(Received December 5, 1995; revised January 6, 1997)
Abstract. Consider a linear process Xt = \sumi=0\infty ci Zt-i, where the innovations Z's are i.i.d. satisfying a standard tail regularity and balance condition, viz., P(Z > z) \sim rz-alphaL1(z), P(Z < -z) \sim sz-alphaL1(z), as z \rightarrow \infty, where r + s = 1, r, s > 0, alpha > 0 and L1 is a slowly varying function. It turns out that in this setup, P(X > x) \sim px-alphaL(x), P(X < -x) \sim qx-alphaL(x), as x \rightarrow \infty, where alpha is the same as above, p is a convex combination of r and s, p + q = 1, p, q > 0 and L = |c|alphaalpha L1 where |c|alpha = (Sigma|ci|alpha)1/alpha. The quantities alpha and beta = 2p - 1 can be regarded as tail parameters of the marginal distribution of Xt. We estimate alpha and beta based on a finite realization X1, ...., Xn of the time series. Consistency and asymptotic normality of the estimators are established. As a further application, we estimate a tail probability under the marginal distribution of the Xt. A small simulation study is included to indicate the finite sample behavior of the estimators.
Key words and phrases: Linear processes, heavy tailed distribution, tail parameters, tail probability.
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