16,868 research outputs found
Characterization of the frequency of extreme events by the Generalized Pareto Distribution
Based on recent results in extreme value theory, we use a new technique for
the statistical estimation of distribution tails. Specifically, we use the
Gnedenko-Pickands-Balkema-de Haan theorem, which gives a natural limit law for
peak-over-threshold values in the form of the Generalized Pareto Distribution
(GPD). Useful in finance, insurance, hydrology, we investigate here the
earthquake energy distribution described by the Gutenberg-Richter seismic
moment-frequency law and analyze shallow earthquakes (depth h < 70 km) in the
Harvard catalog over the period 1977-2000 in 18 seismic zones. The whole GPD is
found to approximate the tails of the seismic moment distributions quite well
above moment-magnitudes larger than mW=5.3 and no statistically significant
regional difference is found for subduction and transform seismic zones. We
confirm that the b-value is very different in mid-ocean ridges compared to
other zones (b=1.50=B10.09 versus b=1.00=B10.05 corresponding to a power law
exponent close to 1 versus 2/3) with a very high statistical confidence. We
propose a physical mechanism for this, contrasting slow healing ruptures in
mid-ocean ridges with fast healing ruptures in other zones. Deviations from the
GPD at the very end of the tail are detected in the sample containing
earthquakes from all major subduction zones (sample size of 4985 events). We
propose a new statistical test of significance of such deviations based on the
bootstrap method. The number of events deviating from the tails of GPD in the
studied data sets (15-20 at most) is not sufficient for determining the
functional form of those deviations. Thus, it is practically impossible to give
preference to one of the previously suggested parametric families describing
the ends of tails of seismic moment distributions.Comment: pdf document of 21 pages + 2 tables + 20 figures (ps format) + one
file giving the regionalizatio
Pseudo-nonstationarity in the scaling exponents of finite-interval time series
The accurate estimation of scaling exponents is central in the observational study of scale-invariant phenomena. Natural systems unavoidably provide observations over restricted intervals; consequently, a stationary stochastic process (time series) can yield anomalous time variation in the scaling exponents, suggestive of nonstationarity. The variance in the estimates of scaling exponents computed from an interval of N observations is known for finite variance processes to vary as ~1/N as N for certain statistical estimators; however, the convergence to this behavior will depend on the details of the process, and may be slow. We study the variation in the scaling of second-order moments of the time-series increments with N for a variety of synthetic and âreal worldâ time series, and we find that in particular for heavy tailed processes, for realizable N, one is far from this ~1/N limiting behavior. We propose a semiempirical estimate for the minimum N needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some âreal worldâ time series
Sampling Properties of the Spectrum and Coherency of Sequences of Action Potentials
The spectrum and coherency are useful quantities for characterizing the
temporal correlations and functional relations within and between point
processes. This paper begins with a review of these quantities, their
interpretation and how they may be estimated. A discussion of how to assess the
statistical significance of features in these measures is included. In
addition, new work is presented which builds on the framework established in
the review section. This work investigates how the estimates and their error
bars are modified by finite sample sizes. Finite sample corrections are derived
based on a doubly stochastic inhomogeneous Poisson process model in which the
rate functions are drawn from a low variance Gaussian process. It is found
that, in contrast to continuous processes, the variance of the estimators
cannot be reduced by smoothing beyond a scale which is set by the number of
point events in the interval. Alternatively, the degrees of freedom of the
estimators can be thought of as bounded from above by the expected number of
point events in the interval. Further new work describing and illustrating a
method for detecting the presence of a line in a point process spectrum is also
presented, corresponding to the detection of a periodic modulation of the
underlying rate. This work demonstrates that a known statistical test,
applicable to continuous processes, applies, with little modification, to point
process spectra, and is of utility in studying a point process driven by a
continuous stimulus. While the material discussed is of general applicability
to point processes attention will be confined to sequences of neuronal action
potentials (spike trains) which were the motivation for this work.Comment: 33 pages, 9 figure
Unbiased estimation of risk
The estimation of risk measures recently gained a lot of attention, partly
because of the backtesting issues of expected shortfall related to
elicitability. In this work we shed a new and fundamental light on optimal
estimation procedures of risk measures in terms of bias. We show that once the
parameters of a model need to be estimated, one has to take additional care
when estimating risks. The typical plug-in approach, for example, introduces a
bias which leads to a systematic underestimation of risk. In this regard, we
introduce a novel notion of unbiasedness to the estimation of risk which is
motivated by economic principles. In general, the proposed concept does not
coincide with the well-known statistical notion of unbiasedness. We show that
an appropriate bias correction is available for many well-known estimators. In
particular, we consider value-at-risk and expected shortfall (tail
value-at-risk). In the special case of normal distributions, closed-formed
solutions for unbiased estimators can be obtained. We present a number of
motivating examples which show the outperformance of unbiased estimators in
many circumstances. The unbiasedness has a direct impact on backtesting and
therefore adds a further viewpoint to established statistical properties
- âŠ