Estimating the logarithm of characteristic function and stability parameter for symmetric stable laws
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Let X_1,…,X_n be an i.i.d. sample from symmetric stable distribution with stability parameter α and scale parameter γ. Let φ_n be the empirical characteristic function. We prove an uniform large deviation inequality: given preciseness ϵ>0 and probability p∈ (0,1), there exists universal (depending on ϵ and p but not depending on α and γ) constant r̅>0 so that P(sup_u>0:r(u)≤r̅|r(u)-r̂(u)|≥ϵ)≤ p, where r(u)=(uγ)^α and r̂(u)=-ln|φ_n(u)|. As an applications of the result, we show how it can be used in estimation unknown stability parameter α.
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