ORDER STATISTICS FOR NONSTATIONARY TIME SERIES

LANH TAT TRAN^{1} AND BERLIN WU^{2}

1Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A. 2National Chengchi University, Taipei, Taiwan

(Received October 23, 1991; revised October 6, 1992)

Abstract. Order statistics has an important role in statistical inference. The main purpose of this paper is to investigate order statistics, and also explore its applications in the analysis of nonstationary time series. Our results show that linear functions of order statistics for a large class of time series are asymptotically normal. The methods of proof involve approximations of serially dependent random variables by independent ones. The problems of testing for the existence of a linear trend and the problem of testing randomness versus serial dependence are considered as applications.

Key words and phrases: Nonstationary, autoregressive processes, absolute regularity, empirical distribution functions, order statistics.