lThis research was supported in part by the U. S. Army Research Office,
Durham, under Contract No. DAHC04-7l-C-0042.
ESTIMATION OF RANK ORDER l
by
N. L. Johnson
Department of Statistics
University of North CaroUna at Chapel BiH
Institute of Statistics Mimeo Series Number 931
June 1974
ESTIf4ATION 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.
a
i
is
itself).
If no two
a's
are equal,
t~e
rank
g •
If
Xl' XZ' ...• Xn are continuous random variables corresponding to a
random sample, then with probability one, no two are equal and so each X.1
will have a unique rank order different from that of any other
X.
J
(j
;t
i) .
In this paper we will be concerned with the estimation of the rank order
of a specified
Xi
Xl' XZ' ... ,Xn
where
when only partial information about the sample values
is available.
Xl' XZ' ... , Xn
The discussion will be restricted to cases
are independent and identically distributed contin-
uous random variables with common cumulative distribution function
density function
F(x)
and
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 \'Jill be in a later report.
2.
Population Distribution Unknown; r Sample V?lues Known (2
2'
(Xi, X
If we are given the values
subset of
r(~ 2)
of the
values
n
XV
probability that a randomly chosen
Xl' XZ' ... , Xn
, X
g-th
G=g
(i.e. has rank order
<
n)
we can evaluate the
n
is the
r
of a randomly chosen
Xi)
r
Xl' X2 '
~
smallest among
in this set).
Note that
r-
is
a random variable.
G = g , given that the rank order,
TI1e conditional probability that
,
=g
Pr[G
Since
Pr[G
= g] = n -1
X~)
G' = g']
, Pr[Gi
gi
is
in
= Pr(G=g]Pr[Gi=g' ~
Pr[G'=gt]
= g' ] = r -1
(g
= 1,
... , n
g' = 1, ... , r)
and
(1)
Pr [G ' = g I
I
G = g] =
[g. ~ 1
]
~
~
g -1
[max (1, r + g - n)
g'
[n-g,] / [n -1]
r-g
r-1
min (g, r)]
we have
(2)
Pr[G
=g
(g =
G'
1
o0- ,
Note that (1) implies that
parameters
=e']
=[::~l] [:~:,] / [:]
g' + l, ... , g' + n - r) .
(G7 - 1)
(r - 1), (g - 1), (n - 1)
has a hypergeometric distributicn 1,Aiith
-3-
How
~~~.....-=12.-;!-.;- = ~) (n-r+l+g -g)
(g_gl) (n+l-g)
l
(g
= gl
1, ... , gl + n - r)
+
TI1is ratio is greater or less than
1 , according as
>
(g - l)(n - r
+ 1) < neg I
_
g) ,
i.e. as
< neg! -1)
g >
The value of
Pr[G
=g I
GI
(l"'
~.
g = g
(3)
([]
=
r-l
1J
=
is therefore minimized by
In(g'-l) +
L
denotes ,: integer part of".)
(as it is sure to be if
Pr[G
=~ -
and the common value is
I
G'
r-l
lJ
If n(gl - 1) / (r - 1)
n / (r - 1)
1
+ 1 .
is an 'integer
is an integer) then
= g~] = Pr[G = giG' = glJ
max P[G
= giG' = gIl
.
g
One might consider
g as a IJmaximum probabilityH estimator of g.
does indeed maximize the likelihood (1) (as well as (2)) .
We are thus led to consider the statistic
G=
as an estimator of
g.
In(G'-l) +
r··l
L
IJ
It
-4~~e
if nl (1' - 1)
given
= n(G/-I) +
Ga
is an integer,
r-l
~
G
a
-
1 are easily established.
= G)
The distribution of
(Note that
(G i _ I ) ,
G = g , is hypergeometric (see'(l)) and
(4.1)
I
E[G'
(4.2)
~
properties of
1) I (n - 1)
g] = 1 + (1' - 1) (g
= Un - 1') (n - 2)](1' - l)[(g - l)/(n - 1)][1 - (g - l)/(n-l)]
var[G'lg]
= (n
- 1')(1' - l)(g - l)(n - glen - 1)-2(n _ 2)-1
whence
~
I
E[Ga
(5.1)
[G I
var
(5.2)
g]
a
g]
= [n/(n
=
~
T = (1' - 1)
(5.3)
is an unbiased estimator of
Note that val' [G
a
I
g]
- 1)] (g - 1)
+
1
2
n (~-r)
'. (g-l) (n-g)
(n-l) (n-2)
1'-1
-1
(n - 1) (G; - 1) + 1
g.
=0
if either
g
=1
or
g
=n
, as is to be
expected.
3.
Population Distribution Known: Only One Sample Value Knolrln
1be analysis of Section 2 can be applied only if
the common density function of
Xl' X2 , .. , ,Xn
is
possible to derive a distribution for the rank order
x , even if no other X-values are known.
x=x
, is the probability that
x , and the other
(n - g)
(g - 1)
l' ~
kno~m,
2
However, if
then it is
G of a randomly chosen
The probability that
of the remaining
are greater than
x.
This is
X's
G = g , given
are less than
-5-
(61 Pr[G
= g I x = x] = [n-1]
{F(X)}g-l
{l -
F(x)}n- g (g = 1, 2, ... , n) .
g-l
The value of g which maximizes (6) is
= [nF(x)
~
If nF(x)
+
1]
is an integer then
Pr[G
= nF(x)
I
1
+
= Pr[G = nF(x) I X = x]
X = x]
and the common value maximizes
=g I
Pr[G
X = x] .
\l!e can consider
"
G = [nP (X) + l]
at a maximum probability estimator of
g.
Considering the related statistic
= nF(X)
Ga
we note that
F(X)
+
1
has a beta distribution with parameters
and so
(7.2)
I
E[F(X)
(7.1)
var [F(X)
I
g] = g/(n
= g(n _ g
g]
+
1) (n
1)
+
+
1)-2 (n
+
2)-1
+
2)-1
whence
E[G
(8.1)
(8.2)
var rCa
I
g]
a
I
= g(n
g]
= gn(n
- g
+
+ 1)-1 + 1
1) n 2 (n
+
1)-2 (n
(g, n - g
+ 1)
-6(8.3)
4.
T
=
is an unbiased estimator of
(n + 1) F (X)
Population Distribution Known:
g .
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
the
G of an individual
G'
Xamor.g
n
randomly chosen
among a randomly chosen subset of
2)
r(~
n values and the value of the cumulative distribution function
for the individual concerned.
The exact values of
among
F(X) ,
X for the other
(r - 1)
members of the subset contain no further information beyond the value of
(to which of course they contribute).
G'
(In practice, of course, i f these
values were abnormal they might raise doubts whether the correct function
F(x)
was being used for the cumulative distribution function.)
The conditional distribution of
(9)
Pr[G
G is
= gl(X = x)n(Gi = gil] = fn-r
19_9i
=g
(g
i,
]{F(X)}g-gi{l _ F(x)}n-r
cg g'
+
g' + 1, ... , g I + n - r) •
TIlis is maximized by
g
If
= g = [en
(n - r + 1) F(x)
equal values for
Pr[G
- r + 1) F(x) + gil.
is an integer, then the values
=g I
(X
probabil i ty .
Consider now the estimator
= x)
neG'
= gill
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[Ga
I
= (n
g]
- r +
={ 1
Conditional on
G=g ,
1)
..lL
n+l
+ 2r-n-1}
n2- 1
+ (r - 1) g-l + 1
n-l
n-r
g+
n-- l '
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 [Gl
2
= ~(n_-_r",,"+.l-,)__ g(n _ g + 1) + (n-r) (r-1)
(n+l)2(n+2)
I
g]
(g _ 1) (n _ g) .
(n-l)2(n-2)
Also
(10.3)
We have
(11.1)
var [T
g] = (n - r) (r - 1)-1 en - 2)-1 (g - 1) (n - g)
(11.2)
var (T
g)
(11.3)
var(Tlg] = {
= (n
2)~1
+
g(n - g + 1)
~n-r)
(n_r+1)2 g{n _ g + 1) + (r-l)
(g - 1) en _
(n+1)2 (n+2)
(n-I) (n-2)
x
[1 +
2r~n-lJ-2
n -1
g)}
-3-
If
g, r,
and n
are large, ...l ith
(12.1)
n var [g
-l.r I g]
(12.2)
n var [g
-1 . .
T
n var [g-l.y
(12.3)
I
ew
T
-1
. Y-1
g]
T
g]
T
(1
gin = y , rln
- 1) (y
-1
=w
then
- 1)
- 1
- w)
(y
-1
- 1)
.
Hence
var [T
I
....
g]
var [T
I
var [T
g]
g] ~ w- 1 : (1 _ w)-l
These approximate values do not depend on
on the assumption that
gin
will not apply when
is near
g
y(=
gin) .
1 .
Since they are based
is fixed, and not tending to
0 or
1, they
1 or n .
In large samples, we will have
var [T
according as
w
= rln
< 1
>
2"
I
gJ : var [T
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-event! 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.
l~e
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
since it combines information from both
Gl
and
F(x) .
T or
T
-9-
Table 1 shows some values of efficiencies (reciprocal ratio of variances)
of T and
of
T relative to
I
eff[T
function
g]
F(x)
either (for
= var
[T
I
T for a few values of
g]/var [T Ig]
Even for quite small
nand
r
r.
The values
do not depend on the distribution
n,
fixed), provided
n, g and
t~ey
g
do not vary much with
is not near
1 or
g
n .
Table 1:
:i.e1ative Efficiencies of Some Unbiased Estimators of '"?ank Order
1 ( + 1). ,use eff [T I n - g + 1] = eff [T I g] )
(For g. > 2"
n
n r g
"
2{;
ef£[Tlg]
0.28
0.24
3{;
0.54
0.49
4{~
efffTlg]
0.72
0.76
0.47
..0.51
0.80
~. 75
0.23
".26
5
n r g
r
8
10
0.24
0.21
0.21
0.77
0.78
0.79
1O{~10
0.51
0.47
0.47
0.49
0.52
0.52
IS{:
0.77
0.74
0.74
0.24
0.26
0.26
5
20
10
"
eff[Tlg]
0.11
0.85
0.83
0.89
4H
0.44
0.34
0.33
0.59
0.66
0.67
sp
5
0.57
0.45
0.44
0.47
0.55
0.56
6H
0.68
0.57
0.56
0.36
0.44
0.44
aU
0.90
0.79
0.78
0.17
0.22
0.22
2H
10
e£f(Tlg]
0.16
0.11
-10-
Table 1 (cont. )
Relative Efficiencies of Some Unbiased Estimators of Rank Order
1
(For g > I(n
+ 1) , use eff [T I n - g + 1] = eff[T I g])
A
n r g
eff[Tlg]
0.19
0.18
0.18
0.81
0.82
0.82
t{IO
10 20
25
.0.4p'
'
0.39
20 20
25
5
A
eff[Tlg]
t
t
30 20
25
40 20
25
0.39
0.60
0.61
0.61
0.61
0.60
p.60
0.40
0.41
0.41
p.83
0:82
0.82
0.20
0.21
0.21
n r g
eff[Tlg]
eff[Tlg]
0.20
0.20
0.19
0.19
0.80
0.80
0.81
0.31
0.59
50
0.41
0.40
0.39
0.39
0.60
0.61
0.61
60 20
40
50
0.62
0.60
0.60
0.60
0.39
0.40
0.40
0.40
0.81
0.80
0.30
0.30
0.19
0.20
0.20
0.20
r
r
r
r
20 20
40
50
40 20
40
100
80 20
40
50
Note that if
expected value
T were the linear function of
from this situation is when
or
1.
(See cases
n
n
= 10,
is small, rln
g
=2
1.
which (a) had
TI1e biggest deviation
is near
take functional values.
1
and
gin
is near
in Table 1) .
It should be realized that each of the estimators
integer valued
T
g and (b), subject to (a), had minimum variance, then the
sum of the two efficiencies would be equal to
o
and
T
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-
4.
Population Distribution Unkno\tn: Series of Sinple Observations of the
Unknown Rank in Random Samples of Known Size n
Sam~
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'iie5~'
\"ithoi1t
having some further information about the common distribution.
If we have a series of
N
such values
Yl , Y 2 , ••• , YN and know that
each takes the same rank (g) among a random sample of size
known), we may in some circumstances be able to use the
F(x)
and so obtain some information on
samples of size
(a)
n
(but
g
is not
to estimate
We will suppose the
N random
to be selected independently.
Form of Population Distribution Known
We suppose it is known that
F(x
g.
Y's
n
I
61 , ... , as)
= F(x
I~)
F(x)
of
s
is a
kno~~
parameters
function
61 , ... , as
The joint
function of Yl , Y 2 , ••• , YN is then
(13)
Lg(~)
where
f(t
_ {
nl
}N ~
(g-l)l(n-g)l j=l
I ~) = ~F(t I ~)}
For a given set of values
e_0 of the parameters a_ , Lg_o
(8 ) is
~
maximized by taking
which
g
= g(~o)
~
where
g(~o)
is the greatest integer for
-12i.e. for which
: { F(Yjl~) 1
[~-g+llN
g-l) n=l I-FCY j ~J
1
>
We have
(14 )
= [n{l
g(~o)
+ Q(Y
where
We then have to seek
value,
~,
of
~(~o)
S
_0
The corresponding
is the maximum likelihood estimator of
g.
The
process can be lengthy, but considerable curtailment can be effected by
noting that
g must be an integer, and as
max Lg(e
)(~o)
is approached
_0
the value of
ft(~o)
does not change over many iterations.
Another method, which is especially useful for small
value
of
L
e( ) , maximizing
- g
g and then choose
with respect to
(e)
g -
~
as the value minimizing
requires determination of just
n
e
- Cg
S , for fixed integer values
Lg(~(g)) .
This procedure
"
values
calculations needed to determine each
n, is to find the
... Ln (6 )) , but the
Cn
)
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 -xIS
(x > 0 ; S > 0)
(corresponding to an exponential distribution for
X).
-13-
For
(16)
g fixed,
= K ~ {(l _ e- Yj/6 )g-1
L (6)
g
r
_ {.
tg -
exp {
(n-1) !
(g-l)! (n-g)!
To minimize
Lg (e)
}N
-en -
i.e.
(n - g
e-
J
e
-Y.le}
J
N
)'
g + 1)
Y.
j;l
}
N (
TI
0
L Y.
(n-r+1)
j=l J
J
J
L
j=l l-e
-Y./e
J
-Y./e
=0
J
Y./e
L {I - (g
j=l
or
l)(n - g
6+(g_1)N- 1
(17)
(g-l)
e2
e2
Y.e
N
N
l)N -1
-Y./e)g-l
j=l
J
II
+
e
e we take e equal to the root of
with respect to
(e)
g
~I
= - - +
2e
e
1 -
.
the equation
2L
-(n-g)Y./e 1
g j=l
= Kge -N
where
e
?1
L Y.(e
Y./e
J
1)-1 (e J
+
_1)-1
j=l J
'1=
n-g+l
Since the right hand side of (17) is a continuous increasing function
o<
6 , and tends to
single solution
0,
as
<Xl
e
-+
0,
<Xl
e
for
respectively, the equation has a
e = e(g)
In the present case, since
(18)
E[Y
I
g] = egf
(-l)j
.[g - 1] (n- g
j=O
say, an initial value of
(b)
+
1
+
j)-2 = ea
j
e could be taken as
Distribution Form Known to
In the special case when
De~end
g,n
Yla g, n.
only on Scale and Location Parameters
x = 2 and
-14-
where
H(o)
is an explicit
estimating 81 and
results.
kno~m
function, the following heuristic method of
82 which is relatively simple to apply, may give useful
It is based on the fact that the expected value of the
smallest value in a random sample of size
n
(the
g-th
g-th
order statistic) is
(19.1 )
where
a
g-ta
g,n
is a known constant.
order statistic is
(19.2)
where
b
b g,n
g, we estimate
8
where
Sy
estiJrate
(20.2)
g,n 82
is a constant.
For a given
with
Similarly the standard deviation of the
2
= S
8
2
by
Y / b g,n
is the sample standard deviation of YI , Y2' ... 'YN
e1 by
8
1
= y - a g,n
8
2
and we
-15-
We then insert these values in the likelihood function giving
(21 )
where
h(o) = HI (0) •
This is repeated for
say,
g
= 1,
2, ... ,n
and we find the value of
L
g
As a final step we may calculate the values of
~lich
maxi~ize
g, g
maximizes
L, for
g
=g
- 1, g and g
+
61 and
6 which
2
1 and take our final estimator as
(6
Lg 1 (g)' a2 (g)) is greatest.
In some cases, experience may suggest that this last step might be
that value of the three for which
omitted.
Conversely, in situations where we find that
is greater for
g
=g
desirable to try also
+
1 or
g
=g
(6
6
Lg l (g)' 2 (g))
1+g I = g - 1 than it is for g = g , it may be
+
Z or
g
=g -
2 respectively.
For the case of an exponential distribution we have only one
parameter
6 , and this would be estimated as
e =Y /
a g,r.
unkno~m
(see text
following (18) ).
5.
Further Possibilities
In the circumstances of the last section, it is possible, in principle,
to estimate n, the total
sa~)le
size (if it is unknown) as well as
given a sufficient number of values
g-th
in rank among
g,
Yl , YZ' ... , YN each of which is the
N random values from the same continuous distributions.
Either maximum likelihood or moments can be applied.
-16-
For example in the exponential case we can estimate
e , g and n as
solutions of the moment equations.
mir(Y) = arr(r + 1) g-l
I (_l)J"lgJ-.l] (n - g + 1 + j)-(r+l)
j=O
(22)
(r = 1, 2, 3)
where
mi
values
r
(Y)
YI , Yz'
maximizing
is the
... ,
r-th
YN .
sample moment about zero of the
N observed
Alternately we may seek values of n, g and
e
(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
a .