copulas: an introduction iii - inferencerf2283/conference/3inference (2) seagers.pdf ·...

Copulas: An IntroductionIII - Inference

Johan Segers

Université catholique de Louvain (BE)Institut de statistique, biostatistique et sciences actuarielles

Columbia University, New York City9–11 Oct 2013

Johan Segers (UCL) Copulas. III - Inference Columbia University, Oct 2013 1 / 60


Inference on measures of association

The empirical copula

Inference on parametric copula families

Shape-constrained inference: Extreme-value copulas


Copula models:Separating the margins and the copula

Sklar’s celebrated theorem:

F(x1, . . . , xp) = C(F1(x1), . . . ,Fp(xp)

)Separate assumptions on C and F1, . . . ,Fp:

margins F1, . . . ,Fp

copula C nonparametric parametricnonparametric empirical copula plug-in

shape constraints Archimedean, extreme-value, elliptical, . . .

parametric pseudo-likelihood likelihood


Rank-based inference

If there are no assumptions on the margins (except for continuity), copulamodels are invariant under component-wise increasing transformation.

Same invariance property for the estimators of copula properties?⇒ rank-based inference.

I robust w.r.t. outliersI no need to select models for the margins


Kendall’s tau as a correlation

Let (X1,Y1), (X2,Y2) be iid F, continuous margins. Recall

τ(F) = P[X1 − X2 and Y1 − Y2 have the same sign]

−P[X1 − X2 and Y1 − Y2 have opposite signs]

Ex. Show thatτ(F) = cor

(1(X1 ≤ X2), 1(Y1 ≤ Y2)

)[Hint: P(X1 ≤ X2) = 1/2 and there’s lots of symmetry.]


An extension of Kendall’s tau:Association between two random vectorsLet (X1,Y1), (X2,Y2) be iid (X,Y) in Rp+q.As when p = q = 1, quantify association between X and Y via

τ(X,Y) = cor(1{X1 ≤ X2}, 1{Y1 ≤ Y2})

=pX,Y − pX pY√

pX (1− pX) pY (1− pY)

where

pX,Y = P(X1 ≤ X2, Y1 ≤ Y2),

pX = P(X1 ≤ X2),

pY = P(Y1 ≤ Y2)

I Depends on the law of (X,Y) only through its copula.I Sample version: U-statistic of degree m = 2.


U-statistics: generalizations of sample means

Let PX be the distribution of a random object X taking values in a space X.Suppose we want to estimate a ‘parameter’ θ = θ(PX) of the form

θ = E[g(X1, . . . ,Xm)]

=

∫· · ·∫

g(x1, . . . , xm) dPX(x1) · · · dPX(xm)

where X1, . . . ,Xm are iid X and where g : Xm → R is given.

The U-statistic estimator for θ based on a sample X1, . . . ,Xn is

θm =1

n!/(n− m)!

∑(i1,...,im)∈{1,...,n}m

]{i1,...,im}=m

g(Xi1 , . . . ,Xim)


U-statistics show up everywhere

m = 1 Expectations and sample averages:

θ = E[g(X1)]

θn =1n

n∑i=1

g(Xi)

m = 2 Variance of a real-valued random variable X:

σ2 = var(X) = E[

12

(X1 − X2)2]

σ2n =

1n(n− 1)

∑i 6=j

12

(Xi − Xj)2 =

1n− 1

n∑i=1

(Xi − Xn)2

the (unbiased version of) the sample variance


Hoeffding’s decomposition theorem:Linear expansion of a U-statistic

If E[g2(X1, . . . ,Xm)] <∞, then

√n(θn − θ) =

m√n

n∑i=1

h1(Xi) + op(1)

where

h1(x1) = E[gsym(x1,X2, . . . ,Xm)]− θ

gsym(x1, . . . , xm) =1m!

∑permutations

(i1,...,im)

g(xi1 , . . . , xim)


U-statistics with non-degenerate kernelsare asymptotically normalBy Slutsky’s lemma and the multivariate central limit theorem, Hoeffding’sdecomposition yields joint asymptotic normality of a vector of U-statistics:

√n(θn − θ)

d−→ N(0,m2σ21) (n→∞)

σ21 = var h1(X1)

The asymptotic (co)variance(s) can be estimated consistently byI U-statisticsI jackknifeI the sample (co)variance of

h1,n(Xi), i = 1, . . . , n

h1,n(x) = U-statistic


The estimator of Kendall’s tauis asymptotically normal

Hoeffding’s decomposition yields joint asymptotic normality of

√n

pX,Y,n − pX,YpX,n − pXpY,n − pY

with explicit 3× 3 covariance matrix Σ.

From the delta method, we get asymptotic normality of√

n(τn(X,Y)− τ(X,Y)

)The expression for the asymptotic variance is longish, but explicit,and can be estimated consistently.


European sovereign debt crisis:North and South

I Association between north and south European bond markets changes ascredit-worthiness of countries evolve

I Data: daily returns of Merrill Lynch government bond indicesI North: France, Germany, the NetherlandsI South: Italy, Portugal, SpainI From January 1, 2007 to November 15, 2012I Forward looking moving window of 150 days


With the crisis, association becomes negative

2008 2009 2010 2011 2012

0.0

0.2

0.4

0.6

0.8

1.0

Canonical Correlation

Distance Correlation

RV Coefficient

2008 2009 2010 2011 2012

−0

.20

.20

.61

.0

Kendall Association

Spearman Association


Inference on measures of association:Some literature

El Maache, H. and Y. Lepage (2003). Spearman’s rho and Kendall’s tau formultivariate data sets. In M. Moore, S. Froda, and C. Léger (Eds.), MathematicalStatistics and Applications: Festschrift for Constance van Eeden, pp. 113–130.Beachwood, OH: Institute of Mathematical Statistics.

Grothe, O., J. Schnieders, and J. Segers (2013). Measuring association anddependence between random vectors. Journal of Multivariate Analysis (toappear), arXiv:1107.4381.

Quessy, J.-F., M. Saïd, and A.-C. Favre (2013). Multivariate Kendall’s tau forchange-point detection in copulas. The Canadian Journal of Statistics 41, 65–82.

Schmid, F., R. Schmidt, T. Blumentritt, S. Gaisser, and M. Ruppert (2010).Copula-based measures of multivariate association. In P. Jaworski, F. Durante,W. K. Härdle, and T. Rychlik (Eds.), Copula Theory and Its Applications, LectureNotes in Statistics, pp. 209–236. Berlin: Springer.

Székely, G., M. Rizzo, and N. Bakirov (2007). Measuring and testing dependence bycorrelation of distances. The Annals of Statistics 35(6), 2769–2794.


Why nonparametric inference on a copula?

Use a nonparametric estimate C of C in order to . . .. . . perform goodness-of-fit testing and to assist in model selection

I Compare C with Cθ. . . test for some qualitative property

I Assess (lack of) symmetry by comparing C(u, v) with C(v, u)

. . . have a starting point for estimation of measures of associationI Apply the plug-in principle: e.g. τ(C) := τ(C)

. . .


Empirical distributions: counting points

Points X1, . . . ,Xn. Empirical probability measure

Pn(A) =1n

n∑i=1

1(Xi ∈ A)

Special case: empirical cdf

Fn(x) = Pn((−∞, x]) =1n

n∑i=1

1(Xi ≤ x)

Also multivariate: A = (−∞, x1]× . . .× (−∞, xd]

Ex. Make a plot of the empirical cdf of a univariate sample.

I Where does it jump?I What are the jump sizes?


Ranks and the empirical cdf

Univariate sample X1, . . . ,Xn. Evaluate Fn at the data:

Fn(Xi) =1n

n∑k=1

1(Xk ≤ Xi)

=1n

Ri,n

with Ri,n the rank of Xi among X1, . . . ,Xn

Ex. What is the empirical cdf of the points Fn(X1), . . . , Fn(Xn)?


The empirical copula:Empirical distributions inside and outsideAssume continuous margins F1, . . . ,Fd ⇒ no ties. Recall

C(u) = P[F1(X1) ≤ u1, . . . ,Fd(Xd) ≤ ud]

Sample X1, . . . ,Xn from F = C(F1, . . . ,Fd).

1. Replace population by empirical probability measure:

Cn(u) =1n

n∑i=1

1{F1(Xi1) ≤ u1, . . . ,Fd(Xid) ≤ ud}

2. Replace unknown margins by empirical cdfs: empirical copula

Cn(u) =1n

n∑i=1

1{Fn,1(Xi1) ≤ u1, . . . , Fn,d(Xid) ≤ ud}


Properties of the empirical copula

I Multivariate cdf supported on n points in the grid {1/n, 2/n, . . . , 1}d

I Margins: discrete uniform on {1/n, 2/n, . . . , 1}I Not a copula! But close. . .I Based on ranksI Invariant under component-wise increasing transformations of the data

Ex. Make a picture of the support of the empirical copula of a bivariatesample of size n = 4.


Alternative versions of the empirical copula

The versions below differ from Cn no more than O(1/n):I Avoid boundary problems by dividing by n + 1 rather than by n in the

definition of the marginal empirical cdfsI Obtain a genuine copula by eiter

I smoothing out point masses to obtain uniform (0, 1) margins, i.e. subtractfrom each component an independent Uniform(0, 1/n) random variable:checkerboard copula

I or convoluting Cn with a kernel with standard deviation O(1/n) andtransforming back to Uniform(0, 1) margins

I Simplify asymptotic analysis by using generalized inverses:

u 7→ Fn(F←n,1(u1), . . . , F←n,d(ud)

)


Not knowing the margins makes a difference

If margins were known, we could estimate C(u) by

Cn(u) =1n

n∑i=1

1{F1(Xi1) ≤ u1, . . . ,Fd(Xid) ≤ ud}

By the central limit theorem,

√n(Cn(u)− C(u)

) d−→ N(0,C(u) (1− C(u))

)Is this still true if we replace Cn by Cn, i.e. Fj by Fn,j? — No!

Ex. Compare the variances of Cn and Cn via a simulation study.

I Which of the two has the smaller variance?I Find an intuitive explanation.


The empirical copula process:A key that opens many doors

I View the collection of random variables

Cn(u) =√

n(Cn(u)− C(u)

), u ∈ [0, 1]d

as a stochastic process indexed by [0, 1]d: the empirical copula process.I Knowledge of the limit behavior of Cn helps to find the limit distribution

of any statistic based upon Cn.I Spearman’s rho ρS(C) = 12

∫[0,1]2 C(u, v) d(u, v)− 3, then

√n(ρS(Cn)− ρS(C)

)= 12

∫[0,1]2

Cn(u, v) d(u, v)

I The delta method allows to deal with non-linear functionals in C whichare Hadamard differentiable.


Weak convergence of the empirical copula processfollows from the functional delta method

Consider the copula mapping sending a cdf F to its copula C:

Φ : F 7→ F(F←1 , . . . ,F←d ) = C

Then √n(Cn − C

)=√

n(Φ(Fn)− Φ(F)

)If Φ is (Hadamard-)differentiable at F with derivative ΦF, we find

. . . = ΦF(√

n(Fn − F)︸︷︷︸=αn

)with αn the ordinary empirical process.


The empirical copula process converges weaklyto a Gaussian process with continuous trajectories

In an appropriate function space, jointly in u ∈ [0, 1]d,

√n(Cn(u)− C(u)

) d−→ α(u)−d∑

j=1

α(1, . . . , 1, uj, 1, . . . , 1)∂C(u)

∂uj︸︷︷︸price for not knowing the margins

(limCn)

where {α(u) : u ∈ [0, 1]d} is a collection of zero-mean Gaussian randomvariables with

cov(α(u), α(v)

)= cov

(1(U ≤ u), 1(U ≤ v)

)(covα)

Assumption: the partial derivatives ∂C(u)/∂uj exist and are continuous on thedomain {u ∈ [0, 1]d : 0 < uj < 1}.


The weak limit of the empirical copula process:Understanding its covariance function

Ex. Calculate the covariance on the rhs of (covα).

Ex. Calculate the variance of the random variable on the rhs on (limCn).

Ex. The distribution on the rhs of (limCn) is zero-mean normal with variancestemming from the previous exercise. Compare the asymptoticdistribution of Cn with the finite-sample distribution computed from MonteCarlo simulations.

Ex. Use the central limit theorem to show that for Cn (known margins) ratherthan Cn, the limit in (limCn) would just be α(u).


Empirical copulas: Some literature I

Bücher, A. and S. Volgushev (2013). Empirical and sequential empirical copulaprocesses under serial dependence. Journal of Multivariate Analysis 119, 61–70.

Fermanian, J.-D., D. Radulovic, and M. H. Wegkamp (2004). Weak convergence ofempirical copula processes. Bernoulli 10, 847–860.

Genest, C. and J. Segers (2010). On the covariance of the asymptotic empiricalcopula process. Journal of Multivariate Analysis 101, 1837–1845.

Rémillard, B. and O. Scaillet (2009). Testing for equality between two copulas.Journal of Multivariate Analysis 100(3), 377–386.

Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank orderstatistics. The Annals of Statistics 4(5), 912–923.

Segers, J. (2012). Asymptotics of empirical copula processes under nonrestrictivesmoothness assumptions. Bernoulli 18(3), 764–782.

Stute, W. (1984). The oscillation behavior of empirical processes: The multivariatecase. The Annals of Probability 12(2), 361–379.

Tsukahara, H. (2005). Semiparametric estimation in copula models. The CanadianJournal of Statistics 33(3), 357–375.


Empirical copulas: Some literature II

van der Vaart, A. and J. Wellner (1996). Weak Convergence and Empirical Processes.New York: Springer.

van der Vaart, A. and J. A. Wellner (2007). Empirical processes indexed by estimatedfunctions. In Asymptotics: Particles, Processes and Inverse Problems, Volume 55of IMS Lecture Notes–Monograph Series, pp. 234–252.


Why parametric copula families?

I Avoid curse of dimension: more accurate inference.I Hopefully interpretable parameters in ‘natural’ model.I Allow for covariates.I Can help solving identifiability issues:

I Discrete dataI Censoring


Estimation strategies: there’s choice

Moment-type estimators. For instance, if the map θ 7→ τ(Cθ) isone-to-one, define

θn = θ(τn)

Minimum-distance estimators. For instance w.r.t. an L2 norm,

θn = arg minθ

∫(Cn − Cθ)2

Likelihood based procedures.


Depending on the assumptions on the margins,we obtain a parametric or a semiparametric modelConsider a parametric copula family {cθ : θ ∈ Θ}, with Θ ⊂ Rk.

The final model for f depends on the assumptions on the margins:I If we just assume the margins to be absolutely continuous,

the model is semiparametric: joint pdf

f (x1, . . . , xd) = cθ(F1(x1), . . . ,Fd(xd)

)︸︷︷︸parametric

f1(x1) . . . fd(xd)︸︷︷︸nonparametric

I If we assume parametric models {fj( · ; θj) : ηj ∈ Hj} for the margins,the model is fully parametric: joint pdf

f (x1, . . . , xd; θ, η)

= cθ(F1(x1; η1), . . . ,Fd(xd; ηd)

)︸︷︷︸parametric

f1(x1; η1) . . . fd(xd; ηd)︸︷︷︸nonparametric


The maximum likelihood estimator:maximizing the loglikelihood

X1, . . . ,Xniid∼ f ( · ; θ, η).

Maximum likelihood estimator: Joint optimisation over θ and η

(θn, ηn) = arg maxθ,η

n∑i=1

{log cθ

(F1(Xi1; η1), . . . ,Fd(Xid; ηd)

)+

d∑j=1

log fj(Xij; ηj)

}Asymptotic normality: under regularity conditions,

√n(θn − θ, ηn − θ)

d−→ N(0, I−1(θ, η)

)with I(θ, η) the Fisher information matrix


Avoiding high-dimensional optimization,treat margins and copula separately

1. Perform inference for each margin separately:

ηn,j = arg maxηj

n∑i=1

log fj(xj; ηj), j ∈ {1, . . . , d}

2. Pretend margin parameters are known and estimate θ:

θn = arg maxθ

n∑i=1

log cθ(F1(Xi1; ηn,1), . . . ,Fd(Xid; ηn,d)

)I Easier to compute than full maximum likelihood estimator.I Asymptotically normal too.I A (little) less efficient than the maximum likelihood estimator.


Even simpler computationally: composite likelihoods

Sometimes, even the copula density is hard to computeI High-dimensional extreme-value copulas (spatial extremes)

If the parameter (vector) θ is determined by the pairwise distributions,replace log cθ(u) by a weighted sum of bivariate log densities:

u 7→∑

1≤j1<j2≤d

wj1j2 log cj1j2(uj1 , ujd ; θ)

Resulting estimators are still asymptotically normal,the asymptotic (co)variances depending on the weights wj1j2 ≥ 0

Similar idea for pairwise copula constructions:perform inference pair by pair


Pseudo-likelihood estimator:Estimate margins by empirical cdfsSemiparametric model for a d-variate density f :

I No assumptions on the marginal pdfs f1, . . . , fdI Copula density c belongs to a parametric family {cθ : θ ∈ Θ}

Log-likelihood for θ given iid sample X1, . . . ,Xd ∼ f :

θ 7→n∑

i=1

{log c

(F1(Xi1), . . . ,Fd(Xid); θ

)+

d∑j=1

fj(Xij)

}Maximum pseudo-likelihood estimator:

θn = arg maxθ

n∑i=1

log c(Fn,1(Xi1), . . . , Fn,d(Xid); θ

)with Fn,j the j-th marginal empirical cdf


Properties of the maximum pseudo-likelihoodestimator

+ Based on ranks

+ Asymptotically normal– In general not semiparametrically efficient

I Exception: Gaussian copula models with certain correlation structuresI Efficiency loss, if any, is most of the time rather small

? Semiparametrically efficient procedure?


Borrowing strength from parametric procedures:Sieve estimator

1. For each margins, choose a sequence of nested parametric models thatare dense in the family of all distributions

I E.g. normal mixtures with m components

2. For a finite sample, fix a parametric model for each margin and estimateparameters (margins and copula) by maximum likelihood

3. Asymptotically, let the marginal models change with the sample sizeI E.g. m = mn →∞

+ Asymptotically normal

+ Semiparametrically efficient

– Not rank-based

– Requires potentially influential choice of marginal parametric models

? Semiparametrically efficient rank-based procedure?


Parametric inference: Some literature

Chen, X., Y. Fan, and V. Tsyrennikov (2006). Efficient estimation of semiparametricmultivariate copula models. Journal of the American Statistical Association 101,1228–1240.

Genest, C., K. Ghoudi, and L.-P. Rivest (1995). A semiparametric estimationprocedure of dependence parameters in multivariate families of distributions.Biometrika 82, 543–552.

Hobæk Haff, I. (2013). Parameter estimation for pair-copula constructions.Bernoulli 19, 462–491.

Kojadinovic, I. and J. Yan (2010). Comparison of three semiparametric methods forestimating dependence parameters in copula models. Insurance: Mathematics andEconomics 47, 52–63.

Lawless, J. and Y. Yilmaz (2011). Comparison of semiparametric maximumlikelihood estimation and two-stage semiparametric estimation in copula models.Computational Statistics and Data Analysis 55, 2446–2455.


Shape-constrained inference problemsshow up naturally for copulasCertain copula families are described in terms of lower-dimensional functionssubject to shape constraints:

I Archimedean copulas:

C(u, v) = ψ(ψ−1(u) + ψ−1(v)

)with ψ : [0, 1]→ [0,∞] decreasing, convex, and ψ(0) = 1 andψ(∞) = 0

I Extreme-value copulas:

C(u, v) = (uv)A(t), t =log(v)

log(uv)

with max(t, 1− t) ≤ A(t) ≤ 1 and A is convexI Elliptical copulas, Archimax copulas, . . .

⇒ nonparametric, shape-constrained inferenceJohan Segers (UCL) Copulas. III - Inference Columbia University, Oct 2013 42 / 60

Multivariate extreme-value copulasA copula C is an extreme value copula if

C(u) = exp{−`(− log u1, . . . ,− log ud)}, 0 < uj ≤ 1,

with stable tail dependence function

`(y) =

∫∆d−1

max(y1v1, . . . , ydvd) H(dv),

and spectral measure H satisfying∫∆d−1

vj H(dv) = 1, j ∈ {1, . . . , d}.

Unit simplex:

∆d−1 = {(w1, . . . ,wd) ∈ [0, 1]d : w1 + · · ·+ wd = 1}.


The stable tail dependence function is determinedby the Pickands dependence function

C(u) = exp{−`(− log u1, . . . ,− log ud)}

1. Homogeneity: `(c y1, . . . , c yd) = c `(y1, . . . , yd) for c > 0.

2. Bounds: max(y1, . . . , yd) ≤ `(y1, . . . , yd) ≤ y1 + · · ·+ yd.

It follows that ` is determined by its Pickands dependence function A

`(y1, . . . , yd) = (y1 + · · ·+ yd) A(w1, . . . ,wd−1),

where wj =yj

y1 + · · ·+ yd∈ ∆d−1.


The Pickands dependence function A:An integral transform of a measure

The restriction of ` to the unit simplex ∆d−1 is given by A : ∆d−1 → [1/d, 1]and is known as the Pickands dependence function

A(w) =

∫∆d−1

max(w1 v1, . . . ,wd vd) H(dv)

with spectral measure H as defined before. Necessarily

1. A is convex;

2. max(w1, . . . ,wd) ≤ A(w) ≤ 1;

3. and thus A(ej) = 1, for ej = (0, . . . , 0, 1, 0, . . . , 0).

Except if d = 2, these properties do not characterize the class of Pickandsdependence functions.


Plotting the Pickands dependence function A:Independence copula

Independence: A(w) = 1

C(u) = exp

3∑

j=1

log uj

A (. . .)︸︷︷︸=1

= u1 · · · u3.


Plotting the Pickands dependence function A:Gumbel copula

Gumbel aka logistic

A(w) = (wθ1 + wθ2 + wθ3 )1/θ,

with θ ≥ 1.

C(u) = exp{−((− log u1)θ+

· · ·+ (− log u3)θ)1/θ

}


Plotting the Pickands dependence function A:Fréchet–Hoeffding upper bound

FH upper bound

A(w) = max(w1,w2,w3).

C(u) = exp{−max(− log u1,− log u2,− log u3)}= min(u1, . . . , u3)


The Pickands dependence functionas the rate of an exponential distribution

Suppose for the moment the margins F1, . . . ,Fd are known. Put

Ui = (Ui,1, . . . ,Ui,d)

= (F1(Xi,1), . . . ,Fd(Xi,d))

For w ∈ ∆d−1, define

ξi(w) = min(−

log Ui,1

w1, . . . ,−

log Ui,d

wd

).

The distribution of ξi(w) is exponential with mean 1/A(w):

P[ξi(w) > x] = P[Ui,1 < e−w1 x, . . . ,Ui,d < e−wd x]

= C(e−w1 x, . . . , e−wd x) = e−x A(w).


The exponential representation suggestsnonparametric estimators for A

Let ξi,n(w) be as ξi(w), with Fj replaced by Fj,n: rank-based.I Pickands (1981)

1

AP(w)=

1n

n∑i=1

ξi,n(w).

I Capéraà, Fougères and Genest (1997)

log ACFG(w) = −1n

n∑i=1

log ξi,n(w)− γ

with the Euler–Mascheroni constant γ = 0.5772 . . ..


Connection with the empirical copula

If margins are unknown, estimate them by the empirical distribution functionsand proceed as before. For every w ∈ ∆d−1, we have:

n1/2

(1

APn(w)

− 1A(w)

)=

∫ 1

0Cn(uw1 , . . . , uwp)

duu,

n1/2(log ACFGn (w)− log A(w)) =

∫ 1

0Cn(uw1 , . . . , uwp)

duu log u

.

where

Cn(u1, . . . , ud) =1n

n∑i=1

1(Ui,1 ≤ u1, . . . , Ui,d ≤ ud

)empirical copula

Cn =√

n(

Cn − C), empirical copula process


The estimator satisfies a functional central limittheorem

Under reasonable smoothness conditions on A,

Cnd−→ C in `∞([0, 1]p)

As a consequence, in the space(C(∆d−1), ‖ · ‖∞

)√

n(

APn(w)− A(w)

)d−→ −A2(w)

∫ 1

0C(uw1 , . . . , uwd )

duu

√n(

ACFGn (w)− A(w)

)d−→ A(w)

∫ 1

0C(uw1 , . . . , uwd )

duu log u


How to ensure that the obtained estimatoris a valid Pickands dependence function?

Nonaparametric estimators do not necessarily provide valid estimates for A.I A should be convex;I Bounds: max(w1, . . . ,wd) ≤ A(w) ≤ 1;I If d ≥ 3, the previous conditions do not even characterize the set of

Pickands dependence functions.


Integral representation of Pickands functions

The class of Pickands dependence functions A:The collection of all functions A on ∆d−1 such that

A(w) =

∫∆d−1

max(w1 v1, . . . ,wd vd) H(dv), w ∈ ∆d−1

for some Borel measure H defined on ∆d−1 satisfying∫∆d−1

vj dH(v) = 1, j ∈ {1, . . . , d}.

A is a closed convex subset of the Hilbert space of L2(∆d−1)


Enforce the shape constraints on a pilot estimateby projecting it onto the appropriate set of functions

Initial (nonparametric) estimator An for A.

Projection APr of An on A w.r.t. the norm ‖ · ‖2:

APr = Π(An|A) = arg minA∈A‖An − A‖2.

But A is an infinite-dimensional set⇒ How to implement the projection?

SolutionDense sequence of finite-dimensional subclasses Am ⊂ A

APrm = Π(An|Am) = arg min

A∈Am‖An − A‖2, m ∈ N


Enforce the shape constraints on a pilot estimateby projecting it onto the appropriate set of functions


Finite-dimensional approximations:Discrete spectral measures supported on a grid

Vd,m = {v ∈ ∆d−1 : v = (k1/m, . . . , kd/m), kj ∈ {0, . . . ,m}, k1 + · · ·+ kd = m}

Am(w) ∈ Am if H =∑

v∈Vd,m

H({v}) δv.

Am(w) =∑

v∈Vd,m

H({v}) max{w1v1, . . . ,wpvp}.

v1

v2


Computing the projection on the subclass:Solving a quadratic program

Find h = (hv)v∈Vd,m minimizing the least-squares criterion

arg minh

∫∆d−1

(An(w)− APr

m (w))2 dw1 . . . dwd−1,

whereAPr(w) =

∑v∈Vd,m

hv max(w1v1, . . . ,wdvd), w ∈ ∆d−1,

satisfying the linear constraints

hv ≥ 0, ∀v ∈ Vd,m

APr(ej) =∑

v∈Vd,m

hv max(ej,1v1, . . . , ej,dvd) = 1, ∀j ∈ {1, . . . , d}


Asymptotic distribution of the projected estimator:Project on the tangent cone

If √n(An − A)

d−→ A, in L2(∆d−1),

then, provided m = mn →∞ such that√

n/mn → 0,

√n(APr

m − A)d−→ Π

(A | TA(A)

)in L2(∆d−1), n→∞,

with TA the tangent cone of A at A:

TA = {λ(A− A) : λ ≥ 0, A ∈ A}

Open problem: Limit distribution in(C(∆d−1), ‖ · ‖∞

)?


Shape-constrained inference on copulas:Some literature

Berghaus, B., A. Bücher, and H. Dette (2013). Minimum distance estimators of thePickands dependence function and related tests of multivariate extreme-valuedependence. Journal de la Société Franaise de Statistique 154, 116–137.

Gijbels, I., M. Omelka, and D. Sznajder (2010). Positive quadrant dependence testsfor copulas. Canadian Journal of Statistics 38(4), 555–581.

Gudendorf, G. and J. Segers (2012). Nonparametric estimation of multivariateextreme-value copulas. Journal of Statistical Planning and Inference 142,373–385.

Guillotte, S. and F. Perron (2008). A Bayesian estimator for the dependence functionof a bivariate extreme-value distribution. The Canadian Journal of Statistics 36(3),383–396.

Lambert, P. (2007). Archimedean copula estimation using Bayesian splinessmoothing techniques. Computational Statistics & Data Analysis 51, 6307–6320.


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