A bias correction for cross-validation bandwidth selection when a kernel estimate is based on dependent data (Q2744937)

For a stationary ergodic process \(X(t)\) with bouded one-dimensional probability density \(f\) a kernel-type estimate of \(f\) is considered: NEWLINE\[NEWLINEf_h(x)=T^{-1}\int_0^T K\left((X(t)-x)/h\right)dt,NEWLINE\]NEWLINE where \(K\) is a non-negative bounded and compactly supported kernel, and \(h\) is a bandwidth. A cross-validation algorithm of bandwidth selection is considered in which the \(h\)-dependent part of MISE is estimated by its empirical counterpart NEWLINE\[NEWLINELSCV(h)=\int f^2_h(x)dx -2T^{-1}\int_0^T f_h(X_t)dt. NEWLINE\]NEWLINE The author derives an asymptotic expansion of \(E\{LSCV(h)\}\) and demonstrates that its bias is \(4f_{X'}(o)\log(h)T^{-1}+ o(\log(h)T^{-1}\), where \(f_{X'}(0)\) is the limit at zero of the pdf of the r.v. \((X_t-X_0)/t\) as \(t\to 0\). It is proposed to correct this bias using a kernel estimate for \(f_{X'}(0)\). Results of simulation studies are presented.