Probability and Its Engineering Uses

Density estimation for compound Poisson processes from discrete data

We study the nonparametric estimation of the jump density of a compound Poisson process from the discrete observation of one trajectory over $[0,T]$. We consider the microscopic regime when the sampling rate $\Delta=\Delta_T\rightarrow0$ as $T\rightarrow\infty$. We propose an adaptive wavelet threshold density estimator and study its performance for the $L_p$ loss, $p\geq 1$, over Besov spaces. The main novelty is that we achieve minimax rates of convergence for sampling rates $\Delta_T$ that vanish with $T$ at arbitrary polynomial rates. More precicely, our estimator attains minimax rates of convergence provided there exists a constant $K\geq 1$ such that the sampling rate $\Delta_T$ satisfies $T\Delta_T^{2K+2}\leq 1.$ If this condition cannot be satisfied we still provide an upper bound for our estimator. The estimating procedure is based on the inversion of the compounding operator in the same spirit as Buchmann and Grubel (2003).

Adaptive wavelet estimation of a compound Poisson process

North Anatolian Fault (NAF) is one from the most important strike-slip fault zones in the world and located among regions in the highest seismic activity. The NAFZ observed very large earthquakes from the past to present. The aim of this study; the important parameters of Gutenberg-Richter relationship (a and b values) estimated and this parameters taking into account, earthquakes were examined in the between years 1900-2015 for 10 different seismic source regions in the NAFZ. After that estimated occurrence probabilities and return periods of occurring earthquakes in fault zone in the next years, and is being assessed with Poisson method the earthquake hazard of the NAFZ. The Region 2 were observed the largest earthquakes for the only historical period and hasn’t been observed large earthquake for the instrumental period in this region. Two historical earthquakes (1766, MS=7.3 and 1897, MS=7.0) are included for Region 2 (Marmara Region) where a large earthquake is expected in the next years. The 10 diffe...

A Poisson method application to the assessment of the earthquake hazard in the North Anatolian Fault Zone, Turkey

North Anatolian Fault (NAF) is one from the most important strike-slip fault zones in the world and located among regions in the highest seismic activity. The NAFZ observed very large earthquakes from the past to present. The aim of this study; the important parameters of Gutenberg-Richter relationship ( a and b values) estimated and this parameters taking into account, earthquakes were examined in the between years 1900-2015 for 10 different seismic source regions in the NAFZ. After that estimated occurrence probabilities and return periods of occurring earthquakes in fault zone in the next years, and is being assessed with Poisson method the earthquake hazard of the NAFZ. The 10 different seismic source regions are determined the relationships between the cumulative number-magnitude which estimated a and b parameters with the equation of LogN=a-bM in the Gutenberg-Richter. A homogenous earthquake catalog for M S magnitude which is equal or larger than 4.0 is used for the time period between 1900 and 2015. The database of catalog used in the study has been created from International Seismological Center (ISC) and Bogazici University Kandilli observation and earthquake research institute (KOERI). The earthquake data were obtained until from 1900 to 1974 from KOERI and ISC until from 1974 to 2015 from KOERI. The probabilities of the earthquake occurring are estimated for the next 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100 years in the 10 different seismic source regions. The highest earthquake occur probabilities in 10 different seismic source regions in the next years estimated that the region Tokat-Erzincan (Region 9) 99% with an earthquake occur probability for magnitude 6.5 which the return period 24.7 year, 92% with an earthquake occur probability for magnitude 7 which the return period 39.1 year, 80% with an earthquake occur probability for magnitude 7.5 which the return period 62.1 year, 64% with an earthquake occur probability for magnitude 8 which the return period 98.5 year. For the Marmara Region (Region 2) in the next 100 year estimated that 89% with an earthquake occur probability for magnitude 6 which the return period 44.9 year, 45% with an earthquake occur probability for magnitude 6.5 which the return period 87 year, 45% with an earthquake occur probability for magnitude 7 which the return period 168.6 year.

/pdf/a-poisson-method-application-to-the-assessment-of-the-2halijjaaq.pdf

Compound Poisson process models have been studied earlier for earthquake occurrences, with some arbitrary compounding distributions. It is more meaningful to abstract information about the compounding distribution from the empirical observations on the earthquake sequences. The difinition of a compound distribution can be interpreted as an integral transform of the compounding distribution. The latter distribution can therefore be estimated by inverting the integral transform. Alternatively, from the moments of the observable random variablesviz. (a) the number of earthquakes per unit time or (b) the waiting times for subsequent earthquakes, the moments of the compounding distribution can be obtained. This information can be converted into a statement about the compounding distribution.

/pdf/estimation-of-the-compounding-distribution-in-the-compound-4xsiv6v1a8.pdf

P S Moharir

Papers

Estimation of the compounding distribution in the compound Poisson process model for earthquakes