INFERENCE FOR THE TAIL PARAMETERS OF A LINEAR PROCESS WITH HEAVY TAIL INNOVATIONS

SOMNATH DATTA AND WILLIAM P. MCCORMICK

Department of Statistics, Franklin College of Arts and Sciences, University of Georgia, 204 Statistics Building, Athens, GA 30602-1952, U.S.A.

(Received December 5, 1995; revised January 6, 1997)

Abstract. Consider a linear process X_{t} = ∑_{i=0}^{∞} c_{i}Z_{t−i}, where the innovations Z’s are i.i.d. satisfying a standard tail regularity and balance condition, viz., P (Z > z) ∼ rz^{−α}L_{1}(z), P (Z < −z) ∼ sz^{−α}L_{1}(z), as z → ∞, where r + s = 1, r, s ≥ 0, α > 0 and L_{1} is a slowly varying function. It turns out that in this setup, P (X > x) ∼ px^{−α}L(x), P (X < −x) ∼ qx^{−α}L(x), as x → ∞, where α is the same as above, p is a convex combination of r and s, p + q = 1, p, q ≥ 0 and L = ∥\underline{c}∥_{α}^{α}L_{1} where ∥\underline{c}∥_{α} = (Σ|c_{i}|^{α})^{1/α}. The quantities α and β = 2p − 1 can be regarded as tail parameters of the marginal distribution of X_{t}. We estimate α and β based on a finite realization X_{1}, …, X_{n} 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 X_{t}. 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.