NSF CA

Deconvolution Problem

by

Jian(ling Fan Department of Statistics University of California

Berkeley, CA 94720 Technical Report No. 157

Junie 1988

Researchi suipported Research suppolted

by NSF Grant No. DMS84-51753 by NSF Grant No. DMS87-01426 Department of Statistics

Uniiversity of California

Berkeley, Californiia

ON THE OPTIMAL RATES OF CONVERGENCE FOR

NONPARAMETRIC DECONVOLUTION PROBLEM

Jianqing

Fan

Departenteitt of Statistics Un1iersity of Californiia, Berkeley

Berkeley, CA 94720

Abstract

Suppose

we

have

n

observations fromti

Y =X +

c, wlhere e is

measurement

error with known distribution, and the density f of X is u"known with the non-parametric constraint f

E

f (x): If( '(x)

-

f(')(x + 6)I <.

B

a, If

I

<C C. Suppose the functional of interest is T(f) = f(')(xo) ; for I

=

t, T(f) is Xthe density function

at a

poinlt. Then the optimal

rate

of

m+aC-l

estimatinig T(f) is 0((log ⁱⁱ⁾

¹

) if the tail of thle chtaracteristic function of e is of order

m+az-l

I I

Pbexp(-

It I

P/y)

^{as t -}

^{o and is 0} (11 ^2(m

^+a)+

²¹

⁺

I) if thle ^{tail of} the characteristic func- tioni

is

of order 0 (t- 0). Moreover, tlle optimal rate of convergence of

a

distribution function is also found, wlich is no longer "root-n consistency"

as

in the ordiniary

case. In

addition, the optimal

rate

of es(inmating the functional T(f) = aj f ()(xo) is also addressed.

KEY

WVORDS: Deconvolution; nonparamietric deiisity estimation; Estimation of Distribution;

Optimyial

rates of

convergence; Miiiinasix risk; Kernel estimation; Fourier Transformation;

Smiioothnless of error.

Abblreviated title: Optim1al rates of Decoiivohltiion

1. Introduction

Suppose

we

have observations Y, *** Y, having the

same

distribution

as

that of Y available to estimnate the unknown density f(x) of

a

random variable X, where

Y =X +£ (1.1)

with measurenient error £ of knownt distribution. Assunie furtherimiore that the random vari- ables X and e

are

independent. We will discuss herein l1ow well the unknown density and its distribution functioni can be estimated noniparametrically under certain smoothness conditions.

The usual smootimess condition imnposed onI f is the set with kth bounded derivatives.

More generally, we shall assume f satisfies Lipschhitz condition of order a, i.e. f is in the set

nt,B = (f(x): If "').x) -f(m)(x + 0)1 5 BO¶ If I C) (1.2) whiere B, C, and 0 . a < 1 are constants. Theni ithe optimal rate of convergence will depends on mn, a through nt + a.

Sucih a model of measurements beinig cotitaminiatede with error exists in niany differenit fields and has been widely studied. For examnple, the observation Y is the survival time of an

animal, X is (lie tiune that a tumor occurs, and e is the time from tumor occurring to death.

Some oth1er extniples, described in Liu and Taylor (1987), Carroll and lIall (1987), are Crunip

and Seinfeld (1982), Mendelsohn and Rice (1982), and Medgyessy (1977).

The applications for such a model in tiheoretical settings are mentioned in Carroll aned Hall (1987). For example, our results call be applied to the estimation of the prior of non- parametric empirical Bayes problemii (Berger, 1980). Also, the theory can be aplplied to non- parametric regression estimationi for estimation of regression functiotn and to thie generalized

linlear model ( Stefinski anid Cairoll (1987) ), andt otier moodels suclh as Y

=

Xe.

Carroll and Hall (1987) give tlie optiniial rates of deensity estimation at a point whien tile

error is niornal, and give the result for gamnia distributions omitting thle proof. As far tlhe

author can deterinine from their proof, we don'lt not know exactly why the lower rates should

-

3

-

depend

on

the tail of te. ^In

our

heuristic

argtiment

in section 3, the

reason

for this dependence

are

clearly stated. We will generalize the result

to inore

general setting by assuwling that thle tail of clharacteristic function is eithler of fonn

I 4e(t ^{) I}

⁼

^{O (I}

^{t I}

Poexp(-

^ItI

/y)) (as

t -

oo) (1.3)

or

+£t)

=

0 (C P) (as

^t

-o) (1.4)

Moreover,

we

will give

the

rate of convergence of estimating distribution

,

which is

not n

1/2 conivergence

any more.

We

will address how the difficulty of deconvolition depen(ds heavily

on

both imposed smootlhness conidition

on

density

f and on

the sniootlmness conidition of distribution of error.

By

snioothness of tie distribution

of

error,

we

mean tlhe

order of

characteristic function te(t)

of

£ as

t--*o. The difficulty

can be

explainied inttuitively: onl

the

basis of finiite observations, the tail of t,(t) makes it difficult to identify tlhe tail

of

clharacteristic function

of X,

and

hence

the distribution of

X. It can

also be explained by the fact that tlie empirical characteristic func- tioii of

Y

does not

go to

0 (

as t -- oo

), anid lhence

to use

inversion formula heuristically,

we

lhave

to truncate the integration

somlewlhere.

The

smoother

of the distribution of C, the more we

have

to

tnincate, whicih increases thie variance anid

bias of

the estimator. Tlhus,

we

have

to pay

some extra

cost

for estimaning the density of f in deconvolution form. Therefore, the optimal

rate

depends

on

how nmuch

we pay

for tlhe extra

cost.

To

find the optimial

rate

of cotivergenice,

we usually liave to

linid

a lower

bound and anl upper bound. Suppose

we

wanlt

^to

estiimate

a

functional

T

of f.

In

primiciple,

we

can

apply

lihe

niodulus lower bounid b( b

)

invenite(d by Donolio and Liu (1987), and defined

by

b(e)

=

supl IT(f)

-

T(f2)

1:

f 1i ^f

2 E

Cni,a,B,

H

(fY, fY2)

.

-I (1.5)

int cuirrent seltitig, where

H

(fy 1. f yX2)

is

thie Hellinger metric

between

fy

I

an(l fY2

(see

Lecamil

(1973), (1985)) and( fy, is the denisity of

Y

unider the convolution of no(lel

(1.1).

But

constructing a lower bound in this way inay obscure the essential difficulty inside. Instead of using Hellinger metric, we will use x2 metric. To fomiulate the idea, let

Xkic

f

2) = j 1 -f2)2f- 1d (1.6)

be Ihe x2 metric between two densi(ies, andt

br(e) = SUP( IT(f 1) - T(f2) 1: f 1. f 2 E Cm,a. XVY(If fY2) . e (1.7) Then, the lower bound bT( lW ) will be the attainable lower bound, and the rate cani be com- puted explicitly. In otlher words, when we cannot distinguish between two densities based on n observations of Y. then the change of functional is a lower bound.

To find an achievable upper bounid, we will utse Parzen's (1962) type of denisity estima- tor.

A

similar construction is used by Stefanski and Carroll (1987) and Liu and Taylor (1987).

For a nice kernel function K(x), let K(t) be its Foturier transfonn with tK(O)

=

1. Then the kernel density esliimator is defined by

00 A

f "x)= -! J exp(- itxr) (th). dt (1.8)

for suitable choice of bandwidth h and kerniel function. Note (1.8) can also be written in ker- nel type of density (see (2.3)). Moreover, we will use (1.8) to construct tlie estiinators of the derivatives of thie unknown density and its distribution.

In

section 2,

we will

exhibit the

rates of

kernel

type of

density estimators

, which are

the

optnmal in

terms

of rates of

convergence. In

sectioni 3,

we will

constrtict the lower bounds and give hetuistic argument of the results which allow

us

to

say

the optimal

rate

of

convergence.

hi

section 4,

we will

give somie brief (liscussioni anid coimimlents.

bI

section 5,

we

will give the

proofs of the results.

-

5

-

2. Kernel density estimators

Let's start with the kernel density estimator (1.8) with empirical characteristic function delinied by

A ~~~n

n(= - exp( itY.) (2.1)

and

a

given known funjction tK (t ). Let K(x) be the Fourier inversion of

K

(t ) defined by

+00

K(x)

^{= 2n} ex

er(-

i

tK +(t )dt, (2.2)

a

smooth kernel function. Then (1.8) can be rewritten as a kernel type of estinmator:

fnT.,) =- E

^{- ,}

( )

^(2.3)

wlhere

+ 00

gh,rx)=- I exp(-itKv) ^{dt (2.4)}

2ir

t

4(tlh)(24

Define the maxinuin mean square error (MMSE) of an estimator f to be

MMSE(f)= sip E(f (rx)-f (x0))2 (2.5)

f ⁶ Cm,a

To compute (2.5), first conmpute its bias and see wthat kind of kernel or equivalently its Fourier transformationi 4K(r), ^{we should use.}

+ 00

Ef

n

(xt)

^-

f (xf) = J exp(- itx(,) 4K (tIh )x (t) dt

^-

f (x(.)

h⁰⁰

Tlle last expressioni does not dependl ont thte error (listribution. Thius, tlie niiniilutim conditions

we

hlave to hipose

are

that the Kernel function K satisfies the conditions of those without

convolution. We will

state

them oIn its Fourier domain.

-The conditions

we are

going

to

impose

on

K amd

on

4,

are

Al) *jt)

*

0, for any t.

A2) *K

(t ) is a

symmetric function, havinig

IU + 2

bounded integrable derivatives

on

( , +

oo).

A3)

*K(t)

⁼ 1 +

O(It

Ia'm ), as t -- 0.

Note

that A2) and A3) is inmposed simply

to

make kemel K(.) satisfy the condition of

"classical" (witlhout convolution) kernel function. Additional contditions will be specified below.

More generally, we

can

use f,AI ^{(x0) to estimate} f(')(x0), the 1th derivative of the unk-

IIowiI density at xo. For thle exponenitial decay of te case (we will call this the supersmootd case), we have thle followinig rates of convergence.

Thteorem 1: Under the assumlptions Al)- A3) and El) *K(t)=O for ItI .1.

E2) Ite(t)1 It

I

"'exp( I

t

I1 P/y)

^> c (is t -+ oo)

withi 3, y,

c >

0.

I ^-

'I

Tlhen by clhoosing tlhe bandwidthih = (4/y) P (log i ) P,

f:9

E

(f, ,'o(x)) *lf('

x))2 = ⁰((log ⁿ

)-

2(m +^a-

1)13)

(2.7) f eimal

Remark 1:

When I =

0, f,, (x(p) is the estimator of density functioni itself, which

has

rate of convergence of 0 ((log n ) (m + Wx)/0). The constant I in condition El) is not essential.

It

can be replaced hy any positive conistanlt. Tlhe reason we impose such a condition is simply to make (1.8) converge, and for easy calculation ^{in the} proof.

For the case of geometric decay of tE (called thle smiioothness case), we have the follow-

il)g result.

-

7

-

Tlheorem 2: Under the assunmptions Al)

^-

A3) and GI) 4)+'(t)t1 + -PC, te(t)tp -+ c, (t

^-

+0) wi(h c > O.

+00 +00

Then by choosiiig the bandwidth h

=

0 (n

-

1/[2(m + a) + 21 + 1),

2(m + a-l)

SLIP

E

(f,n,l')(txf) f()(x())2

⁼

0 (ii

2(m +

a)

+ 21 + (2.8)

f e Cmal

Remark 2: If we want to estimate T(f) = z ajf()(xo) in Cm,a,B, then thle kernel den- sity estimator T(f,) = ajf,"(j)(x) (a,

*

0) has the optimlal

rate

of 0 ((log

n

) (m + of

^-

1)11) or

m +a-1

o (fl 2(m + + P + I), depending

on the rate

of the

tail of

4e. The proof of such

a

result fol- lows easily from the proofs given in this awid the next section uider the same assunptions.

Note that

the rate

of convergence given by (2.7) an(d (2.8) is the optimal one, which will be shown in Ile

next

section.

Now,

we

conisider

an

estimator of distribution function. Define thle estiniator of distribu- tioni

F

(@r () by

xo

ft, (,0o) = f f(t) dt (2.9)

where f,

(t)

is the kernel denisity estimnator

given by (2.3).

Theoreim

3: Under

assumptionis

El), E2), Al) of

Theorem

1, stippose

th)at K(t)

is a

symmetric function, having m+3 bound(led initegrable derivatives on ( - 0, +° ), and 4K (t) = I

+

0 ( I

t

I + + a),

as t

-* 0. Tlhemi by choosinig the same bandwidth

as

for Theoremn

I and M,,

=

13

we

lhave:

sup Ef (A, (-x(t) - F (.))2 = 0 ((log

n

2(m + a + 1)/P)

f E cm.aB

whiere C'm,,"

^-

If

r

Cm,0,: FF(-n)

e

D(log n)-(m +2yP.

Remark 3: In the proofs of Theorem 5 & 7, we will see that the rate given in Theorem 3 is the optimal one, because the least favorable pair we choose is in C'm,vi,".

3. Lower bounds

In this section, we will lined lower bounds for estimating densities and distributions.

More generally, suppose we want to estimate T(f) from observation (1.1). Then we have the following lower bound of bT( c )

Theorem 4: If for some sequence of positive constants (an: n 2 1 1, we have limif igf Pf IIf

^-

^T(f)I . a, I=1

n eo t sm,uj

tlien

lim,inf a, lbTf- 2 1/2 (3.1)

if -*00 Nl;

Moreover for any estihmtor tf of T(f), we liave for V c,

sIP ^Ef (f

^-

^T (f ))2

>

Cb

2

(3.2)

for sonie constants C. In other words, no estimator can estimiate better that bT( c )

Remark 4: (3.2) is implied by

the result

of Donolho and Liu (1987) in thle

current set-

(itig. In fact, such

a

theorem

may

be familiar

to

somne

authors. For

the

purpose

of later

use we

state here anid give

a

proof.

Now, let's study the lower botund of b1. (-) To begin with,

suppose

ilte futnctionial of interest is T(f

) =

f (x(,), density

at a

poinit.

Let's give a

heuristic

argunient to see why

the

result

shiould depened

on the

tail of te. ^Rigorous proof

will be

given in section

5,

wlhich

inivolves mathiematical details and nmore careful conistructions.

-9-

Let's assume without loss of generality that xo = 0 by relocating xo

to

origin. To calcu- late the abstract bouInd bT (f ), take a pair f0(x ) e Cm ,aB, f I(x ) 6 Cm ,aB, for which

f l(x) = f (x) + c68H(x,8) (3.3)

+ 00

whiere

k =

nt + a,

H

(0)

^*

0, H(x) dy

=

0 and the m"' derivative of H(x) satisfies Lipschitiz condition of order a. Then by suiitable clhoice of the tail of H(x), fO(x) and con- stant c, f

1

will be a density in Cm ,a,B for small 6. 6 is chosen such that X2-distance

+00

J (fyI -fY2)2ff I'dX

^{S C}_n ^(3.4)

and the lower bound of density estimnation at a point will be hialf of the change of functional

IT(fo)

^-

T(f,)1/2

⁼

2 IH(0)16=0(68k) (3.5)

2

Iliius, we have to lind 8 as larger as l)ossible so that (3.4) holds, or equivalently such that

+00 +00

62k+ I jY.

H(x _-)dF£(6)))2 g0

1-

(6x)dx .O(-)

8) r< I

(3.6)

_ 00 _00

where Fe is the distribution functiont of the randoni variable £, go = f O*FC.

Suppose we

can

prove that as 6

^-4

0,

+ 00 + 00

J [ J H (x

-y

)dF (6y )] g (ex ) dx (3.7)

+00 +00

+00 +00

.C J [ J H(x

^-

y)dF(6v)] dx

_ 00 _0

wlhere C is a constant independient of ni. Then by Parseval's identity, to make (3.5) hold, we havc to chtoose

6

fromn

+00 + C'

62k+12

H(x -y )dF (6y)2 dc =O(-)

⁼

(3.8)

J00 00

or equivalently from

+00

62k + J

|

I *CH(t)# (t/8)

1

2dt

O

I (3.9)

00

~~~~n

where tH is die Fourier transformation of H. Tlius, the result will depend on the tail of £ only. It is not hard to choose 5 front (3.9) anid consequently to get a desired lower bound.

Theorem 5: Suppose thtat the

tail

of

,

satisfies

I4)e(t)l It I POexp(t IP/y) ^.

^c

(as

t

o)

aid

P

ix

+

Ix l'

2:x -

IxI

=

O(Ix I

(a

°))for 0OSo<I (as

x

-±oo) for 0 . ab

<

1,

a

>0.5, then no estimaittor cani estinate T(f) = f,(1)(xO) ^{knowing f e} Cm,,,, faster than 0 ((log n (, + a - P) in the sense tliat if

limilmf inif Pf If(')(xo)

^-

f ('xo) I . a. I = 1 (3.10)

n of e B

thlen

(log n)(m +a-a)/an -c00, (3.11)

Moreover, for any estimator tP,I

sup Ef (T,T

^-

T(f ))2> 0 ((log

n

' +a-l)/P) (3.12)

f Citm,",

From dte result given in Theorenm 1, we klow that the optimal rate of estimtiatilig a den- sity

in

the supersniooth noise case is only of order 0 ((logn )- (m + a)/P)). Specifically, wheii tile

error

is distributed as Cauchy, thieni t(ie optimial rate is 0 ((logn - (t + a)), and witeci tlie error is normal, then the optinmal rate is 0 (logn ot)/2(

+

Th1eorem 6: Suppose that the tail of +£ satisfies tlte conidition GI) of Theorein 2, and

¢"(t )t(a + 2)

-*

a(a

+

1 )c (t

-+

oo), then no estimnator can estimate T(f) = f (1(xu) ^{, under}

m +a-I

tlte constraint thiat ^f e C,a,, faster than 0(11 2n# + 2a + 20 +I) in the sense of (3.1) anid

- i1 ^-

(3.2).

Remark 5: In some cases, *(t) = exp(ite0)4(t), where +(t) satisfies the condition of Theorem 6. Then fe itself doesn't satisfy the conditions of Theorem 6, but by translation the result still holds. Note that the coiistant c can be 0 in Theorem 6.

Remark 6: For estimating the functional T(f) in Remark 2, the lower bounds are exactly the same as those given in Theoreni 5 and 6 using the same constructions.

Thus, we get the optimal rates for the smootlh cases and the supersnmooth cases. In prac- tice, those con(itions are easy to check. The cases of error distributions satisfying Theorem 2

& 6 iniclude ganmina distribution, double exponenitial distribution , etc. Anid the cases of error distributions satisfyinig Tlheoremn 1 & 5 are tiormiial, cauclhy, niixture normal, and niany othler distributions. Now, we state some lower bounds for estimating the distribution function.

Theorem 7: Under the condition of Thleorem 5, then no estimator of estimation the distri-

m +cz+I

bu(ioni function of X at a pohit ui(ler conts(rainit (1.2) can be faster than 0 ((logn) ) in the sense of (3.10)

^-

(3.12).

Tlteoreni 8: Under the condition of Theorem 6, then no esihnator of estimation the dis- tribution function of

X

at a poillt un(der constraint (1.2) caui be faster than

m +aOt I

0 (n 2m + 2a + 21 + I ) when B . 0.5 and 0 (i 112) when ,

<

0.5.

Remark 7:

For

estimating PF (

X

E (a ,b J ), the lower bounds

are

the

same as

those in Theorem

7

& 8. However, the lower bouni given by Theorem 8

may not

be attainable. The

m +a+ I

attainable

one

iight be o (n 2(m + +

P

+1)).

4. Discussion

We hope to decompose the difficulty of deconvolutioi into two parts: the difficulty of deconvolving a functional T, and modulus function without conlvolution (does not depend on the tail of characteristic function). The filrst part tells us how difficult of deconvolution is for a functional, and the second part tells us the difficulty of estimating a functional even though no convolution exists. To formula the idea, let modtulus function

+00

f f I

IT(f 1)

-T(f

2)1: (f

^I

-f2)2fFXI.)

be the difficulty function of estimating T(f) without convolution ( i.e. the lower bound if error

£ = 0 ). And one way to define the difficulty of deconvolving a functional is

+00 + 00

DT(O) =fIf22 (Y'S ^{( J}

^YI

-fY2)fy )f'dx: J (fX f)2jdx 6)

whlere ^C = ((f

^1,

f2): f

I

f

2 C

CtaB, IT(f)

^-

T(f2) b()/2) an) ^d then the lower bound for estimating T(f) is b (Dj ( )). The lower boundi suggests that we find a pair of density functionIs whichi is the least favorable in thle situation without convolution and such that the X2

distance is as small as possible. Of course, we hope to find a difficulty functioin D, which does not depenid on T. But, it is imnpossible simply looking at estimating the me.l and density of the normal error case e

^-

N(0, 1). In this case, D (6)

=

0(6) for estimating t(le mean, while

2(m +a+

1P)

+ I

D (6)

=

0 (6 2(nm

+

a)

+

I) for estimating tlie (lensity.

Wihen error e is unifornily dlistributed on [0,lJ, say, the model (1.1) itself is ideentifiable.

Theorem 6 tells us that no estimator can estimate the density of X at a poinlt faster than

ni +a

0 (it 2(no +

^a

+ 1) +

^I

) for aniy b

<

1. Hlowever, we canmot use kernel density estimsator (1.8) to

estimiate the density, because (1.8) is iiot initegrable alhiost surely.

-

13

-

5. Proofs

Proof of Theorem 1

According to our remtiark in section 2, the function K(t) satisfies thie conditionis of a kernel futnctioni in density estimation in the situationl of no convolution. Tius, we can apply thle result of classical kernel density estimationx (2.6) (see Rao (1983), P 46

^-

47), and it follows that

sup}

IEf,(l'I(x0)

-

f(I)(k()

I f E

+00

t SwB f

(

)(.,o

^-

Y)t K( / ) dy -f (')(xO) ^I

S

Chk-I

for some constanit C, where k

=

ni + a. Now thie varianice of f,,{'kxo) is

+ 00

var(f/)(x02))= ²²

^I

^{| (-it)'} exp(-=it(1o[-

Y

(h dt

+ 00

1 E I

J

it)l exp(- it(x.0

-

Y1)) (fK dt 12

(2i)i

^E

(- 40e(V)

+l

I t 'I

K(t)

1 2

(2it)2u1a2 l4r(t/)l-dt] ^(5.1)

By assumption E2, when Mhi

5

I t I . 1 (for large but fixed M), I e(t l/i ) I

> C

(t/h ) exp(- hF PIy)

2 Moreover, by (Al)

I

Ihe(rh)I

^>.

mini 4e(t)

> 0,

w1hen

It I ^<Mh

ThIus, by (5. 1)

var ( v( )(rO))

^<

0 (exp(2hi -

(2n2ih9ii2

- I

=o(n 3)

by cioosing the bandwidth h

=

0 ((4/y) p (log n) P). Hence, the conclusion follows.

Lemma 5.1: Under the assumiption of Theorem 2,

h g2 [gh(\(.)j2 D2 (uifornly in small h).

x2

for some constant D.

Proof of Lemma 5.1:

By integration by parts,

+00

r 4~~~~~K (t) g(l)(<x) = ± j exp(- itr)[(- it), ] dt

ix ~Jt/h)

Thus, by the usual argument,

+0

if, 2'[gj(I)(.X)j2 ² ( hP 1[

^t

] I dt) X2

^-

41e(tlh/

+ 00

5 21

tKI¢(t)tP+'-'l

+

ICK01)

^{t +lwt}

^,2

(uniformly in sinall

h

) for

some

constant C. Hlence, thie assertioni follows.

Proof of Theorein 2:

By choosinig ^the bandwidth

as

giveni by Theoremii 2 and by the calculation of Tlieorem 1,

we

haive

k-I

Sup

IEI '(v 00) -f(')(x()I .O(/hkl)=O(n 2k+2af+1)

f tm.ai.8

whlere k =

ni +

a. Now,

we

ineed onily

^to

comipute the variance of the estimnator.

Let

ghl

=

g1(1). Then by (2.3),

-

15

-

var(f,)'xo)) h 221 Eg 2(

h

)

nh~~~ ^(5.2)

Let fy(y) be tlhe density of

Y = X +

e, (hen

+00

Eg2( ⁰

^I

) = g28(Y )fyZ (xo-y) dy

h

h_

^o

(5.3)

Note that Ify(x )

I

. C for all f e Cm a, Hence, the following result follows uwformly in f 6 C,,,f,B. For any small rq,

+00 +00

I J fy(x0y) lg2(.Y)dy- fy(xO) JO g(y)dyI

_ 00 h 00

+ 00

=

I |_ (fy(x_-y)- f(0)) g3() dy ^I

-00_1h

^oo

.5 max Ify(.v()

- y

)

-

fy(.T) I B 23Y ) dy

H.vI yg-l II I

sVIn

h

+

j'2 . 1.

2(

) dy + fy (x,)

yh ghh |O ghl,(2h) ^dy

IyI:1 h h

m

inax

IfYy(XO-Y)-fY(xo)

^I +

12

+I3

It is

easy

to

chieck by delinition that

I

g(')(y )

I

. Ch- 2p unformly for small

h

and

some con-

stant C. By Leinma 5.1,

1,

⁼ |

l 2(_i) ^dy

IyI

.1h

ll

. Oghl(') dv

+

IV I X,,,.

J g,h(y) dy

IV I

<

Dh- 20 _

1 dv +

2Ch

-2

1.w2 I 2 -

(5.5)

(5.4)

Applying Lemma 5.1 again, we have

2 f (o

^-

y) y ^{2(Y dy}

I2= y

h g"hl'h

1 'lg Yg3) ²

< 1

sl_ly Dt_-2

Ti

IyI21t 1/hy

-

o(1- 2)

(5.6)

Sinmilar

reason

shows

13 Ify(x0)

I

|O &2g(, )dy =o(h 21) (5.7)

1v .- 11/h h

Note thiat

(5.4)

and (5.5) implies that

+0

g,(y)

2

dy = O(1h-21) (5.8)

_00

Combining (5.5)

-

(5.8), we conclutde fronm (5.3) that 2

I E

gl,/ 2( )(-

2(k ^-I)

I 0

(h

^- 2P^-21 -

1)=

0(1t 2k +

2p

+ I

n

Henice, we get the desired conclusion.

Proof of Thteoremn 3

Xt +00

EP,,(x(o)

⁼

^f(u -y)-L K(2)dy

dii

+00

=

(F(x0u- y) -F(-ni

y

)K(x)d

J- hi h

- 17 ^- Now

by the stanldard proof,

we cait show

that

su EFn (x0)

-

F (x0)I f Cm,

+00 +00

.

I

f F(xo-y)i!KK(Y-)dy -F(x0)l

⁺

^J F(-n"3-hy)IK(y)ldy

-pgl1/M3

:5Clim +a1+

1 +

O(F(- n-l"312))

⁺

|IK(y)l dy

mI

^{+a+ I}

=o((log)

^{p )}

On the other hand, the variance of

F,

(x.0) is

+00

var

(F (x0))5(

/3 +

x) (27r)2nit 2

2

-2

|

lK(t)l/l d h )lt]

. 0 ("i 1'3 2 exp(2

h-

/yI)) The coniclusio follows.

Proof of Theorem 4

Take

a

pairff, f

2

e CC,,a, stuch

that it satisfies (3.4) and

bT(C)

^<

^{IT(f ,)- T2)1}

⁺

^{o(a,) (5.9)}

Then

Efn

[.fY2(Y)1)

fY2ynY ) 2

f I

fy i(y) . fy (Y)

+00

=(1+ J (fYf-Y2)2/fy dIx)" .e (5.10)

wherefy

¹

is the convolution of f , witll the (listribution

of

e. Henice

by

the Cauchy-Schwartz

inequality,

2

[Pf2 (IPi - T4f2)I . aj ] . ec Pf1 1,4 - T(f2)I . an) (5.11) On the other hand, by (5.11)

we

liave

p,e I ITn

^-

T(f2)I .k 2a,)

2 Pf, IIT(f 1)

^-

4n 1

:5

an, ITV 2) fn I

^<

an)

=Pflz IT(f2) -Tn

I :! an ) +

o(1)

2 e- c (as ii -+ oo)

Hence,

we

conclude that

I

T(f)-T(f

2)1 .

2an

and thie conclusion follows.

We need tlie following Leiniia in order to prove tlheorem 5

^-

8.

Lemma 5.2: Suppose that F is a distributioni function, then the convolution density

+00

g0(x) = j Cr

go(x) |212,. dF+()

satisfies

go(x) 2 D Ix 1- 2r as

x -+

00 willh D > 0.

Proof: Choose M large enough such that

F(A)

-

F(-M)> 0 Then when lxl is large,

Af

Cr

g(()

JI

^{+ -}

y)2), dFXy)

^I2r

- 19 -

Lemma 5.3: Suppose P

x +

I

x

° 2x

-

I

x

I O

=

O(lx I

(a

%)) for 0 . oO

< 1

and H(x) is bounded with H(x)

=

o(lx

I

m) (as

x

± oo ). Then there exists a large M and a constait C such that when

I

6x 1 2 M,

+ c

J

H(x -y)dFF(y)

<

C(6Ix I)-.5-(a

-0.5)/2

if nzo(a

-

0.5)

>

1.5

+

(a-0.5)/2.

Proof: Divide the real line into two parts:

I

I:I Y618

^<

^IX

1Ia, 12⁼ Y:

IX: Y/891

> IX

I")

Theni, by simple algebra,

+00

J

H(Yr-y)dFI(6y)

< + H

H(x

^-y

/8) dF,(y)

0(61X 01) °)0¹ + o(lX I-ta

Now

clioosing a

=

(a

-

0.5)/2, thie coniclusion follows.

Proof of Thteorem 5:

By relocating x( to the origin, without loss of generality assume that x(p= 0. Denote k

=

iti

+ a. Take a real function

funiction

H(.)

satisfying

the following

contditions:

1.

H(9)(O)

*

0.

2. H(k)(x) is bounded

continuous for each k.

3. H

(r) =

0

(x-

f),

as x --

oo, for soime

given

mi11.

+ 00

4. JH()dx =0.

0

5. J H(xdx *0.

6.

*l(t)

=

0, when It I is outside [1, 2], where tH is the Fourier transfonnation of H.

To

see

why such

a

function H(.) exists, let's take

a

nonnegative syinmetric function 4(t) whiichi vanishes outside [1, 21 whent

t

. 0 aii(d has conttiniuous first m(p bounlded derivatives (

m(7 is large enough such that Lemma 5.3 holds). Moreover, 4(t) satisfies

h\)(0) . h0(11) (5.12)

aid

'!sin

^t

+(t ) dt

;&

0 where h(x) is the Fourier iniversion of t(t) defined by

2

h1x) = if[cos (tx) (t)dt (5.13)

Such

a

4(.) exists because all func(itis satisfying thie

above

conditions are infinite dimiensionial.

Let H(x)

=

h (x)

^-

ht (x

+

1),

then

its

Fourier

transformation 4H(t)

=

(I

- e-

i)4(t), and H(x) satisfies the condlitions 1

^-

6.

Now take a pair of densities

Cr

2

r'ad f=

f +

cSk H(0/8)

Thlen, by Lemnma 5.2, f

I

is a denisity whien 8 is small, anid by clioosing

r

close to 0.5 and c close to 0, f r, and f

1

EC,a,B.

Denote go

=

fo*FC.

Now

the x2 (listance between the

two

denisities in conivolution

space

is

of

order (c.f. (3.6))

+00 + 00

82kt

J

(

J

H(x

^-

y)dFW(y)) g2o ^{(Sx) dx}

_ 00 _ 0

-

21

^-

j +00 +00

. 62k + J ( ^J H(x -y)dFe(8y))2dJr

_ 00 _00

x / |~ (J|H(x -y)dFe(5y)/g (&t))2 dx (5.14)

_00 _00

Note that by Parseval's identity the first terni of (5.14) is

+00

2

=2 IOH(t)

12

1 o(tI8) 12 dt

. 0 (56 2poexp(- 2- P/y))

uniiformly in small &. Let the minimiiuim value of go(0x) over [-M, M] be mg, which is bigger than 0. By Lemmna 5.2 and 5.3, the second tern is bounded by

+00 +"

t-

J [ H((x -y))dFe(6y)J2dx +

. -(a^-0.5)/2

dX ( )

-00 __

~~~~~~~~16x1'>M D(8xY j22

⁼

Consequiently, when 5 - 0,

+00

(fI

-

fY2)2(f yI)-'

^{dx <}

^C6cexp(-

^-

P/y)

(5.15)

_00

for soine constanits c and C. Taking

S = (log

it

+ (C + l)log(log 1i)) r

(5.15). C

⁼

1I

log i 1

amid thle change of tlie functional is

If f (0))

^-f

(l) (0)

I = O

(8(k

-

1)

Thus,

bT(!) -0 ((log n)f(k

-

1)) and the conclusion follows.

Proof of Theorein 6:

Use the

same

notation

as

in the proof of Theorem 5. Take the

same

0(t)

except

only the first

two

continuous derivatives

are

reqtuired in this

case. Now take a

pair of denisities

Cr and f

I

= f0 + c8 aH(xl6)

Let

H

(t)

=

(1

- e-i

)t(t) be the

Fourier

transformation of H(rx), and define O6(t)

=

(OH(t) O£(r/6))""

and

O,l(t)

⁼

litn 8-

P

O8(t)

8-H (

wlhichi

converges

uniformly hi It I e (1, 2J. Now, by Fourier inversion formula,

(5.16)

+ 00

J H(x

-

y)dF e(8y)

_ 00

+00

1 J exp(- itr )Oj(t)te(t/8) ^dt

_00

1 | e

-

IO(t) dt

Is I,1 2

Let N=

I S 11l 2

I&-+ (t)

-

t(t)Idt, wliichi

goes

toO

( as

8

-+

0

). Then by (5.17),

+ 00

I 8- J H(x-y)dFe(&1') x2I

S

N+ ___

1

-00 ~ ~ -x

²

2ytx2I.I2

^I

o(t) I dt

Now, we are reazdy to compute (3.6). By Parseval's indentity, when 8 is small,

(5.17)

(5.18)

- 23 -

+00

I, ^ H (x

Ix I _ 0

+00 +00

c ( H(x

-y

)dFE(8y)) g4

I

(6x) dx

-

y)dFc£(8y )) dx

+go

SC | I

t,j(t) +r((/8)12

dt

_ oo

(5.19)

=

Q (82 P)

wlhere g =fo*Fe does not vanish, and henlce C is a finite constant. By Lemma 5.2, and (5.18)

+00

22-^6p

⁽

^H

^{(x -y}

^{)Fe8y ))}

^{2 _}

'(fx) dx

2~~~|

^{2 go o}

N

1 5I2 (t) I dt ]g (6x ) dx

lx

I.

^{X + I 5}

=

0(1) (5.20)

Consequenitly, the X7-distaice

in

tihe Y variable ( see (3.6) ) is

;2(m

⁺a)+I

(I,

⁺

82 PI2)= 0(n

-)

and the change of the functional is

m +a-l

I

T(f1)

-

T(f0)

I = &m + a- 11t

(/)(1)

^-

ht()(0)

I = 0

(i

2(m +a

+)+ I) Hience the conclusion follows.

Proof of Thteoreim 7 & 8:

By

ranslation, witlhout loss of

geiierality

assulne tlhat x0

=

0.

Take ille

same

least favorable

pairs as

use(d

in

Theoreim 5

and 6. Then the

chanige

of

func-

tional is

0

IF1(O)-FO(O)I =6m+a I J H(xl6)dxl

=

O(6' + a+ 1)

Hence the result follows.

ACKNOWLEDGEMENTS

The author gratefully acknowledges Prof. Peter J. Bickel for proposing the problem and

fruitful suggestions and discussions. The author would like to express his sincere thanks to

Toinas Billings for careful reading of the manuscript and useful suggestions.

-

25

-

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

47. DABROWSKA, D. M. andDOKSUM, K.A. (1985,revisedMarch1987). Partiallikelihoodintransformation models with censored data.

48. HAYCOCK,K. A. and BRILLINGER, D. R. (November 1985). LIBDRB: A subroutinelibraryforelementary time series analysis.

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

50. BRILLINGER, D. R. (November1985). What do seismology andneurophysiology have in common? -Statistics!

ComptesRendusMath. p Acad. Sci. Canada. January, 1986.

51. COX, D.D.andO'SULLIVAN, F. (October 1985). Analysis of penalized likelihood-type estimators with application to generalizedsmoothing in Sobolev Spaces.

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3

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52. O'SULLIVAN, F. (November 1985). Apractical perspectiveonill-posed inverse problems: A reviewwith some new

developnents.

To appearinJoumal of Statistical Science.

53. LECAM,L. andYANG,G.L (November

1985, revised

March 1987).

On

^the

preservation

of localasymptotic

normality

under information loss.

54. BLACKWELL, D.(November 1985).

Approximate

nomlity of large

products.

55. FREEDMAN,D. A. (Jume 1987). Asothrsseeus: A casestudy inpath analysis. Joumal of Educational

Statistics. 12,

101-128.

56. LECAM, L. andYANG, G. L. (January 1986). Replaced by No. 68.

57. LECAM, L. (February 1986). On theBernstein- vonMises theorem.

58. O'SULLIVAN, F. (January 1986). Estimation ofDenties and HazardsbytheMethod ofPenalizea likelihood.

59. ALDOUS, D. andDIACONIS, ^P. (February 1986). Strong UniformTimesandFinite RandomWalks.

60. ALDOUS, D. (March 1986). On theMarkov Chain simulation Method forUniforn Combatorial Distributions and SimulatedAnnealing.

61. CHENG,C-S. (April 1986). AnOpdtmizationProblemwithApplications toOptimalDesign Theory.

62. CHENG, C-S.,MAJUMDAR, D., STUFKEN,J. &TURE, T. E. (May 1986,revisedJan1987). Optimalstep trpe

design fo

^compangtesttreatnents

with

acontroL

63. CHENG, C-S.(May 1986,revisedJan. 1987). AnApplicationofthe Kiefer-WolfowitzEquivalenceTheorem.

64. O'SULLIVAN, F. (May 1986). Nonparametric Estimationin theCox

Proportional

HazardsModel.

65. ALDOUS, D.(JUNE 1986). Finite-TimeImplications ofRelaxationTimes forStochasticalyMonotoneProcesses 66. PiTMAN, J. (JULY 1986,revisedNovember 1986). Stationary Excursions.

67. DABROWSKA,D. andDOKSUM, K. (July 1986,revisedNovember 1986). Estimatesandconfidence intervals for median and meanlifein the

proportional

hazardmodelwithcensored data.

68. LECAM, L andYANG, G.L. (July 1986). Distinguished Statistics, Loss ofinformationand atheorem of Robert B.

Davies(Fourthedition).

69. STONE,CJ. (July 1986). Asymptoticpropertiesoflogsplinedensityestimation.

71. BICKEL, PJ. andYAHAV, J.A. (July 1986). Richardson ExtrapolationandtheBootstrap.

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

73. STONE, C.J. (August 1986). Anonparametric frameworkforstatisticalmodelling.

74. BIANE,PH. andYOR, M. (August 1986). Arelation between

L6vy's

stochasticareaformula,Legendrepolynomial, andsomecontinued fractions ofGauss.

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

76. O'SULLIVAN, F.

(September

1986). Relativerisk

estmation.

77. O'SULLIVAN, F. (September 1986). Deconvolution ofepisodic hormone

data.

78. P1TMAN,J.&YOR, M.

(Septmber

1987). Furtherasymptotic laws of planarBrownian motion.

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

To appearinStatisticalScience.

80. BRILLINGER, D.R. (October 1986). Maximum likelihood

analysis

ofspike trains ofinteracting nervecells.

81. DABROWSKA,D.M.(November 1986). Nonparametricregressionwithcensordsurvival

time data.

82. DOKSUM, K.J. andLO, A.Y. (Nov 1986,revisedAug 1988). Consistentand robustBayes Procedures for

Location

basedonPartial

Information.

83. DABROWSKA,D.M., DOKSUM,KA. andMIURA,R.(November 1986). Rankestimates in a class of

semiparametric

two-samplemodels.

84. BRILLINGER, D.(December 1986). Somestatistical methods for randomprocess data fromseismology and neurophysiology.

85. DIACONIS,P. andFREEDMAN, D. (Decenber1986). A dozen deFinetti-styleresults in searchofatheory.

Ann.Inst. HenriPoincar6, 1987, 23, 397-423.

86. DABROWSKA, D.M. (January 1987). Unifornconsistency of nearest

neighbour

andkemelconditional Kaplan -Meier estimates.

87. FREEDMAN, DA.,NAVIDI, W. andPETERS,S.C. (February 1987). Ontheimpactofvariableselection in fittingregressionequations.

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

89. DABROWSKA, D.M.andDOKSUM,KA. (March 1987,revisedJanuary 1988). Estimating andtesting inatwo

sample generalized

oddsratemodel.

90. DABROWSKA, D.M.(March 1987). Ranktestsformatchedpair experimentswithcensored data.

91. DIACONIS, P andFREEDMAN, D.A. (April 1988). Conditionallimittheorems forexponentialfamilies and finite versionsof de

Finetti's dteorem.

To appear in the

Journal

of

Applied Probability.

92. DABROWSKA, D.M. (April 1987,revisedSeptember 1987). Kaplan-Meier estimateontheplane.

92a. ALDOUS, D.(April 1987). TheHarmonicmeanformula forprobabilitiesof Unions: Applicationstosparse random graphs.

93. DABROWSKA, D.M. (June 1987,revisedFeb 1988). Nonparametric quantile regressionwithcensored data.

94. DONOHO, D.L. & STARK,P.3. (June 1987). Uncertainty principles andsignalrecovery.

95. CANCELLED

96. BRILLINGER, D.R.(June 1987). Someexamples of the statisticalanalysis ofseismological data. Toappear in

Proceedings,

Centennial

Anniversary Symposium, Seismographic

Stations,

University of California,

Berkeley.

97. FREEDMAN, DA. andNAVIDI, W.(June 1987). Onthemulti-stagemodel forcarcinogenesis. Toappearin Enviromnental Health

Perspectives.

98. O'SULLIVAN, F. andWONG, T. (June 1987). Determining afunctiondiffusion coefficient in the heatequation.

99. O'SULLIVAN, F. (June 1987). Constrained non-linearregularization with application to some systemidentification

problems.

100. LECAM, L. (July 1987, revised Nov 1987). On the standardasymptoticconfidence ellipsoids of Wald.

101. DONOHO, D.L. andLIU, R.C. (July 1987). Pathologiesofsome minimum distance estimators. Annalsof

Statistics, June,

1988.

102. BRILLINGER, D.R., DOWNING, K.H. andGLAESER, R.M. (July 1987). Some statistical aspectsoflow-dose electron

imaging

of

crystals.

103. LECAM, L. (August 1987). HaraldCramerandsumsof independent randomvariables.

104. DONOHO, A.W., DONOHO, D.L. andGASKO,M. (August1987). Macspin: Dynamic graphics on a desktop computer. IEEEComputer Graphicsandapplications, June, 1988.

105. DONOHO, D.L. andLIU, R.C. (August 1987). On minimax estimation oflinearfunctionals.

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

107. CHENG,C-S. (Aug 1987,revisedOct1988). Someorthogonalmain-effectplans for asymmetricalfactorials.

108. CHENG, C-S.and JACROUX, M. (August 1987). On the construction oftrend-freerun orders of two-level factorial

designs.

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

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

AnnalsofStatistics,June, 1988.

111. BICKEL, PJ. and GHOSH, J.K. (August 1987, revised June 1988). A decomposition for the likelihood ratio statistic andtheBartlettcorrection-a Bayesian argument.

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5

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112. BURDZY, K., PFrMAN,J.W. andYOR,M.(September1987). Some asymptotic lawsforcrossingsand excursions.

113. ADHIKARI, A. andPfITMAN, J.(September 1987). Theshortestplanararcofwidth 1.

114. RITOV,Y.(September 1987). Estimationinalinearregression modelwithcensored data.

115. BICKEL, PJ. andRlTOV, Y.(Sept. 1987,revisedAug1988). Largesampletheory of estimationinbiasedsampling regressionmodels L.

116.

RlTOV,

Y. and

BICKEL,

P.J.

(Sept.1987,

revisedAug. 1988). Achieving

information bounds

innonand semiparametricmodels.

117.

RITOV,

Y.

(October 1987).

On the convergenceofamaximal correlationalgorithmwith alternating

projections.

118.

ALDOUS,

D.J.

(October

1987).

Meeting

timesfor

independent

Markovchains.

119. HESSE,^C.H. (October 1987). Anasymptotic expansion^forthemeanofthepassage-timedistribution ofintegrated

Brownian Motion.

120. DONOHO,D. andLIU, R.(Oct. 1987,revisedMar. 1988,Oct. 1988). Geometrizingratesofconvergence,H.

121. BRILLINGER, D.R. (October 1987). Estimating thechances oflargeearthquakesby radiocarbon

dating

andstatistical modelling. To appear in StatisticsaGuidetotheUnknown.

122. ALDOUS, D.,

FLANNERY,

B.andPALACIOS,J.L. (November 1987). Two applications ofumprocesses: Thefiinge analysis of searchtreesand thesimulation ofquasi-stationay distributionsof Markov chains.

123. DONOHO, D.L., MACGIBBON,B. andLIU,R.C.(Nov.1987,revisedJuly 1988). Minimax risk for hyperrectangles.

124. ALDOUS,D. (November 1987). StoppingtimesandtightnessI.

125. HESSE,C.H.(November 1987). Thepresentstateofastochastic model forsedimentation.

126.

DALANG,

R.C. (December 1987,revisedJune 1988). Optimal stoppingoftwo-parameterprocesses on nonstandardprobabilityspaces.

127. SameasNo. 133.

128. DONOHO,D.andGASKO,M. (December1987). Multivariate generalizationsof themedian andtrmmedmean

EI.

129. SMITH,D.L. (Decenber1987). Exponentialbounds in

Vapnik-tervonenkis

classesofindex 1.

130. STONE,C.J. (Nov.1987,revised

Sept.

1988). Uniformerrorboundsinvolving logspline models.

131. SameasNo. 140

132. HESSE, C.H. (December 1987). ABahadur-Typerepresentationforempirical quantiles ofalarge class ofstationary, possiblyinfinite-variance, linearprocesses

133. DONOHO, D.L. andGASKO,M.

(December

1987). Multivariate generaIlizationsofthemedian and

trimmed

mean, I.

134. DUBINS, L.E. andSCHWARZ,G.(December 1987). Asharp inequality for martingalesand

stopping-times.

135.

FREEDMAN,

DA.andNAVIDI,W.(December 1987). Ontheriskoflungcancerforex-smokers.

136. LECAM, L.(January 1988). Onsomestochastic models of the effects of radiationoncellsurvival.

137.

DIACONIS,

P.and

FREEDMAN,

DA. (April 1988). On the uniformconsistencyofBayesestimatesfor multinomial probabilities.

137a. DONOHO, D.L.andLIU,R.C. (1987). Geometizingratesof convergence, I.

138. DONOHO, D.L. andLIU,R.C.(January 1988). Geometrizingratesofconvergence, m.

139. BERAN,R. (January 1988). Refining simultaneousconfidence sets.

140. HESSE,C.H. (December 1987). Numericalandstatisticalaspects of neural networks.

141. BRILLINGER, D.R.(January 1988). Two reports on trendanalysis: a) An Elementary Trend Analysis of Rio Negro LevelsatManaus, 1903-1985

b)

Consistent Detection ofaMonotonicTrend

Superposed

^{on a}

Stationary

Time

Series

142. DONOHO,D.L. (Jan. 1985,revisedJan. 1988). One-sidedinference aboutfunctionalsof a density.

143. DALANG, R.C.(Feb. 1988,revisedNov. 1988). Randomization in the two-armedbanditproblem.

144. DABROWSKA, D.M., DOKSUM,KA. andSONG, J.K. (February 1988). Graphical comparisons ofcumulative hazards fortwo

populations.

145. ALDOUS, D.J. (February 1988). Lower bounds for covering timesforreversibleMarkov Chains and randomwalkson

graphs.

146. BICKEL, PJ. andRITOV, Y. (Feb.1988, revised August 1988). Estimatingintegrated

squared

density derivatives.

147. STARK, P3. (March 1988). Strict bounds andapplications.

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

149. NOLAN, D. (March 1988). Asymptoticsforamultivariate location estimator.

150. SEILLIER, F. (March 1988). Sequentialprobabilityforecasts and theprobability integral transform.

151. NOLAN, D. (March 1988). Limit theorems forarandomconvex set.

152. DIACONIS, P. andFREEDMAN, DA. (April 1988). Onatheorem of Kuchler and Lauritzen.

153. DIACONIS, P. andFREEDMAN,DA. (April 1988). Ontheproblem oftypes.

154. DOKSUM,KA. (May 1988). On thecorrespondencebetweenmodels inbinaryregressionanalysis and survivalanalysis.

155. LEHMAN, E.L. (May 1988). JerzyNeyman, 1894-1981.

156. ALDOUS, D.J.(May 1988). Stein's methodinatwo-dimensionalcoverageproblem.

157. FAN,J. (June 1988). On theoptimalratesofconvergencefor

nonparametic

deconvolutionproblem.

158. DABROWSKA,D.(June 1988). Signed-ranktestsfor censored matchedpairs.

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

160. BERAN, R.J. andMILLAR, P.W. (June 1988). Tests of fitforlogisticmodels.

161. BRELMAN,L. andPETERS, S. (June 1988). Comparing automaticbivariate smoothers(Apublic serviceenterprise).

162. FAN, J.(June1988). Optimalglobalratesofconvergencefornonparametricdeconvolutionproblem.

163. DIACONIS, P. and FREEDMAN, DA.(June 1988). Asingularmeasurewhichislocally unifonn. (Revised by TechReport No. 180).

164. BICKEL, P.J. and KRIEGER, A.M. (July1988). Confidencebands for adistibutionfunction using thebootstrap.

165. HESSE, C.H. (July 1988). New methodsin the analysisofeconomictimeseries I.

166. FAN,

J[ANQING

(July 1988). Nonparametricestimation ofquadratic functionalsin Gaussianwhite noise.

167. BREIMAN,L.,STONE, C.J. andKOOPERBERG, C. (August 1988). Confidence bounds for extremequantiles.

168. LECAM, L. (August 1988). Maximum likelihood anintroduction.

169. BREIMAN, L. (August 1988). Submodelselection andevaluationinregression-Theconditional case andlittlebootstrap.-

170. LECAM, L. (September 1988). On the Prokhorov distancebetween theempiricalprocess and the associatedGaussian bridge.

171. STONE, C.J. (September 1988). Large-sampleinference for logsplinemodels.

172. ADLER, R.J. andEPSTEIN,R. (September1988). Intersection local times for infinitesystems ofplanarbrownian motions and for the

brownian density

process.

173. MILLAR, P.W.(October 1988). Optimal estimation in thenon-parametricmultiplicativeintensity model.

174. YOR,M.(October1988). Interwinings ofBessel processes.

175. ROJO, J.(October 1988). Ontheconcept oftail-heaviness.

176. ABRAHAMS, D.M. ^andRIZZARDI, F. (September 1988). BLSS -TheBerkeley interactive statisticalsystem:

Anoverview.