On the Estimation of Tails Parameters for Pareto and Exponential Distributions

Abstract

with μ = Lnσ and θ = α−1. Thus, estimation of scale and shape in the model PARσ,α is equivalent to estimation of location and scale in Ex(θ, μ). Robust estimation of θ is treated in a very important paper [3]. In their paper, the authors introduced new estimators of “generalized quantile” type and compared them with several well-established estimators (corresponding to methods of trimming, least squares, and quantiles). The authors examined asymptotic relative efficiency with respect to the (efficient but nonrobust) maximum likelihood estimator and breakdown point. Consider the original statistical model M1 = (R+,B+, {Pθ,μ,1, θ > 0,−∞ < μ < +∞}) where R+ is the real positive half-line, B+ is the family of the Borel subsets of R+, and Pθ,μ,1 is the exponential distribution Ex(θ, μ) with p.d.f. fθ,μ(x), given in (2), and its extension