Inference for the truncated exponential distribution

Raschke, Mathias

doi:10.1007/s00477-011-0458-8

Cited by 19 publications

(12 citation statements)

References 34 publications

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“…If the sub-maxima of not overlapping sub-sections of the time history are log-normally distributed, then the maximum of the entire time history cannot be log-normally distributed (exception: all sub-maxima are identical). Log-normal assumptions would also contradict all our experiences with extreme values , Raschke 2011, 2012, 2013.…”

Section: The Distribution Of the Maximum Of A Random Sequencementioning

confidence: 77%

“…Beside this, truncation of the log-normal distribution was suggested to avoid overestimations, but choosing the truncation point is difficult according to Strasser et al (2008). Therein, statistical estimation methods for truncation points (Raschke 2011) have not been considered. We generally note a lack of consideration of current knowledge of stochastics and statistics in the research of GMR.…”

Section: Introductionmentioning

confidence: 99%

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Statistical modeling of ground motion relations for seismic hazard analysis

Raschke¹

2013

J Seismol

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We introduce a new approach for ground motion relations (GMR) in the probabilistic seismic hazard analysis (PSHA), being influenced by the extreme value theory of mathematical statistics. Therein, we understand a GMR as a random function. We derive mathematically the principle of area-equivalence; wherein two alternative GMRs have an equivalent influence on the hazard if these GMRs have equivalent area functions. This includes local biases. An interpretation of the difference between these GMRs (an actual and a modeled one) as a random component leads to a general overestimation of residual variance and hazard. Beside this, we discuss important aspects of classical approaches and discover discrepancies with the state of the art of stochastics and statistics (model selection and significance, test of distribution assumptions, extreme value statistics). We criticize especially the assumption of logarithmic normally distributed residuals of maxima like the peak ground acceleration (PGA). The natural distribution of its individual random component (equivalent to exp(ε 0 ) of Joyner and Boore 1993) is the generalized extreme value. We show by numerical researches that the actual distribution can be hidden and a wrong distribution assumption can influence the PSHA negatively as the negligence of area equivalence does. Finally, we suggest an estimation concept for GMRs of PSHA with a regression-free variance estimation of the individual random component. We demonstrate the advantages of event-specific GMRs by analyzing data sets from the PEER strong motion database and estimate event-specific GMRs. Therein, the majority of the best models base on an anisotropic point source approach. The residual variance of logarithmized PGA is significantly smaller than in previous models. We validate the estimations for the event with the largest sample by empirical area functions, which indicate the appropriate modeling of the GMR by an anisotropic point source model. The constructed distances like the Joyner-Boore distance do not work well for event-specific GMRs. We discover also a strong relation between magnitude and the squared expectation of the PGAs being integrated in the geo-space for the event-specific GMRs. One of our secondary contributions is the simple modeling of anisotropy for a point source model. ground motion relation, probabilistic seismic hazard analysis, area-equivalence, regression analysis, extreme value statistics, model selection, statistical test, random function * *

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Section: The Distribution Of the Maximum Of A Random Sequencementioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Statistical modeling of ground motion relations for seismic hazard analysis

Raschke¹

2013

J Seismol

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“…We explain the proposed technique in different examinations. e execution of BGΓM is compared with the WM [16], RM [19], EM [14], LM [22], GM [15], GGM [25], ΓM [1], GΓM [2], BWMM [26], BRM [27], BEM [28], BLM [22], BGM [29,30], BGGM [22], and BΓM [31]. To measure the fitting precision of every model, we use the corresponding −2 Log-likelihood (−2L) values of models fitted to data.…”

Section: Methodsmentioning

confidence: 99%

“…To maximize the likelihood function in (28), we consider the derivation of L with the location u at the (t + 1) iteration step. We have…”

Section: Location Parameter Estimationmentioning

confidence: 99%

A New Double Truncated Generalized Gamma Model with Some Applications

Bakery

Zakaria

Mohamed

2021

Journal of Mathematics

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The generalized Gamma model has been applied in a variety of research fields, including reliability engineering and lifetime analysis. Indeed, we know that, from the above, it is unbounded. Data have a bounded service area in a variety of applications. A new five-parameter bounded generalized Gamma model, the bounded Weibull model with four parameters, the bounded Gamma model with four parameters, the bounded generalized Gaussian model with three parameters, the bounded exponential model with three parameters, and the bounded Rayleigh model with two parameters, is presented in this paper as a special case. This approach to the problem, which utilizes a bounded support area, allows for a great deal of versatility in fitting various shapes of observed data. Numerous properties of the proposed distribution have been deduced, including explicit expressions for the moments, quantiles, mode, moment generating function, mean variance, mean residual lifespan, and entropies, skewness, kurtosis, hazard function, survival function, r th order statistic, and median distributions. The delivery has hazard frequencies that are monotonically increasing or declining, bathtub-shaped, or upside-down bathtub-shaped. We use the Newton Raphson approach to approximate model parameters that increase the log-likelihood function and some of the parameters have a closed iterative structure. Six actual data sets and six simulated data sets were tested to demonstrate how the proposed model works in reality. We illustrate why the Model is more stable and less affected by sample size. Additionally, the suggested model for wavelet histogram fitting of images and sounds is very accurate.

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“…There are different estimation methods for the upper bounds of a truncated distribution including the corresponding confidence regions (Kijko and Singh 2011;Raschke 2012). We apply the simple method by Robson and Whitlock (1964) with the estimation b m max ¼ M n þ M n −M n−1 ð Þ, where the two largest observations are M n and M n−1 .…”

Section: The Exponential Distributionsmentioning

confidence: 99%

Modeling of magnitude distributions by the generalized truncated exponential distribution

Raschke¹

2014

J Seismol

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The probability distribution of the magnitude can be modeled by an exponential distribution according to the Gutenberg-Richter relation. Two alternatives are the truncated exponential distribution (TED) and the cutoff exponential distribution (CED). The TED is frequently used in seismic hazard analysis although it has a weak point: when two TEDs with equal parameters except the upper bound magnitude are mixed, then the resulting distribution is not a TED. Inversely, it is also not possible to split a TED of a seismic region into TEDs of subregions with equal parameters except the upper bound magnitude. This weakness is a principal problem as seismic regions are constructed scientific objects and not natural units. We overcome it by the generalization of the abovementioned exponential distributions: the generalized truncated exponential distribution (GTED). Therein, identical exponential distributions are mixed by the probability distribution of the correct cutoff points. This distribution model is flexible in the vicinity of the upper bound magnitude and is equal to the exponential distribution for smaller magnitudes. Additionally, the exponential distributions TED and CED are special cases of the GTED. We discuss the possible ways of estimating its parameters and introduce the normalized spacing for this purpose. Furthermore, we present methods for geographic aggregation and differentiation of the GTED and demonstrate the potential and universality of our simple approach by applying it to empirical data. The considerable improvement by the GTED in contrast to the TED is indicated by a large difference between the corresponding values of the Akaike information criterion.

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Inference for the truncated exponential distribution

Cited by 19 publications

References 34 publications

Statistical modeling of ground motion relations for seismic hazard analysis

Statistical modeling of ground motion relations for seismic hazard analysis

A New Double Truncated Generalized Gamma Model with Some Applications

Modeling of magnitude distributions by the generalized truncated exponential distribution

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