....

Q;(x, dy) = P;(x_1. dy;)E,_;(dY-.;). An introduction to such Markov chain Monte Carlo procedures is the monograph Gilks et a!. ( 1996). The Gibbs sampler with deterministic sweep uses the transition distribution Q = Q1 · • • Qk, the one with random sweep uses Q = (I jk) L Q;. If we denote the simulations from the corresponding Markov chain by X0 , . .. , xn, the expectation 1rj can be estimated by the empirical estimator (1/n) z=;~d(Xi). Its asymptotic variance is studied, e.g., by Peskun (1973), Frigessi, Hwang and Younes (1992), Green and Han (1992), Liu, Wong and Kong (1994) and (1995) and Clifford and Nicholls (1995). Does the empirical estimator make effective use of the simulations? Greenwood et a!. ( 1997) view 1r as the unknown parameter and calculate variance bounds for estimators of 1r.f. It turns out that it is best not to use the usual empirical estimator but to use the empirical estimator for deterministic sweep which considers the update of a single component as a new observation:

This estimator is close to efficient. D

Q(x, dy) = J p(y-(a:+ z)x) dG(z) dy. The model is studied in the monographs by Nicholls and Quinn (1982) and Tong (1990). Efficient estimators for a: are constructed in Koul and Schick (1996). Weighted least squares estimators and local asymptotic normality for generalized random coefficient autoregressive models are studied in Hwang and Basawa ( 1996), (1997). 0

Q(ui-1 ,Yi-1• du;, dy;) = g(u;)p(y; - (3u; - O:(Yi-1 - (3u;_t)) du; dy; and four parameters, a:, (3, p, g. Local asymptotic normality for deterministic U; is proved in Swensen ( 1985), for moving average X; in Garel ( 1989). For the additive regression model Y; = (3 U; +I'( V;) + X; of Engle et al. ( 1986), efficient estimators for a: and (3, respectively, are constructed in Schick (1993) and (1996). Quite different efficient estimators are obtained for long range stationary Gaussian errors X;: see Dahlhaus (1995). 0

REFERENCES Akahira,M.(1976).Ontheasymptoticefficiencyofestimatorsinanauto-

regressiveprocess.Ann.Inst.Statist.Math.28:35-48.