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Zhang, Chenhua; McCormick, William P.

Asymptotic properties of the tail distribution and Hill’s estimator for shot noise sequence. (English) Zbl 1329.62241

Extremes 15, No. 4, 407-435 (2012).
Summary: Asymptotic properties of a shot noise sequence, \(\{X_j=\sum_{i\leq j}h(\tau_j-\tau_i)A_i\}\), are considered, where the marginal distribution of the \(A_i\)’s is regularly varying at infinity with negative index \(-\alpha\). The tail distribution of the \(X_j\)’s is studied with regard to higher order tail area expansions, validity of the Von Mises condition and the inheritability of the second order regular variation property. The Hill’s estimator for the tail index is shown to be asymptotically normally distributed provided the impulse response function \(h\) satisfies a mild integrability condition and the distribution of the \(X_j\)’s satisfies some regularity conditions.

MSC:

62G32 Statistics of extreme values; tail inference
62G20 Asymptotic properties of nonparametric inference
60G70 Extreme value theory; extremal stochastic processes

Keywords:

shot noise; heavy tail; regular variation; von Mises condition; second-order regularly varying; higher-order expansion; tail index; Hill’s estimator
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