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INFERENCE FOR THE TAIL PARAMETERS OF A LINEAR

PROCESS WITH HEAVY TAIL INNOVATIONS

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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} = \sum_{i=0}^{\infty} *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*) \sim *rz*^{-alpha}*L*_{1}(*z*),
*P*(*Z* < -*z*) \sim *sz*^{-alpha}*L*_{1}(*z*), as *z* \rightarrow \infty,
where *r* + *s* = 1, *r*, *s* __>__ 0, *alpha* > 0 and *L*_{1} is a slowly
varying function. It turns out that in this setup,
*P*(*X* > *x*) \sim *px*^{-alpha}*L*(*x*), *P*(*X* < -*x*) \sim
*qx*^{-alpha}*L*(*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*|_{alpha}^{alpha} *L*_{1} where
|*c*|_{alpha} = (*Sigma*|*c*_{i}|^{alpha})^{1/alpha}. The
quantities *alpha* and *beta* = 2*p* - 1 can be regarded as tail
parameters of the marginal distribution of *X*_{t}. We
estimate *alpha* and *beta* 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.

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