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From Reviews: 0MR4085751 62G10 60E15 62G07 62G20
Amini-Seresht, Ebrahim (IR-BASU-S) ; Milošević, Bojana (SE-BELGM)
New non-parametric tests for independence. (English summary)J. Stat. Comput. Simul. 90 (2020), no. 7, 1301–1314.

In this article, two nonparametric tests of independence against a particular form ofdependence are proposed. The alternative considered here is a stochastically increasing
property (also known as positive regression dependence), which is defined as follows:
For a pair of positive random variables (X, Y ), Y is said to be stochastically increasingin X if, for all y > 0, Pr(Y > y|X = x) is increasing in x > 0.
For a given y > 0, let δ(s, t; y) := Pr(Y > y|X = s) − Pr(Y > y|X = t) whenevers ≥ t > 0. The test statistics are built on the following measure of deviation:∆(F,G) (y) := E { δ (X2 , X1 ; y)| X2 ≥ X1 } ,where F and G are marginal distribution functions of X and Y , respectively. Given theconditional survival function G(y|x), ∆(F,G) (y) can be expressed asZ∆(F,G) (y) = G ( y| x) {2F (x) − 1} dF (x) .
The sup measure ∆∗(F,G) := supy>0 ∆(F,G) (y) and the density-average measureR∆∗∗∆(F,G) (y) dG(y) are employed to gauge the deviation from independence.
(F,G) :=Because these measures include a few unknown quantities, the corresponding test
statistics can be constructed by replacing the quantities with their consistent estimates. Specifically, using n i.i.d. observations, {(Xi , Yi )}ni=1 , G(y|x) can be estimatednonparametrically byPn
j=1 k {(x − Xj ) /an } 1 (Yj > y)Pn,Gn ( y| x) :=j=1 k {(x − Xj ) /an }where k(·) is a kernel function, an (> 0) is the sequence of bandwidths, and 1(·) is anindicator function. By further replacing (F, G) with their empirical measures (Fn , Gn ),a natural estimator of ∆(F,G) (y) can be obtained as
1 X k {(Xi − Xj ) /an } 1 (Yj > y) {2(1(Xi ≥ X` )) − 1}Pn.
∆(Fn ,Gn ) (y) := 2n
j=1 k {(Xi − Xj ) /an }i,j,`In the end, the test statistics become∆∗(Fn ,Gn ) := sup ∆(Fn ,Gn ) (y) and ∆∗∗(Fn ,Gn ) :=y>0Z∆(Fn ,Gn ) (y) dGn (y) .

In particular, ∆∗∗(Fn ,Gn ) can be simplified as
1 X k {(Xi − Xj ) /an } (Sj − 1) (2Ri − n)Pn,n3 i,j
j=1 k {(Xi − Xj ) /an }PnPnwhere Ri := Rank(Xi ) = `=1 1(Xi ≥ X` ) and Sj := Rank(Yj ) = h=1 1(Yj ≥ Yh ).
Convergence properties of the test statistics ∆∗(Fn ,Gn ) and ∆∗∗(Fn ,Gn ) can be established∆∗∗(Fn ,Gn ) =by the following weak convergence of the empirical process {∆(Fn ,Gn ) (y), y > 0}:√nan ∆(Fn ,Gn ) (y) − ∆(F,G) (y) ⇒Z
1/2−1/2{f (x)}{2F (x) − 1} B {G ( y| x)} dF (x)Q (y) := k0
Rin D(0, ∞), where k0 = k 2 (u) du and B is a standard Brownian bridge on the unit
interval [0, 1]. This yields the limiting null distributions of ∆∗(Fn ,Gn ) and ∆∗∗(Fn ,Gn ) asno√
1/2nan ∆∗(Fn ,Gn ) − ∆∗(F,G) ⇒ k0 Γ sup B {G (y)} , andy>0
no√1−1/2 −1∗∗∗∗nan ∆(Fn ,Gn ) − ∆(F,G) ⇒ N 0,,k0
Γ
12R
where Γ := {f (x)}−1/2 {2F (x) − 1} dF (x). The null of independence is rejected if test
statistics take large values.
Observe that the above convergence results depend on unknown marginal distributions of X and Y . Then, bootstrapping is adopted in the Monte Carlo study. However,the bootstrap algorithm does not look fully operational in the sense that it is subject toprior knowledge of the joint distribution of (X, Y ). When implementing each test, we
may have to either (i) estimate the unknown quantities nonparametrically and rely on
its limiting null distribution or (ii) explore a nonparametric bootstrap test.
Masayuki Hirukawac Copyright American Mathematical Society 2020