206 research outputs found
Inference for double Pareto lognormal queues with applications
In this article we describe a method for carrying out Bayesian inference for the double
Pareto lognormal (dPlN) distribution which has recently been proposed as a model for
heavy-tailed phenomena. We apply our approach to inference for the dPlN/M/1 and
M/dPlN/1 queueing systems. These systems cannot be analyzed using standard
techniques due to the fact that the dPlN distribution does not posses a Laplace transform
in closed form. This difficulty is overcome using some recent approximations for the
Laplace transform for the Pareto/M/1 system. Our procedure is illustrated with
applications in internet traffic analysis and risk theory
Modeling Teletraffic Arrivals by a Poisson Cluster Process
Modeling Teletraffic Arrivals by a Poisson Cluster Proces
More "normal" than normal: scaling distributions and complex systems
One feature of many naturally occurring or engineered complex systems is tremendous variability in event sizes. To account for it, the behavior of these systems is often described using power law relationships or scaling distributions, which tend to be viewed as "exotic" because of their unusual properties (e.g., infinite moments). An alternate view is based on mathematical, statistical, and data-analytic arguments and suggests that scaling distributions should be viewed as "more normal than normal". In support of this latter view that has been advocated by Mandelbrot for the last 40 years, we review in this paper some relevant results from probability theory and illustrate a powerful statistical approach for deciding whether the variability associated with observed event sizes is consistent with an underlying Gaussian-type (finite variance) or scaling-type (infinite variance) distribution. We contrast this approach with traditional model fitting techniques and discuss its implications for future modeling of complex systems
On the accuracy of phase-type approximations of heavy-tailed risk models
Numerical evaluation of ruin probabilities in the classical risk model is an
important problem. If claim sizes are heavy-tailed, then such evaluations are
challenging. To overcome this, an attractive way is to approximate the claim
sizes with a phase-type distribution. What is not clear though is how many
phases are enough in order to achieve a specific accuracy in the approximation
of the ruin probability. The goals of this paper are to investigate the number
of phases required so that we can achieve a pre-specified accuracy for the ruin
probability and to provide error bounds. Also, in the special case of a
completely monotone claim size distribution we develop an algorithm to estimate
the ruin probability by approximating the excess claim size distribution with a
hyperexponential one. Finally, we compare our approximation with the heavy
traffic and heavy tail approximations.Comment: 24 pages, 13 figures, 8 tables, 38 reference
Wavelet and Multiscale Analysis of Network Traffic
The complexity and richness of telecommunications traffic is such that one may despair to find any regularity or explanatory principles. Nonetheless, the discovery of scaling behaviour in tele-traffic has provided hope that parsimonious models can be found. The statistics of scaling behavior present many challenges, especially in non-stationary environments. In this paper we describe the state of the art in this area, focusing on the capabilities of the wavelet transform as a key tool for unravelling the mysteries of traffic statistics and dynamics
The extremogram: A correlogram for extreme events
We consider a strictly stationary sequence of random vectors whose
finite-dimensional distributions are jointly regularly varying with some
positive index. This class of processes includes, among others, ARMA processes
with regularly varying noise, GARCH processes with normally or
Student-distributed noise and stochastic volatility models with regularly
varying multiplicative noise. We define an analog of the autocorrelation
function, the extremogram, which depends only on the extreme values in the
sequence. We also propose a natural estimator for the extremogram and study its
asymptotic properties under -mixing. We show asymptotic normality,
calculate the extremogram for various examples and consider spectral analysis
related to the extremogram.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ213 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Weak Convergence of the function-indexed integrated periodogram for infinite variance processes
In this paper, we study the weak convergence of the integrated periodogram
indexed by classes of functions for linear processes with symmetric
-stable innovations. Under suitable summability conditions on the
series of the Fourier coefficients of the index functions, we show that the
weak limits constitute -stable processes which have representations as
infinite Fourier series with i.i.d. -stable coefficients. The cases
and are dealt with by rather different
methods and under different assumptions on the classes of functions. For
example, in contrast to the case , entropy conditions are
needed for to ensure the tightness of the sequence of
integrated periodograms indexed by functions. The results of this paper are of
additional interest since they provide limit results for infinite mean random
quadratic forms with particular Toeplitz coefficient matrices.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ253 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Cache Miss Estimation for Non-Stationary Request Processes
The aim of the paper is to evaluate the miss probability of a Least Recently
Used (LRU) cache, when it is offered a non-stationary request process given by
a Poisson cluster point process. First, we construct a probability space using
Palm theory, describing how to consider a tagged document with respect to the
rest of the request process. This framework allows us to derive a general
integral formula for the expected number of misses of the tagged document.
Then, we consider the limit when the cache size and the arrival rate go to
infinity proportionally, and use the integral formula to derive an asymptotic
expansion of the miss probability in powers of the inverse of the cache size.
This enables us to quantify and improve the accuracy of the so-called Che
approximation
Bayesian modelling of skewness and kurtosis with two-piece scale and shape distributions
We formalise and generalise the definition of the family of univariate double
two--piece distributions, obtained by using a density--based transformation of
unimodal symmetric continuous distributions with a shape parameter. The
resulting distributions contain five interpretable parameters that control the
mode, as well as the scale and shape in each direction. Four-parameter
subfamilies of this class of distributions that capture different types of
asymmetry are discussed. We propose interpretable scale and location-invariant
benchmark priors and derive conditions for the propriety of the corresponding
posterior distribution. The prior structures used allow for meaningful
comparisons through Bayes factors within flexible families of distributions.
These distributions are applied to data from finance, internet traffic and
medicine, comparing them with appropriate competitors
Estimating Quantile Families of Loss Distributions for Non-Life Insurance Modelling via L-moments
This paper discusses different classes of loss models in non-life insurance
settings. It then overviews the class Tukey transform loss models that have not
yet been widely considered in non-life insurance modelling, but offer
opportunities to produce flexible skewness and kurtosis features often required
in loss modelling. In addition, these loss models admit explicit quantile
specifications which make them directly relevant for quantile based risk
measure calculations. We detail various parameterizations and sub-families of
the Tukey transform based models, such as the g-and-h, g-and-k and g-and-j
models, including their properties of relevance to loss modelling.
One of the challenges with such models is to perform robust estimation for
the loss model parameters that will be amenable to practitioners when fitting
such models. In this paper we develop a novel, efficient and robust estimation
procedure for estimation of model parameters in this family Tukey transform
models, based on L-moments. It is shown to be more robust and efficient than
current state of the art methods of estimation for such families of loss models
and is simple to implement for practical purposes.Comment: 42 page
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