Sample splitting with Markov chains

Sample splitting techniques play an important role in constructing estimate s with prescribed influence functions in semi-parametric and nonparametric models when the observations are independent and identically distributed. T his paper shows how a contiguity result can be used to modify these techniq ues for the case when the observations come from a stationary and ergodic M arkov chain. As a consequence, sufficient conditions for the construction o f efficient estimates in semi-parametric Markov chain models are obtained. The applicability of the resulting theory is demonstrated by constructing a n estimate of the innovation variance in a nonparametric autoregression mod el, by constructing a weighted least-squares estimate with estimated weight s in an autoregressive model with martingale innovations, and by constructi ng an efficient and adaptive estimate of the autoregression parameter in a heteroscedastic autoregressive model with symmetric errors.