THE ROBUSTNESS AGAINST DEPENDENCE OF NONPARAMETRIC TESTS FOR THE TWO-SAMPLE LOCATION PROBLEM

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1 APPLICATIOES MATHEMATICAE,4 (1995), pp P. GRZEGORZEWSKI (Warszawa) THE ROBUSTESS AGAIST DEPEDECE OF OPARAMETRIC TESTS FOR THE TWO-SAMPLE LOCATIO PROBLEM Abstract. onparametric tests for the two-sample location problem are investigated. It is shown that the supremum of the size of any test can be arbitrarily close to 1. one of these tests is most robust against dependence. 1. Introduction. Situations with some kind of dependencies for the Mann Whitney Wilcoxon test were investigated by Hollander, Pledger and Lin [3], Pettit and Siskind [4], Serfling [6], Zieliński [8], [9]. In this paper we consider the robustness against dependence of a large family of nonparametric tests for the two-sample location problem, including the test mentioned above. We take advantage of a new description of dependence, called Rüschendorf s ε-neighbourhoods, proposed in [].. Problem and notation. Let 1,..., m and Y 1,..., Y n denote two independent random samples from populations with continuous distribution functions F (x) = F (x ) and F Y (y) = F (y) respectively. We verify the hypothesis H : = 0 against K : > 0 by means of a test φ of size α. We assume that φ belongs to some family Φ of tests (see Sec. 3 for the definition of Φ). Let P(F ) = {P : P (Z i z) = F (z), i = 1,..., m + n} describe all possible violations of independence. We denote P(F ) briefly by P. Let P C P be the subfamily of all continuous distribution functions. Moreover, let C ε P C be a family of all c.d.f. which correspond to small dependencies (for more details see Sec. 4) Mathematics Subject Classification: Primary 6G35; Secondary 6G10. Key words and phrases: robustness of tests, robustness against dependence, nonparametric tests for the two-sample location problem, size of test.

2 470 P. Grzegorzewski (A) (B) (C) The following problems are considered in this paper: Given any φ Φ, compute the supremum of the size of φ under all kinds of dependencies, i.e. sup P P φ dp, where denotes a sample space. Given any φ Φ, evaluate the robustness of the size of φ against small dependencies. We use the oscillation of the size over C ε as a measure of robustness (see [7]): r ε (φ) = sup P C ε φ dp inf P C ε φ dp. Find the most robust test in Φ, i.e. a test φ 0 such that r ε (φ 0 ) r ε (φ) ( φ Φ) for all ε. 3. The family Φ. We restrict our consideration to a family Φ of one-sided nonparametric tests for the two-sample location problem given as follows: Definition. φ Φ if and only if (i) φ(x 1 + τ,..., x m + τ, y 1 + τ,..., y n + τ) = φ(x 1,..., x m, y 1,..., y n ) τ, (ii) φ(x 1,..., x i 1, x i + δ, x i+1,..., x m, y 1,..., y n ) φ(x 1,..., x m, y 1,..., y n ) ( δ 0), i = 1,..., m, (iii) if 1:m > Y n:n then φ(x 1,..., x m, y 1,..., y n ) = 1, if m:m < Y 1:n then φ(x 1,..., x m, y 1,..., y n ) = 0, where i:m and Y i:n denote the ith order statistics from the first and the second sample respectively. The conditions (i) (iii) seem to be quite natural. The Mann Whitney Wilcoxon test, the Fisher Yates test, the Rosenbaum test and many other tests for the two-sample location problem belong to the family Φ (see [1]). 4. A description of dependence. By the Rüschendorf theorem (see [5]) we know that h is the density of a probability measure on [0, 1] r with uniform marginals and continuous w.r.t. the Lebesgue measure dµ on [0, 1] r if and only if h = 1 + Sf where f L 1 ([0, 1] r ) and S : L 1 L 1 is the linear operator given by r Sf = f f dz 1... dz i... dz r + (r 1) f dz 1... dz r i=1 [0,1] r 1 [0,1] r and Sf 1. Without loss of generality we assume that F is the uniform distribution on [0, 1].

3 Two-sample location problem 471 Moreover, define R = {f L 1 ([0, 1] r ) : Sf 1}, and let [0, u] r = {x [0, 1] r : 0 x i u i, i = 1,..., r}. Basing on the above theorem and assumptions we may write that P C = {P } : P (u) = (1 + Sf) dµ, f R. [0,u] r In [] a new description of dependence, called Rüschendorf s ε-neighbourhoods, was proposed and motivated. Following that paper let R ε = {f L 1 : Sf ε, Sf 1}, where is the L 1 norm. Then C ε = {P } : P (u) = (1 + Sf) dµ, f R ε [0,u] r describes the family of distributions which correspond to small departures from independence, so called ε-dependence, i.e. if ε is sufficiently small then the dependence measured by ϱ-spearman s and many other meassures is small as well, and conversely. In order to solve our problems (A) and (B) it will be necessary for any φ Φ to compute: (A) (B) sup f R [0,1] r (1 + Sf)φ dµ, r ε (φ) = sup f R ε f R ε [0,1] r (1 + Sf)φ dµ inf [0,1] r (1 + Sf)φ dµ. It is easy to show (see Sec. 7) that the above expressions are equivalent to the following, more convenient in further investigations: (A) (B) sup g G [0,1] r (1 + g)φ dµ, r ε (φ) = sup [0,1] r (1 + g)φ dµ inf [0,1] r (1 + g)φ dµ, where G = {g L 1 ([0, 1] r ) : g 1, [0,1] r g dµ = 0, [0,1] r 1 g dz 1... dz i dz r = 0, i = 1,..., r}, and G ε = {g G : g ε}. 5. Results. ow we can state the solutions of our problems (A) (C). Theorem 1. Let φ Φ. Suppose that all kind of dependencies between samples and among observations in samples are allowed. Then the size of the test φ can be arbitrarily close to 1, i.e. sup P P φ dp = 1.

4 47 P. Grzegorzewski Theorem. The robustness of any test φ Φ against ε-dependence equals ε/, i.e. r ε (φ) = ε/. From this theorem we get immediately: Corollary. In the family Φ of one-sided nonparametric tests for the two-sample location problem, no test is most robust against dependence. 6. Proofs P r o o f o f T h e o r e m 1. Let {G } = be the sequence of r-dimensional subsets of [0, 1] r, r = m + n, given by G = [( i, i + 1 ) m ( j 1, j ) n ], where the union is extended over all (i, j) of the form (k mod, k), for k = 1,...,. Let {g } = be the sequence of real functions on [0, 1]r defined as follows: { g (z) = r 1 1 for z G, 1 for z G. We show that g G ( ): (a) (b) (c) g 1 by the definition, ( + ( 1) g dµ = r 1 1 r [0,1] r [0,1] r 1 g dz 1... dz i... dz r = r 1 1 r r ) = 0, ( + ( 1) 1 1 r 1 ) = 0 for i = 1,..., r. So g G for every. ow take φ Φ and denote by α its size. Suppose that φ is a non-randomized test with a critical region. It is easy to check that for each, [( G \ 0, 1 ) m ( ) n ] 1, 1 {(z 1,..., z r ) : 0 z j z i 1, i = 1,..., m; j = m + 1,..., m + n} for every φ Φ (see conditions specified in Sec. 3). So we get

5 sup P P φ dp sup P P C = sup g G = α + Two-sample location problem 473 φ dp = sup (1 + g)φ dµ g G [0,1] r (1 + g) dµ (1 + g ) dµ [ ( 1) r 1 ( 1 r + ( 1) α ( 1) 1 r = α + 1 α = 1. Choosing large enough one can come arbitrarily close to 1. R e m a r k. For simplicity we have assumed in the proof that φ is a non-randomized test. The theorem is true for randomized tests as well. P r o o f o f T h e o r e m. We take a non-randomized test φ Φ and denote by α its size and by its critical region. Let {G } be as in the proof of Theorem 1. Consider the sequence {g } of real functions on [0, 1]r defined by ε g (z) r 1 for z G, = ε r 1 r 1 for z G, 1 for 0 = (/( ε)) 1/(r 1). It is easily seen that g G ε ( 0 ). So we get sup [0,1] r (1 + g)φ dµ = sup = α + (1 + g) dµ [ ( 1) ε r 1 1 r + α + ε (1 α) as. (1 + g ) dµ ( ε )] r 1 )( r 1 α ( 1) 1 )] 1 r In order to show that also sup g Gε (1 + g)φ dµ α + [0,1] ε r (1 α), we consider the operator T g = g dµ. It is a bounded linear operator, so we get T g T g ( g G). By the proof of Theorem 1, T g T = sup g G g = 1 α

6 474 P. Grzegorzewski (this is evident, because ε cannot be greater than ). So we get and therefore Hence by sup ( g G ε ) T g 1 α g 1 α ε (1 + g) dµ = sup (α + T g) α + 1 α ε. sup (1 + g)φ dµ = α + ε (1 α). [0,1] r ow we consider inf g Gε [0,1] r (1+g)φ dµ. Let us define a sequence {G } G = [( ) m ( i 1, i j, j + 1 ) n ] [0, 1] r, where the union is extended over all (i, j) of the form (k, kmod ) for k = 1,...,. Let {g } be the following sequence of real functions on [0, 1]r : ε g (z) r 1 for z G, = ε r 1 r 1 for z G 1, where 0. It is easily seen that g G ε ( 0 ). So inf (1 + g)φ dµ = inf (1 + g) dµ (1 + g g G [0,1] r ε ) dµ [ ε = α + r 1 1 r ε r 1 ( r 1 α 1 )] 1 r α(1 ε/) as. Similarly, we can prove the opposite inequality: and therefore inf inf [0,1] r (1 + g)φ dµ α(1 ε/) [0,1] r (1 + g)φ dµ = α(1 ε/).

7 Thus finally r ε (φ) = sup P C ε which completes the proof. Two-sample location problem 475 φ dp inf P C ε = sup (1 + g)φ dµ inf [0,1] r = α + ε ( (1 α) α 1 ε ) φ dp [0,1] r (1 + g)φ dµ = ε, As before, the theorem is also true for randomized tests. 7. Complements. In Section 4 we have stated that in our problem we could consider the families G and G ε instead of R and R ε. This follows from Lemma. Let S be the operator defined in Section 4. Then S(R) = G and S(R ε ) = G ε. P r o o f. It suffices to show that S(R) = G. The same proof remains valid for the second assertion. Suppose f R. Then Sf L 1 ([0, 1] r ), Sf 1 and Sf dµ = [0,1] r Sf dµ = 0. So S(R) G. [0,1] r 1 ow take any g G. Then r Sg = g gdz 1... dz i... dz r + (r 1) g dz 1... dz r = g i=1 [0,1] r 1 [0,1] r and hence S(R) G. References [1] P. G r z e g o r z e w s k i, onparametric tests for two-sample location problem, Mat. Stos. 34 (1991), (in Polish). [], The infinitesimal robustness of tests against dependence, Zastos. Mat. 1 (3) (199), [3] M. Hollander, G. Pledger and P. E. Lin, Robustness of the Wilcoxon test to a certain dependency between samples, Ann. Statist. (1974), [4] A.. Pettit and V. Siskind, Effect of within-sample dependence on the MWW statistic, Biometrica 68 (1981), [5] L. Rüschendorf, Construction of multivariate distributions with given marginals, Ann. Inst. Statist. Math. 37 (1985), [6] R. J. Serfling, The Wilcoxon two-sample statistic on strongly mixing processes, Ann. Math. Statist. 39 (1968), [7] R. Zieliński, Robust statistical procedures: a general approach, in: Lecture otes in Math. 98, Springer, 1983,

8 476 P. Grzegorzewski [8] R. Zieliński, Robustness of two-sample tests to dependence of the observations, Mat. Stos. 3 (1989), 5 18 (in Polish). [9], Robustness of the one-sided Mann Whitney Wilcoxon test to dependency between samples, Statist. Probab. Lett. 10 (1990), PRZEMYS LAW GRZEGORZEWSKI ISTITUTE OF MATHEMATICS WARSAW UIVERSITY OF TECHOLOGY PL. POLITECHIKI WARSZAWA, POLAD Received on