"

a

•

THE MOMENTS OF THE SAMPLE MEDIAN

by

J. T. Chu and Harold Ho~elling

Special report to the Office ofNaval Research of work at ChapelHill under Contract NR 042 031 ,Project N7-onr-284(02)1 for re­search in Probability and Statistics.

Institute of StatisticsMimeograph Series No. 116August1 1954

·e

•

THE MCMENTS OF THE S1U-IPLE MEDIANl , 2

by

John T. Chu and Harold HotellingInstitute of Statistics, University of North Carolina

1. Summary. It is shown that under certain regularity conditions, the moments

about its mean of the sample median tend, as the sample size increases indefinitely,

to the corresponding ones of the asymptotic distribution (which is normal). A

method of approximation, using the inverse function of the cumulative distribution

function, is obtained for the moments of the sronple median of a certain type of

parent distribution. An advantage of this method is that the error can be made as

small as is required. Applications to normal, Laplace, and Cauchy distributions are

discussed. Upper and lower bounds are obtained, by a different method, for the

variance of the sample median of normal and Laplace parent distributions. They are

simple in form, and of practical use if the sample size is not too small.

2. Introduction. Let a population be given with cdf (cumulative distribution

function) F(x) and pdf (probability density function) f(x), and median ~ which we

assume to exist uniquely. Let xdenote the sanlple median of a sample of size

2n + 1. Then the pdf g(x) of xand the pdf h(x) of the asymptotic distribution of

-x are respectively

where C = (2n + l)!/(n! n!), andn

1. This paper was presented at the summer meeting of the Institute ofMathematical Statistics at Montreal, P.Q., Canada, on September 11, 1954.

2. Sponsored by the Office of Naval Research under Contract NR 042 031 forresearch in Probabilj.ty and Statistics.

(2) h(x)

2

where il2

=

A question that follows naturally is: Can the moments of the asymptotic dis­

tribution of xbe used as approximations to the corresponding moments of i, and if

not, how to find better anproximations? When the parent distribution is normal,

this question has been answered by various authors, e.g., T. Hojo ~6_7, K. Pearson

~8, 9_7 and more recently, J. H. Cadwell ~3_7. It has been stated, e.g., in ~3_7,

that experiments showed that the distribution of xtends rapidly to normality, but

the variance of x(as ofquantiles in general) tends only slowly to the variance of

the asymptotic distribution. For this reason of slow convergence, approximations...

were derived for the variance of x when the sample size is small. 1Nhile different

methods were used by different authors, their results agree fairly well with each

other. In fact, the problem should be considered as completely solved but for the

unknown error cowaittcd in using such approximation....

One of us ~4_7 recently proved that the distribution of x, for a normal

parent distribution, does tend to normality rather "rapidly". In g 6 we shall

-confirm another experimental result that the variance of x tends flslowly" to the

variance of the asymptotic distribution (actually not "very slowly"). Upper and

lower bounds are obtained (§ 6, Theorem 4) for the variance of~. A slightly better

lower bound is obtained, by a different method, in ~ 5, formula (49), or ~ 6,

formula (74). It seems that even for sample sizes around 10 to 20, the asymptotic

-variance is not a bad approximation to the true variance of x. It becomes a very

good approximation if the sample size is large. However, for large samples, an even

better approximation is obtained in ~ 5, formula (56).

Before further discussion, the following notations will be introduced. If

f(x) and g(x) are functions of x, then Ef(g) denotes the expectation of g(x) with

3

00

~ respect to f(x), i.e., J g(x) f(x) dx. We use, where f, g, and h are given by

-00

(1) and (2),

(3) ,

and for any integer k ~ 2,

,

( 4) ... ( ... )k~k = Ee x - ~l '

- ( -)k~k = Eh x - ~l '

It should be pointed out that, although the pdf g(x) of i tends to h(x), the moments

~k of i in general do not necessarily tend to ~k. In fact ~k may never exist ~2_7.

Nevertheless, if the parent pdf satisfies certain conditions, then it can be shown...

that ~k tends to ~k as sample size tends to infinity ( §3, Theorem 1). Therefore

under such circumstance, it is justifiable, at least for large samples, to use ~k

as an approximation to ~.

If the parent distribution satisfies certain conditions, a general method is

obtained in ~ 4 for computing ~k' k = 1, 2, ..•• The method is based on the Taylor

expansion of x(F), the inverse function of F(x). For example, if x(F) - ~

00 1 m k= ~ ~(F - 2) converges for 0 < F < 1, a = O(ZUm ) where k ~ 0 and f(x) is sym-~l m

metric with respect to x = ~J then when n > 2k + 3,

4

1

ii2 * j S~ Cn F"(I - F)n dF ,

o

m 1 rwhere C is given by (1) and S = ~ a (F - -2) (~4, Theorem 3). Error in such

n m r=l r

approximation can be computed, and it tends to 0 as m tends to 00. If the parent

pdf is not symmetric, similar approximations can be obtained ( ~ 4, formula 26).

Applications are given to the variances of the sMQple medians of Laplace and Cauchy

parent distributions (§ 4, Examples 1 and 2).

Finally upper and lower bounds are derived in ~ 7 for the variance of x of a

Laplace parent distribution. It then can be seen that for estimating the mean of a

Laplace distribution, the sample median is a "betterB estimate than the sample mean,

not only for large smnples, but for small samples as well.

3. Large Sample Moments.

Lemma 1. If 0 ~ c ~~, then for m, n = 1, 2, ... ,

(6)

12 + c(I

1 )2 - c

4c 2

m 1 m+2n+l I ()/

I u - ~ I unCI - u)n du - ( "2 ) ,J t m-I 2 (1 - t)n dt

o

1In particular, if c = 2 ' ~d Cn = (2n + l)l/nl nl , we have for fixod m,

1

(7) {Cn

I u - ~ 1m

unCI _ u)n du = O(n-m/2) ,

o

1()It 1)2m n( )n 1)2rn 1 • 3 .•• (2m - 1)8 Cn(u - 2 u 1 - u du = (? (2n + 3)(2n + 5) ... (2n + 2m + 1) ,

a

These formulae are easily proved using transformations v = ~ (u - ~), etc.

Theorem 1. Let a population be given with cdf F(x) and pdf f(x). Suppose

that the median ~ of the given population exists uniquely and f(~) r 0, and fl(x)

exists and is bounded in some neighborhood of x =~. If xis the sample median of

a sample of size 2n + 1, and ~ and ~k' as defined by (4), are respectively the

kth moment of x about its mean and the corresponding one of its asymptotic dis-

tribution, then

(10)

(11)

Lim - -lJ.2k-l = lJ.2k-ln ~oo '

Lim ~2k/ ~?k = 1 ,n ....;> 00

k = 1, 2, ... ,

provided that in each case, lJ. 2k- l and ~2k are finite for at least one n.

Proof. We will prove (11) as an illustration of the method we usc. (10)

can be shown in the same way. Obviously

2k-l(I?) ~2k = lJ.~k + Z (2k) (_1)2k-j l-Li2k

-j l-Lj ,

j=O j

where t~k) = (2k)!/ {jl(2k - j)l} • We say that if ~2k is finite for a certain

6

n == no' then ~2k is finite for all n ~ nO' and

(13)

(14)

, _ (-m)!J.2m_l - 0 n ,

( a1)2m 1 • 3 ••• (2m - 1) ( -m-1/2)p.L = "?I'" ( + 0 nelll ~ (2n + 3) 2n + 5) ... (2n + 2m - 1) ,

for m = 1, 2, •.• , k,

where a1 = l/f(~). On combining (12), (13), and (lh), it follows that

(IS)... a1 2k!J. 2k = (2)

1 • 3 •.• (2k - 1)( 2n + 3)( 2n + 5) ... ( 2n + 2k + 1) ( -k-l/2)+ 0 n •

Since ij:2k == 1·3 ••• (21( -1) ij:~, where ij:2 is dofined by (?), we have (11).

To complete the proof, it remains to establish (13) and (14). Now for

example,

00

(16) !J.' = (x - ~)2m C rF(x) 7n rl - F(x) 7n f(x) dx2m n- -...... -

-00

a b 00

= + i + f = II + 12 + I) , say,

-00 a b

where a < ~ and b > ~ will be chosen later. For 0 ~ F ~ 1, the function F(l - F) is

t " h"t" 1 t F 1 d"" "f 0 F 1 dnon-nega ~ve, reac os ~ s maxmum '4 a = "2 ' an. ~s IDcreas~ng or ::: ~ '2 an

decreasing for ~ ~ F ~ 1. Lot

(17)

7

then 0 < r < 1. Since C = O(22nnl / 2), it follows thatn

On the other hand, if a and b are so chosen that, e.g., F(b) - ~ = ~ - F(a) = c is

small, then for ~ - c ~ F ~ ~ + c, x(F), the inverse function of F(x~ is uniquely

defined and may be expanded, by Taylorrs method, into

x(F) - l;

where al = l/f(~) and R2 is the remainder. Substituting (19) for x - ~ in I 2 of

(16), it can be shown, using Lemma 1, that I 2 is equal to the RHS (right hand side)

of (14). Combining this fact with (16) and (18), we obtain (14).

Regarding the above theorem, we make

Remark 1. A sufficient condition for ~1 being finite for some n = nO (hence.(

all n ~ no) is that ~k bo finite. This condition, however, is not necessary. For

example, the variance of the sample median of a Cauchy parent distribution is finite

if the sample size 2n + 1 ~ 5, though the variance of the parent distribution is

infinite ( ~ 4, Example 2).

Remark 2. lbeorem 1 states only some sufficient conditions under which (10)

and (11) are true. For a Laplace parent distribution, f'(l;) does not exist, yet

(10) and (11) hold ( ~ 4, Example 1).

The above theorem provides a justification, at least for large samples, for

using ~k as 3n approximation to ~k. In the next section we will proceed to show that

if tho parent pdf satisfies some additional conditions, then satisfactory approxDua­

tions can be obtained for ~k for samples of smaller sizes as well.

8

4. Approximations.

Lenuna 2. If Ie is rcal, then the follOl'1ing series is convergent for every

positive integer n > kl

(20)00

Em=l

o

Proof. Use Lemma 1 and the fact iI, p. 33_7 that if am ~ 0 and m( ~/am+l-l)

00

approaches r > 1, then E a is convergent; or apply the Stirling's approximation,m=l m

with m large, to the gamma functions obtained by putting c = ~ in (6).

'lbeorem 2....

Let F(x) be the cdf of a given distribution and ~ and x be

respectively the median and the sample median of a sample of size ?n + 1. Suppose

that x(F), the inverse function of F(x), is for 0 < F < 1 uniquely defined and equal

1to a convergent series of powers of F -?; let

(21)

v.Trite

(22)

x(F) - ~00 1 m

= E am(F - '2) •m=l

00 1 rand R = E ar(F - 2)

m r=rn+l

If there exists a sequence {b ) such thatm

(23)00

Em=l

2(a /b) < 00 ,m m

for 0 < F < 1, and

(25)00 2Z b

mm=l

1

f

1 2m(F - -) C pn(l - F)n dF < 00 ,

2 n

o

9

- -for some positive integer value nO of n, and ~2' tho variance of x, is finite for

n = nO' then for all integers n > nO'

( ?6) -1J.2 = Lim• m-.;> 00

1

J82 C ~(l - F)n dF - (m n

o

1

)

1 8 C ~(l _ F)n dF/ Jm n

o

Further, if f(x) is symmetric with respect'to~, then the second term in the bracket

should be omitted.

Proof. For simplicity we assume that f(x) is ~mmetric with respect to x = ~.

In this case ~l :: ~ and

-IJ. =2

00

( (x - ~)2C ;-F(x) 7n r-l - F(x) 7nf(x)dx =) rr- _L -

-00

aI

Ij

-00

+

b

ja

+

00

ib

where a < £ and b > t will be chosen later. It can be shown that

(28 )

where.r is defined by (17). 1 1Choose a, b, and c such that 0 < 2 - F(a) = F(b) - 2

= c <~. Using (21) and (22), we have

where

(30)

(31)

(J3)

1

( 2) SmCnF'l(l - F)ndFI ~ J1 + J 2 + J 3 + J 4 '

o

1

J1 = I S;Cnrn(l - F)n dF ,

1 '~ + c

1~ - c

J 2 = ( S2C F'l(l - F)n dF ,I mno

1'2 + c

J4 = I 21sR ICrn(l-F)n dF .mm n

1 .'2 - c

10

By Schwarz IS inequality, we get

(34)1 00 a 2

J + J < 6(- - c) ~ (~)1 2 - 2 m=l bm

00 2~ b

m=1 m

12m

(F _~) C ~-l(l _ F)n-l dF2 n-l

o

By (23) and (25), the two series on the RHS are convergent for n > nO. Hence if

n > nO' J1 + J 2 tends to 0 as c tends to i .Further, from (32),

00 a 2J < 1: (.....!:)3 - r=m+l br

00

1:r=m+l

1. 2rI b2(F _ 1) C pn(l _ F)n dFj r 2 n

o

11

(36)00 a 2

J4 ~ 2£ 1: (if)r=l r

•00

1:r=m+l

a 2 .1(if) _7 2 •

r

1I

00 2 I

1: b I

r=l r )o

As m tends to infinity, J3

+ J 4 tends to O. Consequently, for any fixed n > nO' .

~ = Lim2 m--;. 00

1(

)' S2C pn(l _ F)n dF

mn

o

An ~~cdiate consequence of Lemma 2 and Theorem 2 is

Theorem 3. If, in the preceding theorem, a = O(znmk) for some integer k,m

then (?6) holds for every n > 2k + 3.

Proof. Choose b = 2ffimk+l

m •

(38)

Thus we have found an approximation for

1I

. S2C ~(l - F)n dFm n '

o

for all integers n which are not too small. The integral on the RHS of (38) can be

evaluated by formulns given in Lemma 1. An upper bound for the error committed in

such approximation is given by the sum of the RHS of (35) and (36). Finally we

note that the same method can be used to obtain the moments of Xin general.

E 1 1 L 1 n· t ·but· Let f(x) = ~ e- Ixl , then F(x)xamp c • ap ace J.S rJ. J.on. c

1= 1 - ""2

-xe if x ~ 0, and F(x) .1 x= ? e if x ~ 0. Hence

12

1

~2 = 2 ) x2Cn~(1 - F)n dF •

1'2

1 00 1 1 1If '2 ~ F < 1, then x = - log 2(1 - F) = L m- ~2(F - 2)_Jffi. So am = znm-. It

m=l

follows that for n > 1,

1

(40)

( 41)

~2 = Lim . ". m-.> 00

1"2

1

00 J.= L wm=l m

o

m+112(F -~) I Cn~(l- F)n dF,

If we usc

m mw = ~ r-1(m _ r + 1)-1 = 2(m + 1)-1 ~ r-1

ill r=l r=l

(43).-!J.2 ...

1

2k-1 IL w

mm:::lo

then thG error ccrmnittod is bounded by

(43a) 2n-l / 2w2k

(1 + --21n)(n + k)-1/2 2 • 4···2k(2n+l)(2n+3) ••• (2n+2k-1J·

In deriving (43a), we used the facts that w is a monotonically decreasing sequencem

13

of m~l, P. 85_7 and the Wallis product ~7, p. 385_7

1 • 3 ••• (211-1) 1/22> • 4 ••. (211) n

-1/2n

is a monotonically increasing sequence of n. Sinularly if

1

(44) -1J.2

?k2: w )

m=l no

then the error is bounded by

(44a)

Example 2. Cauchy distribution. Let f(x) = l/n(l + x 2), then

F(x) = n-l~tan-lx + ~_7, for -00 < x < 00, so x(F) = tan n(F - ~) for 0 < F < 1.

It can be shown that the variance of the sample dedian of a samplo of sizG 211 + 1 ~ 5

is finite:

(45)

1

~2 = ( tan2 n(F - ~) CnJ!1(l - F)n dF)

o

It is known ["7, pp. 204, 237_7 that

(46)

where

00

tan x = 2:m=l

( _l)m-l 22m(22m - 1) ?m-l(2m)! B2mx , for Ix I < ~ ,

14

1~e see that a = O(ZU), hence by Theorem 3, (37) holds if n > 3.m

25. Normal distribution. Throughout this section, f(x) = (?n)-1/2 e-x /2 and

x

F(x) = ) f(t) dt, and x(F) is the inverse function of F(x). No simple goneral

-00

form of the derivatives of x(F) at F • ~ is known. But tho first few derivatives

of x(F) can be obtained by direct differentiation, a.g.,

dx _ 1 d2x - 1 -dxd (f(lxJ) , ....<iF - rrxJ' dF? - f{XJ

For finite m and 0 < F < 1, let

) ( 1) (1)2 (l)m l)m+l(47) x(F = ~ F - 2 + a2 F - 2 . + ••• + am F - 2 + Rm+1(F - 2 '

then

(48)

a = (?n)1/2,1 , ... ,

15

A. A Lower Bound. Take the integral (39), let the range of integration be divided

into two: III 12 to 2 + c, and 2 + c to 1, where 0 < c < 2 . If we neglect the last

integral; in the first integral, replace x(F) by its expansion (47) with m = 6, and

then neglect all terms containing the remainder term Rr (Which is non-negative), and

finally let c approach ~ and use Lemma 1, we arc able to obtain a lower bound for

the variance ~2 of x of a normal parent distribution with unit variance, i.o.,

where

(50)

Incidentally, for n ~ 3, we may expand the RHS of (49) into a power series in

(2n + 1)-1 and obtain an approximation for

where ii2 is given by (2). In terms of standard deviations, and with the numerical

values of the coefficients computed, (51) is equivalent to

This agrees with a formula obtained by K. Pearson for the same purpose ~8, P.363_7.

16

B. Approximations for large samples.

00

(53) ~2 = 2 jo

a2 n I

x CJF(x)_7 ["1 - F(xt'flf(x)dx = 2 )

o

00,

a

Since F(x)rl - F(x) 7 < F(a)rl - F(a) 7 if x > a > 0 and C < (2n)-1/2rl- - - - - - - n- -

+ (?n)-1_7(2n + 1)1/222n+l , it follows that

(54)

00I

12~2 ~ (2/n)3/2(1 + 3/2n)(2n + 1)3/2~4F(n) (1-F(a))_7"2)

a

In II' use F as the indepondent variable, and replace x(F) by al(F - ~) + a3(F _ %)3+ M

5(F - ~)5 where M

5= Max R

5= (2n)5/2

A/51 and A = (7+46a2+24a4)e5a2/2, thenO<x<a

(55)

+ (~)3 2A 3 • 5 . 72 3!5! (2n+5){2n+7)(2n+9)

+ (1!2)4 (.{;,!)2 3· 5-~ 7 • 9='; (2n+5) ... (2n+ll) •

Combining (49), (53), (54), and (55), we conclude:

(56) First Approximation:

Second Approximation:

n""2 = 2(2n + 3)

If the second approximation is used, an upper bound for the proportional error (de­

fined to be I (True value / Approximation) 1- 1 ) is given by the sum of tho RHS

17

of (54) and the last three terms of that of (55). If tho first approximation is

used, then there is an additional error n/2(2n+5), the second term in the br3cket

of the second approxliaation.

The following table is given for illustration. We choose successively for

a: .35, .50, .65, .75. It is to be noted that the RHS of (54) is a decreasing

function of n for, e.g., n ~ 25, a = .75. The mrs of (55) is obviously also a de-

creasing function of n. Therefore what Table 1 means is that: e.g., for sample

sizes ~ 51 (not just = 51), if the first approximation is used, then the proportional

error is < .092, or explicitly: 1 ~ ~2/A2 ~ 1.092.

TABLE 1

Proportional Error

Sample Size

501

201

101

51

First Approximat~on

3.2 x 10-3

8.5 x 10-3

2.2 x 10-2

-29.2 x 10

Second Approximation

6.8 x 10-5

7.8 x 10-4

6.9 x 10-3

6 -2.3 x 10

18

6. Normal distribution -- a different approach. In this section a different method

is used to derive upper and lower bounds for the variance of xof a normal parent

distribution with unit variance. We state

Theorem 4. Let ~ and ~2 be respectively the sample median and its variance

of a sample of size 2n + 1 drawn from a normal distribution with unit variance. If

~2 = n/2(2n + 1) is the variance of the asymptotic distribution of i, then

1 3/2 1 3/2B (1 - 2) < ~2/ ~2 < D (1 + -2)n 2n + - . - n n

all practical purpose and n ~ 4,

(58)

or

1 7n+3 1 11 + ~n - 2 < Bn < 1 + en + 16n(8n-i) ,

on 24n (2n+l)

1Bn ... 1 + 8n •

Proof. By using the following transformations consecutively,

(60) u = F(y) , 1v=u--2

,y 2where F(y). J (21<) -1/2 e-x /2 dx , we obtain

-00

(61)... 1 2nI.J. = 2C (-)

2 n 2

1/2

f2 2 n

y (1 - 4v) dv

o

19

Let

(62)

then

(63)

2/ 1/2v = ~ (1 _ e-t (2n+l» ,

00

~ = 2B J (2n)-1/2 y2 e-(n+l)t2/(2n+l) h

1(t/(2n+l)1/2) dt ,

,2 n

o

t2 -1/2where hl(t) = tel - c-) ~ 1 for all t ~ O. Further, it is known ~10_7 that

(64)

y

J (20)-1/2 e-x2

/ 2 dx ~

o

2 - 2 )Using (64), it can bJ shown that y ~ ~2t. Therefore it follows from (63 that

(6,)

On the other hand, we have from (63),

(66)

00

- - f (2n)-1/2t 2 e-nt2/{2n.H)1J.2 = 2BnIJ.2

o

,

t 2 t 2 1/2 2 1/2whore h2{t) = 0- /t(l - e- ) . If we can show that (2/n)y h2{t/{2n+1) ) ~ 1

for all t ~ 0, thon

20

(68) ) 2 2/2eO(Y = Y (1 - 4v ) 4v •

It can be seen that Lim €O(y) = n/2. Hence it suffices for our purpose to showy~O

that go(y) is decreasing. Let",,, denote differentiation with respect to y. Thon,

where

( 71)

Y2 2/2 J 2/2g3(y) = (n/6)y eY - eY e-x dx.

o

I t is known flO_7 that

Y - ..2/2 J. 2/2 00 2 +1eY e-x dx 2 ~ Y n /1.3 ••• (2n+l) •

n=Oo

00 n 1Hence g3(y) = z {6nl - 1.3••• (2n+l)

n=O

2n+ly

21

It can be shown, by a similar

argument used in ~lo_7 for a similar purpose, that

( 73)

where aO < 0 and ai > 0, i = 1, 2, •••• Hence there exists a YO > 0 such that

g3(y) ~ 0 if 0 ~ Y~ YO and g3(y) ~ 0 if Y ~ YO. So as y increases from 0 to 00,

g2(y) decreases steadily from 0 to a minL~um and then increases steadily to 00.

Consequently gl(y) first decreases steadily and then increases steadily. As

Lim gl(y) = Lim gl(y) = 0, it becomes clear that gl(Y) ~ 0 for all Y~ O.Y --:p 0 y:-:> 00

Therefore gO(y) is a decreasing function of y. This completes the proof.

1 / )-1Finally we note that (58) is obtained by using n! _(~)2nn+l 2e-n+(12n •

Remark 1. The lower bound for ;2 given by (49) is better than the one given

by (57) if we use (59) for B. This is so even if the last term at the RHS of (49)n

is omitted. For

n + n2 n r (8-2n)n + 20 - n 7(74) 2(2n+3) 4(2n+3)(?n+S) = 2(2n+l) _ 1 -. 2(2n+3)(?n+S) -

22

Now if n ~ 2, the last term in the bracket of (74) is smaller in absolute value

than (2n+2}-1 and (1 + ~) ~l - l/(2n+2}_~ ~ 1. Therefore the quantity in the

bracket of (74) is greater than

31 112

1 - 2n + 2 ~ (1 + 'Sii) (1 - 2n + 2) •

For n = 1, direct comparison shows also that the LHS of (74) is greater than that

of (57).

Remark 2. Since both lower bounds for ~2' (49) and (57) are smaller than

~2' while the upper bound is greater than ~2 (1 + 1/2n), we cannot be sure, just

by comparing these bounds, that in using ~2 as an approximation to ~2' the propor-

tional error is less than 1/2n. Therefore the second approximation given by (56)

is better than ~2' for large samples. (Table 1).

7. Laplace distribution. We shall now employ the same technique, used in ~ 6,

to derive upper and lower bounds for the variance ~2 of the sample median xof a

sample of size 2n + 1 drawn from a Laplace distribution with pdf

(75)1 -Ixl

f(x} =2 e

....Clearly, the variance in this case of the asymptotic distribution of x is

11J.2 = 2n + 1

We state

Theorem 5. If ~2 and ~2 are as defined above, then

(77)3 3

Bn(l - 2~2)2 < ~2/~2 < 1.51 Bn(l + ~)2 ,

23

where B is given by (57) and (59).n

-Proof. It can be seen easily that 1J.2 1s equal to the RHS of (63) if v and

t satisfy (62) and

o

(78) v = ( 1 -x2 e dx.

We proved J:4_7 that for all y ~ 0,

2.1.1 Y 2

(1 - e- )v~2

From (62) and (79), we have y2 ~ ~2 t 2 • Hence it follows from (63) that ~2/~2 has

a lower bound given by (77).

Further, from (63)

00

(80) ~2 = 2Bn~2 J (2IIl-4 t 2 ._nt2/(2n+l) ( lh2(t/(2n+ll~l dt l,

where h2(t) is given by (66). We say that

(81)1

y2h2(t / (2n+l)2) S 1.51 •

If this is true, then the RHS of (77) 1s an upper bound of ~2/~2. Thus the

proof is completed.

24

To establish (81), we introduce, as in (68),

(82) 22/ 2gO(y) = y (1 - 4v ) 4v

where y and v satisfy (78). For all y ~ 0, gO(y) is not smaller than the LHS of

(81) and

(83)1

oas y->

o

00

Let "I" denote differentiation with respect to y, then

(84) , where

(85)

(86) gl'(y) =1 e-y g (y)2 2 , where

(87)

(88)

= -

= -

If f(x) is a function of x, and if as x increases from 0 to 00, f(x) varies from,

e.g., positive to negative, and then back to positive, we will write, for

simplicity, As x: 0-> 00, f(x): +,-,+. Now

25

I > >g,(y) ~ 0 according as y ~ log 2 ,

So as y:

Now g2(0) = 0, while g2(00) =00. We say that as y: 0 ---> 00, g2{y): +,-,+.

I

Otherwise g2{y) ~ 0 for all y ~ 0, so gl(y) ~ 0 and Sl(y) ~ 0 for all y ~ 0, as

gl(O) = O. Hence gO{y) is steadily increasing. This, however, contradicts (8,).

I

It follows that as y: 0 ---> 00, gl{y): +,.,+. Now gl(O) =gl(oo) = 0, hence as

y: 0 ---> 00, gl(y): +,-. Therefore we conclude that as y: 0 ---> 00, SO(y)

increases steadily from 1 ,to a maximum, and then decreases steadily to O. To

find the maximum of BO(y), we first solve Bl(y) = 0, which is equivalent to

2v(1+2V) • y = O. Using table ~12_7, we obtain an approximate solution y = 1.15.

The maximum of gO(y) is then found to be 1.51.

Remark. The variance of the sample mean (of a sample of size 2n+l) drawn

from a Laplace distribution with pdf given by (75) is 2/(2n+l). It follows, from

Theorem 5, that the sample median has smaller variance than the sample mean for

sample size 2n+l ~ 7. In a recent paper, A. E. Sarhan 111_7 found that for

sample sizes equal to 2, 3, 4, and 5, the variance of sample median is also

smaller than that of the sample mean.

\

26

l References

)I ~1-7 T. J. Ila. Bromich, An Introduction to the theory of Infinite Series,. MacMillan and Co., London, 1908•

.. 1:2_7 G. W. Brown and J. W. Tukey, Some distributions of sample means", Ann. Math.Stat., Vol. 17 (1946), pp. 1-12.

£:3_7 J. H. Cadwell, "The distribution of quantiles of small samples", Biometrika,Vol. 39 (1952), pp. 207-211.

["4_7 J. T. Chu, "On the distribution of the sample median", to be published.

£:5_7 W. Feller, "On the normal approximation to the binomial distribution", Ann.Math. Stat., Vol. 16 (1945), pp. 319-329. ----

£:6_7 T. Hojo, "Distirbution of the median, etc.", Biometrika, Vol. 23 (1931),pp. 315-360.

L7_7 K. Knopp, Theory and Application of Infinite Series, Hafner PublishingCompany, New York.

£:8_7 K. Pearson, "On the standard error of the median, etc.", Biometrika 23 (1931),pp. 361-363.

L9_7 K. Pearson, "On the mean character and variance of a ranked indiVidual, etc."Biometrika, Vol. 23 (1931), pp. 364-397.

POlya, "Remarks on computing the probability integral in one and twodimensions", Proceedings of the Berkeley Symposium on MathematicalStatistics and Probability, University of California Press, Berkeley,1949, pp. 63-78.

Lll_7 A. E. Sarhan, "Estimation of the mean and standard deViation by orderstatistics", Ann. Math. Stat., Vol. 25 (1954), pp. 317-328.

1:12-7 Tables of the Exponential Function eX, Applied Mathematics Series, 14,National Bureau of Standards, Washington, D. C., 1951.