•
THE 5PEARM\N FOOTRm.E AND A MARJ<QV CHAIN J?ROPERTY
by
Pranab Kumar Sen and IbrahJm A. Salama
Department of Bi,ostatist:tcs
University of Nortll Carolina. at Chapel Hill
Institute of Statistics Mimeo Series No. 1431
February 1983
•
1HE SPEARMAN FOOTRULE AND A MARKOV CHAIN PROPERTY
.
*
by
Pranab Kumar Sen and Ibrahim A. Salama
Department of Biostatistics, University of North Carolina,
Chapel Hill, NC 27514 USA
SUMMARY
An equivalent representation of the Spearman footrule is considered
and a characterization in terms of a Markov chain is established.
A
martingale approach is thereby incorporated in the study of the asymptotic
normality of the statistics.
AMS Subject Classifications:
Key words:
62ElO, 62E20, 60J99
Asymptotic normality; Markov chain; martingale; Spearman
footrule; uniform (permutational) distribution.
*Work supported partially by the National Heart, Lung and Blood Institute,
Contract NIH-NHLBI-7l-2243-L and partially by the Grant 5-732-ES070l8-0l
from the National Institute of Environmental Health Sciences .
.
1. INTRODUCTION
Let
S
n
Dn (TI~n ,0~n ) = ~n1.·=lIOn(i)-TIn(i) I
where
of
integers
by
S
n
•
n
As in Diaconis and Graham (1977), we may define a metric (D )
{I, ... ,n}.
on
be the set of all (n!) permutations of the first
n
(=
0
~n
S.
D
n
n
n (n)
(0 (1), ... ,0
n
and
(1.1)
TI~n
(= (TIn (1), ... , TIn (n))
is known as the Spear.man (1904) FootruZe.
are elements
Its relationships
with other commonly used nonparametric measures of association (such as the
Kendall tau and Spearman rho coefficients) and its asymptotic normality
(under the assumption that
tributed uniformly in
o
~n
and
TI
"",n
are chosen independently and dis-
S ) have been studied by Diaconis and Graham (1977).
n
The object of the present investigation is to consider an equivalent rep•
resentation of
to characterize a Markovian struoture for the same, and
D,
n
to incorporate a martingaZe approaoh in the proof of the asymptotic normality
of the statistics.
The representation is considered in Section 2.
Along
with the Markovian structure, some distributional results are presented in
Section 3.
The concluding section deals with the asymptotic normality result.
2.
Note that for every
A REPRESENTATION FOR D
n
cr
eS ,
...n n
..
= ~~1.= l~~J= lli-j Iw 1.J
where
1
~n
= (l, .•. ,n)
and
we have
(2.1)
a (j)
l
wij =
0',
{
= i,
for
n
i,j
=
l, ... ,n
(2.2)
otherwise;
•
Define then
Tn,~.
for
=T
= Tn , i(ln'£n)
= E~_lI(a
_
Jn (j)<i)
-
. (a-n )
n,~
= l,2, ... ,n,
i
T
n
•
and let
= T (a ) = T (1 ,a )
n -n
(2.3)
n _n _n
= E.n
~=
1T
(2.4)
.
n,~
Then, we have the following.
Theorem l.
For every
a_n ESn
and n (~l) ,
(2.5)
n
J=
Note that
Proof.
so that on letting
E. 1 w..
u(t)
~J
=1
(Vi),
n
E.lw .. =l
be equal to 1 or 0, according as
~J
1=
t
(\1j),
is > or < 0,
we have
T (a )
n -n
(2.6)
where
avb
=
max(a,b)
and
aAb = min(a,b).
-2-
Therefore,
•
= n(n+l) -
1
",n
",n
~~j=l~k=l Wjk
(k .
+J)
n . rn
1 {L:
= n (n+ 1) - ~
j=l] k=l Wjk
= n(n+l) -
1
{L: n
. +
]= 1]
~.
+
n
n
4k=1 k 2: j =l Wjk }
4~=1 k} = n(n+l) -
~n(n+l)
= (n+l)
(2.7)
2
Q.E.D.
By virtue of Theorem I,
of
D.
T
n
may be regarded as a complementary part
Also, note that the distribution of
D (a
,7T )
n -n -n
are independent and each distributed uniformly over S
n
n
when
a
~n
and
7T
_n
is the same as the
has a uniform distribution
D (a ) (= D (l,a )) when a
n - ~n
n -n
-n
The same is true for T (a ) (and T (a ,7T )). Hence, we shall
on S
n
n -n,-n
n -n
consider only the case of D (a ) and T (a ).
n -n
n -n
distribution of
•
T
A MARKOVIAN PROPERTY OF THE
3.
.
n, l.
The main result of this section is the following
Theorem 2.
S,
n
{T
For every
.;i<n}
n,l.-
whenever a is distributed unifo~Zy on
-n
is a Markov chain, i.e., for every k(~n-l) and
n (~l) ,
P{Tn, k + l=rk + liTn,].=r.,j<k}
] (3.1)
Proof.
Let
P be the set of all permutations (of
{I, ... ,n}) satisfying
T
n,k+l
~
then among the set
{al, ... ,ak },
we have
-3-
k-r k
elements of the set
{k+l, ... ,n}.
If we denote this set by
A,
then, we may have either of
the following:
(i) k+l
A.
E
This happens with the (conditional) probability
(3.2)
f
and (ii) k+l
1 -
A.
This happens with the (conditional) probability
(k-rk)/(n-k) = (n-2k+r )/(n-k)
k
In case (i),
can assume only the values
T
n,k+l
probabilities
(3.3)
{(n-k) - (k-rk)}/(n-k)
while in case (ii),
T
n,k+l
respective probabilities
and
rk+l
(k-rk)/(n-k),
T
n,k+l
r
k
assumed by
(viz.,
Tn, k'
with
respectively,
or
rk+l
with
and
and r +2) and
k
their respective (conditional) probabilities (given the
only on the value
r +2
k
can assume only the values
{(n-k) - (k-r ) - l}/(n-k)
k
Thus, the assumable values of
or
T .,
n,l
i~k)
depend
•
Q.E.D.
Actually, by very similar (elementary) arguments, we have the following:
Lemma 3.
k
(1 ~k~n)
If
cr
~n
and
has a uniform distribution on
r,
for
P{Tn, k=r}
and~
for every
sn , then for every
k<q,
P{T
=r T =S}
n,k ' n,q
r = ov (2k-n) , ••• , k ,
(3.4)
r~s,
n-q
k q-k u q-u
= (q-s)~u>r(u)(s-u)(r)(k-r)
(3.5)
nJ{kJ(q-k)J(n-q)J}-l
For our subsequent analysis, we may note that by (3.4) and (3.5),
"
(3.6)
P{Tn, k + l=slTn, k=r} =
-4-
for
s>r
(and 0, for
s<r), so that
.(n-.2k+r) (n-2k+ r -l)/(n-k)2,
s=r
2
s=r+l
= ~ (n-2k+r) (2k-2r+l)/(n-k) ,
P{Tn, k+ l=slTn, k=r}
(k-r)2/(n-k)2,
l
0,
s~r+3
or
s=r+2
s<r
(3.7)
Hence, from (3.7) we have
I k} =
E{T kIT
n, + n,
for
k
0,1, ... ,n-l,
=
(n-k-l) 2
2k+l
k
2 T k + n-k 2
(n-k)
n,
(n-k)
(3.8)
T
n,O
where, conventionally, we let
=
O.
Finally,
by (3.4) and (3.5), we have
(3.9)
2
Yn,kq
2
(3.10)
CoV(Tn,k,Tn,q) = (n(kAq)-kq) /n (n-I)
for every
k,q = l, ... ,n,
so that
vn = ETn = n(2n+I)/6
and
Var(T)
n
=
2
n(2n +7)/180
(3.11)
We also write
\)
n,k
=
2
O<k<n
ET
n,k
4.
ASYMPTOTIC PERMUTATIONAL NORMALITY OF
(3.12)
T
n
Motivated by the Markovian property in (3.1), we would like to prove
the following result via a martingale central limit theorem.
'I'heorom 4.
~
then as
If for every
n,
(J
~n
has the uniform distribution over
n~,
-5-
S ,
n
(4.1)
Proof.
For every
d
nk
k
(l,2.k,2.n-l), we let
= {(n-k+l)(2k-l) - (k-l)}/{(n-k)(n-k+l)}2;
d~k
=
L~=l
d
nk
. (4.2)
Then, by (3.8), we have on letting
Y
nk
for
= (n-k)-2 T
- d* ,
n,k
nk
k = 0,1. .. ,n,
B
= B(T o' JO_<k)
n,J'
nk
(4.3)
that
(4.4)
Therefore, if we let
l<k<n-l
then the
Znk
(4.5)
are martingale differences.
iy (2.4), (4.2), (4.3) and
(4.5), we have
T
n
-]J
n-l
. 2
n = L.1= 1 (n -1) Ynl.
n-l
n-l. 2
Lk=l Znk (Lj=k (n-J) )
n-l{
}
= Lk=l (n-k) (n-k+l) (2n-2k+l)/6 Znk
(4.6)
Thus, if we let
C
°
nl
U.
nl
= (n-i)(n-i+l)(2n-2i+l)
2
6 {(n+l) (2n +7)/180} ~ ,
c.Z.
nl nl
i = l , ... ,n-l;
l<i<n-l
UnO=O
-6-
(4.7)
(4.8)
then, by (3.11), (4.6), (4.7) and (4.8), we have
(4.9)
where by (4.4), (4.5) and (4.8),
(4.10)
and
L:~1== 1 E J.
n1 == 1,
Thus,
T*
n
'if
(4.12)
n> 1
relates to a martingale array, normalized by (4.12), and to
establish (4.1), we need only to verify the following:
~
U
n-1 ~
n-12
L:.1= 1 Un1. = L:.1= 1 E (lJn1.
n
and, for every
n
n1
Since the
T k
n,
p
. 1)
n1-
-+
(4.13)
1
s>O,
L: · --1 E{J. I
1
IB
(I Un1. I>s) IBn1. 1} ~
are nonnegative,
(4.14 )
0
T k<k,
n, -
'if k~n,
by using (3.7), (4.2),
(4.3), (4.5), (4.7) and (4.8), it can be easily seen that
IUn1·1
for every
with probability 1
i:
1~~n-l,
where
(4.15)
C does not depend on
(4.15) ensures that (4.14) holds for
n
n
(O<C<oo).
adequately large.
Hence,
Further, by
virtue of (4.12), to prove (4.13), it suffices to show that
E(U _1)2
n
-+
0,
as
n~
Towards this, note that for every
(4.16)
i:
•
-7-
l~i~n-l,
unl. = E (t?nl·18 nl-. 1)
2
= a . T.
nl nl~1
B . T.
nl nl-l
+
+ l! .
Inl
say,
,
(4.17)
where, by (4.7),
a .
nl
for every
= 0 (n -3 )
i
(= 1, ... ,n-l).
~
~
U -l=U
n
n
Further,
~
n
2
1: {a . (T
i=1
(4.18)
-EU
n-l
=
= O(n -1)
and
nl
. I-V
n,l-
. 1) + b . (T
n,l-
nl
.
n,l-1
-]..I
•
n,l-1
)
(4.19)
so that on noting that (by virtue of (3.4)-(3.5))
2
E(T . I-V . 1 )2 = 0(n3)
n,ln,lE(Tn,l. 1-]..1n,l. 1)
2
(4.20)
(4.21)
= O(n)
2
2
E(Tn,l. I-V n,l. l)(T . 1-]..1 . 1) = 0(n )
n,ln,l-
(4.22)
E(Tn,l. 1-]..1n,l. l)(Tn,J. 1-]..1n,J. 1) = O(n)
(4.23)
for every
i
(= 1, ... ,n-l),
and hence, (4.16) holds.
we obtain from (4.19)-(4.23) that
This completes the proof of the theorem.
We conclude this section with the remark that the martingale approach
also enables us to prove an invariance principle relating to the partial
sequence
k
{E._l(T
1n,l.-]..1n,l.)/y,
n k~n},
-8~
where the earlier method of Diaconis
•
and Graham (1977) may not lead to a simple solution.
This invariance prin-
ciple will particularly be useful to extend (4.1) to the case where the
sample size is itself a (positive integer valued) random variable .
.
REFERENCES
Persi Diaconis and R.L. Graham (1977). Spearman footrule as a measure of
discovery. J. Roy. Statist. Soc. B. 39, 262-268.
Charles Spearman (1904). The proof and measurement of association between
two things. Amer. J. PsychoZogy 15, 72-101 .
.
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