The Selection of the Domain of Attraction of an Extreme Value Distribution from a Set of Data

1989 ◽  
pp. 181-190 ◽  
Author(s):  
E. Castillo ◽  
J. Galambos ◽  
J. M. Sarabia
Keyword(s):  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Selection Of

Convergence rates for the ultimate and pentultimate approximations in extreme-value theory

1982 ◽  
Vol 14 (04) ◽  
pp. 833-854 ◽  
Author(s):  
Jonathan P. Cohen
Keyword(s):  
Extreme Value Theory ◽  
Convergence Rates ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Type I ◽  
Type Ii

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.

Limit laws for maxima of a sequence of random variables defined on a Markov chain

1970 ◽  
Vol 2 (2) ◽  
pp. 323-343 ◽  
Author(s):  
Sidney I. Resnick ◽  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Extreme Value Distributions ◽  
Limit Laws ◽  
Limiting Behavior ◽  
Normalizing Constants

Consider the bivariate sequence of r.v.'s {(Jn, Xn), n ≧ 0} with X0 = - ∞ a.s. The marginal sequence {Jn} is an irreducible, aperiodic, m-state M.C., m < ∞, and the r.v.'s Xn are conditionally independent given {Jn}. Furthermore P{Jn = j, Xn ≦ x | Jn − 1 = i} = pijHi(x) = Qij(x), where H1(·), · · ·, Hm(·) are c.d.f.'s. Setting Mn = max {X1, · · ·, Xn}, we obtain P{Jn = j, Mn ≦ x | J0 = i} = [Qn(x)]i, j, where Q(x) = {Qij(x)}. The limiting behavior of this probability and the possible limit laws for Mn are characterized.Theorem. Let ρ(x) be the Perron-Frobenius eigenvalue of Q(x) for real x; then:(a)ρ(x) is a c.d.f.;(b) if for a suitable normalization {Qijn(aijnx + bijn)} converges completely to a matrix {Uij(x)} whose entries are non-degenerate distributions then Uij(x) = πjρU(x), where πj = limn → ∞pijn and ρU(x) is an extreme value distribution;(c) the normalizing constants need not depend on i, j;(d) ρn(anx + bn) converges completely to ρU(x);(e) the maximum Mn has a non-trivial limit law ρU(x) iff Qn(x) has a non-trivial limit matrix U(x) = {Uij(x)} = {πjρU(x)} or equivalently iff ρ(x) or the c.d.f. πi = 1mHiπi(x) is in the domain of attraction of one of the extreme value distributions. Hence the only possible limit laws for {Mn} are the extreme value distributions which generalize the results of Gnedenko for the i.i.d. case.

Rate of convergence for the generalized Pareto approximation of the excesses

2003 ◽  
Vol 35 (04) ◽  
pp. 1007-1027 ◽  
Author(s):  
J.-P. Raoult ◽  
R. Worms
Keyword(s):  
Distribution Function ◽  
Uniform Convergence ◽  
Rate Of Convergence ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Generalized Pareto

Let F be a distribution function in the domain of attraction of an extreme-value distribution H γ. If F u is the distribution function of the excesses over u and G γ the distribution function of the generalized Pareto distribution, then it is well known that F u (x) converges to G γ(x/σ(u)) as u tends to the end point of F, where σ is an appropriate normalizing function. We study the rate of (uniform) convergence to 0 of F̅ u (x)-G̅γ((x+u-α(u))/σ(u)), where α and σ are two appropriate normalizing functions.

Convergence rates for the ultimate and pentultimate approximations in extreme-value theory

1982 ◽  
Vol 14 (4) ◽  
pp. 833-854 ◽  
Author(s):  
Jonathan P. Cohen
Keyword(s):  
Extreme Value Theory ◽  
Convergence Rates ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Type I ◽  
Type Ii

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.

Tail equivalence and its applications

1971 ◽  
Vol 8 (01) ◽  
pp. 136-156 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Distribution Functions ◽  
Value Distribution ◽  
Normalizing Constants ◽  
Independent Identically Distributed

If for two c.d.f.'s F(·) and G(·), 1 – F(x)/1 – G(x) → A, 0 &lt;A &lt;∞ , as x → ∞, then for normalizing constants an &gt; 0, bn, n &gt; 1, Fn (anx + bn ) → φ(x), φ(x) non-degenerate, iff Gn (anx + bn )→ φ A−1(x). Conversely, if Fn (anx+bn )→ φ(x), Gn (anx + bn ) → φ'(x), φ(x) and φ'(x) non-degenerate, then there exist constants C &gt;0 and D such that φ'(x) =φ(Cx + D) and limx→∞ 1 — F(x)/1 — G(x) exists and is expressed in terms of C and D, depending on which type of extreme value distribution φ(x) is. These results are used to study domain of attraction questions for products of distribution functions and to reduce the limit law problem for maxima of a sequence of random variables defined on a Markov chain (M.C.) to the independent, identically distributed (i.i.d.) case.

Limit laws for maxima of a sequence of random variables defined on a Markov chain

1970 ◽  
Vol 2 (02) ◽  
pp. 323-343 ◽  
Author(s):  
Sidney I. Resnick ◽  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Extreme Value Distributions ◽  
Limit Laws ◽  
Limiting Behavior ◽  
Normalizing Constants

Consider the bivariate sequence of r.v.'s {(J n , X n ), n ≧ 0} with X 0 = - ∞ a.s. The marginal sequence {J n } is an irreducible, aperiodic, m-state M.C., m &lt; ∞, and the r.v.'s X n are conditionally independent given {J n }. Furthermore P{J n = j, X n ≦ x | J n − 1 = i} = p ij H i (x) = Q ij (x), where H 1(·), · · ·, H m (·) are c.d.f.'s. Setting M n = max {X 1, · · ·, X n }, we obtain P{J n = j, M n ≦ x | J 0 = i} = [Q n (x)] i, j , where Q(x) = {Q ij (x)}. The limiting behavior of this probability and the possible limit laws for M n are characterized. Theorem. Let ρ(x) be the Perron-Frobenius eigenvalue of Q(x) for real x; then: (a)ρ(x) is a c.d.f.; (b) if for a suitable normalization {Q ij n (a ijn x + b ijn )} converges completely to a matrix {U ij (x)} whose entries are non-degenerate distributions then U ij (x) = π j ρ U (x), where π j = lim n → ∞ p ij n and ρ U (x) is an extreme value distribution; (c) the normalizing constants need not depend on i, j; (d) ρ n (a n x + b n ) converges completely to ρ U (x); (e) the maximum M n has a non-trivial limit law ρ U (x) iff Q n (x) has a non-trivial limit matrix U(x) = {U ij (x)} = {π j ρ U (x)} or equivalently iff ρ(x) or the c.d.f. π i = 1 m H i π i(x) is in the domain of attraction of one of the extreme value distributions. Hence the only possible limit laws for {M n } are the extreme value distributions which generalize the results of Gnedenko for the i.i.d. case.

Distributions that are both subexponential and in the domain of attraction of an extreme-value distribution

1988 ◽  
Vol 20 (4) ◽  
pp. 706-718 ◽  
Author(s):  
Charles M. Goldie ◽  
Sidney Resnick
Keyword(s):  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Convolution Power

When does a distribution F have the property of both being in the domain of attraction of exp {–e–x} and having a second convolution-power tail equivalent to the first: Sufficient conditions and examples are given.

Rate of convergence for the generalized Pareto approximation of the excesses

2003 ◽  
Vol 35 (4) ◽  
pp. 1007-1027 ◽  
Author(s):  
J.-P. Raoult ◽  
R. Worms
Keyword(s):  
Distribution Function ◽  
Uniform Convergence ◽  
Rate Of Convergence ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Generalized Pareto

Let F be a distribution function in the domain of attraction of an extreme-value distribution Hγ. If Fu is the distribution function of the excesses over u and Gγ the distribution function of the generalized Pareto distribution, then it is well known that Fu(x) converges to Gγ(x/σ(u)) as u tends to the end point of F, where σ is an appropriate normalizing function. We study the rate of (uniform) convergence to 0 of F̅u(x)-G̅γ((x+u-α(u))/σ(u)), where α and σ are two appropriate normalizing functions.

Tail equivalence and its applications

1971 ◽  
Vol 8 (1) ◽  
pp. 136-156 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Distribution Functions ◽  
Value Distribution ◽  
Normalizing Constants ◽  
Independent Identically Distributed

If for two c.d.f.'s F(·) and G(·), 1 – F(x)/1 – G(x) → A, 0 <A <∞, as x → ∞, then for normalizing constants an > 0, bn, n > 1, Fn(anx + bn) → φ(x), φ(x) non-degenerate, iff Gn(anx + bn)→ φ A−1(x). Conversely, if Fn(anx+bn)→ φ(x), Gn(anx + bn) → φ'(x), φ(x) and φ'(x) non-degenerate, then there exist constants C >0 and D such that φ'(x) =φ(Cx + D) and limx→∞ 1 — F(x)/1 — G(x) exists and is expressed in terms of C and D, depending on which type of extreme value distribution φ(x) is. These results are used to study domain of attraction questions for products of distribution functions and to reduce the limit law problem for maxima of a sequence of random variables defined on a Markov chain (M.C.) to the independent, identically distributed (i.i.d.) case.

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