404,015 research outputs found

    Residual analysis methods for space--time point processes with applications to earthquake forecast models in California

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    Modern, powerful techniques for the residual analysis of spatial-temporal point process models are reviewed and compared. These methods are applied to California earthquake forecast models used in the Collaboratory for the Study of Earthquake Predictability (CSEP). Assessments of these earthquake forecasting models have previously been performed using simple, low-power means such as the L-test and N-test. We instead propose residual methods based on rescaling, thinning, superposition, weighted K-functions and deviance residuals. Rescaled residuals can be useful for assessing the overall fit of a model, but as with thinning and superposition, rescaling is generally impractical when the conditional intensity λ\lambda is volatile. While residual thinning and superposition may be useful for identifying spatial locations where a model fits poorly, these methods have limited power when the modeled conditional intensity assumes extremely low or high values somewhere in the observation region, and this is commonly the case for earthquake forecasting models. A recently proposed hybrid method of thinning and superposition, called super-thinning, is a more powerful alternative.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS487 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

    A recursive online algorithm for the estimation of time-varying ARCH parameters

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    In this paper we propose a recursive online algorithm for estimating the parameters of a time-varying ARCH process. The estimation is done by updating the estimator at time point t1t-1 with observations about the time point tt to yield an estimator of the parameter at time point tt. The sampling properties of this estimator are studied in a non-stationary context -- in particular, asymptotic normality and an expression for the bias due to non-stationarity are established. By running two recursive online algorithms in parallel with different step sizes and taking a linear combination of the estimators, the rate of convergence can be improved for parameter curves from H\"{o}lder classes of order between 1 and 2.Comment: Published at http://dx.doi.org/10.3150/07-BEJ5009 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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