On functional records and champions · On functional records and champions 8th International Conference of the ERCIM WG on Computational and Methodological Statistics (ERCIM 2015),

On functional records and champions

8th International Conference of the ERCIM WG onComputational and Methodological Statistics (ERCIM 2015),

London, UK

Clément Dombry, Michael Falk and Maximilian Zott

Université de Franche-Comté, Besançon, France,University of Wuerzburg, Germany

December 12, 2015

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Concurrence, champions and records

Let X1, . . . ,Xn be iid random vectors in Rd , resp. stochasticprocesses with sample paths inC(S) = {f : S → R : f continuous}, S being a compact metricspace.

Dombry, Ribatet and Stoev (2015): Extremes are sampleconcurrent, if maxk=1,...,n Xk = Xi for some i ∈ {1, . . . ,n}.

In the sample concurrence case, we call Xi a champion amongX1, . . . ,Xn.

Xn is a complete record, if Xn > max (X1, . . . ,Xn−1).

Πn(X ) := P (Xn is a complete record)

Xn is a simple record, if Xn 6≤ max (X1, . . . ,Xn−1).

πn(X ) := P (Xn is a simple record)

All operations such as > or max are meant componentwise.

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All operations such as > or max are meant componentwise.2 / 21

Some comments

X1 is a record by definition.Each champion defines a record.Possibly more than one record among X1, . . . ,Xn, but notmore than one champion.

Let X1, . . . ,Xn iid copies of X , then the sample concurrenceprobability is given by

pn(X ) := P (sample concurrence in X1, . . . ,Xn)

= nP (Xn is a complete record) = nΠn(X ).

Different to that, nπn(X ) can not be interpreted as a probability.

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Some comments






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Some comments






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Example: Simple and complete records, n = 5

5 observations X1, . . . ,X5:

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Univariate champions and records

Univariate records have been studied extensively, cf. Chandler(1952), Rényi (1962) and Resnick (1987). If X ,X1, . . . ,Xn areiid rv on the real line with continuous df, thenΠn(X ) = πn(X ) = 1/n.

In the univariate case, pn(X ) = 1, and the champion coincideswith the largest order statistic max(X1, . . . ,Xn).

Multivariate records have been studied by e. g. Goldie andResnick (1989, 1995), Gnedin (1993,1998) or Hashorva andHüsler (2005). −→ Much more difficult to handle!

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Standard max-stable processes

Let η = (ηs)s∈S be a standard max-stable process (SMSP) withsample paths in C−(S) := {f ∈ C(S) : f ≤ 0}, i. e.

η =D n maxi=1,...,n

ηi ,

where η1, . . . ,ηn are iid copies of η.

Theorem (Giné et al. (1990), de Haan and Ferreira (2006))

η SMSP iff there is a process Z = (Zs)s∈S inC+(S) := {f ∈ C(S) : f ≥ 0} with E(Zs) = 1, s ∈ S, andE (sups∈S Zs) <∞ such that

P(η ≤ f ) = exp(−‖f‖D) := exp(−E

(sups∈S

(|f (s)|Zs)

)),

for all f ∈ E−(S):= set of bounded functions f : S → (−∞,0]with only finitely many discontinuities.

Univariate margins of η: P(ηs ≤ x) = exp(x), x ≤ 0, s ∈ S.

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ηi ,




P(η ≤ f ) = exp(−‖f‖D) := exp(−E

(sups∈S

(|f (s)|Zs)

)),


Univariate margins of η: P(ηs ≤ x) = exp(x), x ≤ 0, s ∈ S.

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ηi ,




P(η ≤ f ) = exp(−‖f‖D) := exp(−E

(sups∈S

(|f (s)|Zs)

)),


Univariate margins of η: P(ηs ≤ x) = exp(x), x ≤ 0, s ∈ S.10 / 21

Copulas in the domain of attraction of an SMSP

Suppose a copula process U = (Us)s∈S (continuous samplepaths, uniform univariate margins) is in the max-domain ofattraction of an SMSP η (U ∈ D(η)) with D-norm ‖·‖Dgenerated by Z = (Zs)s∈S, i. e.

n(

maxi=1,...,n

Ui − 1)→D η, (1)

where U1,U2, . . . are iid copies of U.

Eq. (1) implies forf ∈ E−(S)

nP (n(U − 1) 6≤ f )→n→∞ E(

sups∈S|f (s)|Zs

)= ‖f‖D , (2)

see Aulbach et al. (2013). Also, if (1) holds,

nP (n(U − 1) > f )→n→∞ E(

infs∈S|f (s)|Zs

)=: oo f ooD, (3)

for f ∈ E−(S), see Dombry, Falk and Z. (2015).

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n(

maxi=1,...,n

Ui − 1)→D η, (1)

where U1,U2, . . . are iid copies of U. Eq. (1) implies forf ∈ E−(S)

nP (n(U − 1) 6≤ f )→n→∞ E(

sups∈S|f (s)|Zs

)= ‖f‖D , (2)

see Aulbach et al. (2013).

Also, if (1) holds,

nP (n(U − 1) > f )→n→∞ E(

infs∈S|f (s)|Zs

)=: oo f ooD, (3)

for f ∈ E−(S), see Dombry, Falk and Z. (2015).

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n(

maxi=1,...,n

Ui − 1)→D η, (1)

where U1,U2, . . . are iid copies of U. Eq. (1) implies forf ∈ E−(S)

nP (n(U − 1) 6≤ f )→n→∞ E(

sups∈S|f (s)|Zs

)= ‖f‖D , (2)

see Aulbach et al. (2013). Also, if (1) holds,

nP (n(U − 1) > f )→n→∞ E(

infs∈S|f (s)|Zs

)=: oo f ooD, (3)

for f ∈ E−(S), see Dombry, Falk and Z. (2015).11 / 21

Dual D-norm functions

For a given D-norm ‖f‖D = E (sups∈S |f (s)|Zs), the mapping

oo · ooD : E(S)→ R, f 7→ oo f ooD := E(

infs∈S|f (s)|Zs

),

is called dual D-norm function.

The mapping ‖·‖D → oo · ooD is well-defined, but notone-to-one.oo · ooD ≡ 0 if at least two components of the SMSP η in (1)are independent.Finitedimensional setup: dual D-norm function is the tailcopula, cf. Schmidt and Stadtmüller (2006).

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infs∈S|f (s)|Zs

),


The mapping ‖·‖D → oo · ooD is well-defined, but notone-to-one.

oo · ooD ≡ 0 if at least two components of the SMSP η in (1)are independent.Finitedimensional setup: dual D-norm function is the tailcopula, cf. Schmidt and Stadtmüller (2006).

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infs∈S|f (s)|Zs

),


The mapping ‖·‖D → oo · ooD is well-defined, but notone-to-one.oo · ooD ≡ 0 if at least two components of the SMSP η in (1)are independent.

Finitedimensional setup: dual D-norm function is the tailcopula, cf. Schmidt and Stadtmüller (2006).

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infs∈S|f (s)|Zs

),


The mapping ‖·‖D → oo · ooD is well-defined, but notone-to-one.oo · ooD ≡ 0 if at least two components of the SMSP η in (1)are independent.Finitedimensional setup: dual D-norm function is the tailcopula, cf. Schmidt and Stadtmüller (2006).

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Complete Records: Extremal concurrence probabilityof U

Remember

Πn(U) = P (Un is a complete record) = P(

Un > maxi=1,...,n−1

Ui

).

Functional version of the Dombry-Ribatet-Stoev theorem, seealso Hashorva and Hüsler (2005):

Theorem

Let U,U1,U2, . . . be iid copula processes with U ∈ D(η), whereη is an SMSP with D-norm ‖·‖D. The sample concurrenceprobability fulfills

pn(U) = nΠn(U)→n→∞ E (oo η ooD) .

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Complete Records: Extremal concurrence probabilityof U

Remember

Πn(U) = P (Un is a complete record) = P(

Un > maxi=1,...,n−1

Ui

).

Functional version of the Dombry-Ribatet-Stoev theorem, seealso Hashorva and Hüsler (2005):

Theorem

Let U,U1,U2, . . . be iid copula processes with U ∈ D(η), whereη is an SMSP with D-norm ‖·‖D. The sample concurrenceprobability fulfills

pn(U) = nΠn(U)→n→∞ E (oo η ooD) .

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Complete Records: Comments and examples

The number E (oo η ooD) ∈ [0,1] is called extremalconcurrence probability.

If ‖·‖D = ‖·‖∞ (complete dependence case), thenE (oo η ooD) = 1.If ηs, ηt are independent for some s, t ∈ S, thenE (oo η ooD) = 0.More examples in Dombry, Ribatet and Stoev (2015).(logistic model, max-linear model, extremal process etc.)

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The number E (oo η ooD) ∈ [0,1] is called extremalconcurrence probability.If ‖·‖D = ‖·‖∞ (complete dependence case), thenE (oo η ooD) = 1.

If ηs, ηt are independent for some s, t ∈ S, thenE (oo η ooD) = 0.More examples in Dombry, Ribatet and Stoev (2015).(logistic model, max-linear model, extremal process etc.)

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The number E (oo η ooD) ∈ [0,1] is called extremalconcurrence probability.If ‖·‖D = ‖·‖∞ (complete dependence case), thenE (oo η ooD) = 1.If ηs, ηt are independent for some s, t ∈ S, thenE (oo η ooD) = 0.

More examples in Dombry, Ribatet and Stoev (2015).(logistic model, max-linear model, extremal process etc.)

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The number E (oo η ooD) ∈ [0,1] is called extremalconcurrence probability.If ‖·‖D = ‖·‖∞ (complete dependence case), thenE (oo η ooD) = 1.If ηs, ηt are independent for some s, t ∈ S, thenE (oo η ooD) = 0.More examples in Dombry, Ribatet and Stoev (2015).(logistic model, max-linear model, extremal process etc.)

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Simple records: Limit probability

Now focus on the multivariate case, i. e. U,U1,U2, . . . are iidobservations following a copula on Rd . Remember

πn(U) = P (Un is a simple record) = P(

Un 6≤ maxi=1,...,n−1

Ui

).

Theorem (Dombry, Falk and Z. (2015))

Let U,U1,U2, . . . be iid random vectors in Rd with U ∈ D(η),where η is standard max-stable with D-norm ‖·‖D, i. e.P(η ≤ x) = exp (−‖x‖D), x ≤ 0 ∈ Rd . Then

nπn(U)→n→∞ E (‖η‖D) .

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Simple records: Limit probability

Now focus on the multivariate case, i. e. U,U1,U2, . . . are iidobservations following a copula on Rd . Remember

πn(U) = P (Un is a simple record) = P(

Un 6≤ maxi=1,...,n−1

Ui

).

Theorem (Dombry, Falk and Z. (2015))

Let U,U1,U2, . . . be iid random vectors in Rd with U ∈ D(η),where η is standard max-stable with D-norm ‖·‖D, i. e.P(η ≤ x) = exp (−‖x‖D), x ≤ 0 ∈ Rd . Then

nπn(U)→n→∞ E (‖η‖D) .

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Simple record times

Take an iid sequence X ,X1,X2, . . . following a continuous df Fon Rd .

Let N(n) be the (random) index at which the n-th simple recordoccurs, N(1) := 1. Furthermore, ∆n := N(n)− N(n − 1), n ≥ 2,∆1 := 1 (inter record waiting times). For every n ∈ N

P(XN(n+1) ≤ x

∣∣XN(n) = y)

= P(X ≤ x

∣∣X 6≤ y),

∆n+1∣∣XN(n) = x ∼ Geom(1− F (x)).

Indeed,(XN(n)

)n∈N and

((XN(n),∆n

))n∈N are homogenous

Markov chains!

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Simple record times



P(XN(n+1) ≤ x

∣∣XN(n) = y)

= P(X ≤ x

∣∣X 6≤ y),

∆n+1∣∣XN(n) = x ∼ Geom(1− F (x)).

Indeed,(XN(n)

)n∈N and

((XN(n),∆n


Markov chains!

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Simple record times



P(XN(n+1) ≤ x

∣∣XN(n) = y)

= P(X ≤ x

∣∣X 6≤ y),

∆n+1∣∣XN(n) = x ∼ Geom(1− F (x)).

Indeed,(XN(n)

)n∈N and

((XN(n),∆n


Markov chains!

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Simple record times



P(XN(n+1) ≤ x

∣∣XN(n) = y)

= P(X ≤ x

∣∣X 6≤ y),

∆n+1∣∣XN(n) = x ∼ Geom(1− F (x)).

Indeed,(XN(n)

)n∈N and

((XN(n),∆n


Markov chains!

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Simple record times: Integrability of N(2)

Take an iid sequence U,U1,U2, . . . following a copula C on[0,1]d .

Now focus on N(2). Easy to see:

E(N(2)) =

∫[0,1]d

11− C(u)

C(du) + 1.

d = 1 :

E(N(2)) =

∫ 1

0

11− u

du + 1 =∞.

d = 2, independent margins: In that case C(u1,u2) = u1u2, and

E(N(2)) =

∫ 1

0

∫ 1

0

11− u1u2

du1du2 + 1 =π2

6+ 1 <∞.

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E(N(2)) =

∫[0,1]d

11− C(u)

C(du) + 1.

d = 1 :

E(N(2)) =

∫ 1

0

11− u

du + 1 =∞.


E(N(2)) =

∫ 1

0

∫ 1

0

11− u1u2

du1du2 + 1 =π2

6+ 1 <∞.

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E(N(2)) =

∫[0,1]d

11− C(u)

C(du) + 1.

d = 1 :

E(N(2)) =

∫ 1

0

11− u

du + 1 =∞.


E(N(2)) =

∫ 1

0

∫ 1

0

11− u1u2

du1du2 + 1 =π2

6+ 1 <∞.

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E(N(2)) =

∫[0,1]d

11− C(u)

C(du) + 1.

d = 1 :

E(N(2)) =

∫ 1

0

11− u

du + 1 =∞.


E(N(2)) =

∫ 1

0

∫ 1

0

11− u1u2

du1du2 + 1 =π2

6+ 1 <∞.

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Natural question: When is E(N(2)) finite?

Suppose C ∈ D(G) for some standard max-stable df G withD-norm ‖·‖D, dual D-norm function oo · ooD.

Proposition (Part 1, Dombry, Falk and Z. (2015))

If oo 1 ooD > 0, then E(N(2)) =∞.

Note that oo 1 ooD = 0 if G has at least two independent margins!

Proposition (Sketch of Part 2, Dombry, Falk and Z. (2015))Basically, if G has at least two independent margins, thenE(N(2)) <∞.

Corollary

E(N(2)) <∞ for multivariate normal rv unless all componentsare completely dependent.

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If oo 1 ooD > 0, then E(N(2)) =∞.



Corollary


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If oo 1 ooD > 0, then E(N(2)) =∞.



Corollary


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If oo 1 ooD > 0, then E(N(2)) =∞.



Corollary


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If oo 1 ooD > 0, then E(N(2)) =∞.



Corollary


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If oo 1 ooD > 0, then E(N(2)) =∞.



Corollary


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Simple record times: Integrability of N(n)

Similar to Goldie and Resnick (1989): If F ∈ D(G) for somebivariate max-stable df G, then the number of complete recordsin a sequence X1,X2, . . . following the continuous df F is eitherfinite a. s. or infinite a. s., depending on whether G hasindependent margins or not.

What about E(N(n)), n > 2? Clearly, E(N(n)) =∞ ifE(N(2)) =∞.

Proposition (Dombry, Falk and Z. (2015))

(i) If E(N(2)) =∞, then E(N(n + 1)− N(n)) =∞ for alln ∈ N.

(ii) If E(N(2)) <∞, then E(N(n + 1)− N(n)) <∞ for alln ∈ N.

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Thank you very much for your attention!

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Some sources

AULBACH, S., FALK, M., HOFMANN, M. (2013). On max-stableprocesses and the functional D-norm. Extremes.

CHANDLER, K. N. (1952). The Distribution and Frequency ofRecord Values. J. R. Statist. Soc. B.

DOMBRY, C., FALK, M., ZOTT, M. (2015). On functional recordsand champions. arXiv:1510.04529

DOMBRY, C., RIBATET, M., STOEV, S. (2015). Probabilities ofconcurrent extremes. arXiv:1503.05748

GINÉ, E., HAHN, M., AND VATAN, P. (1990). Max-infinitelydivisible and max-stable sample continous processes. Probab.Theory Related Fields.

GOLDIE, C. M. AND RESNICK, S. (1989). Records in a partiallyordered set. Ann. Probab.

GOLDIE, C. M. AND RESNICK, S. (1995). Many multivariaterecords. Stochastic Process. Appl.

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Some sources

GNEDIN, A. (1993). On Multivariate Extremal Processes. J. Mult.Anal.

GENDIN, A. (1998). Records from a multivariate normal sample.Stat. Probab. Letters

DE HAAN, L., AND FERREIRA, A. (2006). Extreme Value Theory:An Introduction. Springer, New York.

HASHORVA, E. AND HÜSLER, J. (2005). Multiple maxima inmultivariate samples. Stat. Probab. Letters

RÉNYI, A. (1962). Théorie des éléments saillants d’une suited’obervations. Ann. scient. de l’Univ. Clermont-Ferrand

RESNICK, S. (1987). Extreme Values, Regular Variation, andPoint Processes. Springer, New York.

SCHMIDT, R. AND STADTMÜLLER, U. (2006). Non-parametricEstimation of Tail Dependence. Scand. J. Stat.

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Documents

On functional records and champions · On functional records and champions 8th International Conference of the ERCIM WG on Computational and Methodological Statistics (ERCIM 2015),