4,401 research outputs found
Rare-event analysis of mixed Poisson random variables, and applications in staffing
A common assumption when modeling queuing systems is that arrivals behave
like a Poisson process with constant parameter. In practice, however, call
arrivals are often observed to be significantly overdispersed. This motivates
that in this paper we consider a mixed Poisson arrival process with arrival
rates that are resampled every time units, where and a
scaling parameter. In the first part of the paper we analyse the asymptotic
tail distribution of this doubly stochastic arrival process. That is, for large
and i.i.d. arrival rates , we focus on the evaluation of
, the probability that the scaled number of arrivals exceeds .
Relying on elementary techniques, we derive the exact asymptotics of :
For we identify (in closed-form) a function
such that tends to as .
For and we find a partial
solution in terms of an asymptotic lower bound. For the special case that the
s are gamma distributed, we establish the exact asymptotics across all . In addition, we set up an asymptotically efficient importance sampling
procedure that produces reliable estimates at low computational cost. The
second part of the paper considers an infinite-server queue assumed to be fed
by such a mixed Poisson arrival process. Applying a scaling similar to the one
in the definition of , we focus on the asymptotics of the probability
that the number of clients in the system exceeds . The resulting
approximations can be useful in the context of staffing. Our numerical
experiments show that, astoundingly, the required staffing level can actually
decrease when service times are more variable
Analytical and numerical approach to corporate operational risk modelling
Although The New Basel Accord gives the methodology for managing operational risk in financial institutions, corporate risk seems not to be recognized enough. In this Ph.D. thesis we make an attempt to put some insight into operational risk measurement in a non-financial corporation. The objective is to apply suitable results from insurance ruin theory to build a framework for measuring corporate operational risk and finding required capital charge.Corporate risk management; Operational risk; Actuarial risk theory; Ruin probability; Operational reserves;
Building Loss Models
This paper is intended as a guide to building insurance risk (loss) models. A typical model for insurance risk, the so-called collective risk model, treats the aggregate loss as having a compound distribution with two main components: one characterizing the arrival of claims and another describing the severity (or size) of loss resulting from the occurrence of a claim. In this paper we first present efficient simulation algorithms for several classes of claim arrival processes. Then we review a collection of loss distributions and present methods that can be used to assess the goodness-of-fit of the claim size distribution. The collective risk model is often used in health insurance and in general insurance, whenever the main risk components are the number of insurance claims and the amount of the claims. It can also be used for modeling other non-insurance product risks, such as credit and operational risk.Insurance risk model; Loss distribution; Claim arrival process; Poisson process; Renewal process; Random variable generation; Goodness-of-fit testing
Fractal time random walk and subrecoil laser cooling considered as renewal processes with infinite mean waiting times
There exist important stochastic physical processes involving infinite mean
waiting times. The mean divergence has dramatic consequences on the process
dynamics. Fractal time random walks, a diffusion process, and subrecoil laser
cooling, a concentration process, are two such processes that look
qualitatively dissimilar. Yet, a unifying treatment of these two processes,
which is the topic of this pedagogic paper, can be developed by combining
renewal theory with the generalized central limit theorem. This approach
enables to derive without technical difficulties the key physical properties
and it emphasizes the role of the behaviour of sums with infinite means.Comment: 9 pages, 7 figures, to appear in the Proceedings of Cargese Summer
School on "Chaotic dynamics and transport in classical and quantum systems
Economic Inequality: Is it Natural?
Mounting evidences are being gathered suggesting that income and wealth
distribution in various countries or societies follow a robust pattern, close
to the Gibbs distribution of energy in an ideal gas in equilibrium, but also
deviating significantly for high income groups. Application of physics models
seem to provide illuminating ideas and understanding, complimenting the
observations.Comment: 7 pages, 2 eps figs, 2 boxes with text and 2 eps figs; Popular review
To appear in Current Science; typos in refs and text correcte
Tail index estimation, concentration and adaptivity
This paper presents an adaptive version of the Hill estimator based on
Lespki's model selection method. This simple data-driven index selection method
is shown to satisfy an oracle inequality and is checked to achieve the lower
bound recently derived by Carpentier and Kim. In order to establish the oracle
inequality, we derive non-asymptotic variance bounds and concentration
inequalities for Hill estimators. These concentration inequalities are derived
from Talagrand's concentration inequality for smooth functions of independent
exponentially distributed random variables combined with three tools of Extreme
Value Theory: the quantile transform, Karamata's representation of slowly
varying functions, and R\'enyi's characterisation of the order statistics of
exponential samples. The performance of this computationally and conceptually
simple method is illustrated using Monte-Carlo simulations
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