Inference for the Tail Parameters of a Linear Process with Heavy Tail Innovations

Abstract

Consider a linear process $X_t = \sum\nolimits_{i = 0}^\infty {c_i Z_{t - 1} }$ where the innovations Z's are i.i.d. satisfying a standard tail regularity and balance condition, vis., P(Z > z) ∼ rz-αL1(z), P(Z < -z) ∼ sz-αL1(z), as z →∞, where r + s = 1, r, s ≥ 0, α > 0 and L1 is a slowly varying function. It turns out that in this setup, P(X > x) ∼ px-αL(x), P(X < -x) ∼ qx-αL(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 = $\left\| {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{c} } \right\|_\alpha ^\alpha L_1$ where $\left\| {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{c} } \right\|_\alpha = \left( {\sum {\left| {c_i } \right|^\alpha } } \right)^{1/\alpha }$. The quantities α and β = 2p - 1 can be regarded as tail parameters of the marginal distribution of Xt. We estimate α and β 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.