25,605 research outputs found
A multivariate piecing-together approach with an application to operational loss data
The univariate piecing-together approach (PT) fits a univariate generalized
Pareto distribution (GPD) to the upper tail of a given distribution function in
a continuous manner. We propose a multivariate extension. First it is shown
that an arbitrary copula is in the domain of attraction of a multivariate
extreme value distribution if and only if its upper tail can be approximated by
the upper tail of a multivariate GPD with uniform margins. The multivariate PT
then consists of two steps: The upper tail of a given copula is cut off and
substituted by a multivariate GPD copula in a continuous manner. The result is
again a copula. The other step consists of the transformation of each margin of
this new copula by a given univariate distribution function. This provides,
altogether, a multivariate distribution function with prescribed margins whose
copula coincides in its central part with and in its upper tail with a GPD
copula. When applied to data, this approach also enables the evaluation of a
wide range of rational scenarios for the upper tail of the underlying
distribution function in the multivariate case. We apply this approach to
operational loss data in order to evaluate the range of operational risk.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ343 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Heavy-Tailed Features and Empirical Analysis of the Limit Order Book Volume Profiles in Futures Markets
This paper poses a few fundamental questions regarding the attributes of the
volume profile of a Limit Order Books stochastic structure by taking into
consideration aspects of intraday and interday statistical features, the impact
of different exchange features and the impact of market participants in
different asset sectors. This paper aims to address the following questions:
1. Is there statistical evidence that heavy-tailed sub-exponential volume
profiles occur at different levels of the Limit Order Book on the bid and ask
and if so does this happen on intra or interday time scales ?
2.In futures exchanges, are heavy tail features exchange (CBOT, CME, EUREX,
SGX and COMEX) or asset class (government bonds, equities and precious metals)
dependent and do they happen on ultra-high (<1sec) or mid-range (1sec -10min)
high frequency data?
3.Does the presence of stochastic heavy-tailed volume profile features evolve
in a manner that would inform or be indicative of market participant behaviors,
such as high frequency algorithmic trading, quote stuffing and price discovery
intra-daily?
4. Is there statistical evidence for a need to consider dynamic behavior of
the parameters of models for Limit Order Book volume profiles on an intra-daily
time scale ?
Progress on aspects of each question is obtained via statistically rigorous
results to verify the empirical findings for an unprecedentedly large set of
futures market LOB data. The data comprises several exchanges, several futures
asset classes and all trading days of 2010, using market depth (Type II) order
book data to 5 levels on the bid and ask
Tests based on characterizations, and their efficiencies: a survey
A survey of goodness-of-fit and symmetry tests based on the characterization
properties of distributions is presented. This approach became popular in
recent years. In most cases the test statistics are functionals of
-empirical processes. The limiting distributions and large deviations of new
statistics under the null hypothesis are described. Their local Bahadur
efficiency for various parametric alternatives is calculated and compared with
each other as well as with diverse previously known tests. We also describe new
directions of possible research in this domain.Comment: Open access in Acta et Commentationes Universitatis Tartuensis de
Mathematic
Modelling the distribution of day-ahead electricity returns: a comparison
This paper contributes to characterizing the probability density of the price returns in some European day-ahead electricity markets (NordPool, APX, Powernext) by fitting some flexible and general families of distributions, such as the alpha-stable, Normal Inverse Gaussian (NIG), Exponential Power (EP), and Asymmetric Exponential Power (AEP), and comparing their goodness of fit. The alpha-stable and the NIG systematically outperform the EP and AEP models, but the tail behaviours and the skewness are sensitive to the definition of returns and to the deseasonalization methods. In particular, the logarithmic transform and volatility rescaling tend to dampen the extreme returns.Electricity prices, alpha-stable, Normal Inverse Gaussian, Exponential Power,Asymmetric Exponential Power, goodness-of-fit
The art of fitting financial time series with Levy stable distributions
This paper illustrates a procedure for fitting financial data with
-stable distributions. After using all the available methods to
evaluate the distribution parameters, one can qualitatively select the best
estimate and run some goodness-of-fit tests on this estimate, in order to
quantitatively assess its quality. It turns out that, for the two investigated
data sets (MIB30 and DJIA from 2000 to present), an -stable fit of
log-returns is reasonably good.Comment: 17 pages, 10 figures, 2 tables. Paper presented at the DDAP4
conference, Pohang, Korea, July 2006. Submitted to Journal of Korean Physical
Societ
Goodness-of-Fit Tests for Symmetric Stable Distributions -- Empirical Characteristic Function Approach
We consider goodness-of-fit tests of symmetric stable distributions based on
weighted integrals of the squared distance between the empirical characteristic
function of the standardized data and the characteristic function of the
standard symmetric stable distribution with the characteristic exponent
estimated from the data. We treat as an unknown parameter,
but for theoretical simplicity we also consider the case that is
fixed. For estimation of parameters and the standardization of data we use
maximum likelihood estimator (MLE) and an equivariant integrated squared error
estimator (EISE) which minimizes the weighted integral. We derive the
asymptotic covariance function of the characteristic function process with
parameters estimated by MLE and EISE. For the case of MLE, the eigenvalues of
the covariance function are numerically evaluated and asymptotic distribution
of the test statistic is obtained using complex integration. Simulation studies
show that the asymptotic distribution of the test statistics is very accurate.
We also present a formula of the asymptotic covariance function of the
characteristic function process with parameters estimated by an efficient
estimator for general distributions
A two-step approach to model precipitation extremes in California based on max-stable and marginal point processes
In modeling spatial extremes, the dependence structure is classically
inferred by assuming that block maxima derive from max-stable processes.
Weather stations provide daily records rather than just block maxima. The point
process approach for univariate extreme value analysis, which uses more
historical data and is preferred by some practitioners, does not adapt easily
to the spatial setting. We propose a two-step approach with a composite
likelihood that utilizes site-wise daily records in addition to block maxima.
The procedure separates the estimation of marginal parameters and dependence
parameters into two steps. The first step estimates the marginal parameters
with an independence likelihood from the point process approach using daily
records. Given the marginal parameter estimates, the second step estimates the
dependence parameters with a pairwise likelihood using block maxima. In a
simulation study, the two-step approach was found to be more efficient than the
pairwise likelihood approach using only block maxima. The method was applied to
study the effect of El Ni\~{n}o-Southern Oscillation on extreme precipitation
in California with maximum daily winter precipitation from 35 sites over 55
years. Using site-specific generalized extreme value models, the two-step
approach led to more sites detected with the El Ni\~{n}o effect, narrower
confidence intervals for return levels and tighter confidence regions for risk
measures of jointly defined events.Comment: Published at http://dx.doi.org/10.1214/14-AOAS804 in the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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