apenalty method for nonparametric estimation of the logarithmic

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THE LOGARITHMIC DERIVATIVE OF ADENSITY FUNCTION. BY ..... (x,y) sa x}, and. Put. O~j
APENALTY METHOD FOR NONPARAMETRIC ESTIMATION OF THE LOGARITHMIC DERIVATIVE OF ADENSITY FUNCTION

BY DENNI S D. COX

TECHNICAL REPORT NO. 38 JULY 1983

DEPARTMENT OF STATISTICS UNIVERSITY WASHINGTON SEATTLE; WASHINGTON

A PENALTY METHOD FOR NONPARAMETRIC ESTIMATION OF THE LOGARITHMIC DERIVATIVE OF A DENSITY FUNCTION

BY • COX DEPARTMENT OF STATISTICS UNIVERSITY OF WISCONSIN MADISON" WISCONSIN

53706

2

1,2

A

OF

C

OF A

lTV FUNCTION

by is O. Cox

ABSTRACT Given a random sample of size

n from a density

real line satisfying certain regularity Par amet r i c estimator for of a where rou

w = -fl/f. '0 0 o:

conditions~

f

on the o we propose a non-

The estimate is the minimizer

functional of the form AJ($) + f[$2-2$JJd F A > 0

is a smoothing parameter,

e s s pe na l t y , on of

and F is the n es

n

~

J(.)

is a

rical c.d.f. of the sample.

A more of

case

is

ven~

i rabl e

since it

i 'Ie

i

te 1

es

s e

1i

[a ~

in

1i (x)

ng

ne~

=-

runct.ton on an i

ne

and d

1

F

ative distribution

o

following

on~

we

identity:

(1)

= -f

o

=f

and

the DOLtnCliarv

cancel.

If

f

I

d

b

+ f ~I dF

0

dF0 '

provided certain regularity conditions hold, rts either

or

are i.i .d. random variables with distribution

imate any

o by

a

in theAntegration

Xl,X2, ••• ~Xn

F ' then we can

CI C,

~

valid for all functions

(x)~(x)l

L (F)

2

product of

form

n

= n-1 i

is

on

te

rhmonc-

on5.

s

recover an es

e

a

a on; can reveal much a cross; (l i near i t y of w) '0 '

x

tends to 0 as (i i i)

W o

i ness

and -+

s of

)

tails

(x)

1 ike 1i

es

as

.

remains

or

too) ; and

can be used in

on of location,

so an estimate of W can be used for adaptive maximum likelihood o estimation of location (see Beran's paper or Stone, 1975). We will make use of equation (1) to develop an estimator of Beran's estimator.

but with a very different form

~o'

First, we need

to spell out the regularity conditions. Let

ASSUMPTION Al.

distribution function

f

o

be a probability density function with Let

F .

o

[a,b],

~

_00

a < b

~ +

00,

Assume: (i) ( i i) (i i i) (iv) (v )

(vi)

if and onl y if

f (x ) > 0

o

f (b) o

f 0 (a)

=

y

is

=

0

is absol

= -1

d

;


0 on

b) •

holds," we shall mean either Al or A2 holds. PROPOSITION 1.

(a)

where

W varies

is the unique minimizer of

over functions satisfying f

{3}

W o

Under assumption Al,

dF

0




The proof is given in Section 4. ourselves to

The main reason for restricting

periodic case

on A2) is t

we can

assert the existence of constants c 1 and c 2 for whi ch (7)

We conjecture that a modified version of Theorem 5 holds under Al, but it is then necessary to deal with eigenvalues of singular differential operators (see

WnA

1). An interesting asymptotic representation of

comes out of the proof of Theorem 5, namely

G{X.YiA)

is a Green's

ret ton for a boundary value

-t-r ..

(which ) =--

, we see 1

G(x is

ly

)

. est

t

are as i 5,

'';:

1

!

(

r

Recalling that

~

o

= 0 (n-(q-r)/(2q+3)) p

already involves one derivative, we see that

this rate of convergence in probability is probably the best possible. This is certainly the best rate of convergence in probability implied the

ts in Wahba (1975), Bretagnolle and Huber (1979), and

11

s

e 1)

i

es

i-

(x y) - -

y roucnness

is a

si

y

1

xy course,

t

must involve

vatives in

y as y

a IInOlrlPclrametrl to satisfies certain constraints, e.g.

the method to cases where

$o(t») = o.

s

1.

a II

As in as o

a

values of

Al is a function me

only a """.JI'\I... 1'l

, ••• , Xn• An .... o ,b ], under A2 is a function ~ E Hm ,bJ. o from a in M~:)Ulli~ on A is to ta

~

'tIe

Xl '

= _00, and similarly n

If

2.

i

and b. is a unique minimizer of (5) w.p.l

m,

2

(with probability 1), and it is the unique function b

1 n (L~)(x)(L~)(x)dx + - I ~(Xk)~(Xk)

J0

ao

n

for a11 admissab1e varia t ions

~

satisfying

= -1n

k=l

~.

The first and second variation of the objective function

PROOF.

are b

)

= 2A

Jo

::: 2A

f

(

)(x)dx + 2 f

)(x)(

a0 b

+ 2

0

ao )

=0

i es

so

0

ve is s

- 2

1s

N(L),

J~

I

dJF

n

e

s

o

on B. - 0 on

n

J

s

es

dJF

s

on

) )

d

1'0,

if

.1'.1,

zer

at e

=0

is a

on.

In ing

to

given in

)

1

,bO ) ' 1 Lis a

ally

ion we

a

(x,y)

sa

x}, and O~j 0,

ent B·('1

luenvdlues (1)

1

1)

all

>

y

( if

construct

::

2, ••• , i

)

t as

2

...

=1

are ea 1y seen to value exi

cj

since

(b), pa

y

v

It

1

regularity of

odic

62 of Na

)

rnnC:T::t

Hence, the

ves ) ,

(1967) and

(

ve

bounoa r

ic

64-65 of

posi

C

f"11:::> ... "1 ('ttl

ues of

are

1 v

1

also satisfies such bounds. Q.E.D.

LEMMA 2.

Define G(X,Y;A)

=l

(1

v

are as in

) =J

1.

For a co

nuous

G

so ) 1

x

,

tion

u on

.b],

o

r

~

-(

K D

u 1 •

n o

F.

We show

G(x,·

)

, for each fixed

E

x

E

[a,b].

ti N

GN(X,Y;A)

= I

(x

(y)

v=l

we have for each fixed

x (x)Z ,

and it is only necessary to show the series remains bounded as N + 00, o since A"A(u,u) 1/2 is equivalent to Hm norm. By Sobolov's inequality (page 32 of Agmon, 1967), we have for any r ~ 1 and all x

K

is a constant

1i

ng on [a ,b] and m. ue bounds of

1, and (7) , and

1, the fact

norm (x)

nce

)

~

K'

V

~

2,

ng

r

=(

t

valence, we

1,2, .•.

)
0,

.

0

,

)

in

note

1

levIs i

is p =

h

) H = L rst of all O 2( 0,1,2, ... , an a ication of

i

p +

K

o -