Kernel estimation and interpolation for time series containing missing observations (Q1062717)

Let \(\{X_ t\); \(t=0,\pm 1,...\}\) be a strictly stationary and ergodic process, observed for \(t=1,...,T\). Let \(\nu_ j(x)=E(X_ t| X_{t- j}=x)\) be the best predictor (in the LS sense) of \(X_ t\), given \(X_{t-j}=x\). Let \(b_ t=1\), if \(X_ t\) is observed and \(b_ t=0\), if \(X_ t\) is missed. A conditional expectation estimator \({\hat \nu}{}_ j(x)\) of \(\nu_ j(x)\) is proposed, which depends on \(b_ t\) and on a real-valued integrable kernel K(x). Under several assumptions (including the strong mixing condition for \(X_ t)\), a central limit theorem is proved for \({\hat \nu}{}_ j(x)\). Consistency of the interpolator \({\hat \nu}{}_ j(X_ m)\) for a missing \(X_{m+j}\) is also proved. Numerical illustrations are given.