•
IThis research was supported in part by the U. S. Army Research Office,
Durham, under Contract No. DAHC04-71-C-0042 .
'
ESTIMATION OF RANK ORDER!
by
N. L. Johnson
Department of Statistics
University of North Carolina at ChapeZ Hill
Institute of Statistics Mimeo Series Number 931
June 1974
•
•
ESTIMATION OF RANK ORDER
By N. L. Johnson
University of North Carolina at Chapel Hill
1.
Introduction The rank order of a member a.1 of a set of n numbers
(aI' a Z' •.• an)
ai
is the number of members of the set less than or equal to
(including, of course,
of the
g-th smallest a.1
ai
itself).
is
g.
If no two a's are equal, the rank
If Xl' X2, ••. , Xn are continuous random variables corresponding to a
random sample, then with probability one, no two are equal and so each Xi
will have a unique rank order different from that of any other
Xj
(j
~
i) .
In this paper we will be concerned with the estimation of the rank order
of a specified
Xi
when only partial information about the sample values
Xl' XZ' ••• , Xn is available. The discussion will be restricted to cases
where Xl' XZ' ... , Xn are independent and identically distributed continuous random variables with common cumulative distribution function F(x) and
density function
f(x) .
Throughout the paper there are two different sampling schemes under
consideration.
In one an individual value is chosen at random and so its
rank order G is a random variable, while in the other it is supposed that
the rank order g is fixed, though remaining observed values (if any) are
chosen at random.
Tests relating to rank order will be in a later report •
•
-2-
e;
2.
Population Distribution Unknown; r Sample Values Known (2 s r
Z'
If we are given the values
(Xi, X
XI)
r
<:
n)
of a randomly chosen
,Xn we can evaluate the
Xl' X2 ,
probability that a randomly chosen XI is the g-th smallest among
subset of
r(~
2)
of the n values
Xl' X2 , ... ,Xn (i.e. has rank order G = g in this set).
a random variable.
z,
(Xi, X
Pr[G
Since Pr[G
=
, X;)
g
= g] = n-1
is
G is
G = g , given that the rank order,
The conditional probability that
GI , among
Note that
in
gl
GI - I] _ ~[G=g]Pr[G'=g'IG=gl
- g Pr[GI=g']
.
Pr [G'
= gI]
= r
-1
(g = 1, •.. , n
g' = 1, ... , r)
and
(1)
Pr[GI = gf
I G = g]
=
19-1 ) [n- g )
g'-l
[max (1, r + g - n)
~
g'
r-g'
~
/ [n-l)
r-l
min (g, r))
we have
(2)
(g
=g
I,
Note that (1) implies that
parameters
•
g I + 1, ... , g' + n - r) .
(G' - 1)
has a hypergeometric distribution with
(r - 1), (g - 1), (n - 1) .
-3-
•
Now
= (g-1){n-r+1+g'-g)
~~~~~,-
(g-gi)(n+1-g)
(g = g' + 1, ... , g I + n - r)
This ratio is greater or less than
(g - 1) (n - r
1 , according as
>
+ 1) <
neg' - g) ,
Le. as
g
n{g'-l) + 1 .
r-1
<
>
The value of Pr[G
= giG' = g'l
is therefore minimized by
part of';.)
If n (g' - 1) I (r - 1)
(3)
([]
denotes
i~integer
{as it is sure to be if n I (r
Pr[G
=g -
and the common value is
1
I
G'
1)
is an 'integer
is an integer) then
= gil = Pr[G = giG' = gil
max P[G = giG' = gil •
g
One might consider
g as a Ilmaximum probability" estimator of g.
does indeed maximize the likelihood (1) (as well as (2)) .
We are thus led to consider the statistic
G= ,n(G' -1)
L
•
as an estimator of
g •
r-1
+
1J
It
•
-4-
.
-
The properties of G = n~;~i1) + _I. are easily -established. (Note that ...
a
if n/(r - 1) is an integer, Ga = G) . The distribution of (GY - 1) ,
~
~
given G = g , is hypergeometric (see "(1)) and
(4.1)
(4.2)
I
E[G'
var[G'lg]
=
=1
g]
+
(r - l)(g - 1) I (n - 1)
[en - r) (n - 2)](r - 1)[(g - 1)/(n - 1)][1 - (g - 1)/(n-l)]
= (n
- r)(r - 1)(g - l)(n - g)(n - 1)-2(n _ 2)-1
whence
~
E[Ga
(5.1)
(5.2)
g]
[G I
var
a
~
(5.3)
I
T
= (r
g]
= [n/(n
=
- 1)
- 1)] (g - 1) + 1
2
n (~-r)
(n-l) (n-2)
-1
0
(n - 1) (G I
(g-l) (n-g)
r-1
-
1) + 1
is an unbiased estimator of g.
Note that var [Ga
expected.
3.
I
g]
=0
if either g
=1
or g
=n
, as is to be
Population Distribution Known: Only One Sample Value Known
The analysis of Section 2 can be applied only if r
~
2.
However, if
the common density function of Xl' X2, ... , Xn is known, then it is
possible to derive a distribution for the rank order G of a randomly chosen
X , even if no other X-values are known.
X = x , is the probability that
x , and the other
•
(n - g)
(g - 1)
The probability that
G = g , given
of the remaining X's are less than
are greater than x.
This is
-5-
--e
(61 -Pr (G =g--' X = x] = .
n-l] {F(x)}~-1 {l.- F(x)}n- g (g = 1,2,_ L.L' n) .
[g-1
The value of g which maximizes (6) is
~
If
nF(x)
= [nF(x)
+
1] .
is an integer then
Pr[G = nF(x) + 1
I X=
= Pr[G
xl
and the common value maximizes Pr[G
= nF(x)
= g I X = xl
I X = x]
•
We can consider
G
= [nF(X)
+ 1]
at a maximum probability estimator of g.
Considering the related statistic
'"
Ga
we note that
F(X)
= nF(X)
+
1
has a beta distribution with parameters
an'ct so
E[F(X)
(7.1)
(7.2)
var [F(X)
I
I
g]
= g/(n
g] = g(n _ g + 1) (n
+ 1)
+ 1)-2
(n + 2)-1
whence
(8.1)
•
(8.2)
']
var [G"
a g
= g(n
- g
+
1) n 2 (n
+
1) -2 (n + 2) -1
(g, n - g
+ 1)
-6(8.3)
T
= (n
is an unbiased estimator of_ g ._
+ l) F (X)
4. Population Distribution Known:
2) sample Values Known
r(~
In this case we have information of each of the kinds described in
Sections 2 and 3.
The two sources of information may be combined to give
information on the rank
values, given its rank
G of an individual
G'
X ·amor.g
n randomly chosen
among a randomly chosen subset of
r(~
2)
among
the n values and the value of the cumulative distribution function
F(X) ,
for the individual concerned.
(r - 1)
The exact values of
X for the other
members of the subset contain no further information beyond the value of G'
(to which of course they contribute).
(In practice, of course, if these
values were abnormal they might raise doubts whether the
F(x)
cor~ect
function
was being used for the cumulative distribution function.)
The conditional distribution of G is
(9)
Pr[G
= g/(X = x)n(GI = gl)] = In-r
g_gl
(g
= gI
j
gI
+
1
j
•••
eg gl
]{F(X)}g-gl{l _ F(x)}n-r +
,
gI
+
n - r) .
This is maximized by
g
If
(n - r
+
=g
1) F(x)
equal values for
Pr[G
= [(n - r + 1) F (x) + g I] .
is an integer, then the values
=g I
(X
probabil i ty •
•
Consider now the estimator
= x)
neG'
= gl)]
g
=g
and
g
=g
and thus maximize this
give
-7-
•
Ga
If G
=g
= (n
1) F(X)
- r +
+
G'
then
~
(10.1)
E [G
a
I
g] = (n - r
={1
+ 1)
+
-L
n+I
+ (r - 1)
.&:!.
n-1
+ 1
Zr-n-1}
2
n-r
g +
n -1
n-l .
Conditional on G = g , the variables
F(X)
and
G'
are mutually
independent, and so
(10.2) var [Ga
I g] = (n
=
- r + 1)2 var [F(X)
I g]
+ var [G'
I
g]
(n r) (r 1)
(n-r+I)2
g(n - g + 1) + - 2 (g - 1) (n - g) .
(n+l) 2(n+2)
(n-I) (n-2)
Also
(10.3)
T
=
I
I
Ga - n-rJ
n-I
1 + 2r-n-1]-1
2 1
is an unbiased estimator of g.
n -
We have
~
.
g]
= (n
- r) (r - 1)
var (T
g)
= (n
+
varrTlg]
=
(11.1)
var [T
(11.2)
(11.3)
"
-1
2);-1 g(n - g
-1
(g - 1) (n - g)
+ 1)
{(n-r;1)2 g{n - g + 1) + (r-1)~n-r) (g _ l)(n _
(n+I) (n+2)
(n-I) (n-2)
x
[1
+
2r-2~-lJ-2
n -1
•
(n - 2)
g)}
-8-
-.~
If g, r,
and n
are
(12.1)
n var [g -rT
(12.2)
n var fg
l~rge,
with
I
(00
g] ~
-1 '"
= y , rln
1) (y
-
-1
= 00
then
- 1)
g] • yo-:-! - 1
T
n var [g-lr
(12.3)
-1
gin
g]
~
(1 -
(y
00)
-1
- 1)
Hence
.
var [T
I
g]
var
..
rT I g]
var [T
.
g]
-1
TOO:
(1 -
These approximate values do not depend on y(= gIn) .
on the assumption that
will not apply when
00)
-1
1 .
Since they are based
gIn is fixed, and not tending to 0 or 1, they
g is near
1 or n .
In large samples, we will have
var [T
according as
00
= rln
< 1
> 2'
I
>
'"
g] < var [T
I
g]
It is to be expected that as
estimator based on rank among a subset of r
r
increases the
values will become more
accurate. It appears that a "break-even" point (as compared with an estimator
based on knowledge of F(X)) is attained when about half of the sample values
are included in the subset.
We note, also, that this estimator can be
computed, even when we have no knowledge of F(X) , except that it is
continuous.
As might be expected,
T has a smaller variance than either T or
since it combines information from both Gl
•
and
F(x) .
T
-9-
Table 1 shows some values of efficiencies
u(re~Jp!,ocal
ratio_of
Y~:ria:l1ce_sl __
~
of T and T relative to
of eff[T
I
g]
= var
function F(x).
[T
I
T for a few values of n, g and
g]/var
[1
Ig]
Even for quite small
either (for nand r
r.
The values
do not depend on the distribution
n, they do not vary much with g
fixed), provided
g is not near 1 or n .
Table 1:
Relative Efficiencies of Some Unbiased Estimators of Rank Order
(For 'g > ~(n+ I). ,use eff [T I n - g + 1] = eff [T 1 g] )
,.
,.
n r g
2{;
5
3{;
4{;
eff[Tlg]
0.28
0.24
0.54
0.49
0.80
0.75
eff[Tlg]
0.72
0.76
0.47
0.51
.'
0.23
0.26
n r g
2H
4H
10
1:
10
20
lOP
lS{:
10
10
•
0.24
0.21
0.21
0.51
0.47
0.47
0.77
0.74
0.74
0.77
0.78
0.79
0.49
0.52
0.52
0.24
0.26
0.26
sH
6H
s{1
eff[Tlg]
0.16
0.11
0.11
0.44
0.34
0.33
0.57
0.45
0.44
0.68
0.57
0.56
0.90
0.79
0.78
eff[Tlg]
0.85
0.88
0.89
0.59
0.66
0.67
0.47
0.55
0.56
0.36
0.44
0.44
0.17
0.22
0.22
•.
-10Table 1
-
(~oJlt.)_
Relative Efficiencies of Some Unbiased Estimators of Rank Order
(For
g
1
I(n + 1) , use
>
eff [T
In
- g + 1]
= eff[T I
,.
,.
n r g
eff[Tlg]
eff[Tlg]
0.19
0.18
0.18
0.81
0.82
0.82
.0.40.
.
0.39
0.60
0.61
0.61
0.40
0.41
0.41
fO
10 20
25
5
t
t
t
20 20
25
0.39
0.61
0.60
p.60
30 20
25
P. .83
40 20
25
n r g
{O
2 20
40
50
r
r
r
40 20
40
50
100
60 20
40
50
0.20
0.21
0.21
0:82
0.82
80 20
40
50
effrTlg]
eff[Tlg]
0.20
0.20
0.19
0.19
0.41
0.40
0.39
0.39
0.62
0.60
0.60
0.60
0.81
0.80
0.80
0.80
0.80
0.80
0.81
0.81
0.59
0.60
0.61
0.61
0.39
0.40
0.40
0.40
0.19
0.20
0.20
0.20
~
Note that if
expected value
,.
were the linear function of T and T which (a) had
T
g and (b), sUbject to (a), had minimum variance, then the
sum of the two efficiencies would be equal to
1 • The biggest deviation
from this situation is when n
is near 1 and
o
or 1.
(See cases
n
= 10,
is small, rln
g
=2
take functional values.
,.
integer valued
gIn
is near
in Table 1) .
It should be realized that each of the estimators
•
g])
T, T and
T can
It
is not intended that they should replace the
,.
G, G and G; but they have been used to facilitate comparison .
-11-
•
~.
Population Distribution Unknown: S.eries of Single Observations of-tbe-Same
Unknown Rank in Random Samples of Known Sizen
If we are given the observed value of a random variable and are told
merely that it is selected at random from a set of n independent identically
distributed continuous random variables, it is not possible to obtain a useful
estimator of the rank of the chosen value among the n-original Val'i1es; \lri't1\oirt
having some further information about the common distribution.
If we have a series of N such values Yl , Y2 , ••• , YN and know that
each takes the same rank (g) among a random sample of size n (but g is not
known), we may in some circumstances be able to use the Y's to estimate
and so obtain some information on g.
F(x)
We will suppose the N random
samples of size n to be selected independently.
Form of Population Distribution Known
(al
We suppose it is known that F(x)
F(x
I
= F(x
I~)
of s parameters 6 1 , ... , as
function of YI , YZ' ... , YN is then
(13)
61' •.• , as)
is a known function
L .a
g(~) -
where f(t
{
n.I
(g-l)!(n-g)!
}N N
II
j=l
I ~) = k<F(t I ~)}
For a given set of values
~o
_
of the parameters a,
(a)
_ Lg_o
"-
maximized by taking
g
= g(~o)
where
g(a)
_0
L (6) > L
g -
is
"-
which
•
The joint
1 (6)
g--
is the greatest integer for
-12-
.___
i.e. for which
We have
= [n{l
(14)
+
Q(Y_
I e_0 )}-1
+
N
F(Y.le)
}l/N
1]
where
Q(~ I ~o) =.{ j~l
We then have to seek
value,
~,
of
~o
I-F(?ji;o)
to maximize
•
L~(~o)(~o) .
The corresponding
is the maximum likelihood estimator of g.
~(~o)
The
process can be lengthy, but considerable curtailment can be effected by
noting that
the value of
~
must be an integer, and as
~(~o)
max
L~(~o)(~o)
is approached
does not change over many iterations.
Another method, which is especially useful for small n, is to find the
value e() , maximizing L
- g
with respect to e , for fixed integer values
(e)
g -
of g and then choose
~
'"
as the value minimizing
Lg(~(g))
.
This procedure
A
requires determination of just
n
values
calculations needed to determine each
~(g)
LI(~(I))
..• Ln(e(n)) , but the
may be heavy.
For this reason
the procedure based on (14) may be preferable for larger values of n •
As a simple example, let us take
(15)
F(x
I
e)
=1
- e- x/e
(x > 0 ;
e
(corresponding to an exponential distribution for
•
> 0)
X) •
-13-
.~
For
g fixed,
N {(
=K
(16)
IT
1-e
-v./e).g-l
J
e
g j=l
-(n-g)Y./e -1
J
r y.}
= K e- N exp {-en - g + 1)
g
j=l
where
~
e
-y./e}
J
(1 _ e-
Yj/e
)g-l
j=l
J
Kg = {( g-.
i);~)!
.
n-g )I}N
.
To minimize
Lg(a)
with respect to
the equation
a we take
N
2Lg (a)
2e
Le.
0
a
(n - g
+
L Y.
(n-r+1)
N
=- -
e
+
N
l)N -1
j=1 J
2
e
L
j=l
or
{I - (g
N
l
equal to the root of
-Y./e
J
(g-l)
J
L
2
e
j=l l-e -Y.la
J
l)(n - g
e+(g-1)N- l
+
=0
Y./e
1)-1 (e J
Y./e
Y.(e J _1)-1
j=1 J
Y=
(17)
-
N Y.e
e
n-g+l
Since the right hand side of (17) is a continuous increasing function
o<
a , and tends to
single solution
0,
as
00
a
~
0,
00
a
for
respectively, the equation has a
a = a(g) .
In the present case, since
(18)
E[Y
I
g] =
egt
(-l)j .!g - 1] (n- g + 1 + j)-2 = ea
g,n
j=O
. j
say, an initial value of
(b)
e could be taken as Y/a n .
g;
Distribution Form Known to Depend only on Scale and Location Parameters
In the special case when x = 2 and
•
-14-
where
H(o)
is an explicit known function, the following heuristic method of
estimating 61 and e2 which is relatively simple to apply, may give useful
results. It is based on the fact that the expected value of the g-th
smallest value in a random sample of size n
(the
g-th order statistic) is
(19.1)
where ag,n is a known constant.
g-th order statistic is
(19.2)
b
Similarly the standard deviation of the
e
g,n 2
where b g,n is a constant.
For a given g, we estimate 6 2 by
62
where
Sy
estiwate
/ b g,n
is the sample standard deviation of Yl' Y2' •.. , YN
61 by
(20.2)
with -Y = N-1
N
. ~ 1 Y. .
Jill
•
= Sy
J
and we
-15-
We then insert these values in the
~~kelihoEd funct~on
giving
(21)
where h ( 0) = HI (0) •
This is repeated for
g
= 1,
2, ..• , n and we find the value of
g, g
say, which maximizes
Lg
As a final step we may calculate the values of 81 and 82 which
maximize L, for g = g - 1, g and g + 1 and take our final estimator as
(6
8
that value of the three for which Lg 1 (g)' Z(g)) is greatest.
In some cases, experience may suggest that this last step might be
omitted.
Lg (8 l (g)' 82 (g)J
Igi = g - 1 than it is for g = g , it may be
Conversely, in situations where we find that
is greater for
g
=g
desirable to try also
+
1 or
g
=g +
2 or
g
=g -
2 respectively.
For the case of an exponential distribution we have only one
parameter
unkno~m
e , and this would be estimated
as e = Y / a g, n- (see text
.
following (18) ).
5.
Further Possibilities
In the circumstances of the last section, it is possible, in principle,
to estimate n , the total sample size (if it is unknown) as well as
given a sufficient number of values
g,
Yl , YZ' ... , YN each of which is the
g-th in rank among N random values from the same continuous distributions.
Either maximum likelihood or moments can be applied .
•
-16-
For example in the el'ponential case we canestimate a , g and n as
solutions of the moment equations.
(22)
m'r(Y)
= arr(r
r
+ 1) g-l (_I)J. [gJ-.I] (n _ g
j=O
+
I
+
j)-(r+l)
(r = I, 2, 3)
where m' r (Y) is the r-th sample moment about zero of the N observed
values YI , Y2, ..• , YN . Alternately we may seek values of n, g and e
maximizing (16).
Apart from technical difficulties, however, it seems likely that very
large values of N would be needed to obtain reasonably accurate estimators
of the three parameters n, g and
•
e .