Inference for the limiting cluster size distribution of extreme values

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

Tails of random sums of a heavy-tailed number of light-tailed terms

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.Comment: Accepted for publication in Insurance: Mathematics and Economic

Geometric ergodicity for some space-time max-stable Markov chains

Max-stable processes are central models for spatial extremes. In this paper, we focus on some space-time max-stable models introduced in Embrechts et al. (2016). The processes considered induce discrete-time Markov chains taking values in the space of continuous functions from the unit sphere of $\mathbb{R}^3$ to $(0, \infty)$. We show that these Markov chains are geometrically ergodic. An interesting feature lies in the fact that the state space is not locally compact, making the classical methodology inapplicable. Instead, we use the fact that the state space is Polish and apply results presented in Hairer (2010)

Preferencing, internalization and inventory position

We present a model of market-making in which dealers differ by their current inventory positions and by their preferencing agreements. Under preferencing, dealers receive captive orders that they guarantee to execute at the best price. We show that preferencing raises the inventory holding costs of preferenced dealers. In turn, competitors post less aggressive quotes. Since price-competition is softened, expected spreads widen. The entry of unpreferenced dealers, or the ability to route preferenced orders to best-quoting dealers, as internalization does restore price competitiveness. We also show that a greater transparency may negatively affect expected spreads, depending on the scale of preferencing.Internalization; Inventory Control; Market Microstructure; Preferencing; Transparency

Estimating the efficient price from the order flow: a Brownian Cox process approach

At the ultra high frequency level, the notion of price of an asset is very ambiguous. Indeed, many different prices can be defined (last traded price, best bid price, mid price,...). Thus, in practice, market participants face the problem of choosing a price when implementing their strategies. In this work, we propose a notion of efficient price which seems relevant in practice. Furthermore, we provide a statistical methodology enabling to estimate this price form the order flow

Estimating the multivariate extremal index function

The multivariate extremal index function relates the asymptotic distribution of the vector of pointwise maxima of a multivariate stationary sequence to that of the independent sequence from the same stationary distribution. It also measures the degree of clustering of extremes in the multivariate process. In this paper, we construct nonparametric estimators of this function and prove their asymptotic normality under long-range dependence and moment conditions. The results are illustrated by means of a simulation study.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ145 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm