Joint Estimation of Discretely Observed Stable Lévy Processes with Symmetric Lévy Density

Abstract

Consider a real-valued non-Gaussian stable Lévy process X such that $\mathcal{L}$(Xt-γt)=Sα(t1/ασ), and suppose that we observe a discrete-time sample (Xihn)ni=0. Under the condition hn→0 at an appropriate rate, the corresponding statistical experiments governed by the parameter θ=(α,σ,γ) exhibit the LAN property at the unusual rate of convergence diag√—n log(1/hn),√—n, √—n hn1-1/α, but the Fisher information matrix is constantly singular as soon as both α and σ are unknown. This implies that the standard asymptotic behavior of the maximum likelihood estimator breaks down, and also that it is in no way obvious whether or not existing results concerning estimators of the stable law in the usual case where hn≡ h>0 can maintain the same asymptotic behaviors. In this note we will provide easily computable full-joint estimators of the parameters, which possess asymptotic normality with a finite and nondegenerate asymptotic covariance matrix, thereby enabling us to construct a joint confidence region of the three parameters: the rate of convergence of our estimators of θ is diag(√—n,√—n,√—n hn1-1/α). Especially, we clarify that a suitable sample-median type statistic $\hat{γn}$ serves as a rate-efficient estimator of the location γ, and that our procedure of estimating the remaining two parameters is not asymptotically influenced by plugging in $\hat{γn}$, even if the convergence rate of <$\hat{γn}$ is slower than the other two (namely, even if α∈(1,2)). Finite-sample behaviors of our estimators are investigated through several simulation experiments.

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