigr)/P (Z>r)\to 1 \] uniformly over the range \(r\in(0,r_n)\) for appropriate \(r_n\to\infty\). Here \(Z\) denotes the standard normal random variable and \(b\) is an explicit variance coefficient depending on \(f\) and \(\{e_j\}\). In particular, for continuous \(f\in L_2[-1,1]\), Legendre system \(\{e_j\}\) and \(M=O(n^{2/9})\) the result holds with \(r_n=o (n^{1/27})\). An analogous large deviation result is proved for the bootstrap approximation of the distribution of \(L_{nM}-EL_{nM}\).