The Full Tails Gamma Distribution Applied to Model Extreme Values

Castillo, Joan del; Daoudi, Jalila; Serra, Isabel

doi:10.1017/asb.2017.9

Cited by 11 publications

(10 citation statements)

References 33 publications

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“…Nevertheless, the lognormal fit is not preferred for the tail. Other works have fit a truncated gamma distribution (which contains an exponential tail) to this kind of data (Corral & Turiel, ; del Castillo et al, ). When fitting a truncated power law, the results are in agreement with the original reference (Corral et al, ), with an exponent β in the range 1.1–1.2, for about 2 orders of magnitude.…”

Section: Data and Resultsmentioning

confidence: 99%

“…These constitute a loose family of distributions (Farmer & Geanakoplos, ) with the characteristic that, for a certain range of x , the distribution resembles in some undefined way a power law. Consider the so‐called full‐tails gamma (ftg) distribution (del Castillo et al, )

f_{ftg} (x) = \frac{1}{θ_{2} Γ (1 - β, θ_{1} / θ_{2})} {(\frac{θ_{2}}{θ_{1} + x})}^{β} \exp (- \frac{θ_{1} + x}{θ_{2}}),

with θ 1 > 0, θ 2 > 0, and − ∞ < β < ∞ and with Γ(1 − β , θ 1 / θ 2 ) the (upper) incomplete gamma function (Abramowitz & Stegun, ). This distribution is a truncated gamma distribution (Serra & Corral, ) extended to 1 − β < 0 and shifted to have support in the interval [0, ∞ ).…”

Section: Power Law Distributions and Power Law‐like Distributionsmentioning

confidence: 99%

See 1 more Smart Citation

Power Law Size Distributions in Geoscience Revisited

Corral

González

2019

Earth and Space Science

106

103

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The size or energy of diverse structures or phenomena in geoscience appears to follow power law distributions. A rigorous statistical analysis of such observations is tricky, though. Observables can span several orders of magnitude, but the range for which the power law may be valid is typically truncated, usually because the smallest events are too tiny to be detected and the largest ones are limited by the system size. We revisit several examples of proposed power law distributions dealing with potentially damaging natural phenomena. Adequate fits of the distributions of sizes are especially important in these cases, given that they may be used to assess long‐term hazard. After reviewing the theoretical background for power law distributions, we improve an objective statistical fitting method and apply it to diverse data sets. The method is described in full detail, and it is easy to implement. Our analysis elucidates the range of validity of the power law fit and the corresponding exponent and whether a power law tail is improved by a truncated lognormal. We confirm that impact fireballs and Californian earthquakes show untruncated power law behavior, whereas global earthquakes follow a double power law. Rain precipitation over space and time and tropical cyclones show a truncated power law regime. Karst sinkholes and wildfires, in contrast, are better described by truncated lognormals, although wildfires also may show power law regimes. Our conclusions only apply to the analyzed data sets but show the potential of applying this robust statistical technique in the future.

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Section: Data and Resultsmentioning

confidence: 99%

f_{ftg} (x) = \frac{1}{θ_{2} Γ (1 - β, θ_{1} / θ_{2})} {(\frac{θ_{2}}{θ_{1} + x})}^{β} \exp (- \frac{θ_{1} + x}{θ_{2}}),

Section: Power Law Distributions and Power Law‐like Distributionsmentioning

confidence: 99%

Power Law Size Distributions in Geoscience Revisited

Corral

González

2019

Earth and Space Science

106

103

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“…or by a mixture model where the components are densities of the same family, such as a mixture of gamma densities (del Castillo et al, 2012;Venturini et al, 2008) or normal densities.…”

Section: Discussionmentioning

confidence: 99%

Bayesian estimation of the threshold of a generalised pareto distribution for heavy-tailed observations

Villa

2016

TEST

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In this paper, we discuss a method to define prior distributions for the threshold of a generalised Pareto distribution, in particular when its applications are directed to heavy-tailed data. We propose to assign prior probabilities to the order statistics of a given set of observations. In other words, we assume that the threshold coincides with one of the data points. We show two ways of defining a prior: by assigning equal mass to each order statistic, that is a uniform prior, and by considering the worth that every order statistic has in representing the true threshold. Both proposed priors represent a scenario of minimal information, and we study their adequacy through simulation exercises and by analysing two applications from insurance and finance.

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“…Using the R programming language, a Monte Carlo simulation is performed with different sample sizes and fixed values of the neutrosophic parameters λ n � [5,8] and p n � [1,1]. An imprecise dataset is produced with defined parametric values, and simulation analysis is replicated N � 10 5 times with sample sizes of m � 5, 15, 30, and 60, respectively.…”

Section: Sample Estimationmentioning

confidence: 99%

“…Wang and Wu [2] presented valuable information on the gamma distribution and its uses. ree types of gamma distribution functions, namely, one, two, and three parameter gamma densities, are employed as wellsuited models for many real datasets [3][4][5].…”

Section: Introductionmentioning

confidence: 99%

On Statistical Development of Neutrosophic Gamma Distribution with Applications to Complex Data Analysis

et al. 2021

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In the absence of a correct distribution theory for complex data, neutrosophic algebra can be very useful in quantifying uncertainty. In applied data analysis, implementation of existing gamma distribution becomes inadequate for some applications when dealing with an imprecise, uncertain, or vague dataset. Most existing works have explored distributional properties of the gamma distribution under the assumption that data do not have any kind of indeterminacy. Yet, analytical properties of the gamma model for the more realistic setting when data involved uncertainties remain largely underdeveloped. This paper fills such a gap and develops the notion of neutrosophic gamma distribution (NGD). The proposed distribution represents a generalized structure of the existing gamma distribution. The basic distributional properties, including moments, shape coefficients, and moment generating function (MGF), are established. Several examples are considered to emphasize the relevance of the proposed NGD for dealing with circumstances with inadequate or ambiguous knowledge about the distributional characteristics. The estimation framework for treating vague parameters of the NGD is developed. The Monte Carlo simulation is implemented to examine the performance of the proposed model. The proposed model is applied to a real dataset for the purpose of dealing with inaccurate and vague statistical data. Results show that the NGD has better flexibility in handling real data over the conventional gamma distribution.

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The Full Tails Gamma Distribution Applied to Model Extreme Values

Cited by 11 publications

References 33 publications

Power Law Size Distributions in Geoscience Revisited

Power Law Size Distributions in Geoscience Revisited

Bayesian estimation of the threshold of a generalised pareto distribution for heavy-tailed observations

On Statistical Development of Neutrosophic Gamma Distribution with Applications to Complex Data Analysis

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