No abstract

No abstract

A local linear kernel estimator of the regression function x\mapsto g(x):=E[Y_i|X_i=x], x\in R^d, of a stationary (d+1)-dimensional spatial process {(Y_i,X_i),i\in Z^N} observed over a rectangular domain of the form I_n:={i=(i_1,...,i_N)\in Z^N| 1\leq i_k\leq n_k,k=1,...,N}, n=(n_1,...,n_N)\in Z^N, is proposed and investigated. Under mild regularity assumptions, asymptotic normality of the estimators of g(x) and its derivatives is established. Appropriate choices of the bandwidths are proposed. The spatial process is assumed to satisfy some very general mixing conditions, generalizing classical time-series strong mixing concepts. The size of the rectangular domain I_n is allowed to tend to infinity at different rates depending on the direction in Z^N.Comment: Published at http://dx.doi.org/10.1214/009053604000000850 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

For a first-order autoregressive process Y t = $Y t _ x + e ti where the e/s are i.i.d. and belong to the domain of attraction of a stable law, the strong con-' sistency of the ordinary least-squares estimator b n of 0 is obtained for 0 = 1 , and the limiting distribution of b n is established as a functional of a Levy process. Generalizations to seasonal difference models are also considered. These results are useful in testing for the presence of unit roots when the e/s are heavy-tailed. INTRODUCTIONConsider the first-order autoregressive process (AR (1)) where $ is the unknown parameter to be estimated from the data {Y t } and {e,} is the random disturbances. Customarily, /? is estimated by its leastsquares estimate /> -[ V y 2 11 V y i Y°nWhen the disturbances f e r } are independent and identically distributed (i.i.d.) or form a martingale difference sequence with (2 + b) moments 9 b > 0, the asymptotic properties of b n such as strong consistency and weak convergence are well-studied both for |/?| < 1 and |/S| = 1. See for example, Lai and Wei [14] and the references therein. Recently, there has been considerable interest in the asymptotic properties of b n when the e/s are heavy-tailed. Heavy-tailed distributions are useful in modeling certain economic variables and stock price changes. Some references are Fama [6], Mandelbrot [15], McCullough [16], and DuMouchel [5]. When the e/s are i.i.d. with common distribution function F which belongs to the domain of attraction of a stable law with index a < 2, Hannan and Kanter [9] and Knight [12] establish the strong consistency of the leastsquares estimates b n of fi for a stationary AR(/?) model. This includes (1) 'Research supported in part by the National Science Foundation through Grant DMS-8503358 and ECE-8513980. We wish to thank the editor and two referees for helpful discussions and comments on an earlier draft. 354

No abstract