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.

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 (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.

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.

DIscrepancy of Gear Tooth Allowable Bending Stress Using the Extreme Value Distribution From Different Test Methods

1991 ◽  
Author(s):  
Chienann A. Hou ◽  
Shijun Ma
Keyword(s):  
Gear Tooth ◽  
Extreme Value Distribution ◽  
Test Methods ◽  
Extreme Value ◽  
Distribution Functions ◽  
Value Distribution ◽  
Test Method ◽  
Bending Stress ◽  
Gear Design ◽  
Experimental Approaches

Abstract The allowable bending stress Se of a gear tooth is one of the basic factors in gear design. It can be determined by either the pulsating test or the gear-running test. However, some differences exist between the allowable bending stress Se obtained from these different test methods. In this paper, the probability distribution functions corresponding to each test method are analyzed and the expressions for the minimum extreme value distribution are presented. By using numerical integration, Se values from the population of the same tested gear tooth are obtained. Based on this investigation it is shown that the differences in Se obtained from the different test methods are significant. A proposed correction factor associated with the different experimental approaches is also presented.

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.

scholarly journals Tail Properties and Asymptotic Expansions for the Maximum of the Logarithmic Skew-Normal Distribution

2013 ◽  
Vol 50 (3) ◽  
pp. 900-907 ◽  
Author(s):  
Xin Liao ◽  
Zuoxiang Peng ◽  
Saralees Nadarajah
Keyword(s):  
Asymptotic Expansion ◽  
Convergence Rate ◽  
Normal Distribution ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Skew Normal Distribution ◽  
Skew Normal

We discuss tail behaviors, subexponentiality, and the extreme value distribution of logarithmic skew-normal random variables. With optimal normalized constants, the asymptotic expansion of the distribution of the normalized maximum of logarithmic skew-normal random variables is derived. We show that the convergence rate of the distribution of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/(log n)1/2.

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.

On asymptotic independence of the partial sums of positive and negative parts of independent random variables

1971 ◽  
Vol 3 (02) ◽  
pp. 404-425
Author(s):  
Howard G. Tucker
Keyword(s):  
Stable Distribution ◽  
Characteristic Exponent ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Distribution Functions ◽  
Independent Random Variables ◽  
Asymptotic Independence ◽  
Partial Sums ◽  
Common Distribution Function ◽  
Independent Identically Distributed

The aim of this study is an investigation of the joint limiting distribution of the sequence of partial sums of the positive parts and negative parts of a sequence of independent identically distributed random variables. In particular, let {Xn} be a sequence of independent identically distributed random variables with common distribution functionF, assumeFis in the domain of attraction of a stable distribution with characteristic exponent α, 0 &lt; α ≦ 2, and let {Bn} be normalizing coefficients forF. Let us denoteXn+=XnI[Xn&gt; 0]andXn−= −XnI[Xn&lt;0], so thatXn=Xn+-Xn−. LetF+andF−denote the distribution functions ofX1+andX1−respectively, and letSn(+)=X1++ · · · +Xn+,Sn(-)=X1−+ · · · +Xn−. The problem considered here is to find under what conditions there exist sequences of real numbers {an} and {bn} such that the joint distribution of (Bn-1Sn(+)+an,Bn-1Sn(-)+bn) converges to that of two independent random variables (U, V). As might be expected, different results are obtained when α &lt; 2 and when α = 2. When α &lt; 2, there always exist such sequences so that the above is true, and in this case bothUandVare stable with characteristic exponent a, or one has such a stable distribution and the other is constant. When α = 2, and if 0 &lt; ∫x2dF(x) &lt; ∞, then there always exist such sequences such that the above convergence takes place; bothUandVare normal (possibly one is a constant), and if neither is a constant, thenUandVarenotindependent. If α = 2 and ∫x2dF(x) = ∞, then at least one ofF+,F−is in the domain of partial attraction of the normal distribution, and a modified statement on the independence ofUandVholds. Various specialized results are obtained for α = 2.

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