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A note on the estimation of Lp-norms
Statistics & Probability North-Holland
Letters
28 September
15 (1992) 131-133
A note on the estimation Michael
1992
of P-norms
Weba
Institut fiir Mathematische Stochastik, Unicersitiit Hamburg, Germany Received
August
1991
Abstract: The Lp’-norm of a random variable with unknown distribution is determined Under a mild moment condition, expansions of bias and variance are established.
Keywords: L”-norm;
statistical
Let X be a random
of the substitution
estimator.
point estimation.
variable
on a probability
Ilxllp=(EIIXI”])“p plays an important problem: we want to of X is unknown or (possibly simulated) same distribution as
by means
space (0,
&, P> with values
in [w. The LP-norm
(l
with error bounds. Consider the following role in stochastics, e.g., in connection make use of II X IIp but an explicit formula is not available because the distribution too complicated. In that case, 11X I(,, must be determined empirically by means of a random sample. Provided X,, X,, . . . , X,, are i.i.d. random variables having the X, the substitution principle suggests to estimate I1X II p by the quantity
iP,n
is the pth root of z,, = (l/n)Cr=, I X, I ‘. Since z?,, is a ‘reasonable’ estimator of E[ I X I “I the problem of computing bias and variance of 6p,n can be viewed as a problem of statistical point estimation. In order to describe bias and variance of t?p,n in the non-trivial case p > 1 - the case p = 1 will be excluded in the sequel - one may try to utilize well-known results such as the theorem on p. 353 of Cramer (1958) or Theorem 5.la on p. 109 of Lehmann (1983). These theorems need some boundedness conditions which are, in general, violated because the function r(t) = t ‘jp, t 2 0, is rather unpleasant: r and all its derivatives are unbounded on (0, m>, and r is not differentiable at the origin. It is the purpose of this note to establish expansions of bias and variance of 6p,n where X is assumed to fulfil a moment condition. The symbol i; will stand for the jth central moment of I X I “, i.e.,
Correspondence to: Michael Hamburg 13, Germany. 0167.7152/92/$05.00
Weba,
Institut
0 1992 - Elsevier
fiir Mathematische
Science
Publishers
Stochastik,
Universitat
B.V. All rights reserved
Hamburg,
Bundesstrasse
55, W-2000
131
Volume
15, Number
2
STATISTICS
& PROBABILITY
28 September
LETTERS
1992
Theorem. Assume that X has the property P(X = 0) < 1 and satisfies the moment condition E[ I x I 4p+a] < m for p > 1 and some real (Y > 0. Then bias and variance of 6p,n admit the representations E[ ip,n] - IIX IIp = al/n
+ az/n2 + o(l/n2)
(1)
and
VX[ Sp,n] = b,/n
+ b,/n’
+ 0(1/n*),
(2)
respectively, where the constants are given by (1 -a) a, = ~ 2P2 a2 =
. 12. [IX Ilj-2p,
(1 -P)(l-2P)
* !tj.
6P3
1 b1= P2
. &.
11X
b =
(1 -PI
2
P3
1j2-2p P
II x II;-“” +
(l-p)(l-2p)(l-3p)
.[*
IIxII,_4p 2'
8P4
P
'
’
.~3.IIxllp2-3P+
(1-p;f4-5p)
.&
IIx
If-"".
r(t) = t l/p , t > 0, can be represented
Proof. For each fixed C > 0, the function
according
to
r(t)=r(~)+r’(~).(t-~)+~r”(~).(t-5)2+~r”’(5).(t-g)3+~r’iv’(~).(t-~)4 + 4(l, where
t) . (t
- 0”
c#&J, t) satisfies t’el 4(L,
By definition
t> = $(l?
r> = 0.
of r(t), we also have sup I4(L
t)l
I>0
Setting
Zi = I X, I p for i E N, z,, = (l/n)C~zIZj
J!?[(Z~)~‘~] =J1lp + +r”(J)
*
and 5 = EIZil =
Ekzln lj21 +6
l+rr(
+&r(‘“)( 5) . +Zl-l>“] n3 i
++i, In order
to verify (1) it remains lilimn2.E[
132
4(l,
l)
II X II:, one obtains .
+I-
03] n2
3(n-1)(-$~l-i)2])2 +
n3
I
z,)*(Gq4]. to show
.Fn). (Zn -i.)“]
= 0.
(3)
Volume
15, Number
For each It-f\ >6
2
STATISTICS
0, there implies
exists a number
E >
It-51
~ IW> Hence
t)
i
6 > 0 with
28 September
LETTERS
I c#&J,t) 1 < F for
I t - J I < 6. On the other
1992
hand,
a’p i
6
& PROBABILITY
. sup I&L
s) I.
S>O
the relation
SUPl4(57 s)l <&f
S>O
.
p/P
I t -
i
I a’P
is valid for all t a 0, and one can conclude
The inequality
of Marcinkiewicz-Zygmund
E
lz,
-
jiJia”]
=
yields
o(~-~-w’(~P)).
1
(See Chow and Teicher
(1988, p. 3681.1 Therefore,
lim sup n2. IE[~(I,Z,).(Z,,-1)4]I~C.& n~m holds for some constant C 2 0. This proves (3) because E > 0 was arbitrary. Repeating the above considerations with r* instead of r, we obtain an expansion equation (2) follows if this expansion is combined with equation (1). q Under
the assumptions E [ (ip,, -
In the special
of the theorem
the mean
square
error can be represented
of E[(.?fn>2/J’]. Then
by
II x II p,2] = b,/n + (4 + b,)/n2 + 0(1/d).
case p = 2, the leading
coefficient
b, takes the simple
form
b, = f.Var[X*]/E[X*].
References Chow, Y.S. and H. Teicher (1988), Probability (Springer, New York). Cram&, H. (1958), Mathematical Methods of (Princeton, Univ. Press, Princeton, NJ).
Theov
Lehmann, E.L. New York).
(19831,
Theory
of Point
Estimation
(Wiley,
Statistics
133
Letters
28 September
15 (1992) 131-133
A note on the estimation Michael
1992
of P-norms
Weba
Institut fiir Mathematische Stochastik, Unicersitiit Hamburg, Germany Received
August
1991
Abstract: The Lp’-norm of a random variable with unknown distribution is determined Under a mild moment condition, expansions of bias and variance are established.
Keywords: L”-norm;
statistical
Let X be a random
of the substitution
estimator.
point estimation.
variable
on a probability
Ilxllp=(EIIXI”])“p plays an important problem: we want to of X is unknown or (possibly simulated) same distribution as
by means
space (0,
&, P> with values
in [w. The LP-norm
(l
with error bounds. Consider the following role in stochastics, e.g., in connection make use of II X IIp but an explicit formula is not available because the distribution too complicated. In that case, 11X I(,, must be determined empirically by means of a random sample. Provided X,, X,, . . . , X,, are i.i.d. random variables having the X, the substitution principle suggests to estimate I1X II p by the quantity
iP,n
is the pth root of z,, = (l/n)Cr=, I X, I ‘. Since z?,, is a ‘reasonable’ estimator of E[ I X I “I the problem of computing bias and variance of 6p,n can be viewed as a problem of statistical point estimation. In order to describe bias and variance of t?p,n in the non-trivial case p > 1 - the case p = 1 will be excluded in the sequel - one may try to utilize well-known results such as the theorem on p. 353 of Cramer (1958) or Theorem 5.la on p. 109 of Lehmann (1983). These theorems need some boundedness conditions which are, in general, violated because the function r(t) = t ‘jp, t 2 0, is rather unpleasant: r and all its derivatives are unbounded on (0, m>, and r is not differentiable at the origin. It is the purpose of this note to establish expansions of bias and variance of 6p,n where X is assumed to fulfil a moment condition. The symbol i; will stand for the jth central moment of I X I “, i.e.,
Correspondence to: Michael Hamburg 13, Germany. 0167.7152/92/$05.00
Weba,
Institut
0 1992 - Elsevier
fiir Mathematische
Science
Publishers
Stochastik,
Universitat
B.V. All rights reserved
Hamburg,
Bundesstrasse
55, W-2000
131
Volume
15, Number
2
STATISTICS
& PROBABILITY
28 September
LETTERS
1992
Theorem. Assume that X has the property P(X = 0) < 1 and satisfies the moment condition E[ I x I 4p+a] < m for p > 1 and some real (Y > 0. Then bias and variance of 6p,n admit the representations E[ ip,n] - IIX IIp = al/n
+ az/n2 + o(l/n2)
(1)
and
VX[ Sp,n] = b,/n
+ b,/n’
+ 0(1/n*),
(2)
respectively, where the constants are given by (1 -a) a, = ~ 2P2 a2 =
. 12. [IX Ilj-2p,
(1 -P)(l-2P)
* !tj.
6P3
1 b1= P2
. &.
11X
b =
(1 -PI
2
P3
1j2-2p P
II x II;-“” +
(l-p)(l-2p)(l-3p)
.[*
IIxII,_4p 2'
8P4
P
'
’
.~3.IIxllp2-3P+
(1-p;f4-5p)
.&
IIx
If-"".
r(t) = t l/p , t > 0, can be represented
Proof. For each fixed C > 0, the function
according
to
r(t)=r(~)+r’(~).(t-~)+~r”(~).(t-5)2+~r”’(5).(t-g)3+~r’iv’(~).(t-~)4 + 4(l, where
t) . (t
- 0”
c#&J, t) satisfies t’el 4(L,
By definition
t> = $(l?
r> = 0.
of r(t), we also have sup I4(L
t)l
I>0
Setting
Zi = I X, I p for i E N, z,, = (l/n)C~zIZj
J!?[(Z~)~‘~] =J1lp + +r”(J)
*
and 5 = EIZil =
Ekzln lj21 +6
l+rr(
+&r(‘“)( 5) . +Zl-l>“] n3 i
++i, In order
to verify (1) it remains lilimn2.E[
132
4(l,
l)
II X II:, one obtains .
+I-
03] n2
3(n-1)(-$~l-i)2])2 +
n3
I
z,)*(Gq4]. to show
.Fn). (Zn -i.)“]
= 0.
(3)
Volume
15, Number
For each It-f\ >6
2
STATISTICS
0, there implies
exists a number
E >
It-51
~ IW> Hence
t)
i
6 > 0 with
28 September
LETTERS
I c#&J,t) 1 < F for
I t - J I < 6. On the other
1992
hand,
a’p i
6
& PROBABILITY
. sup I&L
s) I.
S>O
the relation
SUPl4(57 s)l <&f
S>O
.
p/P
I t -
i
I a’P
is valid for all t a 0, and one can conclude
The inequality
of Marcinkiewicz-Zygmund
E
lz,
-
jiJia”]
=
yields
o(~-~-w’(~P)).
1
(See Chow and Teicher
(1988, p. 3681.1 Therefore,
lim sup n2. IE[~(I,Z,).(Z,,-1)4]I~C.& n~m holds for some constant C 2 0. This proves (3) because E > 0 was arbitrary. Repeating the above considerations with r* instead of r, we obtain an expansion equation (2) follows if this expansion is combined with equation (1). q Under
the assumptions E [ (ip,, -
In the special
of the theorem
the mean
square
error can be represented
of E[(.?fn>2/J’]. Then
by
II x II p,2] = b,/n + (4 + b,)/n2 + 0(1/d).
case p = 2, the leading
coefficient
b, takes the simple
form
b, = f.Var[X*]/E[X*].
References Chow, Y.S. and H. Teicher (1988), Probability (Springer, New York). Cram&, H. (1958), Mathematical Methods of (Princeton, Univ. Press, Princeton, NJ).
Theov
Lehmann, E.L. New York).
(19831,
Theory
of Point
Estimation
(Wiley,
Statistics
133