Conditional Limit Results for Type I Polar Distributions
نویسنده
چکیده
Department of Mathematical Statistics and Actuarial Science University of Bern, Sidlerstrasse 5 CH-3012 Bern, Switzerland October 8, 2008 Abstract: Let (S1, S2) = (R cos(Θ), R sin(Θ)) be a bivariate random vector with associated random radius R which has distribution function F being further independent of the random angle Θ. In this paper we investigate the asymptotic behaviour of the conditional survivor probability Ψρ,u(y) := P {ρS1 + √ 1− ρS2 > y|S1 > u}, ρ ∈ (−1, 1),∈ IR when u approaches the upper endpoint of F . On the density function of Θ we require a certain local asymptotic behaviour at 0, whereas for F we require that it belongs to the Gumbel max-domain of attraction. The main result of this contribution is an asymptotic expansion of Ψρ,u, which is then utilised to construct two estimators for the conditional distribution function 1−Ψρ,u. Further, we allow Θ to depend on u.
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