Rank tests and regression rank scores tests in measurement error models

Transcription

1 Rak tests ad regressio rak scores tests i measuremet error models J. Jurečková ad A.K.Md.E. Saleh Charles Uiversity i Prague ad Carleto Uiversity i Ottawa Abstract The rak ad regressio rak score tests of liear hypothesis i the liear regressio model are modified for measuremet error models. I some situatios the modified tests are still distributio free; their asymptotic relative efficiecies with respect to tests i model without errors are evaluated. 1 Itroductio Cosider the liear regressio model Y i = β 0 + x iβ + e i, i = 1,..., (1.1) with observatios Y 1,..., Y, idepedet errors e 1,..., e, idetically distributed accordig to a ukow distributio fuctio F ; x i = (x i1,..., x i ) is the vector of covariates, i = 1,...,, β = (β 1,..., β p ), ad β = (β 0, β ) are ukow parameters. We shall suppress the subscript wheever it does ot cause a cofusio. If the covariates are oly measured with radom errors, the istead of x i we oly observe w i = x i + u i, i = 1,...,. Several authors studied the behavior of the (evetually modified) regressio quatiles or of the coditioal media i error-i-variables (EV) model; let us metio He ad Liag (2000), where are other refereces cited. Our primary iterest is to cosider how the evetual measuremet errors i the covariates affect the behavior ad the efficiecy of the rak tests of the liear hypothesis i the liear regressio model. I a more geeral cotext, whe the liear regressio is cosidered as a uisace i the model, but is ot affected by measuremet errors, ad we have aother hypothesis of iterest i mid, we ca use the tests based o regressio rak scores proposed i Gutebruer AMS 2000 subject classificatios. 62G10, 62J05 Key words ad phrases: rak test of liear hypothesis, regressio rak score test of liear subhypothesis, measuremet error. Research of J. Jurečková was supported by Research Projects LC024, MSM ad by Czech Republic Grat 201/05/2340. Research of A.K.Md.E.Saleh was supported by NSERC Grat of Caada.

2 2 Rak tests with measuremet errors et al. (1993). Their criteria are based o the regressio rak scores uder the hypothesis, i.e. uder the model (1.1), ad are asymptotically equivalet to the ordiary rak tests pertaiig to the situatio where β is kow. We shall cosider two models: Y i = β 0 + x iβ + e i, i = 1,..., (1.2) where we wat to test the hypothesis H : β = 0, (1.3) ad Y i = β 0 + x iβ + z iδ + e i, i = 1,..., (1.4) with ukow parameters β R p, δ R q, ad the hypothesis H : δ = 0, (1.5) cosiderig β as a uisace parameter. I the models without measuremet errors, the hypothesis (1.3) ca be tested by a rak test, while for the hypothesis (1.5) we ca use the regressio rak score test. The model (1.4) e.g. appears i the polyomial regressio model, whe we wat to test the hypothesis o the order of the polyomial ad we do ot kow the coefficiets of lower order terms. The regressio rak scores eable us to avoid estimatig the uisace parameters. We shall study how the measuremet errors affect the performace of tests of the above hypotheses. I Sectio 2 we shall study the loss of efficiecy of the ordiary rak tests of hypothesis (1.3) if the regressors are affected by radom measuremet errors. Sectio 3 cosiders the tests of hypothesis (1.5) i the presece of uisace regressio, based o the regressio rak scores, i the situatio that the regressors x i are precisely observable, while the z i are affected by radom errors. I turs out that eve i such situatio we ca costruct distributio free tests, that oly loose efficiecy with respect to tests i models without errors. The questio whether we ca costruct a distributio free test i model (1.4) whe the x i or both the x i ad z i are affected by radom errors is still a ope problem. 2 Liear rak test of radomess agaist regressio alterative To see the effect of the measuremet errors i the covariates, let us first cosider how do they affect the simple liear rak test of the hypothesis of radomess agaist a regressio alterative. Let Y 1,..., Y be idepedet observatios, Y i distributed accordig to a d.f. F (y x i β), i = 1,...,. We assume that F has a absolutely cotiuous desity f, otherwise it is geerally ukow; (x 1,..., x ) is a vector of kow regressors, satisfyig lim max (x i x ) 2( (x j x ) 2) 1 = 0, (2.1) 1 i j=1

3 Rak tests with measuremet errors 3 x = 1 x i; moreover, for the sake of simplicity ad cocetratio o the mai idea, we assume that lim C2 = C 2 > 0, where C 2 = 1 (x j x ) 2. (2.2) The problem is that of testig the hypothesis H 0 : β = 0 i the situatio whe the x i are ot exactly observable; istead of x i we ca oly observe w i = x i + u i, i = 1,...,, where u 1,..., u are i.i.d. radom errors. Their distributio (say G) is ukow, but we assume that it has the fiite secod momet ad p j=1 D 2 D 2 > 0, (2.3) D 2 = 1 (u i ū ) 2, ū = 1 u i. Moreover, we assume that Y i ad u i are idepedet, i = 1,...,. Let R 1,..., R be the raks of Y 1,..., Y. I the models without measuremet errors, the rak test of H 0 is based o the criterio S = 1/2 (x i x )a (R i ) (2.4) with the scores a (1),..., a () geerated by a odecreasig, square itegrable score fuctio ϕ : (0, 1) R 1 i either of the followig two ways: a (i) = Eϕ(U :i ), (2.5) ( ) i a (i) = ϕ, i = 1,...,, + 1 where U :1... U : are the order statistics correspodig to the sample of size from the R(0, 1) distributio. The asymptotic distributios of the liear rak statistics are studied i Hájek ad Šidák (1967). Uder the above coditios o the x i ad o F, the asymptotic distributio of S uder H 0 is ormal N ( 0, C 2 A 2 (ϕ) ), A 2 (ϕ) = 1 0 (ϕ(t) ϕ)2 dt, ϕ = 1 0 ϕ(t)dt. Uder the local alterative H : β = β = 1/2 β, β R 1, (2.6) S is asymptotically ormally distributed N ( C 2 γ(ϕ, f) β, C 2 A 2 (ϕ) ), (2.7) γ(ϕ, f) = 1 0 ϕ(t)ϕ(t, f)dt, ϕ(t, f) = f (F 1 (t)) f(f 1, 0 < t < 1. (2.8) (t)) We reject H 0 i favor of alterative K : β 0 o the asymptotic sigificace level α provided ( 1 A 1 (ϕ) (x i x) 2) 1/2 ( S Φ 1 1 α ). (2.9) 2

4 4 Rak tests with measuremet errors The it follows from (2.7) that the asymptotic power of the test agaist H is ( ( 1 Φ Φ 1 1 α ) ) ( γ(ϕ, f) β ( C + Φ Φ 1 α ) ) γ(ϕ, f) β C. (2.10) 2 A(ϕ) 2 A(ϕ) However, we ca oly measure w i = x i + u i istead of x i, i = 1,..., ad hece our test would be based o S = 1/2 (w i w )a (R i ) istead of S. The asymptotic ull distributio of the criterio A 1 (ϕ) ( 1 (w i w) 2) 1/2 S is ormal N (0, 1), while that uder H is ( C 2 γ(ϕ, f) β ) N (C 2 + D 2 ) 1/2, 1. A(ϕ) Hece, i the measuremet error model we reject H 0 provided ( 1 A 1 (ϕ) (w i w) 2) 1/2 ( S Φ 1 α ) 2 (2.11) ad the asymptotic power of this test agaist H : β = 1/2 β 0 is the ( ( 1 Φ Φ 1 1 α ) C 2 γ(ϕ, f) β ) 2 (C 2 + D 2 ) 1/2 A(ϕ) ( +Φ Φ 1 ( α 2 ) + C 2 γ(ϕ, f) β (C 2 + D 2 ) 1/2 A(ϕ) ). (2.12) Hece, there is a positive loss of the asymptotic power of the test (2.11) with respect to the test (2.9); it is bouded from above by 2C γ(ϕ, f) β. 2π A(ϕ) The asymptotic relative efficiecy of the test (2.11) with respect to the test (2.9) is ) 1 e w,x = (1 + D2 [ C 2 ] 0 as D. (2.13) More geerally, cosider the multiple regressio model Y = Xβ + e (2.14) with the vector of idepedet observatios Y = (Y 1,..., Y ), vector e = (e 1,..., e ) of i.i.d. errors, the regressio matrix X = X of order p with the rows x i, i = 1,..., ad a

5 Rak tests with measuremet errors 5 ukow parameter β R p. We wat to test the hypothesis H 0 : β = 0 agaist the alterative K : β 0. Let X deote the matrix with the rows x i x, i = 1,...,, ad let it satisfy Q = 1 X X Q as, (2.15) 1 { max (xi x ) Q 1 (x i x ) } 0 as 1 i where Q is a positively defiite p p matrix. I the model without measuremet errors, the rak test of H 0 : β = 0 is based o the vector of liear rak statistics S R p, S = 1/2 (x i x )a (R i ) (2.16) where R 1,..., R are the raks of Y 1,..., Y ad a (i) are agai the scores geerated by ϕ. The asymptotic distributio of S uder H 0 has ormal N p (0, A 2 (ϕ) Q). The test criterio for H 0 has the form T 2 = A 2 (ϕ) S Q 1 S, (2.17) ad its asymptotic ull distributio is χ 2 with p degrees of freedom. Uder the local alterative H : β = β = 1/2 β R p, (2.18) has the asymptotic ocetral χ 2 distributio with p degrees of freedom ad with the ocetrality parameter η 2 = β Qβ γ2 (ϕ, F ) A 2 (ϕ). (2.19) T 2 However, while we observe Y i s ad their raks, istead of x i we ca observe oly w i = x i + u i, i = 1,...,, where u 1,..., u are i.i.d. p-dimesioal errors, idepedet of the e i. Their distributio is geerally ukow, but we assume that they satisfy 1 (u i ū )(u i ū ) p U (2.20) as, where U is a positively defiite p p matrix. I the model with measuremet errors, the test of H 0 : β = 0 ca be oly based o the observable statistics; hece we costruct the test based o the vector of liear rak statistics S R p, S = 1/2 (w i w )a (R i ). (2.21) Uder the above coditios, its asymptotic ull distributio is ormal N p ( 0, A 2 (ϕ) (Q + U) ), while uder H it has the asymptotic ormal distributio N p ( γ(ϕ, F ) Qβ, A 2 (ϕ) (Q + U) ).

6 6 Rak tests with measuremet errors The test criterio for H 0 ow would have the form T 2 = A 2 (ϕ) S (Q + U ) 1 S. (2.22) Uder H 0, the asymptotic distributio of T 2 is χ 2 with p degrees of freedom. Its asymptotic distributio uder H will be ocetral χ 2 with with p degrees of freedom ad with the ocetrality parameter η 2 = β (Q + U) 1/2 Q (Q + U) 1/2 β γ2 (ϕ, F ) A 2 (ϕ). (2.23) Hece, the asymptotic relative efficiecy of the test of H 0 based o T 2 with respect to the test based o T 2 is et e,t = β Qβ β (Q + U) 1/2 Q (Q + U) 1/2 β. (2.24) Coclusio We ca costruct a distributio free rak test of hypothesis H 0 : β = 0 i the model (2.14) eve whe the regressio matrix X ca be oly measured with additive errors. The test looses the efficiecy with respect to the rak test uder completely kow X. 3 Regressio rak score tests with measuremet errors uder the alterative Cosider the model (1.4). The hypothesis H 0 of iterest is (1.5). Gutebruer et al. (1993) costructed a class of tests of H 0 based o regressio rak scores (3.1), that are ivariat to the uisace parameters β 0, β. Gutebruer et al. (1993) showed that these tests are asymptotically equally efficiet as the correspodig rak tests of H 0 uder kow β 0, β, uder some coditios o the tails of the distributio F of the errors e i. The regressio rak scores i the model (1.1) without errors are defied as the vector â (τ) = (â 1 (τ),..., â (τ)) of solutios of the parametric liear programmig problem Y i â i (τ) := max subject to (3.1) â i (τ) = (1 τ), x ij â i (τ) = (1 τ) x ij, j = 1..., p, â (τ) [0, 1], 0 τ 1.

7 Rak tests with measuremet errors 7 The regressio rak scores are dual to the regressio quatiles of model (1.1), ad it follows from the duality that { 1 if Y â i (τ) = i > x β(τ) i 0 if Y i < x β(τ), (3.2) i where β(τ) is the regressio τ-quatile for the model (1.1). The other values â i (τ) correspodig to Y i = x i β(τ) are determied by the restrictios i (3.1) (otice that the umber of such compoets is p + 1). It follows from the restrictios i (3.1) that â (τ) is ivariat to the trasformatios Y i Y i + b 0 + x ib, b 0 R 1, b R p. (3.3) Gutebruer et al. (1993) costructed the tests uder some coditios o the tail of F ad o the x i ; for the sake of simplicity, we shall cosider slightly stroger but simpler coditios. We shall assume that (F.1) F has a absolutely cotiuous desity f, that is positive for A < x < B ad decreases mootoically as x A+ ad x A, where A sup{x : F (x) = 0} ad + B if{x : F (x) = 1}. The derivative f is bouded a.e. ad f (x) f(x) c x for x K 0, c > 0. (F.2) F 1 (t) c(t(1 t)) 1 4 +ε for 0 < t t 0, 1 t 0 t < 1, ε > 0, c > 0. (F.3) 1/f(F 1 (t)) c(t(1 t)) 5 4 +ε for 0 < t t 0, 1 t 0 t < 1, ε > 0, c > 0. Coditio (F.2) demads that E e 1 4+δ < for some δ > 0, ad joitly with coditio (F.1) it implies that F has fiite Fisher iformatio. Coditios (F.1) (F.3) are satisfied for distributios with the tails of t distributio with 5 degrees of freedom or lighter; hece they are satisfied e.g. for the ormal, logistic ad Laplace distributios. For the simplicity of otatio, deote ] X = [1. X (3.4) the matrix of order (p + 1), where 1 = (1,..., 1) R, ad deote x i the i-th row of X. We shall assume that (X.1) lim Q = Q where Q = 1 X X ad Q is a positively defiite (p + 1) (p + 1) matrix. (X.2) max 1 i x i = O(1) ad 1 x i 4 = O(1) as.

8 8 Rak tests with measuremet errors Let ϕ : (0, 1) R 1 be a odecreasig square itegrable score-geeratig fuctio such that ϕ (t) exists for 0 < t < t 0, 1 t 0 < t < 1 ad satisfies ϕ (t) c(t(1 t)) 1 δ, δ > 0. (3.5) Notice that the coditio (3.5) covers the iverse ormal distributio fuctio, amog others. Calculate the scores ˆb i geerated by ϕ i the form ˆbi = 1 0 ϕ(t) dâ i (t), i = 1,..., (3.6) ad the q-dimesioal vector of liear regressio rak scores statistics where S = 1/2 (Z Ẑ) b, b = (ˆb 1,..., ˆb ) (3.7) Ẑ = ĤZ, Ĥ = X ( X X ) 1 X is the projectio of Z o the space spaed by the colums of X. The test criterio for the hypothesis H 0 is where T 2 = A 2 (ϕ) S D 1 S (3.8) D = 1 (Z Ẑ) (Z Ẑ) D, as (we assume), (3.9) where D is a positively defiite q q matrix. Uder H 0 has T 2 asymptotically χ 2 distributio with q degrees of freedom. Uder the local alterative H : δ = 1/2 δ R q, (3.10) has asymptotic ocetral χ 2 distributio with q degrees of freedom ad with the ocetrality parameter η 2 = δ D δ γ2 (ϕ, F ) A 2 (ϕ). (3.11) T 2 Let us ow cosider the situatio that we still wat to test the hypothesis H 0, but the vectors z i ca be oly observed with errors, i.e. we observe w i = z i + u i, i = 1,...,, where the u i R q are radom errors. Deote W = w 1... w ad U = u 1... u (3.12) the q matrices ad Ŵ = ĤW ad Û = ĤU their projectios o the space spaed by the colums of X. We shall assume that G = 1 (U Û) (U Û) p G as (3.13)

9 Rak tests with measuremet errors 9 where G is a positively defiite q q matrix. I this situatio, we shall replace the statistic S by S = 1/2 (W Ŵ) b (3.14) ad the test criterio T 2 by T 2 = A 2 (ϕ) S (D + G ) 1 S. (3.15) Uder H 0, it has asymptotic χ 2 distributio with q degrees of freedom, while uder the local alterative H i (3.10) it has asymptotic ocetral χ 2 distributio with q degrees of freedom ad with the ocetrality parameter η 2 = δ (D + G) δ γ2 (ϕ, F ) A 2 (ϕ). (3.16) The relative asymptotic efficiecy of T 2 with respect to T 2 the is Coclusio e e T,T = δ D δ δ (D + G) δ. (3.17) If oly regressors uder the alterative i model (1.4) are affected by additive errors, we ca still costruct the distributio free tests based o regressio rak scores uder the hypothesis, provided the distributio of errors has the tails of t-distributio or lighter. The loss of efficiecy is give i (3.17). 4 Some ope problems Let us tur back to model (1.4); cosider the problem that the x i, too, are observed oly with errors, ad we observe w i = x i + u i istead of x i, i = 1,...,. The the regressio rak scores ca be modified i the followig way: Y i ã i (τ) := max subject to (4.1) ã i (τ) = (1 τ), w ij ã i (τ) = (1 τ) w ij, j = 1..., p, ã (τ) [0, 1], 0 τ 1.

10 10 Rak tests with measuremet errors Similarly as i the model without errors, the regressio rak scores of (4.1) are dual to the regressio quatiles i the liear programmig sese; however, ot to the regressio quatiles of the model without errors i the x i, but rather to those of the model Y i = β 0 + w iβ + e i, i = 1,..., (4.2) with the covariates w i, i = 1,...,. Agai, it follows from the duality that { 1 if Y ã i (τ) = i > w i β(τ) 0 if Y i < w i β(τ), (4.3) where β(τ) is the regressio τ-quatile for the model (4.2). The other values ã i (τ) correspodig to Y i = w i β(τ) are determied by the restrictios i (4.1), ad the umber of such compoets is p + 1. It follows from the restrictios i (4.1) that the vector of regressio rak scores ã (τ) is ivariat to the trasformatios Y i Y i + b 0 + w ib = Y i + b 0 + x ib + u ib, b 0 R 1, b R p ; (4.4) but it is ot geerally ivariat to trasformatios (3.3). I this setup, if we replace S by S = 1/2 (Z Z ) b (4.5) where the scores b = ( b 1,..., b ) are geerated by ϕ from the ã i similarly as i (3.6) ad where Z is the projectio of the matrix Z oto the space spaed by the colums of the matrix [1. W ] with W = w 1... w the the test based o S is ot asymptotically distributio free without additioal coditios o the model. A possible costructio of a distributio free test is still a ope problem. Refereces [1] Gutebruer, C. ad Jurečková, J. (1992). Regressio rak scores ad regressio quatiles. A. Statist. 20, [2] Gutebruer, C., Jurečková, J., Koeker, R. ad Portoy, S. (1993). Tests of liear hypotheses based o regressio rak scores. Nopar. Statist. 2, [3] He, X. ad Liag, H. (2000). Quatile regressio estimate for a class of liear ad partially liear errors-i-variables models. Statistica Siica 10, [4] Hájek, J. ad Šidák, Z. (1967). Theory of rak tests. Academia, Prague. [5] Koeker, R. ad G. Bassett (1978). Regressio quatiles. Ecoometrica 46, ,