Abstract
The established Hoeffding-Blum-Kiefer-Rosenblatt independence test statistic is investigated for partly not identically distributed data. Surprisingly, it turns out that the statistic has the well-known distribution-free limiting null distribution of the classical criterion under standard regularity conditions. An application is testing goodness-of-fit for the regression function in a non parametric random effects meta-regression model, where the consistency is obtained as well. Simulations investigate size and power of the approach for small and moderate sample sizes. A real data example based on clinical trials illustrates how the test can be used in applications.
Acknowledgments
The author wishes to thank the two referees for very helpful comments and suggestions.
Appendix A: proofs
Proof of Proposition 3.1.
First, let us consider the stochastic process defined by
where
The crucial point is that under the null hypothesis of independence
and so
in the null hypothesis case, where
Under the null hypothesis it follows that the stochastic process Un can be rewritten as
where the stochastic process
is given by
and the stochastic process
is defined by
The process Vn is an empirical process based on not necessarily identically distributed random vectors. Limiting results are available by Ziegler (Citation1997). In our situation, the conditions in 4.2 in Ziegler (Citation1997) can be easily verified and the convergence in distribution
follows, where
is a centered Gaussian process with a.s. uniformly d2-continuous sample paths and covariance function
and
denotes a random vector with distribution function
Note that in general does not hold, and the same applies to
For that reason, in contrast to the classical situation of identically distributed data, the process V has in general not the structure of a bivariate Brownian bridge.
It is easily seen that converges uniformly to FY. Applying Slutsky’s theorem, that is Example 1.4.7 in van der Vaart and Wellner (Citation1996), yields
and from the continuous mapping theorem, that is Theorem 1.3.6 in van der Vaart and Wellner (Citation1996), it follows that
Moreover, it follows from Lemma 1.10.2. (iii) in van der Vaart and Wellner (Citation1996) that
in the uniformly sense. The latter combined with Slutsky’s theorem yields
and it follows from the continuous mapping theorem that
where the convergence holds uniformly. From a further application of Slutsky’s theorem, we deduce
and finally
from the continuous mapping theorem again. It is clear that
is a centered Gaussian process with a.s. uniformly d2-continuous sample paths. The covariance function u turns out by simple calculation.
Note that, form the latter one it follows that the stochastic process U has the structure of the well-known Brownian pillow analogous to the classical situation of identically distributed data. Regarding that, in contrast to the classical situation of identically distributed data, V has in general not the structure of a multivariate Brownian bridge, this finding surprises. It is a special feature of the process U. □
Proof of Lemma 3.1.
The function is an empirical distribution function based on not necessarily identically distributed random vectors. Limiting results are available by Gänßler and Ziegler (Citation1994). Corollary 4.1 (i) in Gänßler and Ziegler (Citation1994) implies
where the convergence holds uniformly. It is easily seen that
converges uniformly to
In all, we obtain
This implies the statement. □
Proof of Theorem 3.1.
Because the null hypothesis is true, the convergence in distribution of
to
follows immediately from Proposition 3.1 and Lemma 3.1 combined with the continuous mapping theorem. At first, we consider the special case that k = 1 is fixed. In this situation,
based on independent and identically distributed pairs of bivariate random vectors with underlying distribution function
where the only restriction on
is that the related marginal distributions are continuous. Under the stated null hypothesis of independence,
has the distribution-free distribution of the classical Hoeffding-Blum-Kiefer-Rosenblatt test statistic based on independent and identically distributed bivariate random vectors with continuous marginal distributions. On the one hand, it follows from the results in Blum, Kiefer, and Rosenblatt (Citation1961) that
converges in distribution to a real-valued random variable
say, where
is distribution-free, has a continuous and strictly increasing distribution function, and the characteristic function
On the other hand, it is already shown that converges in distribution to
In all,
and
have the same distribution. Because
is an arbitrary uniformly continuous distribution function with continuous marginal distributions, the latter finding is also valid for general k. □
Proof of Lemma 6.1.
In the null hypothesis case, it is f = g. This implies Because X, η, and ε are independent, it follows that X and Y are independent. Now, we consider alternatives. Let
be true. Well, suppose that X and Y are independent for a moment. It is well-known that the distribution of a random vector W with values in
is uniquely determined by its characteristic function
The independence assumption implies that for all
Because X, η, and ε are independent, the latter is equivalent to
for all
We obtain the independence of X and
Because I is the support of the distribution of X, it follows that the map
is constant. From
we deduce f = g, that contradicts the assumption
and completes the proof. □
Proof of Theorem 6.1.
We define the stochastic process by
and the map
It follows from Lemma 3.1 that
in probability. The latter combined with Lemma 3.1 again yields
in the uniformly sense. Regarding the expressions
the continuous mapping theorem implies the convergence of
to Δ in probability. It remains to show that
Denote by ε a real-valued random variable with distribution function
such that X, η, and ε are independent. Setting
we have
as well as
and under the alternative it follows from Lemma 6.1 that X and Y are not independent. In addition, it is easy to see that
is absolutely continuous. In all,
follows from Yanagimoto (Citation1970). □