Sample size and the accuracy of a consistent estimator

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Sample size and the accuracy of a consistent estimator

Citation for published version (APA):

Laan, van der, P., & Eeden, van, C. (2000). Sample size and the accuracy of a consistent estimator. (SPOR-Report : reports in statistics, probability and operations research; Vol. 200001). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/2000

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TU/e

technlsche universlteit eindhoven I department of mathematics and computing science

SPaR-Report 2000-01

Sample size and the accuracy of a consistent estimator

P. van der Laan, C. van Eeden

SPaR-Report

Reports in Statistics, Probability and Operations Research

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Sample size and the accuracy of a consistent

estimator

Paul

VAN DER

LAAN and Constance

VAN

EEDEN

Eindhoven University of Technology and The University of British Columbia

Key words and phrases: Peakedness, consistency, logconcave densities, strong unimodal-ity.

AMS 1991 subject classifications: 62FlO, 62Fll, 60E15.

ABSTRACT

Birnbaum (1948) introduced the notion of peakedness about () of a random variable T, defined by P(IT -

(}I

<

E), £ > O. What seems to be not well-known is that, for a consistent estimator

Tn of (), its peakedness does not necessarily converge to 1 monotonically in n. In this article some known results on how the peakedness of the sample mean behaves as a function of n

are recalled. Also, new results concerning the peakedness of the median and the midrange are presented.

1 Introduction

Suppose Xl, ... ,Xn are a sample from a distribution with finite variance and one wants to estimate It = £XI based on (Xl, ... , Xn). Then it is, of course, well-known that

Xn

=

(I:i=l

Xi)/n is a consistent estimator of IL, i.e., for all £

>

0,

PxJe) =

P(IX

n

ILl

<

e) -t 1 as n -t 00. (1.1) What seems to be less well-known is that PxJe) does not necessarily converge to one monotonically in n. Thus, judging the accuracy of Xn by PX

_n

(e), e

>

0, a larger n might give a worse estimator.

In this article we first recall in Section 2 some known results on how PX_n(e) behaves as a function of n. Then, in Section 3, we present new results on this question for the case where the median or the midrange are used to estimate the median or the mean of Xl.

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2 Results for

Xn

and some generalizations

Birnbaum (1948) calls

])1'(£)

=

P(IT -

01 <

£) £

> 0

the peakedness (with respect to 0) of T and calls T more peaked than S when ])1'( e)

2:

ps( £) for all e

>

O. He proves several properties of the peakedness and gives, e.g., conditions under which, for the same 0 and the same sample size, one of two sample means is more peaked than the other.

Proschan (1965) gives several results on the behaviour of PTn (e) as a function of n where

Tn is a convex combination of Xl, ... _,Xn,a sample from a distribution F. He supposes that F has a density which is symmetric with respect to 0 and is logconcave on the support of F. In particular, Proschan shows that for such a distribution

pxJe)

is, for each e

>

0, strictly increasing in n (i.e., of course, for those IS

>

0 which are in the interior of the support of Xl 0).

Proschan also gives an example where PXn{e) is not increasing in n. In fact, he gives a distribution for which Xl is more peaked about 0 than (Xl

+

X2)/2. This distribution

is the convolution of a distribution with a symmetric (about zero) logconcave density and a Cauchy distribution with median zero. Then, for 4> strictly increasing and convex on (0,00) with ¢(x) = ¢( -x) for all x,

4>(Xd

is more peaked with respect to zero than (¢(Xt)

+4>(X

2))/2. Of course, for this case

Xn

does not converge to zero in probability, so the result might not be too surprising. However, Dharmadhikari and Joag-Dev (1988, p. 171-172) show that, e.g., for the density

f(x)

3

1 I

_(l

x

l:::;

₁₎

+

_{181(1 :::;}1

Ixl ::;

_4),

Xl is more peaked with respect to zero than (Xl X2)/2. And for this distribution

(1.1) clearly holds.

The results of Proschan (1965) have been extended to the multivariate case by Olkin and Tong (1987) (see also Dharmadhikari and Joag-Dev (1988, Theorem 7.11)). Further, Ma (1998) generalized Proschan's (1965) result to the case where the random variables

Xl, ... ,Xn are independent but not necessarily indentically distributed.

3 The case of the median and the midrange

Assume that Xl, ... ,Xn is a sample from a distribution with a density and that n is

odd. Let Mn be the median of _{Xl,'" ,Xn.} For this case, Karlin (1992) proved that, when the density of Xl is symmetric around f.L, Mn+2 is more peaked around f.L than Mn. We give, in Theorem 3.1 below, a more general and more precise form of this result with a different proof.

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Theorem 3.1 Let XI, ... _{,Xn be a sample from a distribution F with density f. Let} )\.-1

=

{x

I

F(x)

=

1/2} be the set of medians of F. Then, for n odd and m E

M,

the peakedness of Mn - m is strictly increasing in n when F(m - c)

<

F(m +c). For c such that F(m - c) = F(m

+

c)(= 1/2) the peakedness of Mn is independent ofn and equal to

O.

Proof. Assume without loss of generality that m

=

O. First note that, for x E (-00,00),

(n-l)/2 ( ) 1 IoF(X) P(Mn>x)=

L

~

F(x)i(I-F(x))n-i=l- ( ) tn;:l(l_t(;:ldt. • 1. B nH!!±!. 0 t=O 2 ' 2 So, as a function of y

=

F(x), 0

<

y

<

1, d Y¥(l-

y)

dy P( Mn

>

x) = - B (!!±!. !!±!.) 2 ' 2

Putting Qn(Y) = P(Mn

>

x) - P(Mn+2

>

x), this gives

d (n 2)1!!±!( )!!.±.l n! n-l )n-l

dyQn(X)

((nil)!)2

y 2 1-y 2

((n;l)!f

y-2 (1-y-2

n - l n - l

n!

(

(n

+

1)2)

=

y-2 (1-Y)-2

((!!:}l

)!r

(n

+

1)(n

+

2)y(1 - y) - - 2 - . This last expression is, for 0

<

y

<

1,

>

0,

=

0,

<

0 if and only if

n

+

1 1

4(n+2) - 4(n+2) -(y

which is equivalent to

I

y -

~

I {

~

}

c =

~

J(

n

+

2)-',

So, Qn(Y) is increasing on

(! -

c,

!

+

c) and decreasing on (O,! - c) and on

(!

+

c, 1). Combining this with the fact that, for all n,

1 for y

=

0

P(Mn

>

x)

=

!

for y

=

!

o

for y = 1, 3

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shows that

>

0 for x such that ~

<

F( x)

<

1 P(Mn

>

x) P(Mn+2

>

x)

=

0 for x such that F(x)

=

1/2

<

0 for x such that 0

<

F(x)

<

~.

This shows that for x

2::

0, i.e. for x such that F( -x) ::; 1/2 ::; F( x), P(IMn+2 \

<

x) - P(\Mn \

<

x)

=

P(Mn > x) - P(Mn+2 > x) - [P(Mn

>

-x) - P(Mn+2

>

-x)J

>0 ifF(-x) <F(x)

=

a

if F( -x) = F(x), which proves the result. 0

Note, from Theorem 3.1, that the conditions on F for the median to have increasing peakedness in n are much weaker than those for the mean. Other than the obvious condition that not both m

+

€ and m - e are medians of F, all one needs for the median to have increasing peakedness with respect to an m E M is a density, while for the mean a logconcave symmetric density is needed in the proofs. But in order for the median to be a consistent estimator of the population median, one needs a unique median m and a density which is positive in a neighbourhood of m. Under this condition the peakedness of the median with respect to m is strictly increasing in n for all e

>

O.

We do not know whether Theorem 3.1 holds for n even.

Now take the case of a sample Xl, .. . , Xn from a uniform distribution on the interval

[8 - 1,8

+

1] and let Sn be the midrange of this sample, i.e. Sn

=

-2

1

(min _l:5i:5nXi

+

_l:5i:5nmax Xi)' Then the following theorem holds.

Theorem 3.2 The peakedness of Sn with respect to (I is strictly increasing in n for n

2::

2 and each € E (0,1).

Proof. Suppose, without loss of generality, that (I

=

O. Then the joint density of

minl:5i:5n

Ii

and maxI:5i:5n

Ii

at (x,y) is, for n

2::

2, given by n(n-l)( _y-x)n-2

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So, for 1::; t ::; 0, n(n - 1)

jt

1

2t -x mftX

Yi ::;

2t) = 2_n dx (y l:5l:5n _-1 x )n-2d (1

+

t)n x y= 2 and, for 0

<

t ::; 1,

P( min

Yi

+

max

Yi

<

2t)

=

1 - P( min

Yi

+

max

Yi

<

-2t)

=

1 _ (1 - t)n ,

1:5i5n l:5i:5n - l:5i:5n l:5i:5n - 2

which gives, for

It I

<

1,

P(ISnl

<

t)

=

1 - (1 -

tt,

from which the results follows immediately. 0

We have not been able to prove or disprove increasing peakedness in n of the midrange for distributions other than the uniform.

Remark

Note that, in quoting Proschan's (1965) results, we ask for the distribution function F

to have a density

f

which is logconcave on the support of F, while Proschan asks for this density to be a P61ya frequency function of order 2 (PFz). However, it was shown by Schoenberg (1951) that

f

is PF 2 {::::=>

f

is logconcave on the support of F, so the two conditions are equivalent.

Further note that Ibragimov (1956) showed that, for a distribution function F with a density

f,

f

is strongly unimodal {::::=>

f

is logconcave on the support of F,

where a density is strongly unimodal if its convolution with all unimodal densities is unimodal. So, the condition of logconcavity of

f

can also be replaced by the condition of its strong unimodality. For more results on P61ya frequency functions see e.g. Marshall and Olkin (1979, Chapter 18) and Karlin (1968).

ACKNOWLEDGEMENTS

The authors thank Chunsheng Ma for pointing out the Ma (1998) and the Karlin (1992) references.

4 References

Birnbaum, Z.W. (1948). On random variables with comparable peakedness. Ann. Math. Statist., 19, 76-81.

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Dharmadhikari, S. and Joag-Dev, K. (1988). UnimodalitYl Convexity! and Applications,

Academic Press.

Ibragimov, LA. (1956). On the composition of unimodal distributions. Theor. Probab. Appl., 1, 255-260.

Karlin, S. (1968). Total Positivity, Vol. I, Stanford University Press.

Karlin, S. (1992). Stochastic comparisons between means and medians for i.i.d. random variables. The Art of Statistical Science, K.V. Mardia, Ed., John Wiley

&

Sons Ltd, p. 261-274.

Ma, C. (1998). On the peakedness of distributions of convex combinations. J. Statist. Plann. Inference, 56, 51-56.

Marshall, A.W. and Oikin, 1. (1979). Inequalities: Theory of Majorization and its Ap-plications, Academic Press.

aIkin, 1. and Tong, Y. L. (1988). Peakedness in multivariate distributions. Statistical Decision Theory and Related Topics IV, S. S. Gupta and J. O. Berger, Eds. Vol. II, Springer-Verlag, New York, p. 373-383.

Proschan, F. (1965). Peakedness of distributions of convex combinations. Ann. Math. Statist., 36, 1703-1706.

Schoenberg, 1. J. (1951). On P61ya frequency functions 1. J. Anal. Math., 1, 331-374. Department of Mathematics and Computing Science

Eindhoven University of Technology P.O. Box 513 5600 MB The Netherlands e-mail: pvdlaan@win.tue.nl Moerland 19 1151 BH Broek in Waterland The Netherlands e-mail: cve@xs4all.nl