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 -
