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Uniform consistency of nearest neighbourand kernel conditional Kaplan - Meier estimates*

By

Dorota M. DabrowskaCarnegie - Mellon University andUniversity of California, Berkeley

Technical Report No. 86January 1987

*Research supported by the University of California Presidential Fellowship and by theNational Institute of General Medical Sciences Grant SSSY iROl GM35416-02.

Department of StatisticsUniversity of CaliforniaBerkeley, California

Uniform consistency of nearest neighbour

and kernel conditional Kaplan - Meier estimates*

Dorota M. Dabrowska

Carnegie - Mellon University and

University of California, Berkeley

Abstract

We consider a class of nonparametric regression estimates introduced by

Beran (1981) to estimate conditional survival functions in the presence of right

censoring. For both uncensored and censored case, we derive an analogue of the

Dvoretzky - Kiefer - Wolfowitz (1956) inequality for the tails of distributions of

kernel and nearest neighbour estimates of conditional survival function. This ine-

quality is next used to prove weak and strong uniform consistency results. The

developments rest on sharp exponential bounds for the oscillation modulus of

multivariate empirical processes obtained by Stute (1984a).

Key words and phrases: Kernel and nearest neighbour regression, right censoring,

oscillation modulus.

* Research supported by the U'niversity of California Presidential Fel-lowship and by the National Institute of General Ntedical Sciences GrantSSSY lROl GM35416-02.

1. Introduction.

Let T be a nonnegative random variable (rv) representing the survival time of an

individual taking part in a clinical trial or other experimental study, and let

Z = (Z1, * * , Zq) be a vector of covariates such as age, blood pressure, cholesterol

level. The survival time T is subject to right censoring so that the observable rv's are

given by Y = min (T ,X), D = I (T < X) and Z. Here X is a nonnegative rv represent-

ing times to withdrawal from the study. Denote by F (t I z) = P (T > t I Z = z),

Hl(t Iz) = P(Y > t, D = 1lZ = z) and H2(t Iz) =P(Y > t IZ = z) the respective con-

ditional survival functions and let

t

A(t I z) = -JF(s-I z)-1dF (s i z) (1.1)0

be the conditional cumulative hazard function associated with F (t z). In terms of the

cumulative hazard function we have

F (t I z) = exp {(Ac(t Iz)}FJl 1 - AA(s I z)}, (1.2)where AC (s I z) is the continuous component of A(s I z), the product is taken over the

set of discontinuities of A(s z) and s < t, and AA(s z) = A(s z) - A(s - I z). This is

the well known product-integral representation of distribution functions, see for

instance Peterson (1977), Gill (1980) and Beran (1981). To avoid trivial cases it is

assumed throughout that p = P (D = 1) satisfies 0 < p < 1. Furthermore, it is

assumed that T and X are conditionally independent given Z, which is a sufficient

condition to ensure identifiability of A(t I z) and F (t I z). Specifically, for any t such

t

that H2(t Iz)> 0 we have A(t Iz) = -J(H2(S-Iz))-1dH1(s Iz).0

Let (Yj, Dj, Zj), j = 1, ... ,n, be a sample of i.i.d. rv's each having the same

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distribution as (Y,D,Z). Beran (1981) proposed to estimate the subdistribution func-n

tionsH1(tIz) andH2(tIz) by H1 (t Iz)=z I(Y. >t,DY =t B, Dz)=H21(t Iz) =

nY I(Yj > t-)Bnj(z), where Bnj(z) is a random set of probability weights depending onj=1

the covariates only. Beran's estimates of A(t I z) and F (t I z) are provided by

t dHln(s jz)An (t I Z) = -J H2 (s z) (1.3)

and

Fn(t 1z) = H-fl - AAn(SI1)}, (1.4)where the product is taken over s < t. Both An (t z) and Fn (t I z) are right continuous

functions of t, jumps occur at discontinuity points of H in (t I z). Note that in the

homogeneous case, (1.3) and (1.4) are simply the Aalen - Nelson (Aalen (1978), Nel-

son (1972)) and Kaplan - Meier (1958) estimates. Hereafter, we refer to these esti-

mates as conditional Aalen - Nelson and conditional Kaplan - Meier estimates.

In this paper we consider two types of estimates. The first of them is a nearest

neighbour estimate of the form Hin (t z) = H3n (z-'Hin (t ,z) where

n

Hin(t,z) = (naq)-j z I(Y. > t, DG= - G

n ( 1.5)H2n (t,Z) = (nan,)f £ I(Yj > t)K (an n(G (z) - Gn (Zj)))

and

nH3n (z) = (naq)j1 Y K(a71(Gn (z) - Gn (Zj))) (1.6)

Here Gn (z) = (Gn 1(z1), , Gnq (Zq)) and Gnj (zj) is the empirical distribution func-

tion of Zij's, j = 1, . q. Further K is a kernel function in Rq and an is a

sequence of bandwidths. These estimates were discussed in particular by Stute (1984b,

1986a), Yang (1981) and Johnston (1981).

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The second estimate is based on Nadaraya - Watson type weights and takes form

Hin (t jZ) = gn (Z) -Hin (t ,z) where

n

Hin (t,z) = (na I(Yj > t, Dj = -)K(ann (1.7)

H2n (t,z) = (nany-1 £ Ij(Y > t)K(an1(z - Z1))

and gn (z) is the Parzen - Rosenblatt (Parzen (1962), Rosenblatt (1971)) density esti-

mate

ngn (z) = (na,)jY z K(aq (z - Z1)) (1.8)

j=1These estimates were extensively studied in the literature, Nadaraya (1964), Watson

(1964), Rosenblatt (1969), Collomb (1981), Devroye (1981) and Mack and Silverman

(1982) are some references.

For uncensored data, conditional empirical processes were studied among others by

Stute (1986a & b) who considered pointwise consistency and weak convergence results

for estimates based on kernel and nearest neighbour weights. Horv'ath and Yandell

(1986) obtained functional laws of iterated logarithm and rates of Gaussian approxima-

tion. For censored data, the conditional Aalen - Nelson and Kaplan - Meier estimates

were discussed by Beran (1981) and Dabrowska (1987) who showed their pointwise

consistency and, respectively, weak convergence to a time transformed Brownian

motion.

In this paper we are concerned with uniform consistency results. For both uncen-

sored and censored case we develop an analogue of the Dvoretzky - Kiefer - Wol-

fowitz (1956) inequality providing an exponential bound for the tails of the distribution

of nearest neighbour and kernel estimates of the conditional survival function. This

bound is next used to establish weak and strong uniform consistency of these estima-

tors. The results rest on sharp exponential bounds for the oscillation modulus of

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univariate and multivariate empirical processes obtained by Stute (1982, 1984b).

2. Main Results.

Nearest -neighbour and kernel estimates of the subdistribution functions Hi (t I z) are

discussed in Sections 2.1 and 2.2 respectively. The conditional Kaplan - Meier esti-

mates are considered in Section 2.3.

qIn what follows, for any rectangle I = Ili=1 [aj, bj ] in Rq, Is denotes a (small)

8-neighbourhood of I of the form

qIs = r1=1 [aj -8, bj + 8].

Unless specified otherwise, for any real function f (t ,z) = f (t ,z1 ..* , zq ) in R q+1

lIf Ill = sup {If (t ,z) 1: t ER, zEI}.

2.1. Nearest neighbour estimates. Let G denote the joint cdf of Zi and let Gi,j = 1, . , q be its j -th marginal cdf. Further, let

Co(u1 ... ,uq) = P (Gj (Zj) < uj;, i = 1, ,q). If G is a continuous cdf then CO

is a cdf with uniform marginal satisfying CO (G 1 (z 1), ,Gq (Zq)) = G (z1, XZq).

The following assumption on G will be needed.

A.1 The joint cdf G is continuous. The cdf CO has density co with respect to

qLebesgue measure on [ 0,1 ]q. If I = fIl=, [ a1, b1] is a rectangle contained in the

support of G, then for some 0 < 8 < 1,

0 <y= inf{co(u): uij } < sup {cO(u): uEIG } = r< , where IG is a 8-

qneighbourhood of G - IIj=i [Gj (aj ), Gj (bj)].

As for the kernel K, we assume that K is a density with finite support and has

bounded second partial derivatives. Under these assumptions K is function of

bounded variation in the sense of Hardy and Krause ( see e.g. Hildebrandt (1963),

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Clarkson and Adams (1933)). Without loss of generality we assume

B.1 K is a density vanishing outside (-1, l)q. The first and the second partial

a _ a2derivatives K(J)(u) = K(u) and K01)(u) =- K(u) satisfy sup {IK(i)(u) I,aUJ Duj Du,IK(jl)(u)I: ue(_i,1)q, j,l = 1, ,q) <X < oo. The total variation of K is less

than X.

Our final assumption imposes growth rate conditions on an.

C.1 Let 0 < an < 1 and 0 < e < 1 satisfy 2 < -a q+2n(y/8X)min{1, l1(Fq2)},

eo > &1'2max{y/6kX, (712XF)1/2} and an < 6/(1 + 2e0), where e0 is some finite

number and 6, y, r, X are as is assumptions A. 1 and B. 1

Define Hin (t I z) = Hin (t,z) / H3n(z) where

Hin (t,z) = a-q JHi (t I u)K(a1 (G (z) -G (u)))dG (u), (2.1)H3n (z) = a- JK(a-l(G (z) - G (u)))dG (u), (2.2)

and G(u) = (G 1 (u 1) , Gq (Uq)). We shall decompose Hin (t I z) - Hin (t I z) into

two terms R in (t I z) and R2in (t I Z), i = 1,2. The exact definition of these terms is

given in Section 4. The following theorem provides exponential bounds for R lin (t,z)

and R 2in (t,z).

Theorem 2.1. If A.], B.] and C.1 hold then there exist constants di > 0,

i = 1, * , 4 (not depending on n, an, c, or the distribution of (Yi , Di, Z1)) such that

for i = 1,2

P (I Hin -Hin I1I >e-) <P(IRlin IIi >F/2)+P(I R2in III >e12)

P(IIR1in III > Fg/2) < dia-qexpf-d2d582na,q}P(IIR2in III > c1 2) < d3a-q7exp{-d4d6 na,'+l}

where d5 = nmin (X-2F, -2rF2) and d6 = min (d5, 1Yf71).

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Theorem 2.1 can be used to derive uniform consistency results for the estimates

Hin (t I z). To handle the bias term, we assume

A.2. The functions co (u) and Hi (t I G-1(u)), G-1 (u) = (G _ 1 (u 1) , G 1 (uq)),

i = 1,2 have continuous first partial derivatives with respect to uj, uel and

supil a Hi(tIG-1(u))I<oo 0<infla cO(u)I < supi c0(u)l<oo where theauj ~~~~~~~~~~~au*

supremum and infimum extend over u EIG, t E R, i = 1,2 and j = 1, . , q.

Corollary 2.1. Suppose that the conditions A.] and B.] hold and an - 0.

(i) If nan+2 - then IIHin - Hin III ->p 0. (ii) If in addition

Ea, qexp{-pnag+q} < oo for all p > 0 then 1IHin - Hin 1-,> 0 a.s. (iii) If A.2 holds

then IIHin -Hi -I> 0.

Note that if naq+2 - o and logn Inaq+2 -* 0 then the assumption of (ii) is

satisfied. The following corollary deals with the rates of uniform convergence. To

handle the bias term we assume

A.3. The differentiability conditions of A.2 are satisfied by the second derivatives.

B.2. The kernel K satisfies ujuK(u)du = 0, j = 1, ,q.

Corollary 2.2. Suppose that the conditions A.1 and B.1 hold and let an -e 0 and

naq+2 __> 00 Set bn = logaqq/ (naq). (i) If bnan 2 - 0 then

IIH- -Hin II1 = OH(b$2). (ii) If in addition Ea 4 < oo for some 4 > 0 then

IInHi - Hin III = 0 (bn112) as n -e with probability 1. (iii) If A.3 and B.2 hold and

na1+4 - 0 then bn,112 tHin - Hi 1I1 -* 0.

2.2. Kernel estimates. In the case of kernel estimates we require the joint distribu-

tion of Zi's to have a density g with respect to Lebesgue measure. The marginal den-

sities Gj'are denoted by gj, j = 1, ,q. The cdf CO defined in Section 2.1 now

has density

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co(u) = g (Gj71 (u1), ,Gq 1 (uq))IJ=1g1(G1 (uq)).The following assumptions will be needed.

A.4 The joint distribution G has density g with respect to Lebesgue measure. If

q

I =FJ=i [aj,bj1] is a rectangle contained in the support of g, then

O<y=inf{g(z), gj(zj):zeI8}<sup{g(z),gj(zj):zeIS}= r<oo for some 6-

neighbourhood of I. Moreover, 0 < 5F < 1 and sup{cO(u): uE IG,r = 1O <o

where I4r is a 6F-neighbourhood of GG - H1[G1(aj ), Gj (bj)].

As for the kernel K, we assume that K is a density with finite support and K is a

function of bounded variation in the sense of Hardy and Krause. Without loss of gen-

erality, assume

B.3. K is a density supported on (-1, l)q. K is a function of bounded variation

with total variation equal to X < oo.

Finally the following growth conditions on an and e will be needed

C.2. Let O< a < 1 and 0 < e < 1 satisfy 0 < an <6min(l, 1 /F), 4 <. nagqyXand e0 ' cy (2rorqX)-1 for some finite e0 > 0, where 6, y, F, ro and k are as in

assumptions A.4 and B.3.

Define Hin (t z) = Hin (t ,z) / g (z) where

Hn (t,z) = a~JqHi (t Iu)K(a, 1(z - u))dG(u) (2.3)

gn (z) = a q |K (an 1 (z - u)) dG (u) (2.4)

Theorem 2.2. Suppose that the conditions A.4, B.3 and C.2 hold. There exist

constants d7, d8 > 0 (not depending on n,anX, or the distribution of (Yi,Di, Zi)) such

that P (11 Hin - Hin II > E) < d7(Fan )exp {-d8dg _ nang where dg = y2/(F0Fq 2).

Similarly to the case of nearest neighbour estimates, we can derive uniform con-

sistency results. Assume

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A.5. The functions g (z) and Hi (t z), i = 1,2, have continuous first partial deriva-

atives with respect to Z z E I and supi - Hi (t I z) < oo

aziO < inflag (z) I < supi -y g (z) i < oo where the sup and inf extend over t e R,

zE I8, i = 1,2 and j = 1, * * ,9q.

Corollary 2.3. Suppose that the conditions A.4 and B.3 hold and let an -4 0 and

na_- oc. (i) If a -expf- pnag}q- 0 for all p > 0 then IIHi, Hin 1l -,p 0. (ii)

If in addition I -q - pnaq I <oo for all p > 0 then 11 Hin - H Ill- 0 a.s. (iii)If A.S holds then 11 Hin - Hi IIz + 0.

Condition (i) is satisfied if naj+S - 0 for some c > 0. Condition (ii) holds if

log n Inaq _+ 0. The following corollary deals with the rates of uniform consistency.

To handle the bias term, assume

A.6. The differentiability conditions of A.5 are satisfied by the second derivatives.

Corollary 2.4. Suppose that the conditions A.4 and B.3 hold and let an -* 0 and

n- oo. (i) If bn = loga,/nann _< 0 then IHin - Hin III = Op (bn112) (ii) If in

addition Ea 4 <oo

for some 4 > 0 then IIHn - Hiin 11

= 0(b 1'2) as n -o with pro-

bability 1. (iii) If A.6 and B.2 hold and na q+4 then bnT1/211Hin - Hi II. - 0

2.3. Conditional Kaplan - Meier estimates. In what follows, for any 0 < t < 00

and any rectangle I in R q, if f (t,z) is a real function in Rq+1 then

lIf hIII = sup{If(t,Z)I: 0 < t < t, Z E I. To avoid problems with the tails of the

conditional subdistribution functions, we add the following assumptions.

A.7. For O< t < oo, inf{H2(t IG-1(u)): 0 < t < t, U IG} > 0 > 0, where I is

defined as in A.1, and G-1(u) = (G j1 (u 1),I -(u

A.8. For O < X < oo, inf{H2(t Iz): 0 < t < t, zeI} > 0 > 0, where I is defined as

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in A.4.

The exponential bounds for the distribution of a suitably standardized version of

Fn (t I z) will be developed under the following assumptions on an and £.

C.3. Let 0 < an < 1 and 0 < c < 1 satisfy 2 < tO5na q+22(y/432X)min{1, 1/1Iq2},e0 ' e 12( /2)112max{y/(6XF), (y/2kr)112} and an < 6/(1 + 2e0), where e0 is

some finite number and 6,y,F,9 and X are as in assumptions A.1, A.7 and B.1.

C.4. Let 0< an < 1 and O< e < 1 satisfy 0 < an <6min(1, 1 /F),

216 < E05nafqy/X and e0 > 9y(roPqFX)-l/4 for some finite e0 > 0, where

6, y, 17,r0, 8 and X are as in assumptions A.4 and B.3.

Let Hin (t z) be defined as in Sections 2.1 and 2.2. Set

A~(ti) = ~dH1,n (s jz)An (t IZ)=-| ]o Hmn (s-Jz)

and

Fn (t Iz) = exp{-An(t IZ)}Hr (1-AAn (s IZ))s.t

where Anc is the continuous component of An the product is taken over the set of

discontinuities of An and AAn (s I z) = An (s zZ) - An (s - l z). The following theorem

provides exponential bounds for Fn (t IZ) - Fn (t I z). In the case of nearest neighbour

estimates, in analogy to Theorem 2.1, we use a decomposition of Fn (t z) - Fn (t I z)

into two terms Sln (t, z) and 52n (t, z). Their exact definitions are given in Section 6.

Theorem 2.3. (i) Nearest neighbour estimates: Suppose that the conditions A.1,

A.7, B.1 and C.3 hold. There exist constants ci > 0i = 1, , 4 such that

P(IIF - Fn III: >E) < P(IISin 1 >> 8/2) + P(11S2 III > /2)P(IIS1n JI > e/2) < cla, q exp-c2d5 2025 na q1}P(IIS2n ~I> 8/2) < C3an qexp{ c4d68O5nag+l},

where d5 and d6 are as in Theorem 2.1.

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(ii) Kernel estimates: Suppose that the conditions A.4, A.8, B.3 and C.4 hold. There

exist constants c5, c6 > 0 such that P(IIFn - Fn III > £) <

c5(Pa )-q exp{-c6d9E2025 nagq) where d9 is as in Theorem 2.2.

The following corollary deals with uniform consistency of the conditional Kaplan -

Meier estimate.

Corollary 2.5. (i) Nearest neighbour estimates: Suppose that the conditions A.],

A.7 and B.1 hold and an 0. If naq+2 00 then II F - Fn III -p 0. If in addition

Ea-qexp{-pnaq+l} < oo for all p > 0 then IIFn - Fn hf - 0 a.s. Further, if A.2

holds then 11 Fn - F 1f e0.

(ii) Kernel estimates: Suppose that the conditions A.4, A.8 and B.3 hold and let

an 0 and nagq __ If a,-qexp{-pnagq e 0 for all p > 0 then

IFn -Fnl If -*p 0. If in addition Ia -qexp{ pnaqI}< co for all p > 0 then

IIF - F f-I 0 a.s. Further, if A.5 holds then IIjF - F II -0.

The following corollary provides rates of uniform consistency of the conditional

Kaplan - Meier estimate.

Corollary 2.6. (i) Nearest neighbour estimates: Suppose that the conditions A.1,

A.7 and B.] hold and let an -0 and naq+2 - oo*. Set b =logaaqlI(naq). If

b a,-2 - 0 then IIFn - Fn If = Op(b 112). If in additionI

< for some 4 > 0

then II F - F nhf = 0(b 1/2) as n -e with probability 1. Further, if A.3 and B.2

hold and nag 0 then bT-1/2IIFn - F If - 0

(ii) Kernel estimates: Suppose that the conditions A.4, A.8 and B.3 hold and let

an 0 and naq _> 0. If bn = loga-/ (nagn) - 0 then II F - Fn If = Op(b 1/2) If

in addition Ea 4 < oo for some , > 0 then 11IF - Fn If = 0 (b$1/2) as n - oo with pro-

bability 1. Further, if A.6 and B.2 hold and nagq4 -> 0 then bj'2I Fn - F If. - 0.

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3. Preliminaries.

3.1. Local deviations of empirical processes on the unit cube. Our proofs rest on

finite sample tail estimates for the oscillation modulus of univariate and multivariate

empirical processes developed by Stute (1982, 1984a).

Let C(v,u), v = (v1, ,vq1) and u = (u1, ,uq2), q1 . 0, q2 > 1, be a cdf on

[0,1]ql+q2 with uniform marginals and let Cn be the empirical cdf corresponding to a

sample of size n from C. For any rectangle I c [ 0,1 ]q 1+q2 let C(I) and Cn(I) denote

the C- and the Cn- measure of I and let cn (I) = Cn (I) - C (I). Given 0 < vi < 1,

i 1=1, ,q1 and 0 < uJ1 < uj2 < 1, j = 1, * ,q2 let

ql q2

I(v,u)=HIi=l[vi,l]xHIj=l[ujl,uj2]. For a=(a1, ,aq2)E[0,1]q2 define the

oscillation modulus

on(a) = n1/2 supII V(I(v,u)) °: < Vi < 1 2 - Uj11 < aj)where i = 1, ,q1 and j = 1, ,q2, q1 . 0, q2 . 1. The following lemma will

be useful.

Lemma 3.1. Let s satisfy 2 < sn12 and e0n1/2 minaj > s for some finite e0 > 0.

There exist constants e1, e2 > 0 (not depending on n, a, s or the distribution C) such

that

P(cO)n(a) > s) < e1(minaj)q2 exp{-e2S2/minaj} (3.1)

For q1 = 0, the inequality (3.1) corresponds to Theorem 1.7 in Stute (1984a). For

q1 > 0, a simple modification of the proof of this theorem is needed. See Section 7.

The factor min aj appearing in the exponent of (3.1) corresponds to the upper

bound on the C -measure of rectangles involved in the definition of con(a). If no

assumptions on C are made, this bound is equal to mina. If the last q2 marginals of

C have a bounded joint density c0(u) with respect to Lebesgue measure on [0, 1 ]q2,

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then this factor can be reduced to r Ijq2laj, provided 2 < sn112 and eon 1/21 HJ 2l

ai > s, where Fo = sup c (u). Finally, we note that a variant of (3.1) remains valid

when the supremum in the definition of cOn (a) is restricted to rectangles I (v,u) con-

tained in some fixed rectangle I c [ 0, 1 ]ql+q2

3.2. Copula functions. We return to the framework of Section 2. Let

M1(t,z)=P(Di =1,Yi <t,Zi <z), M2(t ,Z)=P(Yi <t,Zi <z) and let M1n and

M2n be the corresponding empiricals. (Here Zi < z means Zij < zj, j = 1,.* ,q.)

Denote by H1 and H2 the first marginal cdf's of M1 and M2, and recall that the mar-

ginal cdf's of Z,j's are Gj, j = 1, * ,q. Under assumptions A. 1 and A.4, the margi-

nals H1 and H2 need not be continuous. Furthermore, M1 is a subdistribution func-

tion, M1(oo, * . - ,*o) = P (Di = 1) = p and 0 <p < 1. We shall find it convenient to

replace M1 and M2 by continuous cdf's.

Define M1(x,t,z) = P (Di < x, Yi < t, Zi < z) and let M1n be the corresponding

empirical cdf. M1 is a proper cdf. Its first marginal is purely discrete assigning mass

1 - p to x = 0 and p to x = 1. The remaining marginals are the same as those of

M2. We shall replace M1 and M2 by continuous cdf's. The idea is to spread the

jumps uniformly over intervals that will be inserted at each jump point (see e.g. van

Zuijlen (1978) for the discussion of the technique involved).

Let {4m: m = 1,2,...) be the set of discontinuities of H2 and let {Pm : m = 1,2,...}

be the corresponding heights of jumps, £Pm < 1. Define transformations

Q*(x) =x + (1 -p)I(x > 0) +pI(x > 1) and 42(t) = t + 2pm I(t > m). Further-

more, let {UA: m = 1,2,..., i = 1, ,n} and {VA: m = 1,2, i = 1, ,n} be

mutually independent sets of unifonn (0,1) rv's independent of the sample (Di, Yi, Zi),

i =1, ,n. Define new variables Di =1 (Di) + (1 -p)Vli (1 -Di) +pV2iDi

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and Y* = 4 (Yi) + YPmUmiI(Yi = 4m) Let M1 and M2 be the joint cdf's of

(Di1Y *IZi) and (Yi Zi), respectively, and let M1f and M2n be the corresponding

empiricals. Clearly, Mi* and M2* have continuous marginals and it can be easily

verified that M1(x,t,z) = M1'(f* (x), 42 (t),z) and M2(t,z) = M2 (42 (t),z), and with

probability one, the same relationship holds among the empiricals.

Following Stute (1984a), we shall use a representation of M1 and M2* in terms of

the so-called copula or dependence function. Let H 1* and H2* be the first marginals

of M1 and M2, respectively. The copula functions pertaining to M1_' and M2* are

cdf's C1 and C2 defined on [0, 1 ]q+2 and [0, 1 ]q+l, respectively, satisfying

M (x*,"t**,z) = C (H*(x*), H2*(t*), G(z)) and M2* (t*,z) = C2(H2 (t*), G(z)) where

G(z) = (G 1(z 1), * *Gq(zq)). Furthermore, if Cin and C2n are empirical cdf's

corresponding to C1 and C2, then with probability 1, M*(x *, z) = Cln(H (x*),

H2*(t*), G(z)) and M2 (t ,z) = C2n(H2 (t )I G(z)).

Let us return now to the original processes M1 and M2. Let in = Min - Mi and

qcin =Cin -Ci, i = 1,2. It follows now that if I = [t1,t2] xH11=l[Z1,Zj2] is a rec-

tangle in Rq+1 then with probability one

PlnV) = aln(JO x J); P2nV() = aC2n(J) (3.2)

where Jo=[H>(p(l/2-)), H> (l(3/2))] and J =[H2>4H(t1-)),H2(p(t2))]xq

I-j=l [ Gj (zj 1), Gj (Zj2)]. In conjunction with Lemma 3.1, this representation will be

used to compute the oscillation modulus Of 3in, i =1,2.

4. Proof of Theorem 2.1 and Corollaries 2.1 and 2.2.

In what follows, we write If to denote the supremum norm of any real valued

function f defined on R, Rq1 or R

- 14 -

Define Ain (t, z) = Hin (t, z) - Hin (t,z) i = 1,2 and A 3n (Z) =

H3n (z) - H3n (z) = A 2n (O,z). Under assumption A.1, for z E I we have

H3 (z) = aR-qK(a,-I(G(z) - G(u))) dG (u)

= JK(v)cO(G(z)-van)dv.2ywhere v = (v1, * ,vq), 0 < vi < 1. Let Qln and Q2n be the events

Qin = {IIG~n - Gj 11 < 1l/2a (q1+l)/2y/(6XF) j = 1, ,q } (4.1)and

2n, = {infH3n(z)> y/2:zeI} (4.2)where e satisfies condition C. 1. For i = 1,2, we have

3A =I(Qf )A I+(1)Bj.. whereAin =I(Qln )Ain + I (Qln )Bj=1in'

Blin(t,z) = an nI(y >t)K(arT(Gn(z) -G(u)))d(Min(y,u) -Mi (y,U))B2in(t,z) =

an q I I (y > t) K( ) (a-,(G(z) -G(u))) (fnj (zj) - fn (uj)) dMi (Y, u)j=1n

B3in(t,z) = 2-1 an I I I (y > t)K(il)(An (u,z))

xj f3j (Z) - (Uj ) (fni (Z))-3n (ul )) dMi (Y ,u)

Here Pnj (uj ) = Gnj (uj ) - Gj (uj ), and

An (U,Z) = (Anl(U1,Z1), * * * , Anq (Uq, ZqA where Anj is a function assuming values

between aj-1(Gnj (zj) - Gnj (uj)) and a-'1(Gj (zj) - Gj (uj )). Further, let

Bj3n(z)= B2n(0z) j = 1,2,3. By assumption C.1, on the event Qln the inequality

IGnj(zj) -Gnj(uj) < a, entails IGj(zj) -Gj(uj)| < a, + 2e1/2a (q+l)/2y/(6XF) < ea,

e = 1 + 2e0. Since K vanishes outside (-1, 1)q, without loss of generality we may

assume that in the case of I(fi1n)Bjin the above integrals are restricted to those u-

values for which I Gj (uj) - Gj (z) J < ean, = 1, ,q.

Finally, we have Hi - Hin = R1 + R22in i = 1,2, where

- 15 -

Rlin = I (Qlin n Q2n )H3_n(Bliin + B2in )-Hin H3_n(B13n +B23n)}1

2in = I (Qin n Q2n ) 3n 3in-HinHnB33n } + I (Q -Hin }i

Lemma 4.1. Let A.1 and B.] be satisfied. If 0 < an < 6/e and 0 < Tl < 1 satisfy

2 < nnagqIX and e0 > il/(XJ) then there exist constants 11, 12>0 such that for i = 1,2

P(I(Qn ) 11B lin 11 > 71) < Ilan q exp 12-I22 na q/ (rX2) }

Proof. Let Din = Min - Mi, i = 1,2. Further, set

qI(t, z; u) = [t,oo] xrIj=l Euj, Gj-l(Gj(zj) + ean)]. Integration by parts (Hildebrandt

(1963), Young (1917)) implies that on the event Qln

inB(t,z) < a -q IinV(I(t,z;u))I IdK (anl(G(u) -G< n XSUP I Pin (I (t, z; u)) I

where the supremum is taken over u such that

G7-1(Gj (zj) - ean) < Uj < G7-1(Gj (zj) + ean), j = 1, ,q. Replacing processes

Pin by cin and using (3.2), we have with probability 1

ln (I(t ,z;u)) = a1n(JxOJ(s,v;w))2n(IV(t,z;u)) = on(J(s,v;w))

qwhere J0 is defined as in (3.2) and J(s,v;w)= s, 1] xIHj11[wj,vj + ean ],

s = H2 (p2 (t-)), vj = Gj(zj) and Vj- ean < wj < vj + ean. Lemma 3.1 and the

remark following it, applied with q1 = 2, q2 = q, a = (a1, ,aq), a1 = ea, and

s = rjn 12a q/k entails P (I (Qln) JIB11 > ) qlk)

-q a -qexp{-m2r2 na q I (X2J7e )} for some constants m 1,m2 > O. Inequality (3. 1)

applied with q1 = 1 and the same choice of q2, a and s yields

P(I (Q.1 )IIB 12n II>ri) . m3qqa,-qexp{- 2na ql (X2FeqI for some constants Mi3,in4 >0. The conclusion follows by setting 13 = max(m 1, Mi3)e-q and

14 = min(Mi2,in4)e&.

Lemma 4.2. Let A.1 and B.] be satisfied. If 0 < an < /e and 0 < r1 < 1 satisfy

- 16 -

2 < -1nan I (EXq) and e0 r1 / (Xr) then there exist constants 13, 14 > 0 such that for

i = 1,2 P(I (Qln 11B)2in1B > r1) < 13a Tlexp{-14 q2nan 1 (1r)2)

Proof. On the event Qln, we have

IB2in(t,Z)l<

an (q+1) 5J I f3nj (zj) - Pnj (Uj ) I K(i)(a, -(G(z) - G(u)) co(G(u)) I Ei=l dGi (ui)

< 1 rqX max sup I a2nj (Gj (zj)) -2nj (vj)Iwhere the suprema extend over vj such that Gj (zj) - vj I < ean . Further, a2nj,

j = 1, ... ,q are the last q marginals of a2n. Lemma 3.1 applied with q1 = 0 q2 = 1

a = ean and s = Tin 112an l (Xrq ) entails P (IK(Q2ln ) IIB 2in 11>'i)

<qP(c(ea 1)> Tin1/2a, /(XEFq)) <qmle-la-lexp{-m2Ti2na / e (Xrq )2} where ml

and m2 are the constants of Lemma 3.1. The choice of 13 = qmle-1 and

14 = M2e-1q-2 completes the proof.

Lemma 4.3. Let A.] and B.] be satisfied. If 0 < an < /e and 0 < c < 1 satisfy

2<. naq(y/ 82) min{ 1, 1 / (q)} and e0 > ey/ (8XE) then there exist constants

d1,d2 > 0 such that for i = 1,2 P(I1Rlin III>g/2) < dla -qexp{ d2d5 qnaI} where

d5 = 7k-2 min(P1, 2).

Proof. We have 11 R lin III < I(Qln)y1 (11 B lin 1I + 11B 2in hII+ 1lB 12n 11, + 11 B 22n 11,). Under condition C. 1, the assumptions of Lemmas 4.1 and 4.2

are satisfied by Ti = vy/8. Furthermore

P (11 R in II > 6/ 2) < 2P (I (Qln)JIB lin 11 > vY/ 8) +2P(I(Q ) 11B 2in 11 > Ey/ 8) < 21 la, q exp{ 2-262 nanqy2 / (64X2r) )

+ 213 af exp- 1462 nany2/ (64%3E2)}.The conclusion follows by setting d1 = 2 max(11, 13) and d2 = min(12,14) / 64.

Lemma 4.4. Let A.] and B.] be satisfied. If 0 < an < 61e and 0 < Ti < 1 satisfy

2 < tin2aq+2 I (Erq2) and eO . 2fr / (XE) then there exist constants 15, 16>0 such that

- 17 -

for i = 1,2 P(I (Q21n)IIB3in 11 > 11) < l5an7exp{-166na,+l/(X)}

Proof. We have

I(Qln) IB3in(t Z) I

<21 ana +2) (Xrq 2) max sup I X2nj (Gj (zj)) - X2nj (vj) X2nil (GI (z1)) -X2nl (V1)

where the suprema extend over vj, v1 such that I Gj(zj) vj <ean and

IGi(zI) - v1 < ean, ],l = 1, ,q. Furthermore, 2nj j = 1, , q are the last q

marginals of X2n. Lemma 3.1 applied with q1 = 0, q2 = 1, a = ean and

s = (2rna 2+q)12 /(XFq 2)l/2 entails P(I(QJn)IIB3in > r1) . q2P((a( )s)

< q 2m le -la jlexp{2m 2e-lrqna 1+ql (%Xq 2)} where m1 and m 2 are constants of

Lemma 3.1. The conclusion follows by setting 15 = q2mle1 and 16 = 2m2e1q 2*

Lemma 4.5. There exist constants 17, 18>0 such that P (Qfl) <

11exp{-126na q,l ,2/(6F)2}.

Proof. The Dvoretzky-Kiefer-Wolfowitz (1956) inequality entails

P(Qln) 1/2a+l)/2/(6))j=1

< qm 1 expl-2 ena,+1q+/ (6XF)2}where m1 is the universal constant of the Dvoretzky-Kiefer-Wolfowitz inequality.

Lemma 4.6. Let A.] and B.] be satisfied. If 0 < an <6le and 0 < & < 1 satisfy

2 < gna~n+2(y/6X)min{l,l/(Fq2)} and eO . &l2max{y/(6XF),^y12/(32X)"2}, then

there exist constants 19, 110>0 such that P(Q2c) < lga -qexp{-110l11cna q+l} where

11 = min{y2I7-122, 72v2x-2 yr'X}-l

Proof. We have

P(Q2C) < P(sup(H3 -(z)-H3(z)) > infH3 -(z)- y/2, z E I) PY12) YF-1/ /6)+ P V (Qln) JIB22n 11 > 78 1/26) + P V (Qln)IIB32n 11 > rz/6)

The assumptions of Lemmas 4.1 and 4.2 are satisfied with q = y1/2 /6 and those of

- 18 -

Lemma 4.4 with rl = y /6. It follows that the right-hand side of the above inequality

is bounded by

7exp{-18 ena q+ / (6rX)2} + 11 a -qexp -12e na q I (36F2X2)}+ 13an exp{-14enan+/(6F)2} + 15a1 exp{-16ena+ y/ (617)}.

The conclusion follows by setting 19 = max(l 1, 13,15,17) and

110 = min(l2/36, 14/36, 1616, 18/36).

Lemma 4.7. Let A.] and B.1 be satisfied. If 0 < an < le and 0 < e < 1 satisfy

2 < £nan+2(y/6k)min{l,1/(Fq2)} and e0 > &'2max{y/(6XF), 71y21(2X%)"2}, then

there exist constants d3,d4>0 such that for i = 1,2

P(IIR2 Ii /2) <d2 qexp.-dq+ where d6 =

min{yP1X-2, y2F"2X-2 y,'1X-}.

Proof. We have IIR2inIIi < I(Qln n Q2n)Y1(IIB3in II + IIB33n III) +I(Qfc 'Q2c )IIHin -HiniI. Lemma 4.4 applied with rq = ey/4 and Lemmas 4.5 and

4.6 entail

P (IIR2in III > e/2) < 2P (IIB 3in ll (Q1,n) > uy/4) + P (Qlc) +1 (Q2c) < 215a71exp{ -l6 na,q'1y/(4rk)} +

l7expI{-l8naq+,1/(6Fk)2j + 19a -qexp{ -10l1enaq+l}.The conclusion follows by setting d 3 = max (215,17,1 9) and d 4 = min (16/4, 18/36, 1 11)

and d6 = 111.

To complete the proof of Theorem 2.1 it is enough to note that if C. 1 holds then

the conditions of Lemmas 4.3 and 4.7 are simultaneously satisfied. To prove Corol-

laries 2.1 and 2.2, we consider first the bias term.

Lemma 4.8. If A.2 holdts thien IIHin - Hi II, = 0 (an). If A.3 and B.2 hold then for

i = 1,2 IHin - HiII. = O (an2)

Proof. We have

- 19 -

HII(t,z) = a (t Iu)K (a-' (G(z) - G(u)))dG (u)

= an qJHi (t Iu)co(G(u))K (an 1 (G(z) - G(u)))FJdGj (uj)= JHi (t IG-1(G(z) + van ))c O(G(z) + van)K (v)rldvj

where v = (v 1, ,vq), -1< v < 1, j = 1, * * * , q. A one term Taylor expansion

shows that for i = 1,2 suplHin (t ,z) - H (t Iz)co(G(z))I = 0 (an). Since

H3n (z) = H2n(OZ) and H2(OIz)cO(G(z)) = co(G(z)), we also have

supIH3n (z) - cO(G(z))I = 0 (an). This implies the first part of the lemma. The second

follows from a two-term Taylor expansion.

Proof of Corollary 2.1. (i) For n sufficiently large, the conditions of Theorem 2.1

are satisfied. Since na 3 -> oo, we have a,-qexp{ pna+l} e* 0 for all p>O. It fol-

lows that IIHin - Hin I -p 0 as n -* oo. (ii) follows from (i) and the Borel - Cantelli

theorem. Lemma 4.8 implies (iii).

Proof of Corollary 2.2. (i) Choose arbitrary 4 >O and set

e = f-n = {bn ( + t)/d2d5l 1/2 For n sufficiently large, the conditions of Theorem 2.1

are satisfied. With this choice of E, we have P (IR uin Ii > ) = d a 4q e O so that

IIn I = Op (b1/2). Furthermore, P (hR 2in >R) .

d3exp {log a -q[ 1 - d (na q+2/log a-q)2]}, where d = d4(1 + 4)1"2I(d 2d5)112 Since

log a-qlna q+2 _* 0, for n sufficiently large P(hR2inqI>a) . -* 0O 0 that

hR2inhR = Op((b 12) as n -> oo. (ii) follows from (i) and the Borel - Cantelli theorem.

Lemma 4.8 implies (iii).

- 20 -


We consider first the processes Ain (t,z) = Hin (t,z) - Hi (t ,z), i = 1,2 where

Hin (t ,z) and Hin (t,z) are defined by (1.7) and (2.3).

Lemma 5.1. Let A.4 and B.3 be satisfied. If rj < 1 and an < 1 satisfy 2 < rlnanqlXand e0 2 r1(Fo1q7X)-l for some finite e0> 0, then there exist constants 11, 12 > 0 such

that for i = 1,2 P(1 Ain >A) . n naq

Proof. Recall Pin = Min - Mi, i = 1,2. Further, define

I (t ,z,u) = [t,o]xljL=1[min(uj,zj), max(uj,zj)]. Integration by parts (Hildebrandt

(1963)) yields, after some algebra

AinA(t,z)I = a jqlfI(y > t)K(a -1(z - u))dPin3(y,u)I (5.1)= aI Jf|in (I (t ,z;u)) dK (an (z-u))

where integration is restricted to those u-values for which Z- an < uj < zj + an,

j = 1, ,q. Set J (s,v;w) = s, 1]xIj?q1v[ Vj + min (O,wj ), vj + max (0,w )] and let

J0 be defined as in (3.2). If s = H2 (4*(t-)) and Vj = G1 (zj) then, after a change of

variable in the right-hand side of (5.1), we obtain from (3.2)

Ain (t ,z) I ' an q x1n (J0x J (s,v;w))i Idx(w) I if i = 1

naJi a2n (J (s ,v;w))i Id V(w)qI if i = 2.

Here integration is restricted to those wv-values for which

Gj (Zj - an) - Gj (zj ) < Wj < Gj (zj + an) - Gj (zj), j = 1, ,q. Further,

V(w) = K(w'), WJ' = (zj - GJ1(wj + G1(z1)))/an, j = 1, ,q.

Under assumption A.4, by the mean value theorem Gj (zj + an) - Gj (zj) < Fan

and Gj (zj - an) - Gj (z1) > -yan, so that w i c Fan, j = 1, , q. Moreover,

CO(w), the joint distribution of Gj (Zij), j = 1, , q has density

co(w) =g(G 1 (w ),I G7G1(wq))/Hf 1g (GA(w1)) with respect to Lebesgue meas-

ure on [0, 1 ]q. Under assumption A.4, co(w) is bounded on the rectangle

- 21 -

q

[ Gj(a1) - 617, Gj (bj) + 617] by ]O. Lemma 3.1 and the remark following it,

applied with q1 = 2, q2 = q, a = (a1, ,aq), aj = Ian and s = l7k-'n1/2agq entails

P (IIA in III > Ti) . P (On (a) > l-ln a2an) . m an expm22O n for

some constants Mln, M2 > 0. The same choice of q2, a and s but q1 = 1, yields

P(IIA2n III > ) < M3 an exp1-M4 nan) for some M3, M4> 0. The

conclusion follows by setting 11 = max(ml,m3) and 12 = min(M2, M4).

Proof of Theorem 2.2. Define A3n (Z) = gn (Z) - n (Z) = A 2n (O,z), where gn and

g are given by (1.8) and (2.4). Under assumption A.4, g (z) 2 y for z I. Consider

the event 23n = {infgn (Z) > y/2, z E I 1. We have

P (Q3n) infgn (Z) - /2, z E I)

 y/2) ' P Q1A 2n III > y/2).Further, on the event Q3n,Hin - Hn III <

1h(linA +II+ 3n II) ' Y'l(IIAin III + hIA2n III). Lemma 5.1 applied with Ti = ey/2

yields

P (1Hin -2Hin1>).< P(1A2n 11 > y/2) + P(1Ain 11 > ey/2)+P (IIA 2n 11 > F-/2) < 31 l an- expI- 12X-2]FO +6na4.

The conclusion follows by setting d, = 311 and d2 = 12/4.

To prove Corollaries 2.3 and 2.4, we consider the bias tern first.

Lemma 5.2. If A.5 holds thten 11 Hin - Hi 11 = 0 (an)- If A.6 and B.2 hold then

11 Hin -Hi Il = 0 (an )

Proof. We have

Hin ( Xz= a qHi (t I u) K(a, 1(z - u))g (u) du= Hi(t Iz + van)g(z + van)K(v)dv

where v = (v1, * * ,vq), -l < vj < 1, j = 1, * . ,q. A one term Taylor expansion

shows that suplHin (t ,z) - Hi (t i z)g (z) I = O (an). Since n (z) = H2n (0,z) and

- 22 -

H2(0 1 z)g (z) = g (z), we also have supl g' (z) - g (z) j = 0 (an ). This implies the first

part of the lemma. The second follows from a two-term Taylor expansion.

Proof of Corollary 2.3. (i) For n sufficiently large, the conditions of Theorem

2.2 are satisfied. It follows that IIHin - Hin III -*p 0 as n -+ oo. (ii) follows from (i)

and the Borel - Cantelli theorem. Lemma 5.2 implies (iii).

Proof of Corollary 2.4. (i) Choose arbitrary > 0 and set

e = En = {bn(1 + t)ld8d9} 12. For n sufficiently large, the conditions of Theorem 2.2

are satisfied. With this choice of £, we have P(IIH - Hin 11j > £) < d7-q a$4q _. 0

so that IIHin -Hin IIl = OpH(b$2). (ii) follows from (i) and the Borel - Cantelli

theorem. Lemma -5.2 implies (iii).


The proof is similar in both cases. Let Qn be the event

Qn = {infH2n(t-Iz)> f/2, 0 < t < X zel}.

Lemma 6.1. Suppose that assumptions of Theorem 2.3 are satisfied. There exist

constants 11,12 > 0, such that P (QC) < 11a -qexp {-12d6klna+1 for nearest neigh-

bolur estimates and P (Q c) < d q,(2a2)exp{-d8d9ek2naqg4} for kernel estimates.

Proof. We have

P(Qn) . P(sup(H2n (t-Iz) -H2n (t-Iz)) > infH2n (t-Iz) -0/2) 0/2).

In the case of nearest neighbour estimates, Lemmas 4.3 and 4.7 entail

P(IIH2n - H2n III > 0/2) < P(IIR 12n III > 0O1F2/4) + p(IIR22n III > 0O/4). dla-qexp{-d2d502kna q/4} + d3an qexp{d4d 60qna1+l/2} .

The conclusion follows by setting 11 = 2max(d1, d3) and 12 = min(d2/4, d4/2).

In the case of kernel estimates, Theorem 2.2 implies

- 23 -

P (11 H2n -H2n III > 0/2) Oe/2)d7(a,an )-q exp {- d8d ek2nan/4}.

We consider next the conditional Aalen - Nelson estimate. Integration by parts

shows that on the event Qln

H1n(t IZ) - H1n(t Iz)An(t Iz)-An(t Iz) = - nt Hn(t z)

Hn(s Iz))- dHs(s-Iz)

O H2n(s-1)t H2n (s-IZ) - H2,,,(s-IZ)2n(5-IZ)H2n( IdHin(s I Z) (6.1)0 H2n(s-Iz)H2n(s-Iz)

with probability 1. To handle the nearest neighbour estimates, set

I (5n )(An - An) = Q1 n + Q 2n where for j = 1,2, Qjn is defined by replacing

Hin - Hin with Rjin in the right-hand side of (6.1). The following lemma is now an

immediate consequence of (6.1)

Lemma 6.2. In the case of nearest neighbour estimates IlQjn 117<

20-2{2II11Rin III + IIRj2n 1 I}. In the case of kernel estimates

I(Qn)11An -An 117 <2n2{I2HIl -H in III + 1IH2n -H2n II 1

We proceed to consider the conditional Kaplan - Meier estimate. First note that

for 0 < t < t and z E J.

Fn (tI

z) = exp- Ac(t I z) l ((1 - AA^(s z))S<t

> exptld (sIZ) }I(1- AH 2(s IZ) = H2 (tIz) (6.2)H2n{S,1z s.t H2n (s-IZ)

where H 2n(t I z) is the continuous component of H2n (t z). The last equality follows

from the product integral representation of survival functions given in (1.1) and (1.2).

Further, from Proposition A.4.1 in Gill (1980, p.153), on the event Qn we have

Fn(t IZ) - Fn(t IZ) = -Fn(t IZ) d fn (s I Z) (6.3)0 Fn (S-Z)

- 24 -

where

t

Vn (t iz) = (1 - An(s IZ))fd(An(s IZ) -A(s IZ))0

with probability 1.

To handle the nearest neighbour estimates, let NVin and XV2n be defined by replacing

An - An with Q in and respectively, Q 2n in the definition of Nn(t I z). Further, let

tF (s -I z)Sin(t Iz) = -I(Qn)Fn(t Iz)f n

dvln (s Iz),0Fn (s -IZ)

and S22n = Fn - Fn - S n = IMD (Fn - Fn) - S3n, where

tFn (s - I Z)S3n (t IZ) = -I (Qn)Fn (t I Z)f F( dXJ2n(S I Z).

0Fn (s - I Z)

Proof of Theorem 2.3. We consider first the kernel estimates. Integration by

parts (see Shorack and Wellner, 1986 p.305) and (6.2) yield on the event Qn

t F, (s - Iz)IFn (t I Z) -Fn (t IZ) I< I | - dRn (S I Z)I

0 Fn (s -z)FnF, (Z)

0 Fn (s Z)t

Vn(tIz)I = IA(t Z)-An(tz) IZ)I+

sp{sFZ) F( z) IAA(s Iz)- AA(s Iz)I<Fn(s-Fn(SIZ))

.30 nA(t IZ) - An(t Iz)I

< 60-'{JIH21 -H2n III + 21l H ln -H ln III ((6.5)

- 25 -

with probability 1. Lemma 6.1, (6.4) and Theorem 2.2 applied with c05/54 yield

P(IIFH F 1>) P(Qc)+3P(IH HnIIH > 95/54)< d7(1an )-q exp{- d8d9O2nan,/4} + 3d7(ran )'q exp {f- d8d9O25nan/29 16).

The conclusion follows by setting C7 = 4d7 and cs = d8/2916.

As for the nearest neighbour estimates, Lemma 6.2 and a similar chain of inequali-

ties as in (6.4) and (6.5) shows that on the event Qn

IlSln 117 < 1805{2IIR11in I +IIR12n Ij} IIS3n 117 < 18095{211R21n 11 + IIRm hi)

Lemma 4.3 applied with C95/54 implies

P(IISin 117"> t/2) < 3P(IIR in 11 > E35/108)< 3d -a7qexp{-d2d5O2k2na2/2916)

Lemma 6.1 and Lemma 4.7 applied with &05/54 entail

PO(IS2n II" > e/2) < P(Q) + P(IS3n 117> e051108)II1a, qexp{-12d6cO2nan} + 3d3an,exp{-d4d605na +1/541.

To complete the proof, set c1 = 3d1, c2 = d212916 C3 = max(11,3d3) and

C4 = min(12,d4/54).

The proof of Corollaries 2.5 and 2.6 is similar to those of Corollaries 2.1 - 2.4.

We consider the bias term only.

Lemma 6.3. (i) Nearest neighbour estimates: If A.2 and A.7 hold then

lFn - F 117 = 0 (an). If A.3, A.7 and B.2 hold then 11Fn - FFn? = 0 (a,2)

(ii) Kernel estimates: if A.5 and A.8 hold then IIFn - F11F = 0 (an) If A.6, A.8

and B.2 hold then 11 Fn - F 11 =-0 (an2).

Proof. The proof is the same for both nearest neighbour and kernel estimates.

Using a decomposition siimilar to (6.1), we have IAn(t IZ) - A(t Jz)I

< 20-2{21H'ln - H1 Ij + IIH2n - H2 III}. A similar chain of inequalities as in (6.4)

and (6.5) gives IFn(t IZ) - F(t Iz) I . 18095{2jHIHn - H1j1I +IIH2n - H211I}. Lem-

- 26 -

mas 4.8 and 5.2 complete the proof.

7. Proof of Lemma 3.1.

The proof of inequality (3.1) rests on a modification of arguments used in the proof

of Theorem 1.7 in Stute (1984a). We consider first the case of q2 = 1.

Choose 0 < 8 < 1/2 and let b1, * * bq1, a < 1/2 be such that C(I(v,u)) < 8/4

where v = (v1, v..,Vq1) and I (v,u) = fl112[ vi, 1 ] x [ 0, u]. By Lemma 1.2 in Stute

(1984a), there exists c = c(6) such that for all s > 0, 2 < sn12C(I(l - b,a))112,1 - b = (1 - bl-, * , - bql) and 32 < s262(1 - 26)2,

P (supn 1/2 i(cnV (v,u )) I > s (C(( - b, a 3))12)< c(8)P(n1121c(X(V-b,a))I . s(l - 28)q1l ((C((l - b a))12).

The supremum on the left-hand side extends over v such that 1 - bi < vi < 1

i = 1, *, q 1 and 0 S) <O<vj<1O s (1 - 28)q, + 1

provided 2 < sn 1/2, 32a < s262(l - 25)2 and a < /4. By Bernstein's inequality, the

right-hand side is bounded from above by

2c (6)exp{-s2(l - 28)2q1+212a (1 + eo(l - 26)ql + 1/3)) < e 1'exp{-e2's2/a }

for some el', e2' > 0 depending on 6 and e0.

We proceed next in a fashion similar to the proof of Lemma 2.4 in Stute (1982).

Let R be the smallest positive integer satisfying 62a /4 > IR . Then

P (con(a) > s) <

R-1I P( sup n112jcnV(I(v,u + i/R)) -Xn(I(v,i/R))I > s /(1+ 6))i=O O<v<1

O<u <a

- 27 -

R-1+ £ P( sup n1/2kla(I(v,u + UiR)) - acx(I(v,ilR))I > s61(l + 6)).

i=O O.v1slO0u 51/R

Since 1/R < a < /4, we can apply the exponential bound to both sums to obtain

P(:On(a) > s) < Rel'exp{-e2's2/a(1 + 6)21 +

Re 1'exp{-e2's262/(4a (1 + 6)2))< 2e,'exp{-e2's262/(4a(1 + 6)2))

provided 2 < sn 1/2, 32a < s262(1 - 26)2 and a < 6/4. Under assumptions of the

Lemma, since e1 and e2 are left unspecified, we can assume without loss of generality

that the last two growth conditions are satisfied. The conclusion follows by setting

e = 2e 1' and e2 = e2'62/(4(1 + 6)2).

For q2 = 2, the proof is similar to that of Theorems 1.5 and 1.7 in Stute (1984a).

An induction argument shows that the Lemma remains valid for arbitrary q2.

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TECHNICAL REPORTSStatistics Department

University of California, Berkeley

1. BREIMAN, L. and FREEDMAN, D. (Nov. 1981, revised Feb. 1982). How many variables should be entered in aregression equation? Jou. Assoc.. March 1983, 78, No. 381, 131-136.

2. BRILLINGER, D. R. (Jan. 1982). Some contrasting examples of the time and frequency domain approaches to time seriesanalysis. me Ihd Hyiosciences (A. H. El-Shaarawi and S. R. Esterby, eds.) Elsevier ScientificPublishing Co., Aisterdam, 1982, pp. 1-15.

3. DOKSUM, K. A. (Jan. 1982). On the performance of estimates in proportional hazard and log-linear models. SurvivalAnalysis. (John Crowley and Richard A. Johnson, eds.) IMS Lecture Notes - Monograph Series, (Shanti S. Gupta, seriesed.) 1982, 74-84.

4. BICKEL, P. J. and BREIMAN, L. (Feb. 1982). Sums of functions of nearest neighbor distances, moment bounds, limittheorems and a goodness of fit test. Ana.Qkb. Feb. 1982, 11. No. 1, 185-214.

5. BRILLINGER, D. R. and TUKEY, J. W. (March 1982). Spectrum estimation and system identification relying on aFourier transforn. e Colloected Worlc of J. W. Tukey. vol. 2, Wadsworth, 1985, 1001-1141.

6. BERAN, R. (May 1982). Jacklmife approximation to bootstrap estimates. Arm. S March 1984, 12 No. 1, 101-118.

7. BICKEL, P. J. and FREEDMAN, D. A. (June 1982). Bootstrapping regression models with many parameters.Lehmam FetsbrifLL (P. J. Bickel, K. Doksum and J. L. Hodges, Jr., eds.) Wadsworth Press, Belmont, 1983, 28-48.

8. BICKEL, P. J. and COLLINS, J. (March 1982). Minimizing Fisher information over mixtures of distributions. Sankhyi.1983, 45, Series A, Pt. 1, 119.

9. BREIMAN, L. and FRIEDMAN, J. (July 1982). Estimating optimal transformations for multiple regression and correlation.

10. FREEDMAN, D. A. and PETERS, S. (July 1982, revised Aug. 1983). Bootstrapping a regression equation: someempirical results. JASA. 1984, 79, 97-106.

11. EATON, M. L. and FREEDMAN, D. A. (Sept. 1982). A remark on adjusting for covariates in multiple regression.

12. BICKEL, P. J. (April 1982). Minimax estimation of the mean of a mean of a nonnal distribution subject to doing wellat a point. Recent Advances in Statistics, Academic Press, 1983.

14. FREEDMAN, D. A., ROTHENBERG, T. and SUTCH, R. (Oct. 1982). A review of a residential energy end use model.

15. BRILLINGER, D. and PREISLER, H. (Nov. 1982). Maximum likelihood estimation in a latent variable problem. Studiesin Econometric Tr Series. and Multivariate Statistics. (eds. S. Karlin, T. Amemiya, L. A. Goodman). AcademicPress, New York, 1983, pp. 31-65.

16. BICKEL, P. J. (Nov. 1982). Robust regression based on infinitesimal neighborhoods. Ann. Statist.. Dec. 1984, 12,1349-1368.

17. DRAPER, D. C. (Feb. 1983). Rank-based robust analysis of linear models. I. Exposition and review.

18. DRAPER, D. C. (Feb 1983). Rank-based robust inference in regression models with several observations per cell.

19. FREEDMAN, D. A. and FIENBERG, S. (Feb. 1983, revised April 1983). Statistics and the scientific method, Commentson and reactions to Freedman, A rejoinder to Fienberg's comments. Springer New York 1985 Cohort Analysis in SocialResearch, (W. M. Mason and S. E. Fienberg, eds.).

20. FREEDMAN, D. A. and PETERS, S. C. (March 1983, revised Jan. 1984). Using the bootstrap to evaluate forecastingequations. L. Qf Forecasting. 1985, Vol. 4, 251-262.

21. FREEDMAN, D. A. and PETERS, S. C. (March 1983, revised Aug. 1983). Bootstrapping an econometric model: someempirical results. IBES. 1985, 2, 150-158.

22. FREEDMAN, D. A. (March 1983). Structural-equation models: a case study.

23. DAGGEIT, R. S. and FREEDMAN, D. (April 1983, revised Sept. 1983). Econometrics and the law: a case study in theproof of antitrust damages. Prc.of the Berkeley Conference. in honor of Jerzy Neyman and Jack Kiefer. Vol I pp.123-172. (L. Le Cam, R. Olshen eds.) Wadsworth, 1985.

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24. DOKSUM, K. and YANDELL, B. (April 1983). Tests for exponentiality. Handbook of ,Statisics (P. R. Krishnaiah andP. K. Sen, eds.) 4, 1984.

25. FREEDMAN, D. A. (May 1983). Comments on a paper by Markus.

26. FREEDMAN, D. (Oct. 1983, revised March 1984). On bootstrapping two-stage least-squares estimates in stationary linearmodels. Am. Sttis. 1984, 12, 827-842.

27. DOKSUM, K. A. (Dec. 1983). An extension of partial likelihood methods for proportional hazard models to generaltransformation models. Ann. S 1987, 15, 325-345.

28. BICKEL, P. J., GOETZE, F. and VAN ZWET, W. R. (Jan. 1984). A simple analysis of third order efficiency of estimateof the Neyman-Kiefer Conference. (L. Le Cam, ed.) Wadsworth, 1985.

29. BICKEL, P. J. and FREEDMAN, D. A. Asymptotic normality and the bootstrap in stratified sampling. Ann. Statist._12 470-482.

30. FREEDMAN, D. A. (Jan. 1984). The mean vs. the median: a case study in 4-R Act litigation. JBES. 1985 Vol 3pp. 1-13.

31. STONE, C. J. (Feb. 1984). An asymptotically optimal window selection rule for kemel density estimates. Ann. Statist..Dec. 1984, 12, 1285-1297.

32. BREIMAN, L. (May 1984). Nail finders, edifices, and Oz.

33. STONE, C. J. (Oct. 1984). Additive regression and other nonparametric models. Ann. Statist.. 1985, 13, 689-705.

34. STONE, C. J. (June 1984). An asymptotically optimal histogram selection rule. Proc. of the Berkeley Conf. in Honor ofJerzy Neyman and Jack Kiefer (L. Le Cam and R. A. Olshen, eds.), II, 513-520.

35. FREEDMAN, D. A. and NAVIDI, W. C. (Sept. 1984, revised Jan. 1985). Regression models for adjusting the 1980Census. Statistical Science. Feb 1986, Vol. 1, No. 1, 3-39.

36. FREEDMAN, D. A. (Sept. 1984, revised Nov. 1984). De Finetti's theorem in continuous time.

37. DIACONIS, P. and FREEDMAN, D. (Oct. 1984). An elementary proof of Stirling's formula. Amer. Math Monthly. Feb1986, Vol. 93, No. 2, 123-125.

38. LE CAM, L. (Nov. 1984). Sur l'approximation de familles de mesures par des familles Gaussiennes. Ann. Inst.Henri Poincare. 1985, 21, 225-287.

39. DIACONIS, P. and FREEDMAN, D. A. (Nov. 1984). A note on weak star uniformities.

40. BREIMAN, L. and IHAKA, R. (Dec. 1984). Nonlinear discriminant analysis via SCALING and ACE.

41. STONE, C. J. (Jan. 1985). The dimensionality reduction principle for generalized additive models.

42. LE CAM, L. (Jan. 1985). On the normal approximation for sums of independent variables.

43. BICKEL, P. J. and YAHAV, J. A. (1985). On estimating the number of unseen species: how many executions werethere?

44. BRILLINGER, D. R. (1985). The natural variability of vital rates and associated statistics. Biometrics, to appear.

45. BRILLINGER, D. R. (1985). Fourier inference: some methods for the analysis of array and nonGaussian series data.Water Resources Bulletin. 1985, 21, 743-756.

46. BREIMAN, L. and STONE, C. J. (1985). Broad spectrum estimates and confidence intervals for tail quantiles.

47. DABROWSKA, D. M. and DOKSUM, K. A. (1985, revised March 1987). Partial likelihood in transformation modelswith censored data.

48. HAYCOCK, K. A. and BRILLINGER, D. R. (November 1985). LIBDRB: A subroutine library for elementary timeseries analysis.

49. BRILLINGER, D. R. (October 1985). Fitting cosines: some procedures and some physical examples. Joshi Festschrift.1986. D. Reidel.

50. BRILLINGER, D. R. (November 1985). What do seismology and neurophysiology have in common? - Statistics!Comptes Rendus Math. Rep. Acad. Sci. Canada. January, 1986.

51. COX, D. D. and O'SULLIVAN, F. (October 1985). Analysis of penalized likelihood-type estimators with application togeneralized smoothing in Sobolev Spaces.

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52. O'SULLIVAN, F. (November 1985). A practical perspective on ill-posed inverse problems: A review with somenew developments. To appear in JornaQfStat1stical Science,

53. LE CAM, L. and YANG, G. L. (November 1985, revised March 1987). On the preservation of local asymptotic normalityunder information loss.

54. BLACKWELL D. (November 1985). Approximate normality of large products.

55. FREEDMAN, D. A. (June 1987). As others see us: A case study in path analysis. Lunin!of EducationalStatistis, 12, 101-128.

56. LE CAM, L. and YANG, G. L. (January 1986). Replaced by No. 68.

57. LE CAM, L. (February 1986). On the Bernstein - von Mises theorem.

58. O'SULLIVAN, F. (January 1986). Estimation of Densities and Hazards by the Method of Penalized likelihood.

59. ALDOUS, D. and DIACONIS, P. (February 1986). Strong Uniform Times and Finite Random WaLks.

60. ALDOUS, D. (March 1986). On the Markov Chain simulation Method for Uniforn Combinatorial Distributions andSimulated Annealing.

61. CHENG, C-S. (April 1986). An Optimization Problem with Applications to Optimal Design Theory.

62. CHENG, C-S., MAJUMDAR, D., STUFKEN, J. & TURE, T. E. (May 1986, revised Jan 1987). Optimal step typedesign for comparing test treatments with a control.

63. CHENG, C-S. (May 1986, revised Jan. 1987). An Application of the Kiefer-Wolfowitz Equivalence Theorem.

64. O'SULLIVAN, F. (May 1986). Nonparametric Estimation in the Cox Proportional Hazards Model.

65. ALDOUS, D. (JUNE 1986). Finite-Time Implications of Relaxation Times for Stochastically Monotone Processes.

66. PITMAN, J. (JULY 1986, revised November 1986). Stationary Excursions.

67. DABROWSKA, D. and DOKSUM, K. (July 1986, revised November 1986). Estimates and confidence intervals formedian and mean life in the proportional hazard model with censored data.

68. LE CAM, L. and YANG, G.L. (July 1986). Distinguished Statistics, Loss of information and a theorem of Robert B.Davies (Fourth edition).

69. STONE, C.J. (July 1986). Asymptotic properties of logspline density estimation.

71. BICKEL, P.J. and YAHAV, J.A. (July 1986). Richardson Extrapolation and the Bootstrap.

72. LEHMANN, E.L. (July 1986). Statistics - an overview.

73. STONE, C.J. (August 1986). A nonparametric framework for statistical modelling.

74. BIANE, PH. and YOR, M. (August 1986). A relation between Levy's stochastic area formula, Legendre polynomial,and some continued fractions of Gauss.

75. LEHMANN, E.L. (August 1986, revised July 1987). Comparing Location Experiments.

76. O'SULLIVAN, F. (September 1986). Relative risk estimation.

77. O'SULLIVAN, F. (September 1986). Deconvolution of episodic hormone data.

78. PITMAN, J. & YOR, M. (September 1987). Further asymptotic laws of planar Brownian motion.

79. FREEDMAN, D.A. & ZEISEL, H. (November 1986). From mouse to man: The quantitative assessment of cancer risks.To appear in Statistical Science.

80. BRILLINGER, D.R. (October 1986). Maximum likelihood analysis of spike trains of interacting nerve cells.

81. DABROWSKA, D.M. (November 1986). Nonparametric regression with censored survival time data.

82. DOKSUM, K.J. and LO, A.Y. (November 1986). Consistent and robust Bayes Procedures for Location based onPartial Information.

83. DABROWSKA, D.M., DOKSUM, KA. and MIURA, R. (November 1986). Rank estimates in a class of semiparametrictwo-sample models.

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84. BRILLINGER, D. (December 1986). Some statistical methods for random process data from seismology andneurophysiology.

85. DIACONIS, P. and FREEDMAN, D. (December 1986). A dozen de Finetti-style results in search of a theory.Ana. Ins. Hemri Poincare. 1987, 23, 397-423.

86. DABROWSKA, D.M. (January 1987). Unifonn consistency of nearest neighbour and kemel conditional Kaplan- Meier estimates.

87. FREEDMAN, DA., NAVIDI, W. and PETERS, S.C. (February 1987). On the impact of variable selection infitting regression equations.

88. ALDOUS, D. (February 1987, revised April 1987). Hashing with linear probing, under non-uniform probabilities.

89. DABROWSKA, D.M. and DOKSUM, K.A. (March 1987, revised January 1988). Estimating and testing in a twosample generalized odds rate model.

90. DABROWSKA, D.M. (March 1987). Rank tests for matched pair experiments with censored data.

91. DIACONIS, P and FREEDMAN, D.A. (April 1988). Conditional limit theorems for exponential families and finiteversions of de Finetti's theorem. To appear in the Journal of Applied Probability.

92. DABROWSKA, D.M. (April 1987, revised September 1987). Kaplan-Meier estimate on the plane.

92a. ALDOUS, D. (April 1987). The Harmonic mean formula for probabilities of Unions: Applications to sparse randomgraphs.

93. DABROWSKA, D.M. (June 1987, revised Feb 1988). Nonparametric quantile regression with censored data.

94. DONOHO, D.L. & STARK, P.B. (June 1987). Uncertainty principles and signal recovery.

95. RIZZARDL F. (Aug 1987). Two-Sample t-tests where one population SD is known.

96. BRILLINGER, D.R. (June 1987). Some examples of the statistical analysis of seismological data. To appear inProceedings, Centennial Anniversary Symposium, Seismographic Stations, University of California, Berkeley.

97. FREEDMAN, DA. and NAVIDI, W. (June 1987). On the multi-stage model for carcinogenesis. To appear inEnvironmental Health Perspectives.

98. O'SULLIVAN, F. and WONG, T. (June 1987). Determining a function diffusion coefficient in the heat equation.99. O'SULLIVAN, F. (June 1987). Constrained non-linear regularization with application to some system identification

problems.

100. LE CAM, L. (July 1987, revised Nov 1987). On the standard asymptotic confidence ellipsoids of Wald.

101. DONOHO, D.L. and LIU, R.C. (July 1987). Pathologies of some minimum distance estimators.

102. BRILLINGER, D.R., DOWNING, K.H. and GLAESER, R.M. (July 1987). Some statistical aspects of low-doseelectron imaging of crystals.

103. LE CAM, L. (August 1987). Harald Cramer and sums of independent random variables.

104. DONOHO, A.W., DONOHO, D.L. and GASKO, M. (August 1987). Macspin: Dynamic graphics on a desktopcomputer.

105. DONOHO, D.L. and LIU, R.C. (August 1987). On minimax estimation of linear functionals.

106. DABROWSKA, D.M. (August 1987). Kaplan-Meier estimate on the plane: weak convergence, LIL and the bootstrap.

107. CHENG, C-S. (August 1987). Some orthogonal main-effect plans for asymmetrical factorials.

108. CHENG, C-S. and JACROUX, M. (August 1987). On the construction of trend-free run orders of two-level factorialdesigns.

109. KLASS, M.J. (August 1987). Maximizing E max S / ES': A prophet inequality for sums of I.I.D. mean zero variates.

110. DONOHO, D.L. and LIU, R.C. (August 1987). The "automatic" robustness of minimum distance functionals.

111. BICKEL, P.J. and GHOSH, J.K. (August 1987). A decomposition for the likelihood ratio statistic and the Bartlettcorrection - a Bayesian argument.

112. BURDZY, K., PITMAN, J.W. and YOR, M. (September 1987). Some asymptotic laws for crossings and excursions.

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113. ADHIKARL A. and PriMAN, J. (September 1987). The shortest planar arc of width 1.

114. R1TOV, Y. (September 1987). Estimation in a linear regression model with censored data.

115. BICKEL, PJ. and RITOV, Y. (September 1987). Large sample theory of estimation in biased sampling regressionmodels I.

116. RITOV, Y. and BICKEL, P.J. (September 1987). Unachievable information bounds in non and semiparametric models.

117. RITOV, Y. (October 1987). On the convergence of a maximal correlation algorithm with alternating projections.

118. ALDOUS, D.J. (October 1987). Meeting times for independent Markov chains.

119. HESSE, C.H. (October 1987). An asymptotic expansion for the mean of the passage-time distribution of integratedBrownian Motion.

120. DONOHO, D. and LIU, R. (October 1987, revised March 1988). Geometrizing rates of convergence, II.

121. BRILLINGER, D.R. (October 1987). Estimating the chances of large earthquakes by radiocarbon dating and statisticalmodelling. To appear in Statistics a Guide to the Unknown.

122. ALDOUS, D., FLANNERY, B. and PALACIOS, J.L. (November 1987). Two applications of urn processes: The fringeanalysis of search trees and the simulation of quasi-stationary distributions of Markov chains.

123. DONOHO, D.L. and MACGIBBON, B. (November 1987). Minimax risk for hyperrectangles.

124. ALDOUS, D. (November 1987). Stopping times and tightness II.

125. HESSE, C.H. (November 1987). The present state of a stochastic model for sedimentation.

126. DALANG, R.C. (December 1987). Optimal stopping of two-parameter processes on hyperfinite probability spaces.

127. Same as No. 133.

128. DONOHO, D. and GASKO, M. (December 1987). Multivariate generalizations of the median and timmed mean II.

129. SMITH, D.L. (December 1987). Exponential bounds in Vapnik-tervonenkis classes of index 1.

130. STONE, C.J. (November 1987). Uniform error bounds involving logspline models.

131. Same as No. 140

132. HESSE, C.H. (December 1987). A Bahadur - Type representation for empirical quantiles of a large class of stationary,possibly infinite - variance, linear processes

133. DONOHO, D.L. and GASKO, M. (December 1987). Multivariate generalizations of the median and trimmed mean, I.

134. DUBINS, L.E. and SCHWARZ, G. (December 1987). A sharp inequality for martingales and stopping-times.

135. FREEDMAN, DA. and NAVIDI, W. (December 1987). On the risk of lung cancer for ex-smokers.

136. LE CAM, L. (January 1988). On some stochastic models of the effects of radiation on cell survival.

137. DIACONIS, P. and FREEDMAN, D.A. (April 1988). On the uniform consistency of Bayes estimates for multinomialprobabilities.

137a. DONOHO, D.L. and LIU, R.C. (1987). Geometrizing rates of convergence, I.

138. DONOHO, D.L. and LIU, R.C. (January 1988). Geometrizing rates of convergence, HI.

139. BERAN, R. (January 1988). Refining simultaneous confidence sets.

140. HESSE, C.H. (December 1987). Numerical and statistical aspects of neural networks.

141. BRILLINGER, D.R. (January 1988). Two reports on trend analysis: a) An Elementary Trend Analysis of Rio NegroLevels at Manaus, 1903-1985 b) Consistent Detection of a Monotonic Trend Superposed on a Stationary Time Series

142. DONOHO, D.L. (Jan. 1985, revised Jan. 1988). One-sided inference about functionals of a density.

143. DALANG, R.C. (February 1988). Randomization in the two-armed bandit problem.144. DABROWSKA, D.M., DOKSUM, K.A. and SONG, J.K. (February 1988). Graphical comparisons of cumulative hazards

for two populations.

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145. ALDOUS, D.J. (February 1988). Lower bounds for covering times for reversible Markov Chains and random walks ongraphs.

146. BICKEL, P.J. and RITOV, Y. (February 1988). Estimating integrated squared density derivatives.

147. STARK, P.B. (March 1988). Strict bounds and applications.

148. DONOHO, D.L. and STARK, P.B. (March 1988). Rearrangements and smoothing.

149. NOLAN, D. (March 1988). Asymptotics for a multivariate location estimator.

150. SEILLIER, F. (March 1988). Sequential probability forecasts and the probability integral transform.

151. NOLAN, D. (March 1988). Limit theorems for a random convex set.

152. DIACONIS, P. and FREEDMAN, D.A. (April 1988). On a theorem of Kuchler and Lauritzen.

153. DIACONIS, P. and FREEDMAN, DA. (April 1988). On the problem of types.

154. DOKSUM, KA. (May 1988). On the correspondence between models in binary regression analysis and survival analysis.

155. LEHMANN, E.L. (May 1988). Jerzy Neyman, 1894-1981.

156. ALDOUS, D.J. (May 1988). Stein's method in a two-dimensional coverage problem.

157. FAN, J. (June 1988). On the optimal rates of convergence for nonparametric deconvolution problem.

158. DABROWSKA, D. (June 1988). Signed-rank tests for censored matched pairs.

159. BERAN, R.J. and MILLAR, P.W. (June 1988). Multivariate symmetry models.

160. BERAN, R.J. and MILLAR, P.W. (June 1988). Tests of fit for logistic models.

161. BREIMAN, L. and PETERS, S. (June 1988). Comparing automatic bivariate smoothness.

Copies of these Reports plus the most recent additions to the Technical Report series are available from the StatisticsDepartment technical typist in room 379 Evans Hall or may be requested by mail from:

Department of StatisticsUniversity of CalifomiaBerkeley, California 94720

Cost: $1 per copy.

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