ISSN:
1572-9052
Keywords:
Linear processes
;
heavy tailed distribution
;
tail parameters
;
tail probability
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1003499300817
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