V. F. Pisarenko

Testing the Pareto against the lognormal distributions with the uniformly most powerful unbiased test applied to the distribution of cities.

We provide definitive results to close the debate between Eec khout (2004, 2009) and Levy (2009) on the validity of Zipf’s law, which is the special Pareto law with tail exponent 1, to describe the tail of the distribution of U.S. city sizes. Because the origin of th e disagreement between Eeckhout and Levy stems from the limited power of their tests, we perform the uniformly most powerful unbiased test for the null hypothesis of the Pareto distribution against the l ognormal. Thep-value and Hill’s estimator as a function of city size lower threshold confirm indubitabl y that the size distribution of the 1000 largest cities or so, which include more than half of the tota l U.S. population, is Pareto, but we rule out that the tail exponent, estimated to be 1.4 ± 0.1, is equal to1. For larger ranks, the p-value becomes very small and Hill’s estimator decays systematically with decreasing ranks, qualifying the lognormal distribution as the better model for the set of smaller citie s. These two results reconcile the opposite views of Eeckhout (2004) and Levy (2009). We explain how Gibr at’s law of proportional growth underpins both the Pareto and lognormal distributions and s tress the key ingredient at the origin of their difference in standard stochastic growth models of ci ties (Gabaix 1999, Eeckhout 2004). JEL classification: D30, D51, J61, R12.

Characterization of the Frequency of Extreme Earthquake Events by the Generalized Pareto Distribution

Recent results in extreme value theory suggest a new technique for statistical estimation of distribution tails (Embrechts et al., 1997), based on a limit theorem known as the Gnedenko-Pickands-Balkema-de Haan theorem. This theorem gives a natural limit law for peak-over-threshold values in the form of the Generalized Pareto Distribution (GPD), which is a family of distributions with two parameters. The GPD has been successfully applied in a number of statistical problems related to finance, insurance, hydrology, and other domains. Here, we apply the GPD approach to the well-known seismological problem of earthquake energy distribution described by the Gutenberg-Richter seismic moment-frequency law. We analyze shallow earthquakes (depth h<70 km) in the Harvard catalog over the period 1977–2000 in 12 seismic zones. The GPD is found to approximate the tails of the seismic moment distributions quite well over the lower threshold approximately M ≅ 1024 dyne-cm, or somewhat above (i.e., moment-magnitudes larger than mW=5.3). We confirm that the b-value is very different (b=2.06 ± 0.30) in mid-ocean ridges compared to other zones (b=1.00 ± 0.04) with a very high statistical confidence and propose a physical mechanism contrasting “crack-type” rupture with “dislocation-type” behavior. The GPD can as well be applied in many problems of seismic hazard assessment on a regional scale. However, in certain cases, deviations from the GPD at the very end of the tail may occur, in particular for large samples signaling a novel regime.

Towards landslide predictions: two case studies

In a previous work (J. Geophys. Res. (2004)), we have proposed a simple physical model to explain the accelerating displacements preceding some catastrophic landslides, based on a slider-block model with a state- and velocity-dependent friction law. This model predicts two regimes of sliding, stable and unstable leading to a critical finite-time singularity. This model was calibrated quantitatively to the displacement and velocity data preceding two landslides, Vaiont (Italian Alps) and La Clapiere (French Alps), showing that the former (resp. later) landslide is in the unstable (resp. stable) sliding regime. Here, we test the predictive skills of the state- and velocity-dependent model on these two landslides with a variety of techniques using (i) a finite-time singularity power law, (ii) the state- and velocity-dependent friction law and (iii) resummation methods extrapolating from early times. For the Vaiont landslide, our model provides good predictions of the critical time of failure up to 20 days before the collapse. Tests are also presented on the predictability of the time of the change of regime for La Clapiere landslide.

Statistical Detection and Characterization of a Deviation from the Gutenberg-Richter Distribution above Magnitude 8

Abstract — We present a quantitative statistical test for the presence of a crossover c0 in the Gutenberg-Richter distribution of earthquake seismic moments, separating the usual power-law regime for seismic moments less than c0 from another faster decaying regime beyond c0. Our method is based on the transformation of the ordered sample of seismic moments into a series with uniform distribution under condition of no crossover. A simulation method allows us to estimate the statistical significance of the null hypothesis H0 of an absence of crossover (c0=infinity). When H0 is rejected, we estimate the crossover c0 using two different competing models for the second regime beyond c0 and the simulation method. For the catalog obtained by aggregating 14 subduction zones of the Circum-Pacific Seismic Belt, our estimate of the crossover point is log(c0)=28.14 ± 0.40 (c0 in dyne-cm), corresponding to a crossover magnitude mW=8.1 ± 0.3. For separate subduction zones, the corresponding estimates are substantially more uncertain, so that the null hypothesis of an identical crossover for all subduction zones cannot be rejected. Such a large value of the crossover magnitude makes it difficult to associate it directly with a seismogenic thickness as proposed by many different authors. Our measure of c0 may substantiate the concept that the localization of strong shear deformation could propagate significantly in the lower crust and upper mantle, thus increasing the effective size beyond which one should expect a change of regime.

On the power of generalized extreme value (GEV) and generalized Pareto distribution (GPD) estimators for empirical distributions of stock returns

Using synthetic tests performed on time series with time dependence in the volatility with both Pareto and Stretched-Exponential distributions, it is shown that for samples of moderate sizes the standard generalized extreme value (GEV) estimator is quite inefficient due to the possibly slow convergence toward the asymptotic theoretical distribution and the existence of biases in the presence of dependence between data. Thus, it cannot distinguish reliably between rapidly and regularly varying classes of distributions. The Generalized Pareto distribution (GPD) estimator works better, but still lacks power in the presence of strong dependence. Applied to 100 years of daily returns of the Dow Jones Industrial Average and over one years of five-minutes returns of the Nasdaq Composite index, the GEV and GDP estimators are found insufficient to prove that the distributions of empirical returns of financial time series are regularly varying, because the rapidly varying exponential or stretched exponential distributions are equally acceptable.

Gibrat’s Law for Cities: Uniformly Most Powerful Unbiased Test of the Pareto Against the Lognormal

We provide definitive results to close the debate between Eeckhout (2004, 2009) and Levy (2009) on the validity of Zipf’s law, which is the special Pareto law with tail exponent 1, to describe the tail of the distribution of U.S. city sizes. Because the origin of the disagreement between Eeckhout and Levy stems from the limited power of their tests, we performthe uniformly most powerful unbiased test for the null hypothesis of the Pareto distribution against the lognormal. The p-value and Hill’s estimator as a function of city size lower threshold confirm indubitably that the size distribution of the 1000 largest cities or so, which includemore than half of the total U.S. population, is Pareto, but we rule out that the tail exponent, estimated to be 1.4 ± 0.1, is equal to 1. For larger ranks, the p-value becomes very small and Hill’s estimator decays systematically with decreasing ranks, qualifying the lognormal distribution as the better model for the set of smaller cities. These two results reconcile the opposite views of Eeckhout (2004) and Levy (2009). We explain how Gibrat’s law of proportional growth underpins both the Pareto and lognormal distributions and stress the key ingredient at the origin of their difference in standard stochastic growth models of cities (Gabaix 1999, Eeckhout 2004).

New statistic for financial return distributions: Power-law or exponential?

We introduce a new statistical tool (the TP-statistic and TE-statistic) designed specifically to compare the behavior of the sample tail of distributions with power-law and exponential tails as a function of the lower threshold u. One important property of these statistics is that they converge to zero for power-laws or for exponentials correspondingly, regardless of the value of the exponent or of the form parameter. This is particularly useful for testing the structure of a distribution (power-law or not, exponential or not) independently of the possibility of quantifying the values of the parameters. We apply these statistics to the distribution of returns of one century of daily data for the Dow Jones Industrial Average and over 1 year of 5-min data of the Nasdaq Composite index. Our analysis confirms previous works showing the tendency for the tails to resemble more and more a power-law for the highest quantiles but we can detect clear deviations that suggest that the structure of the tails of the distributions of returns is more complex than usually assumed; it is clearly more complex that just a power-law. Our new TP- and TE-statistic should also be useful for other applications in the natural sciences as a powerful non-parametric test for power-laws and exponentials.

Distribution of maximum earthquake magnitudes in future time intervals: application to the seismicity of Japan (1923-2007)

We have modified the new method for the statistical estimation of the tail distribution of earthquake seismic moments introduced by Pisarenko et al. (2009) and applied it to the earthquake catalog of Japan (1923–2007). The newly modified method is based on the two main limit theorems of the theory of extreme values and on the derived duality between the generalized Pareto distribution (GPD) and the generalized extreme value distribution (GEV). Using this method, we obtain the distribution of maximum earthquake magnitudes in future time intervals of arbitrary duration τ. This distribution can be characterized by its quantile Qq (τ) at any desirable statistical level q. The quantile Qq(τ) provides a much more stable and robust characteristic than the traditional absolute maximum magnitude Mmax (Mmax can be obtained as the limit of Qq(τ) as q → 1, τ → ∞). The best estimates of the parameters governing the distribution of Qq(τ) for Japan (1923–2007) are the following: ξGEV = −0.19 ± 0.07; μGEV(200) = 6.339 ± 0.038; σGEV (200) = 0.600 ± 0.022; Q0.90,GEV(10) = 8.34 ± 0.32. We have also estimated Qq(τ) for a set of q-values and future time periods in the range 1 ≤ τ ≤ 50 years from 2007 onwards. For comparison, the absolute maximum estimate Mmax-GEV = 9.57 ± 0.86 has a scatter more than twice that of the 90% quantile Q0.90,gev(10) of the maximum magnitude over the next 10 years beginning from 2007.

New Evidence Of Discrete Scale Invariance In The Energy Dissipation Of Three-Dimensional Turbulence: Correlation Approach And Direct Spectral Detection

We extend the analysis of Ref. 16 showing statistically significant log-periodic corrections to scaling in the moments of the energy dissipation rate in experiments at high Reynolds number (≈ 2500) of three-dimensional fully developed turbulence. First, we develop a simple variant of the canonical averaging method using a rephasing scheme between different samples based on pairwise correlations that confirms Zhou and Sornettes previous results. The second analysis uses a simpler local spectral approach and then performs averages over many local spectra. This yields stronger evidence of the existence of underlying log-periodic undulations, with the detection of more than 20 harmonics of a fundamental logarithmic frequency f = 1.434 ± 0.007 corresponding to the preferred scaling ratio γ = 2.008 ± 0.006.

The Estimation of Probability of Extreme Events for Small Samples

The most general approach to the study of rare extreme events is based on the extreme value theory. The fundamental General Extreme Value Distribution lies in the basis of this theory serving as the limit distribution for normalized maxima. It depends on three parameters. Usually the method of maximum likelihood (ML) is used for the estimation that possesses well-known optimal asymptotic properties. However, this method works efficiently only when sample size is large enough (~200–500), whereas in many applications the sample size does not exceed 50–100. For such sizes, the advantage of the ML method in efficiency is not guaranteed. We have found that for this situation the method of statistical moments (SM) works more efficiently over other methods. The details of the estimation for small samples are studied. The SM is applied to the study of extreme earthquakes in three large virtual seismic zones, representing the regime of seismicity in subduction zones, intracontinental regime of seismicity, and the regime in mid-ocean ridge zones. The 68%-confidence domains for pairs of parameter (ξ, σ) and (σ, μ) are derived.