EDGEWORTH APPROXIMATION IN THE AR(1) PROCESS WITH SOME POSSIBLY NONZERO INITIAL VALUE

Abstract

where c1, c2 ≥ 0 (see Ochi (1983)). The estimator θ1,0 is the least-squares estimator of θ which minimizes ∑T t=2(Xt − θXt−1), and for the model (1.1) with initial value X0 = 0, it is also the maximum likelihood estimator of θ. Furthermore, the estimator θ1,1 is the Yule-Walker estimator, and the estimator θ1/2,1/2 is the Burg estimator (see also Daniels (1956)). The random variable θc1,c2 is a ratio of quadratic forms of correlated normal random variables. The methods for calculating the probability distribution of such ratios are given in Mathai and Provost (1992; Section 4.5). The formulae they present are complicated, and one resorts instead to numerical methods such as Imhof (1961) and Helstrom (1996). Imhof (1961) gave a numerical method for inverting the characteristic function of a linear combination of several noncentral chi-squares random variables, and Helstrom (1996) used the saddlepoint integration for the numerical inversion. The saddlepoint approximation method in statistics was originally introduced by Daniels (1954, 1956). Following Daniels (1956), Phillips (1978) and Kakizawa (1996) derived the saddlepoint density for