A semiparametrie estimation procedure of dependence parameters in multivariate families of distributions

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1 Biometrika (1995), 82, 3, pp Primed in Great Britain A semiparametrie estimation procedure of dependence parameters in multivariate families of distributions BY C. GENEST, K. GHOUDI AND L.-P. RIVEST Departement de mathematiques et de statistique, University Laval, Quibec, Canada G1K 7 P4 SUMMARY This paper investigates the properties of a semiparametrie method for estimating the dependence parameters in a family of multivariate distributions. The proposed estimator, obtained as a solution of a pseudo-likelihood equation, is shown to be consistent, asymptotically normal and fully efficient at independence. A natural estimator of its asymptotic variance is proved to be consistent. Comparisons are made with alternative semiparametrie estimators in the special case of Clayton's model for association in bivariate data. Some key words: Asymptotic theory, Clayton's bivariate family; Kendall's tau; Multivariate rank statistic; Pseudo-likelihood; Semiparametric estimation. 1. INTRODUCTION It has been known since the work of Sklar (1959) that the joint behaviour of a random vector (X u..., X p ) with continuous marginals F((x ; ) = pr (X t ^ x,) can be characterised uniquely by its associated copula or dependence function, C, defined for all («!,..., tip) e [0,1]" by C(u lt..., i^,) = pr {F 1 (X 1 ) < u ls..., F p (X p ) ^ u p }. Many multivariate models for dependence can be generated by parametric families (CJ of copulas, typically indexed by a real- or vector-valued parameter a. Examples of such systems are given by Kimeldorf & Sampson (1975a, b), Genest & MacKay (1986), Marshall & Olkin (1988), Oakes (1989) and Joe (1993), among others. The recent monograph by Hutchinson & Lai (1990), which includes an extensive bibliography, constitutes a handy reference to this expanding literature. Copula-based models are natural in situations where learning about the association between the variables is important, since the effect of the dependence structure, represented by C x, is then easily separated from that of the marginals. For example, copulas might be used to model the strength of intra-familial association in order to provide insight into the role of heredity in chronic disease onset. In such situations, there is typically enough data to obtain nonparametric estimates of the marginal distributions, but insufficient information to afford nonparametric estimation of the structure of the association. This is particularly common in the presence of censoring. In such cases, it is convenient to adopt a parametric form for the dependence function C a while keeping the marginals unspecified. To estimate the dependence parameter a, two strategies could be envisaged, depending on the circumstances. If valid parametric models are already available for the marginals,

2 544 C. GENEST, K. GHOUDI AND L.-P. RIVEST then it is straightforward in principle to write down a likelihood function for the data. The resulting estimate for a would then be margin-dependent, just as the estimates of the parameters involved in the marginal distributions would be indirectly affected by the copula. When nonparametric estimates are contemplated for the marginals, however, inference about the dependence parameter a must be margin-free. Also, as argued by Clayton (1978), Hougaard, Harvald & Holm (1992) and Oakes (1994), this requirement is sensible in applications where the focus of the analysis is on the dependence structure. The purpose of this paper is to investigate the properties of a general semiparametric method of estimation of the dependence parameter a of a family of copulas C a with multivariate density c a. Given a random sample {(X lk,...,x pk ):k = l,...,n} from distribution F a (x t,...,x p ) = C^F^x^),..., F p (x p )}, the procedure consists of selecting the parameter value ot n that maximises the pseudo log-likelihood Ua) = log lc a {F lh (X lk ),..., F pn {X pk m A = i in which F^ stands for n/(n + 1) times the marginal empirical distribution function of the ith variable. This rescaling avoids difficulties arising from the potential unboundedness of log {c a {u u..., Up)} as some of the u/s tend to one. Related techniques have already been used in special cases by Clayton & Cuzick (1985), and by Hougaard (1989). This estimation strategy has also been described in broad, nontechnical terms by Oakes (1994), who dates it back to the work of Gumbel. However, neither of these authors has examined in detail the statistical properties of this procedure. In 2, it is shown that & is a consistent and asymptotically normal estimator. An expression for the limiting variance v 2 of n*<$ n is then given, and the asymptotic efficiency of 6t n at independence is established A consistent estimator of v 2 is then proposed in 3. Though these developments are presented for a real and p = 2, they apply equally to multiparameter, multidimensional settings, as outlined in 4. Finally, 5 reports asymptotic calculations and simulation results comparing the behaviour of $. with two other semiparametric procedures for the bivariate model of Clayton (1978). Mathematical developments are relegated to a series of appendices. 2. ASYMPTOTIC PROPERTIES OF THE PSEUDO-LIKELIHOOD ESTIMATOR In view of its close affinity with maximum likelihood estimation, it is not surprising that the semiparametric estimator <2 n should be consistent and asymptotically normal under similar regularity conditions: see, for instance, Chapter 6 of Lehmann (1983). To establish these facts, let l(a, u u u 2 ) = log {c a (u u u 2 )} and use indices a, 1 and 2 to denote partial derivatives of / with respect to a, u u and u 2 respectively. The starting point is the fact that & solves L(a) = - l a {a, F ln (X lk ), F^X^)} = 0. (1) n da. n t=1 Expanding in a Taylor series, one obtains n da a=l " " " "'

3 where Semiparametric estimation of dependence parameters 545 A n = - t / {«, F lh {X lk ), FtoiX*)}, B B =-- / a>a {o, F lb (X lt ), F^C^)}, and / aa is the second derivative of / with respect to a. It then follows from equation (2) that «*(# 0) = ^AJB n, the limiting behaviour of which can be deduced from that of multivariate rank statistics. The required results are described in Appendix 1. Successively taking J equal to l a and / J(J in Proposition Al implies, under suitable regularity conditions, that D ^ P CFl f~ I? /V \ TP (V \\~\ ET/2 ( I? / V \ J? (V \\~\ Bn^P = ~^Ua,n l a > ^U-*l/> ^2l^2j)J = ^L'a l a > ^l(-*lj> ^2V-*2J/J aknost surely, and that ri*a n is asymptotically normal with zero mean, and variance where a 2 = var [/ a {o, F^X,), F 2 (X 2 )} + W^XJ + W 2 (X 2 )\ W((Xi) = lif^xi) ^ u,}/ a ((a, u 1( u 2 ) c a( u i) u z) du x du 2 and 1(/1) generally stands for the indicator of A. An alternative expression for W^Xi) is given by r (a, u u u 2 )/,(a, B 1( u 2 )c a {u u u 2 ) du x du 2 upon integrating by parts with respect to u, (i = 1, 2). The asymptotic properties of & can then be summarised as follows. PROPOSITION 21. Under suitable regularity conditions, the semiparametric estimator 6t n is consistent and n*((2 B a) is asymptotically normal with variance v 2 = tr 2 //? 2. A more explicit formula for v 2 can be obtained, on noting that the conditional expectation of U(X 1,X 2 ) = l a {a,f 1 (X 1 ),F 2 (X 2 )} with respect to either of the X t 's is null, so that U(X 1,X 2 ) is uncorrelated with each W^(X ( ). Accordingly, var {U(X U X 2 )} = f} and a 2 = p + var {W^XJ + W 2 (X 2 )}, so that The latter inequality expresses the obvious fact that 6t n has a larger asymptotic variance than the maximum likelihood estimator 6t* of a computed under the assumption that the marginals are known. Interestingly, equality occurs in equation (3) when C a approaches the independence copula C o. This is made precise in the following proposition, whose proof is relegated to Appendix 2. PROPOSITION 2-2. Assume that the l J t s vanish as a approaches 0, and suppose that there exist M>0 and 5>0 such that, for all a in some neighbourhood of 0, one has I*«(«, «i, u 2 )l t (a, u u u 2 )c a {u u u 2 ) < Mrfa)- 1-5^-!) 0 " 5 *' (i = 1, 2), where r(u) = u(l u). The estimator a n is then fully efficient at independence. The conditions of the above proposition are satisfied by a large number of singleparameter families of bivariate copulas, including the standard bivariate normal, the

4 546 C. GENEST, K. GHOUDI AND L.-P. RIVEST Farlie-Gumbel-Morgenstern system, and copulas of the Archimedean variety such as those of Ali-Mikhail-Haq, Clayton, and Frank, whose properties are reviewed by Genest &MacKay(1986). 3. A VARIANCE ESTIMATOR Suppose estimators &j and $\ could be found for tr 2 and ft 2 respectively. A rough-andready estimator of the asymptotic variance of $. would then be given by $1 = 8\l^\. If the variables U(X U X 2 ), V{X U X 2 ) = U(X U X 2 ) + W^X,) + W 2 (X 2 ) could be observed, one could simply estimate /? and a 2 by their respective sample variances. As this is not possible, pseudo-observations 0 k and V k can be used instead. The latter are defined in terms of C n, the rescaled empirical copula function of the bivariate sample, namely <: ("!, u 2 ) = - t HFMJ^UMF^X^^U,}. n k = l For k e {1,..., n}, let t) k = / {*, F ln (X lk ), F^X^)} and V* = V* ~ t [ HF, u u u 2 ) dc n (u u u 2 ). Note that, if the X lk 's are arranged in increasing order, one can also write t) k = {&n, k/(n+ 1), S k /(n+ 1)} and in terms of the rank S k of X^ within the set of X 2 s. Next, to establish the consistency of $1, it suffices to show that 8* and /? are themselves consistent. To prove that /? (} converges to zero almost surely, for example, express this difference as " t J^iFin&ik), FmiX*)} -E[J a {Fi(X 1 ),F 2 (X 2 m n k = l in terms of JJft^, u 2 ) = ll(a, u u u 2 ). By the triangle inequality, this quantity is smaller than I t + ^ t JaiFrniXnlF^X^-EUAFdXAFMU Mimicking the classical developments for convergence of maximum likelihood estimators (Lehmann, 1983, Theorem 1.1, p. 406), now assume that dj^i^, u 2 )/da is bounded by an integrable function of u t and u 2 in a neighbourhood of the true value of a. The first

5 Semiparametric estimation of dependence parameters 547 summand then converges to zero by the Dominated Convergence Theorem. Since the second term vanishes asymptotically as a consequence of Proposition Al, it follows that $ ->/? almost surely. As the argument for d^ is similar, though somewhat more involved, it will not be presented here. 4. MULTIPARAMETER, MULTIVARIATE CONTEXTS The previous developments extend more or less automatically to situations where it is desired to estimate semiparametrically a multidimensional dependence parameter a = {a u..., <x q ) e si ^di q from a multidimensional family F a {x u...,x p ) with associated copula C a {u u..., u p ). Letting {(X lk,..., X pk ): k = 1,..., n} represent a random sample from distribution F a, the semiparametric estimator <x n of a would then be obtained as a solution of the system t ~ log lc a {F ln (X lk ),..., F^X^m = 0 (1 ^J *S q). (4) k = i dtx j Under the standard regularity conditions for consistency of multidimensional maximum likelihood estimators, see, for instance, Lehmann (1983, 6.4), and the multivariate versions of the assumptions of Proposition Al, it can be seen that this procedure yields a consistent and asymptotically normal estimator. The limiting variance-covariance matrix of n*# n is then B~ 1 'LB~ 1, where B is the information matrix associated with c a and is the variance-covariance matrix of the q-dimensional random vector whose ;th component is given by with W i} (X t ) = [ 1{F,.(JQ ^ «,} -? log {c a (u u..., u p )} dc a (u u..., u p ). j j i As the parallel with the case q = 1 and p = 2 described earlier is transparent, an estimator of the variance-covariance matrix of & could be found by repeating, mutatis mutandis, the procedure described in COMPARISONS WITH OTHER PROCEDURES IN CLAYTON'S FAMILY This section describes the results of Monte Carlo simulations and asymptotic calculations carried out in order to examine the performance of the semiparametric estimator & for the bivariate family of Clayton (1978) with associated copula for a > 0. Table 1 reports the sample variance of &, along with those of two competing semiparametric estimators, based on 2000 pseudo-random samples of size 100 from distribution (5) with various parameter values. In the table, a n stands for the estimator based on Kendall's tau (Oakes, 1982), while a* represents the estimator constructed by Clayton (1978) and investigated by Oakes (1986). To assess the loss in efficiency associated with absence of knowledge of the marginals, results for the maximum likelihood estimator 6t*

6 548 C. GENEST, K. GHOUDI AND L.-P. RIVEST Table 1. Sample variance (x 100) of four estimators of the association parameter a. in Clayton's family ofbivariate distributions (5), tabulated as a function ofy = log (a + 1) L-l 1-2 L-3 L computed with given margins were also included. As figures concerning the bias for the four estimation procedures revealed no major difference or trend, they are not reported here. Beyond confirming the obvious dominance of the infeasible solution represented by the maximum likelihood estimator and the poor performance of the Kendall-based estimator, the results in Table 1 indicate that the semiparametric procedure & is comparable to Clayton's for small parameter values, and increasingly superior to it as a gets larger. Although the gain in efficiency may seem modest, approximately 15% for large values of y = log (a + 1), the improvement is noteworthy, given that $. is an all-purpose estimator and that Clayton's procedure is comparable to the one-step approximation to the efficient semiparametric estimator recently developed by Maguluri (1993) for that family. From a theoretical point of view, it follows from Proposition 2-2 and the work of Oakes that the estimators & and &.* are both fully efficient at independence. Furthermore, it is shown in Appendix 3 that, for large values of a, the asymptotic variance of n*& n approaches (n )oc 2 /(n 2 + 3), which is smaller than the corresponding value for n*5*, reported as (5 7t 2 /3)a 2 by Oakes (1986). As a consequence, the difference between the asymptotic variances of these two estimators is roughly equal to 00111a 2 for large a. Although the gain may appear modest, this is the best one could hope for, since the limiting asymptotic variance of n*a\, is precisely that computed by Oakes (1982) for the maximum likelihood estimator of a under the assumption that both marginals are exponential with unknown scale parameters. Consequently, no rank estimator could have a lower asymptotic variance that & at infinity within Clayton's model. In other words, a n is an efficient semiparametric estimator for that family at infinity. Through simulations, it is possible to examine the performance of the variance estimator tf 2 proposed in 3. The percentage relative bias of this estimator is reported in Table 2 for sample sizes n = 50 and n = 100, along with the empirical coverage of the approximate 95% confidence interval for a, computed as 6t H ± l-96v B based on the asymptotic normality of $L n established in Proposition 21. These results are typical of observations made for other a values and other confidence levels. From the results of Table 2, it would appear that the variance estimator for & is reliable for random samples exhibiting small to medium dependence, but that the coverage probability deteriorates as the variables become more correlated. When y = 2, which corresponds to a Kendall's tau of 0-76, the percentage of coverage is around 90% only. In such

7 Semiparametric estimation of dependence parameters 549 Table 2. Estimated percentage relative bias of the variance estimator vj and empirical coverage of the approximate 95% confidence interval for a, ($ B ±l-96v B, based on the asymptotic normality y n = Bias 50 Coverage y n = Bias 100 Coverage circumstances, it may be advisable to use the variance stabilising transformation y = log (a + 1) introduced by Oakes. When y = 2, for example, the probability of coverage of the interval y n ± l-96tf n /exp (y n ) obtained via the delta method was found to be 91-4% when n = 50 and 92-1% for n = 100. An alternative approach for improving the coverage of the confidence intervals would be to adapt the likelihood-based methods of constructing confidence intervals to the pseudo-likelihood context. This could be the subject of future investigations. ACKNOWLEDGEMENT Research funds in partial support of this work were granted by the Natural Sciences and Engineering Research Council of Canada and by the Fonds pour la formation de chercheurs et l'aide a la recherche du Gouvernement du Quebec. APPENDIX 1 Asymptotics of multivariate rank statistics Although the results derived in this appendix are valid in multivariate contexts, they are presented for the two-dimensional case only. Consider a bivariate distribution function F(x i,x 2 ) with associated copula C(ii 1; u 2 ) and marginals F 1 (x 1 ) and F 2 (x 2 ), so that F(x 1,x 2 ) = C{F 1 (x 1 ),F 2 (x 2 )}. Let Fi,,(xi), F 2x (x 2 ) and F H (x 1,x 2 ) represent their rescaled empirical counterparts, based on a random sample (X n, X 21 ),..., (X ln, X^) from distribution F. Also let C n (u u u 2 ) denote the rescaled empirical copula function, as described in 3. Of interest here is the asymptotic behaviour of statistics of the form J{F lm {X lk ),F^X*)} = [ J(u lt u 2 ) J where J(u t,u 2 ) is a continuous function from (0, I) 2 into St. Such statistics are known in the Uterature as multivariate rank order statistics. Their asymptotic properties have been thoroughly investigated by Ruymgaart, Shorack & van Zwet (1972), Ruymgaart (1974) and Ruschendorf (1976). These authors proposed regularity conditions ensuring asymptotic normality. Their techniques can also be used to establish almost sure convergence. PROPOSITION Al. Let r{u) = u(l - u), 5 > 0, p and q positive numbers satisfying l/p + l/q = 1,

8 550 C. GENEST, K. GHOUDI AND L.-P. RIVEST and J(u t, u 2 ) a continuous function from (0, I) 2 into 31 such that 1= [ uu 2 )dc(u 1,u 2 ) exists. (i) IfJ(u u u 2 ) < Mr(u i yr(u 2 ) b with a - (-1 + <5)/p and b = (-1 + <5)/g, f/zen i? B -»/x almost surely. (ii) //JO^, u 2 ) < MriUifrfaf with a = ( )/p and b = ( <5)/g, and if J admits continuous partial derivatives Ji(u u u 2 ) = djiu^ u 2 )/dui on (0, I) 2 suc/i that J^Uj, u 2 ) ^ Mr(u 1 ) li 'r(u 2 ) d3-1 with d 1 =a I and d 2 = b, then n*(i? B (i)-*n(0, a 2 ) in distribution, where a 2 = var [./{F^), F 2 (X 2 )} + j \{X t < xj/jf^xj, F 2 (x 2 )} df(x u x 2 )J. Proof. By the Glivenko-Cantelli Lemma, C B (u!, u 2 ) converges almost surely to C(u, v). Now R H can be rewritten as E(V H ), where F n = J(U ln, U^) and (C/ lb, U^) are distributed according to C H (u 1, u 2 ). Observe that (U ln, U^) converges in distribution to (l^, U 2 ), a pair of random variables distributed according to C{u t, u 2 ). Since J is continuous, V n itself converges to V = J(U i, U 2 ). Thus to estabh'sh (i), it suffices to show that the V H are uniformly integrable. To this end, it will be shown that there exists e > 0 such that E{\ V n \ 1+e ) is bounded. Using the hypothesis together with Holder's inequality, one can derive the following chain of inequalities: E(\ FJ 1+ «) = \J(u u u 2 )\ 1+ < dck, u 2 ) *dc n {u u u 2 ) (-l+i)(l+«n 1/p fi. / U \(-l+s)(l+b))l/q } O(^) ) M - 1+ ')< 1+ «) The last integral is finite for e < <5. This completes the proof of (i). The proof of (ii) mimics the argument used by Ruymgaart et al. (1972) to prove their Theorem 2.1. The formula for a 2 corresponds to their equation (3.10). APPENDIX 2 Proof of Proposition 2-2 First observe that the variance of Wj(X,) is the same as that of Jo Jl/2 By the Dominated Convergence Theorem, Wf{x)-*0 pointwise as a-»0 when F,(x)e(0,1). Furthermore, it follows from the hypotheses of the proposition that Wf(x) is bounded by a square integrable function. Invoking the Dominated Convergence Theorem once more, El{Wt{X { )} 2 ~\ ->() as a->0. Consequently, var {WftJT,)} = var {W^(X t )} ->0.

9 Semiparametric estimation of dependence parameters 551 APPENDIX 3 Calculation of the asymptotic variance of& B This appendix describes the key steps in calculating the limiting value, as <x = 1/T/-KX), of V 2 given by (3) for Clayton's family. Oakes (1982, 4) showed that (A31 > The present discussion is restricted to the evaluation of var {W^XJ + W 2 (X 2 )} as rj-+o. Without loss of generality, assume that the marginals of the X,'s are uniform on the unit interval. The random variables W^Xy) and W 2 (X 2 ) are then exchangeable, so that it suffices to illustrate calculations on one of them, say W^X,). For that purpose, introduce Direct substitution in the definition of W^X,) yields log(l+z) In the hmit as r\ -* 0, X t = X 2 ahnost surely, so that var {WAX,) + W 2 (X 2 )} = 4 var To handle W(X X ), split its domain of integration into three parts, from 0 to 1, from 1 to X, and from Xf lrt to 00. Upon replacing the integrand in each of these terms by its corresponding two-term Taylor series expansion in t] around 0, integration can then be performed term by term. Neglecting constant terms, the first integral is O p {rf), while the second reduces to Finally, the nonconstant contribution of the third integral is Thus up to an additive constant independent of X,. A straightforward calculation then yields J lh, (A3-2) Inserting (A3-1) and (A3-2) into (3) yields VV-CTT )1(1? + 3) as REFERENCES CLAYTON, D. G. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65, CLAYTON, D. G. & CUZICK, J. (1985). Multivariate generalizations of the proportional hazards model (with discussion). J. R. Statist. Soc. A 148, GENEST, C. & MACKAY, R. J. (1986). Copules archimediennes et families de lois bidimensionnelles dont les marges sont donntes. Can. J. Statist. 14,

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