Introduction

Consider X , Y to be two random scalars defined on the same probability space. In various scenarios one is interested in quantifying the strength of association between X and Y as well as determining the validity of the null hypothesis H 0 : X and Y are independent,

(1) both based on a finite sample of size n. These two tasks are usually convoluted, and together play a pivotal role in many statistics and scientific practices (Pearl, 2009;Josse & Holmes, 2016;Maathuis et al., 2019). For handling them, this paper is focused on nonparametric rank correlations that measure associations between X and Y using only the ranks of the data. Rank correlations are particularly appealing for continuous X , Y as then the corresponding tests under H 0 are fully distribution-free. Early proposals include Spearman's ρ (Spearman, 1904), Gini's γ (Gini, 1914), Kendall's τ (Kendall, 1938) and Blomqvist's β (Blomqvist, 1950). However, none of these can be used to consistently test for independence. For the sake of testing consistency, Hoeffding (1948), Blum et al. (1961), Yanagimoto (1970) and Bergsma & Dassios (2014) proposed variants that, not only lead to consistent tests of independence, but are also shown to be rate efficient against many local alternatives; cf. Dhar et al. (2016) and Shi et al. (2022b,c).

In a recent manuscript that has received much attention, Chatterjee (2021) introduced a new rank correlation coefficient that estimates a correlation measure first proposed by Dette et al. (2013). Compared to the existing ones, this new pair of correlation measure and coefficient appears to have some appealing properties, including, in particular, that (i) the measure is between 0 and 1, is 0 if and only if X and Y are independent, and is 1 if and only if Y is a measurable function of X almost surely; (ii) the correlation coefficient has a very simple expression, and is an almost surely consistent estimator of the measure as long as Y is not almost surely a constant.

Because of the above attractive properties, Chatterjee's rank correlation is an appealing choice for measuring bivariate association strength, especially in detecting perfect functional dependence (Cao & Bickel, 2020). On the other hand, Chatterjee's proposal can suffer from a lack of power when testing H 0 in (1); cf. the claim made in Chatterjee (2021, § 7), the analysis conducted in Cao & Bickel (2020) and Shi et al. (2022b), as well as the results in a very recent study (Shi et al., 2022a). As a matter of fact, the critical detection boundary of the test based on Chatterjee's rank correlation was calculated to be at n -1/4 , which is much slower than the usual parametric n -1/2 rate (Auddy et al., 2021). These analyses thus motivate revising Chatterjee's original proposal for, not only detecting perfect functional dependence, but also attaining near-parametric efficiency in testing independence.

In this paper, we make a revision of Chatterjee's rank correlation by incorporating a diverging M = M n many right nearest neighbours, i.e., nearest neighbours that are greater than the target point, into the construction of the correlation coefficient. We show that the resulting estimator is still an almost surely consistent estimator of the measure of dependence of Dette et al. (2013) as long as M /n → 0. One could then regard the revised statistic as an extension of Chatterjee's original one from 1-near-neighbour-based to M -nearest-neighbour-based. Similar ideas were already pursued in Deb et al. (2020, Equations (3.3) and (8.5)); see also Berrett & Samworth (2019) for a related proposal that approximates the mutual information (Berrett et al., 2019). However, our approach to incorporate more nearest neighbours is distinct from theirs; cf. Remark 4 in § 2. In addition, for guaranteeing a normal limiting null distribution, Deb et al. (2020, Theorem 4.1) required a very small M of order Poly-log n. In contrast, the most interesting part of our study is when M scales at nearly the same order as n.

One main ingredient of this paper pertains to a local power analysis of the proposed revised Chatterjee's rank correlation coefficients. In this regard, our attention is restricted to the Gaussian rotation model that is a benchmark in independence testing and has been widely adopted in literature; cf. Konijn (1956, p. 301), Han et al. (2017, § 4), Drton et al. (2020, § 4.2), Shi et al. (2022c, § 5) and Shi et al. (2022b, § 3). Letting the Pearson correlation between X and Y in a local alternative sequence be ρ n , we show that the test based on the revised Chatterjee's rank correlation has power tending to one as long as

here '∨'and '∧'respectively represent the maximum and minimum of the two numbers appearing either side of the notation. In particular, as M /n slowly converges to 0, the denominator in the above fraction can be arbitrarily close to n -1/2 , the well-known parametric detection boundary (Lehmann & Romano, 2005); on the other hand, as M diverges more and more slowly to infinity, the boundary tends to n -1/4 , the critical detection boundary of Chatterjee's original statistic derived in Auddy et al. (2021). Technically speaking, our analysis hinges on a careful, sharp up to some log n terms, calculation of the proposed correlation coefficients' means and variances under both null and local alternatives. A particular focus is on M that can scale fast with and even at a rate close to n. Analogous results in Deb et al. (2020), Auddy et al. (2021), Chatterjee (2021) and Shi et al. (2022a) are not quite helpful in this regime, since they are focused on small M that is either fixed or scaling to infinity at a subpolynomial rate. More specifically, (i) we obtain explicit forms of the statistics'means and variances under the null (cf. Theorem 2 below) via a brute-force combinatorial analysis that is in contrast to existing ones; the latter are only applicable to small M ; (ii) we establish sharp, up to some log n terms, bounds for the statistics' means and variances under Gaussian local alternatives; see Theorem 5 below. The interplay between different units in our formulation of the correlation coefficients is remarkably more sophisticated as M is large, when those units with not enough right nearest neighbours need to be handled carefully, while negligible if M is small.

To complete the story, a central limit theorem of the proposed statistics under the null is also established whenever M scales at a slower rate than n 1/4 , which however is not sufficiently fast for making the corresponding test attain near-parametrical efficiency in the Gaussian rotation model.

In the sequel, a set consisting of distinct elements x 1 , ..

., x n is written as either {x 1 , ..., x n } or {x i } n i=1 . The corresponding sequence is denoted [x 1 , ..., x n ] or [x i ] n i=1 . The notation 1(•) is saved for the indicator function. For a sequence of real numbers a 1 , ..., a n ,weuse(a 1 , ..., a n ) as a shorthand of (a 1 , ..., a n ) T . For any two real sequences {a n } and {b n }, write a n b n ,o r equivalently b n a n , if there exists a universal constant C > 0 such that a n /b n C for all sufficiently large n, and write a n ≺ b n , or equivalently b n a n ,i fa n /b n → 0a sn goes to infinity. Write a

For any random variable Z, P Z represents its law.

Correlation measures and coefficients

Chatterjee's rank correlation

Hereafter, when considering correlation, we use the term correlation measure to represent population quantities and correlation coefficient to represent sample quantities. Denote the joint bivariate distribution function of (X , Y ) by F X ,Y , and the marginal distribution functions of X and Y by F X and F Y , respectively. Let (X 1 , Y 1 ), ..., (X n , Y n ) be n independent copies of (X , Y ). Throughout the rest of this manuscript, we assume (X , Y ) to be continuous, i.e., F X ,Y is a continuous function. This requirement ensures that, with probability 1, there is no tie in the observed data. Denote by P c the family of bivariate probability measures of (X , Y ) such that it is continuous. Dette et al. (2013) introduced the following measure of dependence between X and Y :

Compared to many other popular ones (Hoeffding, 1948;Blum et al., 1961;Yanagimoto, 1970;Bergsma & Dassios, 2014;Shi et al., 2022b), the correlation measure ξ is appealing due to its consistency against dependence (Weihs et al., 2018) as well as its capability to detect perfect functional dependence (Cao & Bickel, 2020). These properties were discussed in § 1, and summarized in Dette et al. (2013, Theorem 2) and Chatterjee (2021, Theorem 1.1).

To estimate ξ , Chatterjee (2021) pioneered an ingenious rank-based idea that, to the authors' knowledge, has not been explored before in the literature. To present his idea in a formal way, let us first introduce some necessary notation. Define

to be the rank of Y i among {Y 1 , ..., Y n }. For any i ∈{1, ..., n} and m ∈{1, ..., n}, define

In other words, j m (i) is the index j ∈{1, ..., n} such that

if there exists such a j; otherwise, let j m (i) = i.

With this notation, we summarize results on Chatterjee's rank correlation as follows.

Proposition 1(Theorems 1.1 and 2.1 of Chatterjee ( 2021)). Chatterjee's rank correlation coefficient, written as

satisfies the following conditions for any P (X ,Y ) ∈ P c .

(i) The coefficient ξ n converges almost surely to ξ .

(ii) Further assuming that Y is independent of X , n 1/2 ξ n converges in distribution to N (0, 2/5), the Gaussian distribution with mean 0 and variance 2/5.

The revised Chatterjee's rank correlation

The formulation of Chatterjee's rank correlation in (2) suggests that it is a 1-nearest-neighbourbased estimator of Dette-Siburg-Stoimenov's correlation measure ξ , and thus intuitively will suffer from a similar efficiency loss to other 1-nearest-neighbour-based estimators in various applications (Wald & Wolfowitz, 1940;Friedman & Rafsky, 1979;Henze, 1988;Liu & Singh, 1993;Henze & Penrose, 1999;Bhattacharya, 2019).

Indeed, recent results have shown that a test of independence based on Proposition 1(ii) is inefficient in common classes of smooth alternatives (Cao & Bickel, 2020;Auddy et al., 2021;Shi et al., 2022b). These results call for variants of ξ n that are able to boost the power of independence tests. Such messages have been made in, e.g., Deb et al. (2020, Remark 4.3).

In this paper we answer this call by introducing the following revised Chatterjee's rank correlation that allows one to take each element's M right nearest neighbours into account:

Several remarks are in order.

Remark 1( Formulation of ξ n,M ). In (3), M = M n ∈{ 1, ..., n} represents the number of right nearest neighbours the proposed correlation coefficient ξ n,M will exploit and is allowed to increase to infinity with n. The denominator, (n + 1){nM + M (M + 1)/4}, is added to ensure that E(ξ n,M ) = 0 under H 0 ; cf. Theorem 2 below. For reasons to be detailed later, we recommend a sufficiently large M for improving testing efficiency against dependence.

Remark 2(Distribution-freeness of ξ n,M under H 0 ). Examining its formulation, it is immediate that, for any M ∈{ 1, ..., n}, the value of ξ n,M depends only on the coordinatewise ranks of

The statistic ξ n,M is thus a rank correlation coefficient and, accordingly, enjoys all the nice properties shared by rank correlations, including in particular the distribution-freeness under H 0 (Sidak et al., 1999;Han et al., 2017;Drton et al., 2020).

Remark 3(Relation between ξ n,M and ξ n ). In contrast to the coefficient ξ n introduced in (2), in ξ n,M we take the minimum instead of absolute difference. This is an idea pursued in Deb et al. (2020) and Azadkia & Chatterjee (2021) as well; cf. the correlation coefficient T n introduced in Azadkia & Chatterjee (2021) as well as those in § 8.3.1 of Deb et al. (2020). When M = 1,

reduces to ξ n with an asymptotically ignorable small order term. This is because |x|+|y|-|x-y|= 2 min{x, y} for any x, y 0, and thus

The derivation details are relegated to the Supplementary Material.

Remark 4(Relation to the proposal of Deb et al. (2020)). Our idea to scale up M for boosting the power of rank-based tests is of course not new, and is particularly related to an earlier proposal made by Deb et al. (2020); see, e.g., Equations (3.3) and (8.5) therein. It is hence helpful to point out our new discoveries. First of all, it was observed that using the right nearest neighbours, in contrast to using nearest neighbours in both directions, is important. In particular, it plays a central role to show that a test based on ξ n,M can provably reach near-parametric efficiency as M is close to n; cf. Theorem 6 below as well as the finite-sample studies in § 5. Secondly, in (3) the normalizing constant (n + 1){nM + M (M + 1)/4} was carefully chosen so that the expectation of ξ n,M is exactly zero under H 0 , which holds for arbitrary M ∈{1, ..., n}. This type of normalization is important if M is large since in our formulation there exists an un-ignorable fraction of is such that j m (i) = i for some m ∈{1, ..., M }.

Remark 5( Relation to spacing statistics). The notion of right nearest neighbours along the real line in the construction of ξ n,M is related to spacing statistics, which are functions of adjacent spacings between order statistics (Darling, 1953;Pyke, 1965;Cressie, 1976;Del Pino, 1979;Hall, 1986). However, these two types of statistics are substantially different; spacing statistics are functions of [X 1 , ..., X n ], while the proposed correlation coefficient ξ n,M is a function of [Y 1 , ..., Y n ] driven by the right nearest neighbours of [X 1 , ..., X n ]. In particular, under independence between X and Y , the distribution of ξ n,M is neither dependent on P X nor related to right nearest neighbours of [Y 1 , ..., Y n ]; see § 3.2 below for an exact distribution of ξ n,M under H 0 . This is clearly distinct from those spacing statistics in the literature.

Remark 6( Extremal properties of ξ n,M ). If Y = f (X ) for some strictly increasing function f (•), one has min{R i , R j m (i) }=R i and thus

In the above two cases, ξ n,M are both equal to 1 up to a bias of order M /n.

Remark 7(Finite-sample range of ξ n,M ). An equivalent form of (3)is

The largest possible value of the correlation coefficient is then

which is attained when Y = X almost surely. On the other hand, since each R i can appear in the summation of (3) for at most 2M times, a straightforward lower bound of ξ n,M is

The finite-sample range of ξ n,M hence belongs to [-1/2, 1] up to a bias of order M /n, which is analogous to that of Chatterjee's correlation coefficient; cf. Remark 9 under Chatterjee (2021, Theorem 1.1). It is still unclear if this lower bound is sharp or not when M 2.

Remark 8(Computational complexity). To compute ξ n,M , one needs to first sort the samples of both {X i } n i=1 and {Y i } n i=1 , and then perform the summation over nM terms. The computational complexity is O(n log n + nM ).IfM is O(Poly-log n), it is nearly linear. On the other hand, the computation of ξ n,M will tend to be quadratic as M is closer to n, which is a seemingly inevitable cost.

We close this section by establishing strong consistency for the proposed rank correlation coefficient ξ n,M that is in parallel to Proposition 1(i).

Theorem 1( Strong consistency of ξ n,M ). Fo r a ny P (X ,Y ) ∈ P c , ξ n,M converges almost surely to ξ as long as M /n → 0.

Tests of independence

Elementary properties under the null

We start with some elementary properties of ξ n,M when Y is independent of X . To this end, we first establish the corresponding mean and variance of ξ n,M .

Theorem 2(Mean and variance of ξ n,M under the null). Assuming that P (X ,Y ) ∈ P c and that H 0 holds, then E(ξ n,M ) = 0. If we further assume that M →∞and M /n → 0 as n →∞, we have

Remark 9. Proposition 1 shows that the asymptotic null variance of n 1/2 ξ n is 2/5 when M = 1. In contrast, Theorem 2 revealed that var(ξ n,M ) scales at a rate of order (nM ) -1 ∨ (M /n 2 ), which is always faster than n -1 under the theorem's conditions. Indeed, the variance will achieve its lowest order n -3/2 if M is of order n 1/2 , and is of order close to n -1 if M is of order n γ with γ close to either 0 or 1.

We then establish a central limit theorem for ξ n,M under the null. It is in parallel to Proposition 1(ii); note that, compared to Theorems 1 and 2, a strong scaling requirement, M ≺ n 1/4 ,i s enforced for guaranteeing its validity.

Theorem 3(Central limit theorem for ξ n,M under the null). Assume that P (X ,Y ) ∈ P c and that H 0 holds. If we further assume that M →∞and M ≺ n 1/4 as n →∞, we have (nM ) 1/2 ξ n,M converges in distribution to N (0, 2/5).

Remark 10 (Technical ingredients of Theorem 3). To establish the central limit theorem of ξ n,M under the null, the following Hájek representation of ξ n,M is the key:

The variance of ξ n,M under H 0 is of order (nM ) -1 as M ≺ n 1/4 ; we show that (nM ) 1/2 (ξ n,Mξn,M ) converges in probability to 0. It then suffices to establish that (nM ) 1/2 ξn,M converges in distribution to N (0, 2/5), which is derived by invoking the normal approximation techniques devised in Chatterjee (2008); assuming that M ≺ n 1/4 is crucial here.

Simulation-based tests of independence

This section focuses on testing H 0 in (1) based on the revised Chatterjee's correlation coefficient ξ n,M that was introduced in § 2.

For a given sample (X i , Y i ) n i=1 , let ξ n,M and ξ - n,M be the correlation coefficients in (3) that are calculated based on (X i , Y i ) n i=1 and (X i , -Y i ) n i=1 , respectively. We are interested in the test statistic

which is invariant to a sign change of the Y i by its construction. For approximating the above test statistic's distribution under H 0 , note that, as X and Y are independent, the joint distribution of (ξ n,M , ξ - n,M ) is distribution-free and a simulation-based test can then be directly implemented. In detail, choosing the number of simulations to be B, in each round b ∈{1, ..., B}, one draws a sample [R

i=1 from the uniform distribution over all possible permutations on {1, ..., n}. We then calculate the value of ξ (b) n,M from (3) as follows:

One could similarly calculate the value of ξ

n,M ) will have the same distribution as (ξ n,M , ξ - n,M ) due to independence between X and Y , and the distribution-freeness of relative ranks; see similar discussions in § A.2 and Lemma C3 of the Supplementary Material to Han et al. (2017). We then consider

For a given significance level α ∈ (0, 1), the proposed simulation-based test is then

whose size validity and power consistency are guaranteed by the following theorem.

Theorem 4(Size validity and consistency). (i) The test T ξ ± n,M α,B is size valid in the sense that, for any fixed P = P (X ,Y ) ∈ P c satisfying H 0 , denoting by P H 0 ≡ P ⊗n the corresponding product measure,

holds for any n 1,B 1 and M ∈{1, ..., n}.

(ii) The test T

α,B is consistent in the sense that, for any fixed P ∈ P c violating H 0 , denoting by P H 1 the corresponding product measure, we have

as long as B = B n →∞and M /n → 0 as n →∞.

Local power analysis

In this section we investigate the local power of the simulation-based test T ξ ± n,M α,B employing the revised Chatterjee's rank correlation ξ n,M introduced in (3). Our attention is restricted to the following Gaussian rotation model that is a benchmark in independence testing (Konijn, 1956). Although the theoretical results then automatically extend to Gaussian copula models, simulation results developed later suggest that power behaviours may alter in other non-Gaussian cases; see § 5 below for more details.

Assumption 1. The bivariate random vector (X , Y ) belongs to the family of Gaussian distributed ones with mean µ and covariance matrix such that

For the local power analysis in this specified alternative set, we examine the asymptotic power along a sequence of alternatives obtained as

Because of the construction of the test statistic (4), it suffices to consider positive sequences of ρ n . We first establish the mean and variance of ξ n,M under (5).

Theorem 5(Mean and variance of ξ n,M under alternatives). Suppose that the considered set of local alternatives satisfies Assumption 1. Then, with any sequence of alternatives given in (5), for any positive sequence ρ n → 0, as long as M / log n →∞and M (log n) 3/2 /n → 0, (i) the mean of ξ n,M under H 1,n satisfies

Remark 11. It is notable that in the above theorem we only establish an upper bound on var H 1,n (ξ n,M ). However, by calculating the coefficient of each term in the proof of Theorem 5(ii), one is also able to show the lower bound

The upper bound in Theorem 5(ii) is therefore rate optimal up to some log n terms, although the upper bound of var H 1,n (ξ n,M ) in Theorem 5(ii) has been sufficient for our purpose.

As a direct consequence of Theorem 5, the following corollary establishes a boundary of ρ n beyond which the mean of ξ n,M under H 1,n shall dominate the standard deviation.

Corollary 1. Suppose that the considered set of local alternatives satisfies Assumption 1. Then, with any sequence of alternatives given in (5), for any positive sequence ρ n → 0, as long as M / log n →∞,M(log n) 3/2 /n → 0 and

Remark 12. In view of Theorem 5(i) and Remark 11, as long as ρ n ≺ ζ n,M , one also has

Combined with Corollary 1,(6) shows that ζ n,M is the critical boundary determining whether the mean of ξ n,M under H 1,n will be dominating or will be dominated by its standard deviation.

Combining Theorem 5 with Theorem 2, we are now ready to establish a sufficient detection boundary of ρ n beyond which the proposed test is of a power tending to 1. Invoking Theorem 3 further, this detection boundary is sharp as long as M ≺ n 1/4 . Theorem 6(Local power analysis). Suppose that the considered set of local alternatives satisfies Assumption 1. Then, with any sequence of alternatives given in (5), for any sequence ρ n → 0, as long as M / log n →∞,M(log n) 3/2 /n → 0,B→∞, and (i) as long as |ρ n | ζ n,M , we have

(ii) as long as |ρ n |≺ζ n,M and further assuming that M ≺ n 1/4 , for any sufficiently small α, it holds that

for some β α < 1 that depends only on α.

Remark 13 (Relation between M and ζ n,M ). Picking M = n γ for some γ ∈ (0, 1), the established boundary in Theorem 6 is

It is easy to check that β is piecewise linear with respect to γ . In particular, (i) if γ → 0 then β → 1/4, corresponding to the detection boundary of ξ n established in Auddy et al. (2021);

(ii) if γ → 1 then β → 1/2, corresponding to the well-known parametric detection boundary (Lehmann & Romano, 2005).

In Fig. 1 we plot the relation between β and γ . Here β is not strictly increasing with γ ; the pattern changes at γ = 1/2 with β(1/2) = 3/8 and γ = 2/3 with β(2/3) = 1/3, indicating an intriguing bias-variance trade-off of the test with regard to the choice of M . Interestingly, similar behaviour was also documented in using spacing statistics for testing uniformity (Hall, 1986). However, the reasons are distinct; see Remark 5.

Remark 14. Deb et al. (2020, Remark 4.3) conjectured that 'allowing for growing [number of nearest neighbours] could potentially lead to information theoretically efficient estimators'. Theorem 6 settles this conjecture in the Gaussian rotation model, and shows that a simulationbased test built on the proposed revised Chatterjee's rank correlation can indeed provably achieve near-parametric efficiency by pushing M to be closer and closer to n; see also Remarks 4 and 13 for related discussions.

Remark 15. We conjecture that ζ n,M is always, regardless of how fast M grows to infinity with n, the critical detection boundary of the proposed test in the sense that there exists a constant C α < 1 depending only on α such that

for any sequence |ρ n |≺ζ n,M . This conjecture is partially supported by Corollary 1 and Remark 12; see also Bhattacharya (2019, Remark 3 and Proposition 1). To fully resolve it, however, one needs to obtain more information on the null distribution of ξ n,M beyond its mean and variance calculated in Theorem 2 for those M that are large. This will be an interesting topic for future research.

Simulation study

In this section we develop additional simulation results to illustrate the finite-sample performance of the developed correlation coefficients and the related tests of independence.

First, we examine the sizes and powers of the proposed tests. To this end, the following four sets of tests of independence are considered:

(T1) the proposed test T In implementing (T4), following Deb et al. (2020) the nearest neighbours are calculated based on the ranks of [X i ] n i=1 , but not the original data, and we select the right one instead of the left if ties exist. Accordingly, as M = 1, we select the right nearest neighbour for all points except for the largest. To implement (T1) and (T4), we use simulation-based tests illustrated in § 3.2 with B = 10 000. On the other hand, asymptotic tests are used to implement (T2) and (T3). Nominal levels are set to be α = 0.05 for all tests.

We perform simulation studies based on the Gaussian rotation model as well as the following non-Gaussian model studied in Chatterjee (2021, § 4.3); see also Shen et al. (2019, § 5) and Shi et al. (2022b, § 5):

where g(t) := cos(2πt), and the X i and i are independent copies of a uniform distribution over [-1, 1] and standard Gaussian, respectively. The Gaussian rotation model satisfies Assumption 1 and is used to validate the theory, while the non-Gaussian model, of a cosine period form, is used to illustrate more intriguing patterns beyond the Gaussian. Throughout, we set n ∈ {1000, 2000, 5000}, ρ n = ρ 0 /n 1/2 and ρ 0 ∈{ 0, 1, 2, 5}. The case ρ 0 = 0 corresponds to the case when H 0 holds, while the remaining three give rise to powers in accordance with different dependence strengths that all shrink to zero.

Tables 1 and 2 illustrate the rejection frequencies for the considered tests over 1000 replicates. For the Gaussian model, we make three observations. (i) All the tests considered have empirical sizes close to 0.05, indicating that they are all size valid. (ii) The power of (T1) increases as M increases, and is close to that of (T2) and (T3) when M is large. However, the power of (T1) decreases for every fixed M , while that of (T2) and (T3) remain stable. Both observations are in line with the theoretical observations made earlier in Theorem 6. (iii) For each selected M , the power of (T1) dominates that of (T4), which echoes Remark 4. These results back up our theory and, since the considered tests, except for (T3), are all rank-based, the same conclusions automatically extend to Gaussian copulas. On the other hand, for the studied non-Gaussian model, generally speaking, the (T4) tests of Deb et al. (2020) now have comparable empirical performance to ours, while it is no longer true that a larger M would always lead to a higher power. However, for such non-Gaussian data, the empirical results still support M →∞for achieving higher powers and favour using our revision of the proposal of Chatterjee (2021). These results should also be compared to Remark 7 of Shi et al. (2022b).

Secondly, we compare the computation times for ξ ± n,M with different n and M chosen as before. To this end, we consider a simple bivariate standard Gaussian model and calculate the averaged computation time for each pair of (n, M ) over 1000 replicates. All experiments are implemented on a laptop with an Apple M1 processor and a 16 GB memory. Table 3 illustrates the computation times for the tests considered, which are compared to these of Hoeffding's D and Pearson's sample correlation coefficient. We observe that the computation time indeed increases, and is approximately linear, with regard to M .

Lastly, we examine the trajectory of ξ n,M as the data generated from a bivariate Gaussian distribution with marginal mean 0, variance 1 and correlation ρ = 0, 0.2, 0.4, 0.6, 0.8 and 1. To this end, Fig. 2 illustrates the boxplots of ξ n,M as n changes from 1000, 2000 to 5000, and M changes from 1, 20, 100 to 200 over 1000 replicates. For comparison purposes, in Fig. 2 we also plot the curve of the population correlation measure ξ as a function of ρ ∈[0, 1]. We make three observations. (i) For any M considered, the averaged ξ n,M gets closer and closer to the correlation measure as n increases, which supports Theorem 1. (ii) As ρ = 0, the empirical variance of ξ n,M first decreases and then increases as M increases, which supports Theorem 5(ii) and Remark 11.

On the other hand, as ρ becomes large, the empirical variance of ξ n,M turns to be stable and unchanged with M . (iii) A bias term exists as M is relatively large compared to n, but will shrink towards 0 for each fixed M as n increases. This is as expected in view of Theorem 1 and is a common occurrence in nonparametric statistics problems.