”Tail Index Estimation for a Filtered Dependent Time Series”

Assumption 1 (Smoothness and Moments). a. Let {=t}t∈Z be a sequence of σ-fields that do not depend on θ and define F := σ(∪t∈Z=t). xt(θ) lies on a complete probability measure space (Ω,F , P ) and is =t-measurable. All functions of xt(θ) satisfy Pollard (1984: Appendix C)’s permissibility criteria. b. xt(θ) is stationary, ergodic and thrice continuously differentiable with =tmeasurable stationary and ergodic derivatives gt(θ) and ht(θ). c. Each wt (θ) ∈ {xt (θ) , gi,t (θ) , hi,j,t(θ)} is governed by a non-degenerate distribution that is absolutely continuous with respect to Lebesgue measure, with uniformly bounded derivatives: supθ∈Θ supa∈R ||(∂/∂θ)P (wt (θ) ≤ a)|| < ∞ and supθ∈Θ supa∈R{(∂/∂a)P (wt (θ) ≤ a)} < ∞. Further E[supθ∈Θ |wt(θ)|] < ∞ for some tiny ι > 0. We assume xt has support [0,∞) and has for each t a common regularly varying distribution tail with tail index κ > 0: