SQUARED COEFFICIENT OF VARIATION OF TAYLOR'S LAW FOR RANDOM ABSOLUTE DIFFERENCES

2017 ◽  
Vol 32 (4) ◽  
pp. 483-494
Author(s):  
Mark Brown ◽  
Joel E. Cohen
Keyword(s):  
Coefficient Of Variation ◽  
Random Variables ◽  
Vector Spaces ◽  
Human Populations ◽  
Population Variance ◽  
Taylor’S Law ◽  
And Migration ◽  
Fluctuation Scaling ◽  
Constant Coefficient Of Variation ◽  
Normed Vector

In a family, parameterized by θ, of non-negative random variables with finite, positive second moment, Taylor's law (TL) asserts that the population variance is proportional to a power of the population mean as θ varies: σ2 (θ) = a[μ(θ)]b, a > 0. TL, sometimes called fluctuation scaling, holds widely in science, probability theory, and stochastic processes. Here we report diverse examples of TL with b = 2 (equivalent to a constant coefficient of variation) arising from a difference of random variables in normed vector spaces of dimension 1 and larger. In these examples, we compute a exactly using, in some cases, a simple, new technique. These examples may prove useful in future models that involve differences of random variables, including models of the spatial distribution and migration of human populations.

Taylor's law, via ratios, for some distributions with infinite mean

2017 ◽  
Vol 54 (3) ◽  
pp. 657-669 ◽  
Author(s):  
Mark Brown ◽  
Joel E. Cohen ◽  
Victor H. de la Peña
Keyword(s):  
Population Density ◽  
Probability Distributions ◽  
Branching Processes ◽  
Population Variance ◽  
Classical Probability ◽  
Birth And Death Processes ◽  
Taylor’S Law ◽  
The Mean ◽  
Fluctuation Scaling ◽  
First Time

Abstract Taylor's law (TL) originated as an empirical pattern in ecology. In many sets of samples of population density, the variance of each sample was approximately proportional to a power of the mean of that sample. In a family of nonnegative random variables, TL asserts that the population variance is proportional to a power of the population mean. TL, sometimes called fluctuation scaling, holds widely in physics, ecology, finance, demography, epidemiology, and other sciences, and characterizes many classical probability distributions and stochastic processes such as branching processes and birth-and-death processes. We demonstrate analytically for the first time that a version of TL holds for a class of distributions with infinite mean. These distributions, a subset of stable laws, and the associated TL differ qualitatively from those of light-tailed distributions. Our results employ and contribute to the methodology of Albrecher and Teugels (2006) and Albrecher et al. (2010). This work opens a new domain of investigation for generalizations of TL.

Understanding change in hydrometeorological extremes with statistical models - the importance of model parametrization

2021 ◽  
Author(s):  
Ilaria Prosdocimi ◽  
Thomas Kjeldsen
Keyword(s):  
Coefficient Of Variation ◽  
Constant Coefficient ◽  
River Flow ◽  
Extreme Value ◽  
Flow Measurements ◽  
Scaling Properties ◽  
Distribution Parameters ◽  
Quantile Estimates ◽  
Constant Coefficient Of Variation ◽  
The Impact

<p>The potential for changes in hydrometeorological extremes is routinely investigated by fitting change-permitting extreme value models to long-term observations, allowing one or more distribution parameters to change as a function of time or some physically-motivated covariate. In most practical extreme value analyses, the main quantity of interest though is the upper quantiles of the distribution, rather than the parameters' values. This study focuses on the changes in quantile estimates under different change-permitting models. First, metrics which measure the impact of changes in parameters on changes in quantiles are introduced. The mathematical structure of these change metrics is investigated for several models based on the Generalised Extreme Value (GEV) distribution. It is shown that for the most commonly used models, the predicted changes in the quantiles are a non-intuitive function of the distribution parameters, leading to results which are difficult to interpret. Next, it is posited that commonly used change-permitting GEV models do not preserve a constant coefficient of variation, a property that is typically assumed to hold and that is related to the scaling properties of extremes. To address these shortcomings a new (parsimonious) model is proposed: the model assumes a constant coefficient of variation, allowing the location and scale parameters to change simultaneously. The proposed model results in more interpretable changes in the quantile function. The consequences of the different modelling choices on quantile estimates are exemplified using a dataset of extreme peak river flow measurements.</p>

Functionals of Bounded Frechet Variation

1949 ◽  
Vol 1 (2) ◽  
pp. 153-165 ◽  
Author(s):  
Marston Morse ◽  
William Transue
Keyword(s):  
Representation Theory ◽  
Spectral Theory ◽  
General Type ◽  
Integrodifferential Equations ◽  
Vector Spaces ◽  
Variational Theory ◽  
Characteristic Equations ◽  
Special Cases ◽  
Jacobi Equations ◽  
Normed Vector

In a series of papers which will follow this paper the authors will present a theory of functionals which are bilinear over a product A × B of two normed vector spaces A and B. This theory will include a representation theory, a variational theory, and a spectral theory. The associated characteristic equations will include as special cases the Jacobi equations of the classical variational theory when n = 1, and self-adjoint integrodifferential equations of very general type. The bilinear theory is oriented by the needs of non-linear and non-bilinear analysis in the large.

scholarly journals Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Messaoud Bounkhel
Keyword(s):  
Optimization Problems ◽  
Time Function ◽  
Vector Spaces ◽  
Directional Derivatives ◽  
Minimal Time Function ◽  
Topological Vector Spaces ◽  
Minimal Time ◽  
Set Valued Mapping ◽  
Bounded Set ◽  
Normed Vector

For a set-valued mappingMdefined between two Hausdorff topological vector spacesEandFand with closed convex graph and for a given point(x,y)∈E×F, we study the minimal time function associated with the images ofMand a bounded setΩ⊂Fdefined by𝒯M,Ω(x,y):=inf{t≥0:M(x)∩(y+tΩ)≠∅}. We prove and extend various properties on directional derivatives and subdifferentials of𝒯M,Ωat those points of(x,y)∈E×F(both cases: points in the graphgph Mand points outside the graph). These results are used to prove, in terms of the minimal time function, various new characterizations of the convex tangent cone and the convex normal cone to the graph ofMat points insidegph Mand to the graph of the enlargement set-valued mapping at points outsidegph M. Our results extend many existing results, from Banach spaces and normed vector spaces to Hausdorff topological vector spaces (Bounkhel, 2012; Bounkhel and Thibault, 2002; Burke et al., 1992; He and Ng, 2006; and Jiang and He 2009). An application of the minimal time function𝒯M,Ωto the calmness property of perturbed optimization problems in Hausdorff topological vector spaces is given in the last section of the paper.

Taylor’s law of fluctuation scaling for semivariances and higher moments of heavy-tailed data

2021 ◽  
Vol 118 (46) ◽  
pp. e2108031118
Author(s):  
Mark Brown ◽  
Joel E. Cohen ◽  
Chuan-Fa Tang ◽  
Sheung Chi Phillip Yam
Keyword(s):  
Scaling Laws ◽  
Higher Moments ◽  
Direct Proportion ◽  
Sample Mean ◽  
Scaling Relationships ◽  
Heavy Tailed Distributions ◽  
Taylor’S Law ◽  
Heavy Tailed ◽  
Fluctuation Scaling ◽  
Sortino Ratio

We generalize Taylor’s law for the variance of light-tailed distributions to many sample statistics of heavy-tailed distributions with tail index α in (0, 1), which have infinite mean. We show that, as the sample size increases, the sample upper and lower semivariances, the sample higher moments, the skewness, and the kurtosis of a random sample from such a law increase asymptotically in direct proportion to a power of the sample mean. Specifically, the lower sample semivariance asymptotically scales in proportion to the sample mean raised to the power 2, while the upper sample semivariance asymptotically scales in proportion to the sample mean raised to the power (2−α)/(1−α)>2. The local upper sample semivariance (counting only observations that exceed the sample mean) asymptotically scales in proportion to the sample mean raised to the power (2−α2)/(1−α). These and additional scaling laws characterize the asymptotic behavior of commonly used measures of the risk-adjusted performance of investments, such as the Sortino ratio, the Sharpe ratio, the Omega index, the upside potential ratio, and the Farinelli–Tibiletti ratio, when returns follow a heavy-tailed nonnegative distribution. Such power-law scaling relationships are known in ecology as Taylor’s law and in physics as fluctuation scaling. We find the asymptotic distribution and moments of the number of observations exceeding the sample mean. We propose estimators of α based on these scaling laws and the number of observations exceeding the sample mean and compare these estimators with some prior estimators of α.

Functionals F Bilinear over the Product A × B of Two P-Normed Vector Spaces

1949 ◽  
Vol 50 (4) ◽  
pp. 777 ◽  
Author(s):  
Marston Morse ◽  
William Transue
Keyword(s):  
Vector Spaces ◽  
Normed Vector
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