On the estimation of the jump activity index in the case of random observation times
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We propose a nonparametric estimator of the jump activity index β of a pure-jump semimartingale X driven by a β-stable process when the underlying observations are coming from a high-frequency setting at irregular times. The proposed estimator is based on an empirical characteristic function using rescaled increments of X, with a limit which depends in a complicated way on β and the distribution of the sampling scheme. Utilising an asymptotic expansion we derive a consistent estimator for β and prove an associated central limit theorem.
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