Gertrude M. Cox
'l'HE "INEFFICIENCY" OF THE SJUviPLE MEDIAN FOR
F(;.R
MANY FAMILIAR SYMMETRIC DISTRIBUTIONS
by
John T. Chu
Department of Statistics
University of North Carolina, Chapel Hill
Special report to the Office of Naval
Research of ~ork at Chapel Hill under
Contract NR 042 031, Project N7-onr284(02), for research in probability
and statistics.
Institute of Statistics
Mimeograph Series No. 125
February 23, 1955
1
THE IlINEFFICIENCyllOF THE SAlt1PLE MEDIAN
FOR MANY FAMILIAR SYMMETRIC DISTRIBUTIONSJ.
By J.T.Chu, University of North Carolina
1. A lower bound. If the reciprocal of the ( asymptotic ) variance of an estimate
is taken as a measure of its ( asymptotic ) efficiency, the s~ple median
is often ( asymptotically ) less efficient than the sample mean
symmetric distributions familiar to statisticians.
x
,for many
In fact, for a symmetric
distribution having its maximum frequency at the point of symmetry, i f
x,
asymptotically less efficient than
as
x
X
then quite often
x
x
is
is not so efficient
at all, with the possible exceptions of very small samples.
To show
these facts, we derive a very simple, yet sharp, lower bound for the variance
of the sample median.
Suppose that F(x) and f(x) are the cdf ( cumulative distribution function ) and pdf ( probability density function ) of a certain continuous distribution, and f(x) is
~ f(x) for all x.
then, where Cn =
(1)
(
var x
Let
x
symmet~ic
with respect to x
= £,
and f(t)
be the s~mple median of a sample of size 2n + 1,
2n + 1 )1 / n1 n~,
=
x - S )2 C /- F(x)
n -
-
7 nr
1 - F(x)
-
-
7 nf(x)
=
~
Equality holds for
Ci
r
1
f(s»)
-2
J (F - 1/2 )2 C
o
Fn (
1 _ F )n dF
n
rectangular distribution,
1 Sponsored by the Office of Naval Research under Contract NR 042 031
dx
-22. Examples. It is well known that .for a normal or rectangular distribution
X
x
is more efficient than
We shall show that this is also true .for
many .familiar symmetric distributions.
(1) Tri~ngular distribution:
i
is less efficient than
=1
f(x)
- lxi, Ixl $ 1.
.for samples of sizes 2n + 1 where n
computation shows that this 1s also true if n
> 1.
For a t-distribution with k degrees o.f freedom,
less efficient than
x
2
2
(kj2) / 4 r ( ( k + 1 )/2 )
= 2m +
>5
-x
=
= r(2p)r- 2 (p)
i-
2, p. 244_7
x p - 1 ( 1 _ x )p - 1,
o < x < 1;
is less efficient than
2
4p
(2n + 3) / (2n + 1).
> 25.
and n
=
>
Computation shows, e.g., that the
(3) Symmetric ~-distribution. The pdf is given by
f(x)
is
1, the LHS ( left hand side ) of the above in-
equality is an incredsing function of m.
inequality holdS if k
x
i f ( not necessarily only if )
te ( k - 2 ) r
k= 2m and k
Direct
= 1.
(2) t-distribution.
For both
x
Following (1),
x
> O.
p
if
2
- 4 ( 2p + 1 ) r\p) / r (2p)
>(
2n + 3 )/( 2n + 1 ) •
The LHS becomes smaller if p is replaced by p + 1, and tends to tej2 as p
tends to
every p
(~)
00.
So it has a lower bound te/2.
> 0 and n
~
2.
Cauchy type distribution.
f(x)
= Cal
( 1 + Ixja ),
It is defined to be one with a pdf of the type
-00
<x <
known Cauchy distribution for
i.
Hence the inequality holds for
00;
~hich
a
-x
> 1. If a
= 2,
is infinitely more efficient than
It would be interesting to examine whether or not
efficient as a increases.
Now
i
x
is less efficient than
x
where x
= teja.
2
sin 3x
I
sin 3 x
i
becomes more
> 3.
C
a
3, p. 118_7.
It
has finite variance only if a
and var i can be obtained by using contour integration
follows that
we obtain the well
>(
i
i-
if
2n + 3 ) j ( 2n + 1 ),
The LHS is a decreasing function of x, so an increasing
,
-3f
.:.::ti'::;'"
:Jf
The least ex's for \.hieh the LHS is equal to
0:..
maximum and minimum of the RHS, are found to be
5/3 and 1, the
4.65 and 3.75 approximately.
3. Remarks.
(~)
Not for all symmetric distributions is
x more efficient than
the parent population has a Laplace distribution, e.g.,
for all samples of odd sizes
x
-
x
When
is more efficient
L· 1.7.
(2) If f(x) satisfies certain continuity conditions,
normal distribution and the asymptotic variance is
x
has an asymptotically
(4' L· f( s).7 2( 2n +
1 )
J .1.
Therefore if the sample size is not too small, the asymptotic variance is for all
practical purposes a lower bound for
efficient than
, then
x
-x
var
x.
And if
is less efficient than
x
X
is asymptotically less
for all samples whose
sizes are not too small.
References
i· 1.7 Chu,J.T.,
and Hotelling, H., The moments of the sample median, to be
published.
L· 2.7 Cramer,
H., Mathematical Methods of Statistics, Princeton University
Press, 1946.
I: 3.7 Whittaker,E.T., and Watson, G.N.,
Press, 1952.
Modern Analysis, Cambridge University