346 research outputs found
On the residual dependence index of elliptical distributions
The residual dependence index of bivariate Gaussian distributions is
determined by the correlation coefficient. This tail index is of certain
statistical importance when extremes and related rare events of bivariate
samples with asymptotic independent components are being modeled. In this paper
we calculate the partial residual dependence indices of a multivariate
elliptical random vector assuming that the associated random radius is in the
Gumbel max-domain of attraction. Furthermore, we discuss the estimation of
these indices when the associated random radius possesses a Weibull-tail
distribution.Comment: 11 pages, case \theta=1 now include
On the asymptotic distribution of certain bivariate reinsurance treaties
Let (X_n,Y_n), n\ge 1 be bivariate random claim sizes with common
distribution function F and let N(t), t \ge 0 be a stochastic process which
counts the number of claims that occur in the time interval [0,t], t\ge 0. In
this paper we derive the joint asymptotic distribution of randomly indexed
order statistics of the random sample
(X_1,Y_1),(X_2,Y_2),...,(X_{N(t)},Y_{N(t)}) which is then used to obtain
asymptotic representations for the joint distribution of two generalised
largest claims reinsurance treaties available under specific insurance
settings. As a by-product we obtain a stochastic representation of a
m-dimensional Lambda-extremal variate in terms of iid unit exponential random
variables.Comment: 11 page
Conditional Limit Results for Type I Polar Distributions
Let (S_1,S_2)=(R \cos(\Theta), R \sin (\Theta)) be a bivariate random vector
with associated random radius R which has distribution function being
further independent of the random angle \Theta. In this paper we investigate
the asymptotic behaviour of the conditional survivor probability
\Psi_{\rho,u}(y):=\pk{\rho S_1+ \sqrt{1- \rho^2} S_2> y \lvert S_1> u}, \rho
\in (-1,1),\in R when u approaches the upper endpoint of F. On the density
function of \Theta 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
\Psi_{\rho,u}, which is then utilised to construct two estimators for the
conditional distribution function 1- \Psi_{\rho,u}. Further, we allow \Theta to
depend on u.Comment: 14 pages, paper submitted to Extremes in 200
Extremes of Aggregated Dirichlet Risks
The class of Dirichlet random vectors is central in numerous probabilistic
and statistical applications. The main result of this paper derives the exact
tail asymptotics of the aggregated risk of powers of Dirichlet random vectors
when the radial component has df in the Gumbel or the Weibull max-domain of
attraction. We present further results for the joint asymptotic independence
and the max-sum equivalence.Comment: published versio
Piterbarg Theorems for Chi-processes with Trend
Let be a chi-process with
degrees of freedom where 's are independent copies of some generic
centered Gaussian process . This paper derives the exact asymptotic behavior
of P{\sup_{t\in[0,T]} \chi_n(t)>u} as u \to \infty, where is a given
positive constant, and is some non-negative bounded measurable
function. The case is investigated in numerous contributions by
V.I. Piterbarg. Our novel asymptotic results for both stationary and
non-stationary are referred to as Piterbarg theorems for chi-processes with
trend.Comment: 22 page
Berman's inequality under random scaling
Berman's inequality is the key for establishing asymptotic properties of
maxima of Gaussian random sequences and supremum of Gaussian random fields.
This contribution shows that, asymptotically an extended version of Berman's
inequality can be established for randomly scaled Gaussian random vectors. Two
applications presented in this paper demonstrate the use of Berman's inequality
under random scaling
- …