The convex hull of a spherically symmetric sample

1981 ◽  
Vol 13 (4) ◽  
pp. 751-763 ◽  
Author(s):  
William F. Eddy ◽  
James D. Gale
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Probability Measure ◽  
Convex Hull ◽  
Random Sample ◽  
Unit Sphere ◽  
Continuous Functions ◽  
Extreme Value ◽  
Spherically Symmetric ◽  
Extreme Value Distributions

Using the isomorphism between convex subsets of Euclidean space and continuous functions on the unit sphere we describe the probability measure of the convex hull of a random sample. When the sample is spherically symmetric the asymptotic behavior of this measure is determined. There are three distinct limit measures, each corresponding to one of the classical extreme-value distributions. Several properties of each limit are determined.

The convex hull of a spherically symmetric sample

1981 ◽  
Vol 13 (04) ◽  
pp. 751-763 ◽  
Author(s):  
William F. Eddy ◽  
James D. Gale
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Probability Measure ◽  
Convex Hull ◽  
Random Sample ◽  
Unit Sphere ◽  
Continuous Functions ◽  
Extreme Value ◽  
Spherically Symmetric ◽  
Extreme Value Distributions

Using the isomorphism between convex subsets of Euclidean space and continuous functions on the unit sphere we describe the probability measure of the convex hull of a random sample. When the sample is spherically symmetric the asymptotic behavior of this measure is determined. There are three distinct limit measures, each corresponding to one of the classical extreme-value distributions. Several properties of each limit are determined.

scholarly journals Continuous solutions to two iterative functional equations

2021 ◽  
Author(s):  
Karol Baron
Keyword(s):  
Probability Measure ◽  
Functional Equations ◽  
Continuous Functions ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Hölder Continuous Functions

AbstractBased on iteration of random-valued functions we study the problem of solvability in the class of continuous and Hölder continuous functions $$\varphi $$ φ of the equations $$\begin{aligned} \varphi (x)=F(x)-\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ),\\ \varphi (x)=F(x)+\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ), \end{aligned}$$ φ ( x ) = F ( x ) - ∫ Ω φ ( f ( x , ω ) ) P ( d ω ) , φ ( x ) = F ( x ) + ∫ Ω φ ( f ( x , ω ) ) P ( d ω ) , where P is a probability measure on a $$\sigma $$ σ -algebra of subsets of $$\Omega $$ Ω .

An Extreme Value Model for U.S. Hail Size

2017 ◽  
Vol 145 (11) ◽  
pp. 4501-4519 ◽  
Author(s):  
John T. Allen ◽  
Michael K. Tippett ◽  
Yasir Kaheil ◽  
Adam H. Sobel ◽  
Chiara Lepore ◽  
...  
Keyword(s):  
Gumbel Distribution ◽  
The United States ◽  
Value Theory ◽  
Extreme Value ◽  
Extreme Value Distributions ◽  
The Past ◽  
Return Levels ◽  
Data Limitations ◽  
Extreme Value Model ◽  
Value Model

The spatial distribution of return intervals for U.S. hail size is explored within the framework of extreme value theory using observations from the period 1979–2013. The center of the continent has experienced hail in excess of 5 in. (127 mm) during the past 30 yr, whereas hail in excess of 1 in. (25 mm) is more common in other regions, including the West Coast. Observed hail sizes show heavy quantization toward fixed-diameter reference objects and are influenced by spatial and temporal biases similar to those noted for hail occurrence. Recorded hail diameters have been growing in recent decades because of improved reporting. These data limitations motivate exploration of extreme value distributions to represent the return periods for various hail diameters. The parameters of a Gumbel distribution are fit to dithered observed annual maxima on a national 1° × 1° grid at locations with sufficient records. Gridded and kernel-smoothed return sizes and quantiles up to the 200-yr return period are determined for the fitted Gumbel distribution. These results are used to illustrate return levels for hail greater than a given size for at least one location within each 1° × 1° grid box for the United States.

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