Pergamon
Nonlinear Analysis, Theory, Methods & Application, Vol. 30, No. 5, pp. 3201-3206.1997 Proc. 2nd World Congress of Nonlinear Analysts
0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved
PII: SO36Z-546X(97)00414-8
0362-546X97 $17.00 + 0.00ON TESTING
INDEPENDENCE
WITH
RIGHT
TRUNCATED
DATA
tiLKii
GiiRLER
Department of Industrial Engineering, Bilkent University, 06533 Bilkent, Ankara, TURKEY.
E-mail:ulku@bilkent.edu.tr.
Lry WC&S and phrases: Right truncation,Test of independence, Reverse hazard, Nonparametric estimation
1.
INTRODUCTION
Inference with bivariate data gained considerable interest recently, See eg.[1],[10],[12]. All of these studies howrver consider estimation of the bivariate distribution function under various bivariate censoring models. Recently (:iirler[7,8] considered estimation of the bivariate distribution and the hazard functions under trun- c.atlon/censoring models. The purpose of this study is to investigate procedures for testing the independence of I hc components of the bivariate vector for truncated data. To this end, further properties of the bivariate functrouals introduced in GiirleQ] are elaborated. Two alternative methods for hypothesis testing are sug- gested aud some large sample properties are derived. The procedures suggested in this paper are applicable to left/right. truncated and left truncated right censored data. However to keep the presentation simple we ~~oufinr t.hr discussion to the right truncated case. Also, to avoid technicalities, it is assumed that all the univariat.e and the bivariate distribution functions are absolutely continuous admitting densities.
2. PRELIMINARIES
In hivaria.te t,runcation model, the triplets (Yi, Xi, ri), i = 1,. , R are observed for which (Yi 5 T;). The interest is in the pair of random variables (I’, X) with distribution function F(y, 2). Here, T is a random V;ll~labl(~. which is assumed to be independent of (Y, X), with d.f. G. The marginal d.f.‘s of Y and S are drnr,trd hy 1;‘,, and Fx respectively. For the identifiability of F, it is assumed that a~, 5 (1~ and b.ur < bG ( he? [IA]). Let F(y, .r) = P(Y > y, X > z) be the bivariate survival function and for a univariate function y, +rtch t,hat 6 < g(e) 5 1, let g(e) = 1 - g(x). The distribution of the observed variables (Y, X, T) are given by:
HY,X,T(Y> x> t) = P(Y<y,X<x,T<tIY<T) J
*
= Q -1 F(Y A u, 2) dG(u) 0
\~her,, 0 = ~(1. 2 T), and y A u = min(y, u). This joint distribution function can be further elaborated to rebult in:
hY,X(Y, xl = ~-lc(Y)f(Y, 2) (1)
C(Y) = a-lc(y)FY (Y) = HY (Y) - MY)
(2)
sVhrrr /I,,,~ and f denote the hivariate densities of H anf F. Considering the foregoing relations, Giirler[S] snggests the following estimator
F,(y, x) = i F g$I(y. > Y, xi > 2) Fy,,(y) = n [1 - s(yi)/ncn(yi)l i.Y.>y s(u) = #{i : y, = u} ncqzr) = #{i : y, < u 5 2-i-i) 3201
‘1‘1~~ Pst.mlator Fy,,(y) is suggested by Lynden-Bell[lO] and th e ar 1 g e sample properties of it for left truncation 1nod~1 art= studied in in [17],[2],[6],[14]. Giirler[B] establishes the following results:
FY,n(Y) - FY(Y) = &p(Y) + Cl,,(Y)
(3)
h&a(Y)
-
%X(Y,x)
=
E2,n(Y, z) + fZ,n(Y, z)
(4)
\Vll,V
w$>a>aFkl,n(Y)I = SWy>a>ar IQ,n(Y, z)I = O(log3 n/n)
;rutl ~l,,~(!//), c?,,?(y, x) as defined in Giirler[B] are mean zero i.i.d. variables. Bivariate reverse-hazard in right truncation model:
Following the representation of Dabrowska[4], of th e b’ Ivariate d.f. in terms of the three component bivariate hazard vrct,or. Giirler[B] suggested a bivariate reverse-hazard vector X(y, Z) for right truncated data as described l)t~low. For any hivariate function #(u, zi)> differentiable in both components, let r$(&, V) denote the partial <IPrivat.ive w.r.t. to first argument with similar notation applying to the other component. Then
Lv~. H(y, 1) = - log F(y, z), then
F(Y, z) = Fx(~)~y(y)e~p{-Ny, ~1)
b,-,
bpx
A(Y,Z)
= J J
R(du, dv)Y =
Cz(Y, z) = G,,(Y, z) - HT,X(Y--, cl
= a
-‘[I - G(Y)lF(Y, 2)
with empirical counterpart given by:
(5)
(6)
(7)
Cz,n(f4 v)
=
n-‘#{i
: Yj 5 u 5 Tj ( xj 5 v}
= +%,Y,X(u, V) - %,T,X(+-, v)].
where H, y,x and Hn,~,X are the empirical bivariate d.f.‘s. For these step functions, let A refer to the difference operator. Then an estimator for the reverse- hazard is obtained as follows:
L(u,v) = {
H n,y,x(A% Au) Hn,v,x(Au, v) G,n(u, Au))
CZ,n(U, v) >
G,n(% v) ’ G,n(U, v)
Second World Congress of Nonlinear Analysts
3203
Observe thatH
n,y,x(AX,Xj)
=
Tl-‘CI(Yk
=I$, Xk <Xj)
k=l
and assuming no ties in the data,
C2,n(yi, AX,)
=
n-‘I(Yj < x < q)H n,Y,X(AYij Xj)
=
n-‘I(Xi
I Xi)
Hence
H n,y,x(AYi,Xj)C2,n(YirAXj)
=nw21(yj
I yi 5 q,Xi
I Xj)
Then t,he following estimator for h(y, z) is obtainedn
I(I;I>Y,Xj>~)I(Yj~Yi~Tj,XjIXj)- $5
n2C~(Y;,Xj)
E
I-II-III
3. TESTING INDEPENDENCE OF Y AND X
‘The foregoing discussions in Section 2 lead to the following approaches for testing the independence of Y and .\-. Let
CZ(Y, 2)
T(Y) xl = c(y) = F(Y> x)
FY(Y)
Sl(Y, x) = F(Y, x) - FY(Y)fqY, x)
which can bc estimated by
Sl,,(Y, x) = &(Y, x) - FY,n(YFn(Y, xl 1-There
r (y x) = I_ C2P(YPX) _ #{i : Y, I Y 5 ri , xi > x} n ,
I :~ldrr I 11~ hypothesis of independence, ~(y, Z) = Fx (x) and Sl(y, Z) = 0. Therefore test statistics can be based ()I, tl~r fuuctionals of Sl,,(y,r), such as C,“=, Sl,,(y- taxi ) or ~upl,~i~kS~,~(yi~si) , where (yi,zi),i = l,..,k
woul(l lx’ prt:-selected points on
R2.
Similarly, from the foregoing discussions on the reverse-hazard, we observe I Ilat if !- a~~cl I are independent, A(y, Z) = 0. H ence an alternative approach could be to use x7=, hl,,(y,, xi)01 ~IIII,~,,,~ ~l~,,,(yi, x,) as a test statistics. A related approach for the censored data can be found in [13] and I II? ~~c+&iices therein. Comparison of these tests and their behavior relating to power, unbiasedness, relative
t+ficictlc,v rt,c. are subject to further investigation. However. we present below the following large sample rtwlth. which would lead to establish such properties. Using the results of [l6],[17], following Lemmas are ,)I,1 ;liJld PK)o& of these results and the theorems below, which are not presented here for space considera.tions cali tiuntl in Giirlrr[9].
LEMMA 1:
a-)
El
&7l (& si) IX = Y> xi = zl = l/da/, x)(1 - [l - C(y, Z)]“} b-)
E[ n2c2(;,
\r,) II: = Y, xi = zl = LZKTY, ~)llWY?
x1
874 I
c-)
a
l(Yjj”< Yi 5 rj , xi 5 x.
fJ2Cr,Z(Yi,
Xj)
3 (I$ = y,Xj = Z)]
= { 1 - I,[C(y, Z,] - [l - C(Y, Z)]“/[4n - lP(Y, Z)lLEMMA 2:
[XldU, w) - AI(U, ~)Az(u, v)ldudv+
B*(Y,z)
- &---&Y,x(~u,
v)Hy,x(u, Bv)ldudu +
&(Y>x)
Z%(U) = Z(Yj 5 u < 2-j) Ia(u,v) = I(Yi 5 U < Ti,Xi < W)
LEMMA 3: Let rE(y, 2) be the Hajek projection of r,(y, 2). Then,
sup l%(Y> x)I = wd n/n) ap<y,o<z<m
Second World Congress of Nonlinear Analysts 3205 and
r;(Y,z) = w + q C[C(Y)&(Y, z) - C,(Y, Z)b(Y)l
=
- s + <3,“(Y, 2)
The above Lemmas, together with the results of Giirler(l996) lead to the following theorems THEOREM 1: Let a~ < y < y and suppose sce2(u)F(du) < co. Then,
+
JJ
u>y u>r
“+@‘& vy’“’ vhy,X(uu, v)dudv + R,(y, x)
where
q h,“(Y/, x) + &,“(Y, x)
(,,~;Pob IMY>~)l = wog3 n/n)
Note t,hat E[<h,,,(y, z)] = 0 and .&(y, z) is a sum of identical but not independent random variables. In fact it is a sum of t.wo U-Statistics and this enables us to study the large sample properties, which is still subject to further investigation. The representations in (3) and (4) now leads to a corresponding representation for the statistic Sl,,(y, z), from which the results for the functionals of it can be derived.
THEOREM 2: Under the conditions of Theorem 1,
SW(Y, ~1 - SI(Y, z) = 52,n(~, x) - F(YI~~,~(Y, x1-t %(Y, z)
= Y,(Y, z) + Rz,n(y, 2)
where
sup IRz,n(y, z)I = O(log3 n/n)
(YF)ET.b
The y,,(y, .c) above is again in the form of sum of mean zero i.i.d. random variables, for which standard large sample results apply.
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