•
•
e
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The Mean-Median-Mode Inequality and
X2 Distributions
by
Pranab K. Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series NO. 1818
•
February 1987
The Mean-Median-Mode Inequality and Noncentral X2 Distributions
BY PRANAB KUMAR SEN
University of North Carolina, Chapel Hill
SUMMARY
The usual ordering of the mean, median and mode for a central chi square
distribution is extended for the noncentral case.
In this context, a subadditive
property of the median and mode is also established.
1. INTRODUCTION
For a distribution F with a density function f, defined on the real line
R, the mean
~
(~),
median (m) and mode (M) are defined by
= JRxdF(X), m = inf{x:F(x)
~ Hand M:f(M)
where, we assume that M is unique.
..
=
~~kf(x),
(1.1)
Pearson (1895) found empirically that
for his Type III distributions,
~
(1. 2)
3 (~ - m),
- M
which led him to the conjecture that for a distribution F,
either
~
>
m > M or
~
<
(1.3)
m < M,
and this is classically known as the mean-median-mode inequality.
Although
(1.2) may not generally hold for all distributions, the ordering in (1.3)
remains of considerable interest, particularly, in the context of measuring
skewness of a distribution (see MacGillivray (1985) and the references cited
there).
•
Though some sufficient conditions ensuring (1.3) have been laid down
by van Zwet (1979) and MacGillivray (1981), among others, their verification
may generally demand considerable manipulations.
A class of distributions arising frequently in problems of statistical
inference is the family of (central and noncentral) chi square distributions.
2
For a central chi square distribution with p degrees of freedom (OF), denoting
these measures by ,,(0) m(O) and M(O), we have
"'p , p
p
l.l(O) = p, M(O) = (p-2)vO and p-l < m(O) < p,
(1.4)
P
p
P
so that (1.3) holds [see Johnson and Kotz (1970, ch. 17)]; actually, for p > 2,
m(O)
varies between p - 1 + ·386 to P - 1 + ·34, as p increases, so that (1.2)
p
holds.
The situation is more complicated for the noncentral case.
For a
noncentral chi square distribution with pDF and noncentrality parameter A(> 0),
the mean is given by
l.l
(A)
P
=p+A=l.l
(0)
p
+A,
(1.5)
so that l.l~A) is additive with the noncentrality parameter.
approximations for the median (m(A»
Though various
have been discussed in Johnson and Kotz
p
(1970, ch. 28), it is not known whether m(A) is additive or not.
Further,
p
it is not precisely known whether the mode (M(A», m(A) and (A) have the
p
p
l.lp
same ordering as in the central case.
The object of the present note is to
focus on (1.3) for the noncentral chi square distributions, and to show that
(1. 6)
so that the mode and median are both sub-additive with respect to the noncentrality
parameter.
The· main results are presented in the next section.
2. THE MAIN RESULTS
We denote the central chi square distribution function with
p
OF by G (x),
p
x > 0, p > 1, and the noncentral chi square distribution function with p DF
and noncentrality parameter A by G(A) (x).
p
qA (r) = e
Also, let
-H (!A) rI'r! ,for r > o.
(2.1)
Then, it is well known that
G(A) (x) = ~
•
p
r>O
q,
A
(r)G 2 (x),
P+ r
x > 0, A >
o.
(2.2)
We also denote the density functions corresponding to G (x) and G(A) (x) by
p
g
p
p
(x) and g (A) (x), respectively, so that
p
g(A)(X)
p
x >
o.
(2.3)
3
We write G (x)
p
1 - G (x) and G(A) (x)
p
p
G 2(x) - G (x)
pp
=
1 - G(A) (x).
P
Then, note that
2g (x), ~ p > 2, x > 0,
p
-
=
(2.4)
so that by (2.2) and (2.4), we have
=
G(A)(x)
p-2
,
G(A)(X) + 2g(A) (x) > G(A)(X),
p
p
P
x > O.
(2.5)
Using (2.5) and (2.4), it is easy to show that for every p > 2,
G(A) is unimodal (with mode M(A) > 0).
p
(2.6)
p
Then, we have the following.
Theorem 1. For every p(> 2) and A(> 0),
-
(A) (M (A»
gp
p
-
g(A)(M(A»,
p-2 p
M(A) < p - 2 + A
p
=
(2.7)
M(0) + A,
p
'(j
A > 0,
(2.8)
(2.9)
Proof.
(2.7) is well-known [viz., Johnson and Kotz (1970, ch. 28, p. 136)].
Nevertheless, it provides the key to the other assertions.
(Cl/Clx)g(A) (x)
E
q, (r) (d/dx) 9
r~O A
p
p+2r
First, by (2.3),
(x)
Er~oqA (r~-!gp+2r(x) + !gp+2r-2(X)}
=-!E r_>oq,A (r)qp+ 2 r
=
(x)
+!E r~ oq,A (r)gp- 2 + 2 r
[g(A) (x) - g(A) (x) ]/2,
p-2
p
(x)
"<S x _> O.
Hence, by the unimodality in (2.6) and (2.10),
(2.10)
(2.7) follows directly.
Next,
we note that (2.10) is positive or negative according as x is < or > M(A),
P
while g(A) (M(A»
p
p
9 (A) (M (A»
p-2 p-2
> g(A) (x),
- p
\l
> 9 (A) (M (A»
- p
p
Next, we note that by (2.3),
E
q, (r)g
(M(A) H1 r>O A
p+2r p
x '! M(A).
p
This implies that for p > 2,
and M(A) < M(A), '(j A > o.
p-2 - P
(2.11)
(2.7) and (2.10),
(p + 2r - 2)/M(A)}
p
=
0,
(2.12)
and this, in turn, leads us to the following implicit equation for M(A) :
p
4
(p-2) +
0:
(p-2) + A{I:
=
,
q (r)q
r~O A
p+2+2r
(M(A»}/g(A)(M(A»
p
p
P
(p-2) + Ag Od (M (A»/g (A) (M (A».
p+2 p
P
P
Making use of (2.11)
(2.13)
(with p replaced by p+2), we obtain that
g(A)(m)/g(A)(m) < 1,
p+2
p
and hence,
2rq (r)g
(M(A»}/g(A)(MOJ)
A
p+2r p
p
p
r>O
\lm <-
M(A)(> M(A»
p+2 - P
,
(2.14)
(2.8) follows directly from (2.13) and (2.14).
Next, we note that for every m > 0,
=
g(A)(m)/g(A)(m)
p+2
p
mO:
q,(r)g 2 (m)(p+2r)-1}/g(A)(m)
r>O A
p+ r
p
> m 0: oq, (r) g
(m) (p+2r) /g (A) (m)} -1
r~
A
p+2r
p
=
m{p +Ag(A)(m)/q(A)(m)}-l.
p+2
p
(2.15)
From (2.15), we readily obtain that
g(A)(M(A»/g(A)(M(A»
p+2 P
p
p
> M(A)/(p + A),
p
(2.16)
so that by (2.13) and (2.16), we have
M(A) > (P-2)+AM(A)/(p + A).
p
-
(2.17)
p
and (2.9) follows readily from (2.17).
This completes the proof of the Theorem.
It may be noted that (2.9) is not sharp for A > 0 (and particularly,
for large A).
It is possible to exploit (2.15) to derive better (lower) bounds
for M(A) for A large.
In fact, we may easily verify that by (2.13) and (2.15),
p
for every A > 0, M(A) satisfies the following inequality:
p
- 2A/[p +
Vp 2
+ 4AM(A) l}
p
~
(p-2).
(2.18)
Next, we consider another inequality on the mode which will be very useful
for the study of the median inequalities.
g (A)(M(A»
p+2 p+2
Note that by (2.7),
= g(A) (M(A»
p
p+2'
so that proceeding as in (2.13), we obtain that
M(A) p+2
(p-2)
> {I:
r>O
=
{I:
r>
oq,(r)g
2(M(A»(2r+2)}/q(A) (M(A»
A
p+2r+
P + 2 P + 2 p+2
(2r+2)-1 q (r)g
(M(A»/g(A) (M(A»}-1
A
~+2r+2
p+2
p+2 p+2
(2.19)
5
(A)
A (A) (M(A»/{ (A) (M(A»
gp+2 p+2
gp
p+2
=
qA(0)gp(M p + )}
2
(A)
A (A)(M(A»/{ (A)(M(A»
gp
p+2
gp
p+2
> A
,V
- QA(0)gp(M + )} [by (2.19) 1
p 2
p, ~ •
(2.20)
Therefore, we obtain that
(A) >
Mp+ 2
P -
2 +
'
\)
1\,
"
,
(2.21)
P , 1\ •
Next, we consider the following theorem on the median.
(A)
d p, m (A) ~s
.
·
Th eorem 2 • For any f ~xe
-
p
and A > 0, and for every p
M(A) <
p
-
P - 2 + A
~
< m(A)
p
~
Proof. Note that Gp(X) is
in A(> 0), m
p
-
2, A
~
0
~ p + A
in x.
(a/;n)G (A) (p + A) = (a/aA)
p
0:
r~
(A)
> m
, for every p > 2
p-2
= J.I (A) •
(2.22)
P
Also, note that
oQ, (r)G
1\
2 (p + A)}
p+ r
•
- (A)
[G p + 2 (p + A)
_ G (A)(p
p
+ A)]/2 _ g(A)(p + A)
P
(A)
_g(A)(p+ A) .
gP+2 (p + A)
p
(A)
Note that by (2.8), M 2 < P + A,
P+
ity holds for m > M(A) .
p+2
rJ
(2.23)
A ~ 0, while in (2.14), the opposite inequal-
Hence, the right hand side of (2.23) is always non-nega-
Therefore, for every A > 0,
tive.
G(A) (p + A) _ G (0) (p + 0) =
p
p
f~{(a/aa)G~a)
(p + a)}dx > O.
-
On the other hand, Gp(0) ( p + 0) = G (p) < G (m (0) ) =
P
P
L
Vp
> l.
(2.24)
Therefore, we
conclude that
G(A) (p + A) is monotone increasing in
p
A(~
0).
(2.25)
6
Further, note that
G(A) (p + A) - lim p{ 2
> P + A}
A->OO P
- A~OO
Xp,A
lim,
Hm
A--7 oo
P{(X
2
p, A
-
(p + A))/(2(p + 2A);
> o}
(2.26)
1/2,
where the last step follows from the asymptotic normality results on the noncentral chi square statistic; it is also possible to use other refined approximations, discussed in Chapter 28 of Johnson and Kotz (1970).
By (2.25) and
(2.26), we conclude that for every p,
G(A) (p + A) < ; for every finite A(> 0).
p
(2.27)
-
On the other hand, by definition,
;, ~ A > 0, p > 1,
so that noting that G(A) (x) is
p
(2.28)
~ in x, we immediately conclude from (2.27)
and (2.28) that
m (A)
p
< p + A,
-
v A
V
_
>
o.
(2.29)
Next, we consider (2.23) at the point p - 2 + A, and get that
(2.30)
Note that by (2.21), P - 2 + A <
M~:~,
so that by (2.14) and (2.30), we obtain
that
(d/dA)G (A) (p _ 2 + A) is ~ 0,
p
."1
A ~ 0,
(2.31)
so that parallel to (2.25) , we obtain that
G(A) (p - 2
p
+ A) is
~ in
A(~
0) ;
G(0) (p - 2)
p
>
;.
(2.32)
Again, proceeding as in (2.26) , we have
j
lim G(A) (p A->OO p
so that
2
+
A)
(2.33)
7
G(A) (p _ 2 + A) ~ L '~ (f inite) A(~ 0).
(2.34)
P
This, along with (2.28), imply that
m
\J.
( A)
o.
> p - 2 + A, ~ A ~
p
(A)
Since Mp
(2.35).
(2.35)
I
~ p - 2 + A, <;f A ~ 0 [(2.8)],
(2.22) follows directly from (2.29) and
Also, note that (2.21) and (2.22) imply that
(A)
(A)
m
> p + A > m
p+2 P
Further,
G~A) (x)
,
?'
is
G(AI) (m(Al»
p
p
,;
~
o.
(2.36)
in A (for any given x and q), so that for A > Al
2
= !
G(A2) (m(AI». G(A2) (m(A2»
p
p
, p
p
<
(A2) > (AI) J
mP
V Al
- mp
,
hence
p, A >
-
_<
~.
0,
(2.37)
,
This completes the proof of the theorem.
A2.
Note that by definition, (O/OA)G(A) (m(A»
p
= 0, and this yields that
p
(A) (m(A»/ (A) (m(A»
gP+2 p
gp
P
= 1 + {g (A) (m (A»
p+2 P
_
(A) (m (A) ) }/g (A) (m (A) )
gp
P
P
P
1 + 2{ (d/dm) log g (A) (m) I
(A)},
p
m=m
•
(2.38)
p
as (d/dx)g(A) (x) = {g(A+ ) (x) - g(A) (x)}/2, ~J x> O.
p
p 2
p.
unique mode at x = M(A) «
p
-
Further, g(A) (x) has a
p
m(A), by (2.22», and hence, we have (dldm)log g(A)
p
p
I
(m)1
(A)
p
m=m
< 0, .~ A > 0, p > 2.
Therefore, by (2.38), we claim that (%A)m(A)
p
< 1, ~ A > 0, so that
m(A) _ m(O) < A,
p
p
V A > 0,
and this verifies (1. 6)
and as (%A)m(A)
p
A > O.
~
•
(2.39)
. >
(A) (m)/ (A) (m) is > 1 according as m ~s
Since gP+2
= M(A) ,
gp
<
< p+2
(A) < M(A)
for every
1, we claim by using (2.38) that mp
- p+2'
Thus, we may even strengthen (2.22) to the following:
M(A) < p - 2 + A < m(A) <:; M(A) < p + A
p
p
- p+2-
-
for all A > O.
=
~
(A)
p
,
(2.40)
Also, while both M(A) and m(A) are subadditive (with respect to
p
p
the noncentrality parameter A), the mean ~~A) is additive.
8
We have considered so far the case of p
general, for all p >
o.
=
For p
2
A
.. P,
=
2, althugh the results hold, in
A, with probability one, so that
0, X
A, while for p
~
1, the normal distribution can readily
be called on to prove the mean-median-mode inequality.
•
(A)
case, M
1
M(O)
2
=
0 <
(A)
o with gl
(0)
= +-00,
I
M~A), ~'l(t A
> 0 and
although m (A)
1
g~A) (M~A»
> 0
'
I
However, in this
A > O.
",.-r
< OO,/A>O.
of other gamma distributions too, parallel results hold.
For p
=
2,
For Poisson mixtures
Finally, we may also
remark that Haldane (1942) has remarked on the adequacy of (1.2) for the gamma
family of densities.
While his treatment may run into complications for the
noncentral case, it appears that by virtue of (1.6) and (2.40)
(and the fact
that for the central case (1.2) holds approximately) that (1.2) holds even
better for the noncentral case.
The author is grateful to Professor Norman L. Johnson for some valuable
J .
e
discussions on this topic.
This work was supported by the National Institute
of Health, Contract No. N01-HV-12243-L •
•
REFERENCES
HALDANE, J.B.S.
(1942).
The mode and median of a nearly normal distribution
with given cumulants.
JOHNSON, N.L. AND KOTZ, S.
Biometrika~,
(1970).
Distributions in Statistics:
Univariate Distributions, I, II.
MACGILLIVRAY, H.L.
(1981).
a class of densities.
MACGILLVARY, H.L.
(1985).
294-299.
Continuous
John Wiley, New York.
The mean, median, mode inequality and skewness for
Austral. J. Statist. 23, 247-250.
The mean, median, mode and skewness.
In Encyclo.
Statist. scs.2, 364-367 (Eds. S. Kotz & N.L. Johnson, John Wiley, New York).
PEAFSON, K.
(1985).
Contributions to the Mathematical Theory of evolution, II.
Phil. Trans. Roy. Soc. London, 186, 343.
VAN ZWET, W.R.
(1979).
Mean, median, mode, II.
Statist. Neerland 33, 1-5.