SIGN TEST OF INDEPENDENCE BETWEEN TWO RANDOM VECTORS

A new a(cid:14)ne invariant extension of the quadrant test statistic (Blomqvist, 1950) based on spatial signs is proposed for testing the hypothesis of independence. In the elliptic case, the new test statistic is asymptotically equivalent to the interdirection test by Gieser and Randles (1997), but is easier to compute in practice. Limiting Pit-man e(cid:14)ciencies and simulations are used to compare the test to the classical Wilks’ test.


INTRODUCTION
In the paper, we consider the interrelations between two sets of variables, the p-vector x (1) and q-vector x (2) .Let (2) 2 , . . ., x (1) n x (2) n , be a random sample from an unknown k variate distribution (k = p + q).The null hypothesis to be tested is H 0 : x (1) and x (2) are independent.
If the observations come from a k-variate normal distribution with partitioned covariance matrix the hypothesis of independence may be formulated as Σ 12 = 0.If S = S 11 S 12 S 21 S 22 .
is the partitioned sample covariance matrix, then the likelihood ratio test statistic (Wilks, 1935) is given by An asymptotically equivalent test statistic is the Pillai's (1955) test statistic T n = T r(S −1 11 S 12 S −1 22 S 21 ).
Note that T n is affine invariant, that is, invariant under the group of transformations Ax (1) 1

Bx
(2) n , where A is nonsingular p × p matrix and B nonsingular q × q matrix.If p = q = 1, the regular squared correlation coefficient is obtained, and T n may be seen as an extension of the multiple correlation coefficient.See Mardia et al. (1997, Section 6.5.4), for a discussion on that.Note that statistic T n can be used to test the hypothesis of independence for nonnormal cases also as independence implies that Σ 12 = 0 (if it exists).Under H 0 (and some general assumptions), nT n → d χ 2 pq .For p = q = 1, nonparametric approaches to the independence problems include classical quadrant test, Spearman's rho and Kendall's tau, see for example Lehmann (1998, Section 7) and references therein.Puri and Sen (1971) considered extensions where the test statistics, analogous to T n , are obtained replacing the regular covariance matrix by the covariance matrix of marginal (centered) signs or ranks (or scored ranks).Unfortunately, these extensions of quadrant test and Spearman's rho miss the desirable affine invariance property.Gieser and Randles (1997) introduced an invariant extension of the quadrant test.In their construction, they first centered separately the x (1) -vectors and the x (2) -vectors (using affine equivariant Oja (1983) median).Next they calculated the interdirection proportions a ij and b ij between pairs of centered x (1) -vectors and centered x (2) -vectors, respectively.Finally, the test statistic is where the average is taken over all possible i, j = 1, . . ., n. Again, under H 0 and under some general assumptions (both x (1) and x (2) elliptically symmetric), nQ n → d χ 2 pq .Gieser and Randles also found asymptotic relative efficiencies of the Q n test as compared to the classical Wilks' test (or the asymptotically equivalent T n test).However, the computation of the interdirection proportions is time consuming, and may cause problems for large sample sizes and in high dimensions.
In the paper, a new invariant test based on multivariate standardized spatial signs is proposed.The standardized spatial signs described in Section 2 are based on the approach launched in Randles (2000) and Hettmansperger and Randles (2002).The test statistic, based on the covariance matrix between the standardized spatial signs of the x (1) -and x (2) -vectors and again analogous to T n , is introduced in Section 3. Also asymptotic theory is developed to provide approximations for the limiting distributions.In Section 4 the limiting Pitman efficiencies are given in the multivariate normal distribution, t distribution and contaminated normal distribution cases and simulations are used to compare finite sample powers.In Section 5 the theory is illustrated by an example and the paper is closed with some comments in Section 6.The proofs are postponed to the Appendix.

STANDARDIZED SPATIAL SIGNS
To construct invariant test procedures for the independence problem, one has to estimate (explicitly or implicitly) the location centres and covariance structures of the marginal multivariate distributions of the x (1) and x (2) .Assume that the distribution of the x is p-variate and elliptic with symmetry centre µ and covariance matrix Σ (if they exist).Matrix V = (k/T r(Σ))Σ is then the related shape matrix.Hettmansperger and Randles (2002) introduced intuitively appealing location vector and shape matrix estimates based on the spatial sign concept.
The spatial sign of vector x is defined as where x = (x T x) 1/2 is the (Euclidean) length of the vector x.Note that S(x) is a unit vector in the direction of x and therefore a natural multivariate extension of the sign concept.Spatial signs S(x i ), i = 1, . . ., n, are not affine equivariant, however.
In Hettmansperger and Randles (2002) estimates µ and V are obtained as follows.( V −1/2 is here taken to be symmetric.)Definition 1.The location vector estimate µ and the shape matrix estimate V are choices such that the spatial signs of standardized observations The vectors S i = S(z i ), i = 1, . . ., n are called the standardized signs.
The resulting shape matrix estimate V is the so called Tyler's M-estimate (1987) and the location estimate vector µ is a transformation-retransformation spatial median utilizing Tyler's transformation matrix.The estimates are naturally affine equivariant.For the transformation retransformation technique, see also Chakraborty et al. (1998).For the properties of these and related location vector and shape matrix estimates, see also Ollila, Hettmansperger and Oja (2002).
We will use the standardized signs in the test constructions and then need the following equivariance property.
Lemma 1.The standardized sign vectors S i are affine equivariant in the sense that if the S * i are calculated from

NEW TEST STATISTIC AND ITS LIMI-TING DISTRIBUTION
We are now ready to introduce our test statistic.As earlier, let (2) n be a random sample from an unknown k-variate distribution (k = p + q) and the null hypothesis to be tested is H 0 : x (1) and x (2) are independent.First, we construct standardized spatial sign vectors separately for the x (1) -and the x (2) -vectors and obtain

S
(2) 2 , . . ., S n The sign vectors S (1) i and S (2) i thus satisfy i } = 0 and ave i {S (1) i S (1 i } = 0 and ave i {S (2) i S (2 Next write i S (2) T i for the covariance matrix between the marginal standardized signs.The new test statistic is then given by the next Definition 2. The standardized spatial sign test statistic for testing the hypothesis of independence H 0 is given by where ||H|| 2 = T r(H T H) is the squared matrix norm (Frobenius norm).
As the standardized signs are affine equivariant, the invariance of the test statistic easilly follows.
Lemma 2. U n is affine invariant.
The null distribution of the test statistic is given by Theorem 1.Under H 0 and for elliptic x (1) and x (2) , the limiting distribution of nU n is a chi-squared distribution with pq degrees of freedom.
Next we derive the limiting distribution of U n under alternative sequences.As U n is affine invariant, we restrict to the spherical case only.Let thus x (1) i and x (2) i be independent with spherical marginal distributions and write where ∆ = δ/ √ n.The sequence of alternatives is similar to that used in Gieser and Randles (1997).If U * n is calculated from these transformed observations in (1), we get Theorem 2. For max(p, q) > 1, the limiting distribution of nU * n is a noncentral chi-squared distribution with pq degrees of freedom and noncentrality parameter where i || −1 ) and c 2 = (q − 1)E(||x For the case p = q = 1, see Taskinen et al. (2003).

LIMITING AND FINITE-SAMPLE EFFI-CIENCIES
We next compare U n to Wilks' likelihood ratio test W n and the test based on interdirections Q n through limiting efficiencies and simple simulation studies.As all the limiting distributions are of the same type, χ 2 pq , the efficiency comparisons (Pitman efficiencies) may be based on noncentrality parameters only.First note that, for elliptic x (1) and x (2) , U n is asymptotically equivalent to the test based on interdirections.See (A.1) in Gieser and Randles (1997).
For the asymptotic efficiencies in the exponential power family, see then Gieser and Randles (1997).Gieser (1993) considered also the t distribution case in his unpublished dissertation thesis.The efficiency comparison is now made in the multivariate normal distribution, t distribution and contaminated normal distribution cases.For simplicity, we assume that M 1 = M T 2 .Since W n has a limiting noncentral chi-squared distribution with pq degrees of freedom and noncentrality parameter δ 2 ||M 1 +M T 2 || 2 , the asymptotic efficiencies are simply with c 1 and c 2 given in Theorem 2. For a k-dimensional standardized t distribution with ν degrees of freedom and the resulting limiting efficiencies for selected degrees of freedom and selected dimensions are listed in Table 1.Note that since limiting multinormality of the regular covariance matrix holds if the forth moments of underlying distribution are finite, W n has a limiting distribution only when ν ≥ 5.In multinormal cases the asymptotic efficiency of U n (and low but gets higher as the dimensions p and q increase.For heavy-tailed distributions the efficiencies are low in the case of small dimensions, but as the dimensions increase, new test outperforms the W n test. We also consider the contaminated normal distribution and assume that p = q and M 1 = M T 2 .Then with probability (1 − ) the x (1) -vectors (and x (2) -vectors) are drawn from a standard normal distribution N p (0, I p ), and with probability from N p (0, c 2 I p ).The cdf is thus F (x) = (1 − ) Φ p (x) + Φ p (x/c), where c > 0 and Φ p is the cdf of N p (0, I p ).Now in ( 2) 1 as a function of c, for p = 5 and for = 0.05, 0.10, 0.20.For heavy-tailed cases (large and/or large c), the new test is much more efficient than the Wilks' likelihood ratio test.
Finally, a simulation study was used to compare the finite sample powers of three test statistics.The critical values used in test constructions were based on the chi-square approximations to the null distributions.The empirical powers were calculated for p = q = 3 (Figure 2) and for p = q = 5  (Figure 3).1500 independent x (1) -and x (2) -samples of sizes n = 50 and 150 were generated from a multivariate standard normal distribution, from a standardized t distribution with ν 1 = ν 2 = 5 and from a contaminated normal distribution with = 0.2 and c = 6.Finally, the transformation in (1) with M 1 = M T 2 = I r , where r = p = q, was applied for chosen values of δ to introduce dependence into the model.
In computations, test statistic W n was multiplied by Bartlett's correction factor 1 − (2r + 3)/2n, and for Q n the data vectors were centered using marginal transformation-retransformation spatial medians (Hettmansperger and Randles, 2002).In most cases, with the exception of n = 50 and r = 3, the interdirection proportions, needed for Q n , were estimated using a sample of 1000 hyperplanes (instead of using all possible n−2 p−1 hyperplanes).Consider first the simulation results in the case r = 3 (Figure 2).In the multinormal case W n is clearly best, as one can expect.In the considered t distribution case W n is slightly better than the other tests, but U n and Q n perform very well in the contaminated normal case.For small samples, Q n seems to be a bit more powerful than U n .As r = 5 (Figure 3), the empirical powers of U n and Q n increase as compared to the W n .As n = 50, Q n does well also in the normal case, but again, as n increases, no significant differences can be seen between U n and Q n .The size of the U n test is reasonably close to the aimed size 0.05 in all cases whereas the Q n and especially the W n test levels often seem to exceed 0.05.

A REAL DATA EXAMPLE
The theory is now illustrated by a simple example.Consider the diabetes data (see Rencher, 1998, p. 17 and references therein), that contains 76 5-variate observations, namely glucose intolerance, insulin response to oral glucose, insulin resistance, relative weight and fasting plasma glucose.We wish to test the hypothesis that the first three variables are independent of the remaining two variables.
First, p-values given by W n , U n and Q n were calculated for the original data set using the chi-square approximation of the null distributions.Then, to illustrate that the test statistics U n and Q n are robust against outliers, the p-values were recalculated for a data set, where the first relative weight measurement 0.81 was replaced by 8.1 (simulating a printing error).The obtained p-values are listed in Table 2.Note that in the original case all statistics produce very small p-values, but the contamination changes the p-value given by W n drastically; only small changes can be seen in the other two cases.

FINAL REMARKS
In the paper we proposed a new robust and affine invariant extension of the quadrant test statistic to test the hypothesis that two random vectors are independent.If the marginal distributions of x (1) and x (2) are elliptic, the new test is asymptotically equivalent with the interdirection quadrant statistic (Gieser and Randles, 1997) but is much easier to compute.The new test is not very efficient in the multinormal case but performs well for heavy-tailed distributions.This is illustrated with limiting and finite sample efficiency studies.
A related testing problem is that of testing the null hypothesis H 0 : Σ 12 = 0, which says that all the canonical correlations between the x (1) -and x (2)variables are zero.For testing H 0 in the elliptic case, any shape matrix V for (x (1) T , x (2) T ) T may be used in the test construction.If sample statistic V is partitioned as then a test statistic, analogous to Pillai's test statistic, may be defined as Again, V n is affine invariant, and nV n has a central (noncentral) chi-squared distribution with pq degrees of freedom under the null hypothesis (alternative sequences of type ( 1)).In the multinormal case, H 0 : Σ 12 = 0 is of course a hypothesis of independence also, and the tests based on shape matrices may be used for testing independency.If V is the Tyler's M-estimate (Tyler, 1987;Hettmansperger and Randles, 2002) and M 1 = M T 2 , the asymptotic efficiency of V n relative to U n is simply where c 1 and c 2 are given in Theorem 2. The V n test then outperforms the U n test in the multinormal case; see asymptotic relative efficiencies listed in Table 3. Appendix: Proofs of the results

Proof of Lemma 1 Using
it is straightforward to show as in Randles (2000) that Proof of Lemma 2 The proof follows from Lemma 1 and the fact that T r(P 2 H T P T 1 P 1 HP T 2 ) = T r(H T H) for any orthogonal matrices P 1 and P 2 .

Proof of Theorem 1
In the proof, we use the general delta method.As the test is affine invariant, it is enough to consider the spherical case with i S (2 Write the standardized location and shape estimates for x (1) as and it is easy to see, considering termwise sums of S (1) i S (2 i S (2 The next step is to note that as H = (h ij ), Finally (3) implies that the limiting distribution of √ nvec(H) is N pq (0, 1 pq I pq ), where vec(H) is obtained by stacking the columns of H on top of each other, and consequently nU n d − → χ 2 pq .

Figure 1 :
Figure 1: ARE(U n , W n ) as a function of c at the contaminated normal model for p = q = 5 and for = 0.05, 0.10, 0.20.
i are the standardized spatial sign vectors of y(1)

Table 2 :
p-values for the original and contaminated data sets.

Table 3 :
ARE(V n , U n ) at different p-and q-variate normal distributions.