The tail of the distribution of a sum of a random number of independent and identically distributed nonnegative random variables depends on the tails of the number of terms and of the terms themselves. This situation is of interest in the collective risk model, where the total claim size in a portfolio is the sum of a random number of claims. If the tail of the claim number is heavier than the tail of the claim sizes, then under certain conditions the tail of the total claim size does not change asymptotically if the individual claim sizes are replaced by their expectations. The conditions allow the claim number distribution to be of consistent variation or to be in the domain of attraction of a Gumbel distribution with a mean excess function that grows to infinity sufficiently fast. Moreover, the claim number is not necessarily required to be independent of the claim sizes.
Any limiting point process for the time normalized exceedances of high levels
by a stationary sequence is necessarily compound Poisson under appropriate long
range dependence conditions. Typically exceedances appear in clusters. The
underlying Poisson points represent the cluster positions and the
multiplicities correspond to the cluster sizes. In the present paper we
introduce estimators of the limiting cluster size probabilities, which are
constructed through a recursive algorithm. We derive estimators of the extremal
index which plays a key role in determining the intensity of cluster positions.
We study the asymptotic properties of the estimators and investigate their
finite sample behavior on simulated data.Comment: Published in at http://dx.doi.org/10.1214/07-AOS551 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
This paper presents a reduced-order approach for four-dimensional variational data assimilation, based on a prior EOF analysis of a model trajectory. This method implies two main advantages: a natural model-based definition of a multivariate background error covariance matrix B r , and an important decrease of the computational burden of the method, due to the drastic reduction of the dimension of the control space. An illustration of the feasibility and the effectiveness of this method is given in the academic framework of twin experiments for a model of the equatorial Pacific ocean. It is shown that the multivariate aspect of B r brings additional information which substantially improves the identification procedure. Moreover the computational cost can be decreased by one order of magnitude with regard to the full-space 4D-Var method.
This paper considers practically appealing procedures for estimating intraday volatility measures of financial assets. The underlying microstructure model accommodates the inherent properties of ultra high-frequency data with the assumption of continuous efficient price processes. In this model, microstructure noise and trading times are endogenous but do not only depend on the prices. Using the (observed) last traded prices of the assets, we develop a new approach that enables to approximate the values of the efficient prices at some random times. Based on these approximated values, we build an estimator of the integrated volatility and give its asymptotic theory. We also give a consistent estimator of the integrated covariation when two assets (asynchronous by construction of the model) are observed.
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