SARA LYNNE STOKES.*
Set Sampling.
An Investigation of the Consequences of Ranked
(Under the direction of NORMAN L. JOHNSON.)
The effects of ranked set sampling on some estimators of location and scale are examined; among these are the MLE's and BLUE's of
~
and
0
where they are parameters of a random variable X which has
.
x-~
c.d.f.
of the form F(---).
o
Estimators of the correlation coefficient,
p, of a bivariate normal distribution are introduced, where these estimators are based on a modified form of ranked set sampling in which
only extreme order statistics are included in the sample.
In both
situations, we find that the precision of the estimators is improved
by the ranked set sampling process over that from estimators based on
the conventional random sample.
Two testing and confidence interval
procedures for p utilizing ranked set sampling are introduced and
compared.
Finally, the cost effectiveness of ranked set sampling is
briefly discussed.
* This research was supported in part by the Army Research Office, under
grant #DAA-G29-74-C-0030.
ACKNOWLEDGEMENTS
I would like to express my appreciation to my advisor, Dr. N.L.
Johnson, for his ideas and assistance during the course of this long
project.
His gentle prodding and encouragement, and his friendship,
played important parts in its completion.
I
also thank the members of
my committee, Dr. R. Carroll, Dr. R. Helms, Dr. P.K. Sen and Dr. G. Simons,
for their careful reading of the manuscript, and the Department of Statistics for financial support throughout my graduate studies,
I
express my gratitude to Ms. June Maxwell for her beautiful and
speedy job of typing, and for her ability to take care of details, both
of which made the last several weeks proceed much more smoothly.
Special thanks go to my classmates for many reasons, both academic
and non-academic; in particular, I would like to thank Chin-Fei Hsu for
his generous help in
progra~ming,
encouragement from the beginning.
and my friend Carlos Segami for his
And finally, to my family and special
friends in Chapel Hill and North Carolina, who have soothed and supported
and listened, I express my deepest gratitude.
ii
TABLE OF CONTENTS
Acknow1edgements---------------------------------------------- ii
1.
INTRODUCTION----------------------------------------------
1
2.
ESTIMATION OF LOCATION AND SCALE PARAMETERS FROM
RANKED SETS-----------------------------------------------
7
3.
4.
5.
Estimators~------------------------------ 7
2.1.1
Efficient
2.1.2
Fisher Information---------------------------------
9
2.2
2.3
Best Linear Unbiased Estimation---------------------- 20
s2- type Estimators of 0 2
36
2.4
Judgment Ordering with a Concomitant Variable-------- 46
ESTIMATION OF PARAMETERS OF ASSOCIATION------------------- 51
3.1.1
Estimation of p with All
3.1.2
Confidence Intervals for p with All Parameters
Known---------------------------------------------- 57
Parameter~
3.2
Estimation of p with
2
x . x Unknown----
68
3.3
Estimation of p with All Parameters Unknown---------
71
HYPOTHESIS TESTING AND RANKED SET SAMPLING---------------
76
~y' 0
2
Known---------- 51
y
Known;
~ ,0
Location------------------------~---------- 76
4.1
Tests of
4.2
Tests of Corre1ation--------------------------------
82
COST CONSIDERATIONS--------------------------------------
98
Appendix----------------------------------------------------- 104
Bibliography------------------------------------------------- III
•
iii
"
e
INTRODUCTION
1.
Ranked set sampling was introduced and applied to the problem of
estimating pasture yields by McIntyre [16].
It has found applicability
in other situations in which observations of a sample are much more
easily and inexpensively ordered than quantified.
The ranked set
sampling process consists of choosing m random samples, each of size m,
from the population.
themselves.
The m elements of each sample are ordered among
Then the smallest observation from the first sample is
measured accurately, as is the 2nd smallest observation from the 2nd
t
sample.
The process continues in this manner until the largest obser-
vation from the m-th sample is measured.
.
This entire cycle is repeated
n times .
Let
\r:m),i
denote the r-th order statistic from the sample of
size m, in the i-th cycle.
A
(1)
1
J1(m)n = mn
Lm
r=l
Takahasi and Wakimoto [21 ] have shown that:
L~=lX(r:m),i'
the arithmetic mean of the mn quanti-
fied observations, is an unbiased estimator of the population mean J1.
var(X)
1
mn
the conventional
('
) ~ -2(m+ 1), where X = Lv~nlx./mn,
1= 1
var J1(m)n
mn
estimator of the mean from a random sample of size mn. (Note that both
(2)
1
~
X and J1(m)n
mn
are calculated from mn quantified observations.)
_
The
A
maximum value for var(Xmn )/var(J1(m)n) of ~(m+l) is attained only for
the rectangular distribution among the class of all continuous distributions with finite variance.
(3)
If N = ab = cd, where a, b, c, d are all positive integers and
a > c, then
2
That is, the ranked set sampling estimator based on N = nm observations
is improved by increasing m and decreasing n accordingly.
Therefore,
the best ranked set sampling estimator based on N quantified observations is
~(N)l.
Unfortunately, the ordering process is likely to be-
come difficult, if not impossible, for large N, and the ranked set
sampling procedure will become impractical.
Dell and Clutter [6] extend the concept of ranked set sampling
to include those situations in which perfectly accurate ordering may
not be possible.
In this case, they suggest that each of the mn
samples be ordered by eye, and the resulting "judgment order statistics"
be used for estimation of the population mean in the same way that the
order statistics were used in the original ranked set sampling procedure.
That is, let
m
A
~[m]n
where X[
r:m
n
=.J:.
I I x[ ] .
mn r=l i=l r:m,l
].
is the r-th judgment order statistic from a sample of size
,1
m in the i-th cycle.
Then they have shown that for any underlying disA
tribution with the appropriate number of moments,
estimator of
A
var(~[
where
~[r:m]
~
~[m]n
is an unbiased
and
1
m] n ) = -mn [a
2
is the mean of the r-th judgment order statistic from a
sample of size m.
So
var(X )
mn
~
1 ,
with equality holding only if judgment ordering is no improvement over
random ordering.
3
In addition to McIntyre's original application, the ranked set
sampling procedure has been implemented successfully by Halls and Dell
r 9],
who used it for estimating forage yields.
Other suggested and
actual applications of the ranked set sampling method in which the
idea of judgment
ordering might be particularly useful and realistic
are:
(1)
Ordering by observation of aerial photographs for estimating
yields of stands of trees.
(2)
Ordering by personal interviews for estimating characteristics
in human populations.
(3)
Ordering by color or appearance for estimating the concentration of a chemical in solution or a pollutant in air.
(4)
Ordering by subjective judgment
as to the degree of illness
of a patient for estimating some measurable characteristic
of the progression of his disease.
In order
to study the effects of possibly imperfect or judgment
ranking on the estimator of the mean, Dell and Clutter propose a model
for ranking error.
In this model, one assumes that the experimenter
ranks the elements from judgment estimates which equal the true value
of the observation plus an error component; i.e.,
x[r:m]
where
E
=
~ N(O,O~) , with X(r:m) and
X
(r:m)
E
+ E
independent.
They examine the
effects of this model on var(X )/var(~[ ] ) by performing computer
mn
mn
simulations, varying the underlying distribution of X, the value of
2
0c' and the size of m.
David and Levine [5 ] verify the accuracy of
some of the computational results by finding the theoretical values of
4
var(Xmn )/var(~[m] n ) for an underlying normal distribution.
Dell and Clutter point out that this model for ranking errors may
be unrealistic in certain situations.
First, judgment errors are likely
to be influenced by the size of m, the number to be ranked.
Secondly,
the independence of the observation and the error component may be a
faulty assumption, since the tendency to misrank may be influenced by
the specific values of the elements themselves, or specific values of
other elements within the sample.
This flaw would be difficult to
correct, however, since those "specific values" may be peculiar to a
given experimental situation.
In Chapter 2, other estimators of the location parameter besides
A
~(m)n
are examined in order to determine if the benefit derived from
ranked set sampling extends to them.' In addition, several estimators
of dispersion parameters and the effect of ranked set sampling on
their variances are studied.
In all cases, we show that at least for
large enough sample sizes, or for large enough values of m, the ranked
s'et sampling estimator has a smaller variance (or mean square error) than
its random counterpart.
In Section 2.4, the concept of judgment ordering is generalized.
I
Suppose we are interested in estimating the mean of X when not only
quantification, but also ordering of the observations of X is difficult.
If the observations of a related variable Y can be easily ordered, then
our goal is to use the information available from the Y's to judgment
order the random sample of X'sand therefore to indicate which set of
X's should be measured.
An example of the type of problem where this
procedure could prove useful is discussed in [20].
Here an estimate of
5
cinchona yield in certain plantation areas of India was desired, both
in terms of dry bark and quinine contents.
an expensive one, however, since it
The measuring process was
involved killing and stripping
the plant, drying and weighing the bark, and chemically analyzing the
bark and roots to ascertain quinine yield.
the course of the study
But it was also found in
that the correlation of dry bark yield with
volume of bark per plant (a figure easily computed from the height,
girth, and bark thickness of a plant)
was quite high.
So bark volume
could have been used as a related variable (Y) for indicating which
plants to include in the sample.
In Chapter 3, the ranked set sampling idea is extended to estimation of parameters of association of bivariate data.
This technique
would be used in situations in which observations of one of the two
variables (which we call X) can be ordered eas'Iy, while the other,
or possibly both, are relatively much more expensive to quantify.
Here
/
Of
we have restricted the investigation to the estimation of p where (X,Y)
has a bivariate normal distribution.
We show that this technique will
improve the estimate of p in certain circumstances.
Useful applications of this
technique can be found in many fields.
Some possibilities are:
(1)
Estimation of correlation between an external characteristic
(X) and some internal characteristic (or one that is simply
expensive, painful, or inconvenient to measure) of a patient
or laboratory animal.
(2)
Estimation of correlation between socio-economic level or
some other observable characteristic, and IQ score, which is
tedious and expensive to obtain.
lbree cases are considered in this chapter.
First we assume that all
6
2 2) are k nown.
parameters except p ( ].1.,].1.,0.,0.
x
that only
].1
x
and
0
2
x
y
x
are known.
y
Secondly, we assume
If Y can be easily measured, it is
conceivable that prior information about its mean and variance is
available, or at least that they can be quickly and accurately estimated.
Finally, we consider the case in which none of the parameters
are known.
Chapter 4 describes how ranked set sampling techniques may be
put to use in testing of hypotheses.
Chapter 5 deals with some
considerations of cost in ranked set sampling.·
Throughout this work, the
e~phasis
in each analysis is on ranked
set sampling when perfect ordering is possible.
However, it often
happens that the presence of errors in ranking does not negate all
its benefits.
When that is the case, it will be noted at that point
in the discussion.
e.
ESTIMATION OF LOCATION AND SCALE PARAMETERS
2.
FROM RANKED SET SAMPLES
2.1.1
Efficient Estimators
Dell and Clutter have considered an estimator of one location
parameter, the mean
~,and
unbiased estimator of
~
have determined in general that it is an
(when it exists) and that its variance is
smaller than that of the conventional sample mean.
In this chapter,
other ranked set sampling estimators of location and scale will be
examined, with particular attention to their efficiencies with respect to those of comparable estimators based on random samples.
-e
First we will determine the amount of i1 formation (in Fisher's
sense) available from the'ranked set sample about the location and
scale parameters.
reasons.
(1)
Fisher Information, 1(8), is important for several
Among them are the following:
Let e, an
estimator of 8, be calculated from a random sample
1).
XI, ... ,X , each Xi having p.d.f.
n
f (x).
8
Then under certain general
conditions (see Appendix, Theorem A)
nI(8)
where Ee
n
=
e+
b (8) and
00
1(8)
=
J(
-00
Consequently, if
en is an unbiased estimator of e,
o
~
1/n}(8)
var(8 )
n
=
e(e ) ~ 1
n
8
0
A
and e(8 n ) will be called the relative efficiency of 8 . If e(8 ) = 1,
n
n
A
A
A
we say 8 is an efficient estimator of 8; if lim e(8) = 1, we say 8
n
n
n
n-+oo
is an asymptotically efficient estimator of 8.
Under fairly general conditions (see Appendix, Theorem B), the
(2)
maximum likelihood estimator (MLE) of 8 is an asymptotically efficient
estimator.
There are extensions to more general situations of both points
(1) and (2) [ 3].
First consider the case of 2 unknown parameters,
A
8 and n.
SUppOSP, for simplicity, that 8
n
mators of 8 and
and
nn
are unbiased esti-
Now under regularity conditions analogous to
n.
those of Theorem A (i. e., the conditions of Theorem A must be satisfied with respect to both parameters 8 and n), we find that the
ellipse
n[E (a .R-n f)2(U_8)2
a8
+
2E(a .R-n f
a8
+
.R-n f) 2 (v-n) 2]
E (a
an
•
a.R-n f
J(u-8) (v-n)
an
A
=4
(2.1)
A
is contained within the concentration ellipse of (8 n ,8 n ), which is
_1_[(u-8)
2
2
l-p
1
°
2
2p(u-8) (v-n)
°1°2
A
2
+
(v-n) ]
2
°2
=4
(2.2)
'
A
where the covariance matrix of (8 n' n)
n is
En
=
A
If the two ellipses (2.1) and (2.2) coincide, we say that 8n
A
and nn are jointly efficient estimators of 8 and n, and
9
where
r E(a ~~.
.!.(e,n)
=
a .l/,n f)
an
E(a .l/,n f
ae
f) 2
E(a .l/,n f • a .l/,n f)
ae
an
E(a .l/,n f) 2
an
I.
J
Furthermore, it is not possible to find a pair of regular unbiased
estimators of (e,n), e* and n*
n
n
say, such that var(e*) ~ var(e ),
n
n
even if we are unconcerned with var(n*).
n
We define the joint efficiency to be
A
A
e(en,nn)
=
[1/ In.!.c e,n) I)J
k
2
~
n
n-+oo
jointly asymptotically efficient-The MLE's of e and n are jointly
asymptotically efficient.
These results ext8nd directly to the case
of several unknown parameters.
The second important extension is that if XI"",Xn is not a set
of iid random variables, but rather has p.d.f.
then results (1) and (2) still hold.
2.1.2
Fisher Information
Let!
=
(XI, ... ,X ) be a random sample from a distribution having
N
x-11
c.d.f. F[(x-11)/O] and p.d.f. O/o)f(o)' (Note that 11 is a parameter
of location and
0
a parameter of scale, although they need not be
expectation and standard deviation of X.)
Let X*
=
(X (l: m) ,1' ... ,
X(l:m),n"'" X(m:m),l" ", X
) be a ranked set sample of size
(m:m),n
mn
=
N from the same distribution.
So the likelihood function of X is
10
L
N
X. -1-1
= -l IT
. 1
aN 1=
f(_l_)
a
and the likelihood function of X* is
l ~
L* = --N
~ f
(X(r:m),i - 1-11)
a i=l r=l (r:m)
a
=
~ ~
~
. 1 r--l
aN 1=
m(m-1Jf[X(r:m),i -1-1) [F(X(r:m),i - 1-1JJr-l
r-l
a
a
x _1 - F fX (r:m)~l. - 1-1)]m-r .
r
l
The Fisher Information of a random sample for such a distribution
is easily found.
(1)
If the scale parameter a is known, then
E d .Q.n f = 0
d1-1
_
N
- "2
a
E
rf'
-
2
(U) 1
feU)
I '
-
X-1-1
where U =
a
(2)
If the location parameter a is known,
d
IN(a) = NI(a)
- ..rJL
- N
d .Q.n
1. f
ada
=!iEruf'(U)
2 I
feU)
a
-
.Q.n f
providing that E da
= 0
e·
11
=
~ E[(lU f' (U))
02
feU)
_
2 2Uf'feU)
(U)
1]
+
+
since
E Uf' (U) = -1 .
feU)
If neither
(3)
matrix
~
nor
is known, then we find the Fisher Information
0
~(~,O) .
E
I-a fu
a~
L.
a JI,n L1
ao J
I-a
=
9,n
N E
o 2 1_
f(x~~)
ao
o~
if
_ -.Ji
-
E
02
f(¥)l
a 9,n
E
-'
a JI,n
ao
I
a .Q.n
=
f
a~
=0
r (f'feU)
(U)) 2
f' (U)]
+ feU)]
Lu
since E
f~~~~ = o.
Then
~(~,o) =
N/o
21
JJ
- (f' CU) )
E 1_u l-f(U)
2
(2,3)
E[(Uf~~~~J2
-
1]
J
Now the same three cases will be considered for a ranked set
sample.
"
( 1* )
Assume
0
kHown and
a Joof(x-~)d
'a' x
a~
_00
=
Joo -1.
a~1
_co
f(x-ll)dx.
0
Th en
12
1*
() - E [d Jl,n L*J 2
(m')n jJ L dlJ
and
n
m
I
Jl,n L* = terms not involving lJ +
I
Jl,n f(
X()
i=l r=l
+
n
m
I
I
(r-l)JI,n F(
i=l r=l
+
I I
x()
r:m
r:m
. - lJ
,1
)
0
. - lJ
,1
)
0
(m-r)JI,n[l - Flx(r:m),i - lJJ]
i=l r=l
(2.4)
0
Thus
n
=
m
I
I
i=l r=l
n
+
'm
L I
(r-l)
i=l r=l
n
+
I
m
I
(m-r)
i=l r=l
and
+
-l
02
_ -l
II
(r-l)
i r
L LCm-r)
o 2 1. r
f'
(U (r:m),l.)
f
cu
. )12}
_ r,
(r:m) ,i
{ F(U(r:m),i)
!(UCr:m),i)]
l
{f CU Cr : m) ,i)
I-FCU r:m ) ,1
C
:r
+
2
rfCUCr:m) ,i) 1 }.
1)-FCu
) .)]
Cr:m ,1
13
=
r
{rft (U(. r:m )) -/2 _ ~~_~.."...
f" (U(r . m)) }
-E. m E.
(i
r=l
- f(U(r:.m)) _
f(U(r:m))
<
ft (U(r:m)) }
F(U(r:m))
f (U(r:m))-/2+f t (U(r:m))}.
Jr
r (m-r)El[I-F(U )J I-F(U(r:m))
n
+
m
2
o
r=I
-
(2.5)
cr:m)
.
_ ft (u) 2
fll (u)
Lettlng h(u) - [~U)~ - feu) , we see that the first term of the
right hand side of (2.5) is
00
n
2
o
~
r
m-I
r-I
J h (u)m(r_I) feu) F (u) [1
L
- F (u)]
n
l'
du
r=I -00
00
n r
. ~ (r_I)F
m-I r-1
=2J
mh(u)f(u)
(u)[1
o
1'=1
L
- F(u)]
m-I'
du
_00
00
Now redefine h(u) =
[~i~~]2
f~i~~.
The second term of the right
hand side of (2.5) is
00
2n ~L
o r=l
m-I.
Jr
(r-l)h(u)m(r_l")~(u)F
(u) [1 - F(u)]
m-r
du
_00
00
=
1'-1
nm
2
o
J
m
h(u)f(u)
_00
00
= nm(m-l)
02
m-I 1'-1
r
(r-I)(r_I)F
(u) [1
1'=1
f h(u)f(u)F(u)du = nm~~-])
_00
F (u):I
m-r
du
2
E{f (U) - flCU)}
F(U)·
14
Similarly, we can show that the third term of the right hand side of
(2.5) is
2
nm(m-l)
a
{f (U)
E I-F(U)
2
+
'}
f (U) .
Then, writing feU) - f and F(U) - F, we have
=
N
a2
El ff' )
2
+
N(m-l)
a2
(
El
2
')
f
J
F(l-F)
f2
Since E F(l-F) ~ 0, we see that I(m)n(~)~IN(~)· This tells us tha~ in
any regular estimation case, an efficient unbiased estimator of
l.l
calculated from a ranked set sample has a smaller variance (by an order
of m) than an efficient unbiased estimator of
l.l
calculated from a random
sample.
(2*)
Assume
~
known.
By the
same method as that employed in (1*)
(see Appendix, p. 106 ), we have
*
_ nm {l-Uf ' (U)12
}
I(m)n(a) - a 2 E
f(U)J - 1
+
nm(m-l)
i
2 2
{U f (U)
}
E F(U) [l-F(U)]
if the p.d.f. of U has the properties
2
2
limuf(u) = lim uf(u) = lim u f(u) = lim u f(u) = 0 .
u-+oo
u+- oo
U-+OO
u+- oo
Here too, we have
Therefore, an efficient unbiased estimator of a based on a ranked set
samp Ie is "better" in the sense described in (l *) than its counterpart
15
from a random sample.
(3*)
Assume lJ and a unknown and the conditions of (2*) hold.
Then the
upper triangle of the matrix
f2
(m-l)E(F(l_F) )
2
f' 2U)+ (m-l)E(F(l_F))
Uf
E((r)
f'II)2
J ( ) ((Uf)2)
E[(~
- 1 + m-l E F(l-F)
Let (~,a) be a pair of jointly efficient unbiased estimators of (lJ,a)
calculated from a random sample, and let (~*,a*) be jointly efficient
unbiased estimators calculated from a ranked set sample in a regular
estimation situation.
Then
2
A
a
J_I_
var(lJ*) = N(m-l)
Im-l
a
2
var(a*) - N(m-l)
where
1
E[(~ U)2
f
f' 2
{m~l E(r-)
(Uf) 2}
IJ + E F(1-F) /A
f2
+ E F(l-F)}/A*
*
l'
16
and
lim N(m-l)var(a*)
=
rn-+oo
2
£2
f2
(Uf) 2
Uf 2
2
a E FCl-F) I(E F(l-F) E F(l-F) - [E F(l-F)] ).
But
lim N(m-l)var(0)
=
rn-+oo
lim N(m-l)var(a)
=
00
rn-+oo
Therefore, if m is large enough,
A
var(].1*)
and
A
::::;
var(].1)
::::;
var(a)
A
A
var(a*)
If the rando~ variable X is symmetric around ].1, then E(~')2U
2
Uf
E F(l-F) = 0, and
o
(
)
1 *-1
(m)~ ].1,a
-1
~ (].1,a)
= 0,
1
l/I*(a)J
= a2
rIll
N
0
(].1)
o
l/ICa)
].
Hence in that case, c0*,0*) has an ellipse of concentration '1hich lies
completely within that of C0,0) for any m; i.e.,
A
varc0*) : : ; varC].1)
and
A
: : ; varCa)
'v'm •
Now we will look at the difference in the appearance of the two
MLE's of the mean ].1 of a normal distribution.
This example is not
meant to suggest that the estimation procedure described here is an
appropriate one for this situation, but rather is included as an
illustration of the modification in the MLE that is caused by the
17
independence of the order statistics.
If the random variable is
known to have a normal distribution, one can obtain more information
about
~,
of course, by selecting the median from each sample of size
m rather than choosing each order statistic in turn.
In fact,
whenever the distribution is known, the estimator might be improved by
selecting a particular order statistic (or a particular set of order
statistics) instead of including n of each in the ranked set sample.
This altered form of sampling will be called modified ranked set
sampling, and we will see examples of its use later.
Let X "."X be iid random variables and Xi ~ N(~,02),
1
N
0
2
known.
/\
the same distribution as above.
equation
The MLE
a .R-n
L*
a~
with
=
~*
is found by solving the
0
.R-n L* as in (2.4), where now fez) = ¢(z), the normal p.d.f. and
F(z) =
the normal c.d.f.
~(z),
n
I
m
n
This equation reduces to
m
n
I r=l
I (r-1)g(Z( r.m,l
.) .)
. - i=l
I z
+
i=l r=l (r:m),l
=
where
Z
I
m
I
i=l r=l
(m-r)g'(Z( .) .)
r.m ,1
(2.6)
0
X
- ~
. = ~(r=....:..::m;:,:)~,:..::i:--_
(r:m),l
g (Z) =
0
¢(Z)
~
(Z)
¢(Z)
g' (Z)
= l-~(Z)
.
A solution for this equation may be found computationally [10].
A method for finding an approximate solution to (2.6) is adapted
from Tiku [22].
This solution is based on his approximation
g (z)
= ¢~(z)
(zl
~
~
rv
v,
+
Bz
(2.7)
18
where
= g (h) - g (R,)
B
h-R,
,
a = g(h) - hB ,
and (R"h) is an interval containing z.
Using this approximation, we
finu that·~* ~ ~*, where ~* is a solution of the equation
n
m
n
m
\'
\ (r-l)(a . + B Z
)
L LZ
. - L
L
rl
rl' (r·.m), l'
i=l r=l (r:m),1
i=l r=l
+
n
m
\
\'
i~l r~l
(m-r) (a'.
rl
+
B'
Z
) = 0
ri (r:m),i
where
a . + B .z(
).,
rl
rl r:m,1
g ' (Z
(r:m),i )
= g( Z
- (r:m),i )
=a'. +B'.Z
.
rl
rl (r:m),1
with a ., B ., a'., B'. as defined by (2.7).
rl
rl
r1
rl
n
m
n
L LX
il*
Thus
m
n
m
L[(l-r)a .+(m-r)a' .]+ I
I {[(l-r)B .+(m-r)B' .]}
i=l r=l
rl
rl i=l r=l
rl
rl
n
m
(2.8)
mn + I
I [(l-r)B .+(m-r)B'.]
i=l r=l
rl
rl
.+0
i=l r=l (r:m),l
=
L
The problem here is that the exact position of each Z( .) . is unknown,
r.m ,1
so that the interval (R, ., h .) which covers Z(
). must be estimated
rl
rl
r:m ,1
'fIr,i.
Let
~(r . m)'
0
2
(r:m) be the mean and variance of the r-th order
statistic from a unit normal distribution, let zp be such that
1 .
In lm X ( ) .•
<P(zp) = p, and let -X = mn 1=1 r=l r:m,1
Then for m fixed and n
large,
x ~ N (~,o
+ I o~r:m)
m n r=l
]
Thus for some small a, (R, .,h .) is likely to cover Z(
). =
rl r1
r:m ,I
(X(
. r:m ),1
~)/0
when
19
x
J/,
ri
=
(r:m),i
(J
- X
Z
l-a/2
~
-1-
m2n
m 2
r~l (J (r:m)
and
X
h
ri
Then
=
(r:m),l.(J
X
- 21
+ zl_a/2
II!
L (J 2(r:m)
m n r=l
Srl.,arl. may be computed by substituting J/,rl., hrl. into (2.7), and
~
~*
may subsequently be calculated from (2.8).
Alternatively, if m is large (a condition which is not as likely
to occur because of practical restrictions, as we have indicated before,
~
but which produces a nice form for
J/,ri
=
h . =
n
~*),
~(r:m)
-KG"
~
+KG"
(r:m)
then we choose
(r:m)
This interval is likely to cover Z( r:m )"
,1
enough.
K
(r:m)
\fi,
if
a constant .
K
is chosen large
Unfortunately, however, unless m is sufficiently large
(causing (J(r:m) to be small), the interval will be too wide to allow
g(z) to be approximated over it accurately by a linear function.
th~s
choice of J/, rl.,h rl., we have J/, rl.
=
-he m-r+ 1)'1 and hrl.
= -J/,(
Therefore
S'
,
(m-r+ 1 :m) , i
=
g(-J/,(m-r+l:m)) - g(-h(m-r+l:m))
J/,
+ h
- (m-r+l:m)
(m-r+l:m)
and
\fi ,
For
m-r+ 1)'1 \fi.
20
since g' (z)
Now suppose that m is even, m = 2k.
g(-z).
m
I [(l-r)arl.
r=l
Then
k
m
+ (m-r)a'.] = I (l-r)a . + I (l-r)a.
rl
r=l
rl
r=k+l
rl
m
(m-r)a'. + I (m-r)a'.
r=l
rl
r=k+l
rl
k
+
k
I
r=l
I
k
k
(l-r)a . + L (r-m)a
I ' + L (m-r)a
.
rl
r=l
m-r+,l
r=l
m-r+l,l
k
+
Similarly,
L
r=l
(r-l)a . = 0 .
r1
,m 1[(I-r)Brl. + (m-r)B'r1.]= O.
Therefore, from (2.8),
Lr =
e(2.9)
where
-
2KO(
KO
(r:m)
)
r:m )
,...,
If m is odd, m = 2k+l, then
2.2
~*
is still given by (2.9).
Best Linear Unbiased Estimation
Let X be a random variable having c.d.f.
(Note again here that
II
X-ll '\
F(-o-:; where F is known.
is not necessarily the mean; a is not neces-
sarily the standard deviation.)
The random variable U =
-aX-~
has a
as U( r:m )'
Parameter-free distribution, and (X( r:m )-~)/a is distributed
.
the r-th order statistic from a sample of size m of a distribution
21
having c.d.f. F(u).
The means, variances, and covariances of the set
of random variables U(l :m)'·· .,U Cm:m)
can be calculated and will be
denoted here as:
= a Cr : m)
E UCr : m)
Var U(r:m)
= a~r:m)
Cov CU ( r:m ) , U( s:m )) = a Crs : m)'
=~
EX
Cr:m)
+
a
(r:m)
a
2 2
Var X
= a a
(r:m)
(r :m)
Cov (XCr:m)'X(s:m))
=a
2
a(rs:m)
Lloyd [15] applied the generalized least squares theorem to these
ordered random variables and developed the best linear unbiased estimators (BLUE's) of
~
and a.
2
If E X = Ae and Var X = w V
Generalized Least Squares Theorem.
(A and V known), then the least squares estimate of e is
e=
CA'V-IA)-IA'V- I!
Var § = w2 CAIV- I A)-1 .
and
.1\
e is also the BLUE.
In Lloyd's application
a(l: N)
A
=
a(2:N)
I
th~
upper triangle of
=
[1-
22
2
O(l:N)
0(12:N)
O(lN:N)
. 2
v
0(2:N)
=
0(2N:N)
2
O(N:N)
2
w2 =
, and -e = [
°
jJ
°
]
.
A
~
A
Thus the estimator e =
has variance-covariance matrix
°
var
For the ranked set sampling case, we again choose mn = N samples
of size m each, and from each sample, we select one order statistic
for quantification.
The quantified elements are therefore inde-
pendent and
2
2
2 * h
D. (2
var X = 0 V w ere V;nxm~ lag 0(1:m)'0(2:m)""'0(m:m)"'"
2
2
O(l:m)" "'O(m:m))
Then the BLUE of e = [
jJ
°
]
is
and
m a
2
2
Var e* = - n~*
A
°
r=l 0
m a
-2
r=l
where
2
(r:m)
2
(r:m)
(r:m)
2
°(r:m)
m a
-I
r=l
(r:m)
2
(r:m)
°
m a2
(r:m)
I
r=l
2
°(r:m)
.;.
23
If the only parameter to be estimated is
the form
A
Var (~*)
, and the c.d.f. is of
then one can show that
F(x-~),
and
~
=
a
2
m
Il'v*- 1
1 =2
a I(n I
2
l/O'( . ))
r=l
r.m
If the only parameter to be estimated is a, and the c.d.f. is of the
form F(x/O'),
then one can show that
2
A
Var(O')
= a IC!:-'V
and
-1
2
Var (0*) = a IC!:-'V*
C!:-
-1
2
m a
C!:- =
0'2 I (n
I
~r :m))
r=l cr
(r:'ll)
If X is a random variable with a distribution symmetrical around
then
U
~,
has a distribution symmetrical around the origin and
a(r:m) = -a (m-r+l :m)
a
2
2
(r:m)
= a (m-r+l :m)
Then
Var (~)
Var (0)
A
Var(~*)
2
2
-1
a IC!:-'V C!:-,
A
= 0;
Cov(~,O')
A
=
m
= a I(n I
1
?
)
Var(O'*)
r=l O'(r:m)
Cov (0 *,
a*)
2
m a (r :m))
a I (n I
2
r=l a
(r:m)
2
A
=
= 0 .
(2.10)
Blom [2] and Jung [12] have suggested alternative linear estimators
of ~ and a which aremore easily computed than
0 and cr,
since they don't
24
require inversion of an NxN matrix.
(Blom calls his estimator the
unbiased nearly best linear estimator.)
Both have shown that their
estimators are asymptotically efficient when certain restrictions on
the p.d.f. of U are fulfilled.
variance can be no
Therefore, the BLUE of J1 or cr, whose
than that of the nearly best estimator,
~arger
must also be asymptotically efficient.
Ogawa [18] used the asymptotic distribution of sample quantiles
to find large sample BLUE's of J1 and cr based on selected quantiles from
a sample of size N.
Their asymptotic distribution is given as follows
by Mosteller [171 .
Theorem.
For k given real numbers
let the A.-quantile of the population be x.; i.e.
1
e-
1
X.
1
J g(t)dt
i=l, ... ,k.
= Ai
-00
Further, assume that the p.d.f. of the population, g(x), is differentiable in the neighborhoods of x = x.1
and that
g.
1
= g(x.)
1
# 0 Vi.
Then the joint distribution of the k order statistics,
where -n.1 = [NA.]
1
+ I,
i= I, ... , k,
tends to a k-dimensional normal distribution with means xl"" ,xk and
variance-covariance matrix
.!. \(l-Aj)))
[[N g.g.
1
J
as N tends to infinity.
Note
here that the joint distribution
of the k order statistics
25
x(m
+ 1'
:m)' ... , X(m :m) from a ranked set sample where m.1 = [rnA.]
1
l
k
i=l, ... ,k, A.'s
as given above, tends to a k-dimensional normal dis1
tribution with means xl' ... ,x ' and variance-covariance matrix
k
Diag
A. (I-A.)
2 1 )
(1
mg
i
The estimators of lJ or
(J
for a particular distribution can be
improved by an optimal choice of the spacings
k.
The optimal spacing for the vector
generalized variance of
e.
e
Al, ... ,A k , for a given
is that which minimizes the
Ogawa has found these optimal spacings for
some small values of k for estimating either lJ or
(J
(but not both
simultaneously except for k = 2) when they are parameters of the normal
distribution, and for
--
(J
from the one-parameter exponential distribution.
Can ranked set sampling be employed to improve unbiased linear
estimators of lJ and
(J
over those from random
amples?
Intuition might
suggest that this would be true for estimators of lJ, since for most
distributions, the BLUE of lJ is of the form
N
lJ
= I
r=l
x
j/,r (r :N)
where
j/,
r
> 0,
r=l, ... ,N.
In such distributions, . if n = 1, then m = N, and
A
Var (lJ) =
Var X(r:N)
Var X
(r:N)
since Cov(X(r:N)'X(s:N) ~ 0
when X(r:N)'X(s:N) are order
statistics from a single sample
N
2
~ L\' c Var X(r:N)
r=l r
where c1, ... ,c N are the coefficients of the BLUE from a ranked
set sample
A
= Var(lJ*)
26
This shows that ranked set sampling does improve the BLUE of
all distributions of this type, when n
=
1.
~
for
However, there are dis-
tributions (such as the U-shaped distribution whose density is given
on p. 43) for which the coefficients of the BLUE in a random sample are
not all positive.
Similarly, intuition might suggest that ranked set sampling would decrease the precision of the estimator of a, since the BLUE's of a tend
to be of the form
A
a
[N/2]
= I 5/,~(x(N_r+I:N)
r=l
with 5/,' > 0
r
~r.
A
Var(a)
2[
- x(r:N))'
Then
[N/2] '2
I 5/,r Var X(r:N) + I I 5/,'5/,' Cov(X(r . N)'X(s . ·N))]
r=l
r<s<[N/2] r s
+ 4
I I5/,' 5/,' Cov (X ( . N)' X(N- s+ 1 .. N)) ]
r<s< [N/2] r s r .
and when n = 1, then m = Nand
N
Var(a*) = 2 I c,2 Var x(r:N)
r
r=l
A
where
ci, ... ,c N are
the coefficients
of the BLUE from a ranked set sample.
Var(a*) lacks the "benefit" of the negative terms which appear in the
expression above for Var(0).
In fact, one can show that intuition is correct in both cases for
n = 1 and m = 2 or 3, at least for symmetric distributions.
cases one can show that
Var (0) > Var (~*)
Var(a) < Var(a*)
In these
e-
27
Despite these examples, however, we will be able to show that for
certain types of distributions, ranked set sampling always improves
the BLUE of
~
and of a, if m is large enough.
Ranked s·et sampling as we have discussed i t so far has involved
choosing for quantification n each of XCl:m)"",XCm:m)' from independent samples of size m, until mn = N observations are quantified.
But note that calculation of a BLUE requires knowledge of the distriX-~
bution of U = -0-'
Having this information may allow us to make a
better choice of which order statistic to measure from each of the
mn samples in order to minimize the generalized variance of the BLUE
vector.
We mentioned this fact in connection with the calculation of
MLE's.
First let us suppose that X is a random variable having a l-para-
cr
meter distribution with c.d.f.
is known.
Then the BLUE of
~
the form FCx-p); Le., we assume a
based on n repetitions of a set of
independent order statistics Xc
. )""'X
.) Ca sampling scheme
rl·m
Crm·m
which we will call modified ranked set sampling) has variance
'
C 2
= Dlag
a
. ), ... ,a 2C , ) " " J
Crl·m
rm·m
where
m
2
=
2
a I (n i~l lla Cr :m) J
i
2
This variance is minimized by choosing every r i so that a Cr :m) is
i
But we know that
minimized over the set {r, : i=l, ... ,m}
1
rna
2
C[Am]+l :rn)
+ A(I-A)
f
2
euA)
as
rn
+00
28
f
where u A is defined by
u
So by minimizing A(l-A)
Af(t)dt = A.
f
-00
2
(u
A)
with respect to A, we can find the quantile to choose from each
sample of size m which will lead, for large m, to the linear estimator with the smallest variance.
AD,
II
Let this optimal A be A* and let
ll'
be the BLUE based on the modified ranked set sample. Then
2
2
AD,
m n Var(l1 )
o A*(l-A*)
1.
__~u__~p~
+
f
2
as m +
00
•
(u )
A
For distributions for which the Cramer-Rao inequality holds, we know
Q()
IT••1
Var(ll)
A
J ff_ feu)
(U)] 2 f(u)du.
>_ 0 2
I
-00
Therefore
AD,
Var (ll (m)n)
+
A
as m
0
+
00
Var(llmn )
for these distributions.
In fact, any sample A-quantile (0 < A < 1)
chosen from each sample would lead to the same conclusion, since
Var X([mA]+l:m) =
O(~)
as m +
00.
A
This fact allows us to show that the BLUE 11* from a conventional
ranked set sample has smaller variance than that from a random sample
for m large enough.
(We note here again, however, that there would,
in practice, be no reason for using the conventional ranked set
sample over the modified ranked set sample in any situation in which
a BLUE could be calculated.)
1
A
Var (11*)
-=--- )
0
2
(r:m)
2 ( [(1- (Oj2))m]
2
o / n
I
1/0
r=[(o/2)m)]
(r:m)
]
for 0 > 0
e-
29
where
a2
2
= max (a (j :m)
(k :m)
But
2
lim su p {ma (o. ): j
m-+oo
[~m], ... ,[(l- ~)m]}
=
J .m
1..* (1-1..*)
=
2
f (u
A
)
*
jf f(x) > 0
maximizes
l o/ 2 ' where 1..* is that value which
2
hl(A) = (A(1-A))/(f (u )) in the interval 0/2 < A < 1-0/2.
A
for
U
o/ 2
< u < u -
If there are finitely many values of u for which feu)
uo/ 2 < u < u l _o/ 2 ' then label these points
=0
wpen
zl" ",Zt ' and let
Z •
Y , ... , \
be such that
l
I J.f(u)du
Y , j=l.
j
=
0
_00
[ (Yl- 3t)m]
\'1
Var(~*) s (a In)
2
{
r=[3 m]
a(r:m)
•
A
2
oL
0
••
,
t.
Then
1
+
a2
(r:m)
[(1- §..)m]
3
+ •.• +
I
21
}
a
~)m]
(r:m)
3t
r=[(y +
t
Let 1..* be that value of A which maximizes hI(A) over the region
+ -
o<
3t
Then for either case,
2
lim m n
A
Var(~*)~
m-+oo
A
=>
Var(~*)
Var(~)
+
0
as m +
00
A<
i)
3
30
if X has a distribution for which the Cramer-Rao inequality holds.
Notice that this proves that for m large enough,
where the upper triangle of the symmetric matrix
r
02
(1 :m)
v=
o
o
2
(12 :m)
2
0
(1m:m)
1
2
0(2m:m)
(2 :m)
0
L
2
(m:m)
Now suppose X has a one-parameter distribution with c.d.f.
of the form
x),
F ((j
Here
'
aga1n
we wou ld b ene f't
1 by
'
Ch
ooslng
a mo d'1-
fied ranked set sample on which to base the BLUE rather than a conventional ranked set sample. In this case, the optimal quantile to
measure from each sample for large m is X([mA]+l:m)' where A
minimizes
Let the optimal A be A;.
"'/1
o
=
mn
\
,L
1=1
and
Then we have
Then
X
/nma
([mA*]+l:m)
([mA*]+l:m)
0
0
31
ill
2
A!J.
as
n Var(a )
ill -r
00
,
and for distributions for which the Cramer-Rao inequality holds,
00
;: : i
mn
fr uf' (u)]2
Jr
_ 00
tL
-l}f(U)dU .
J
feu)
_
Therefore
"'!J.
Var(a(m)n)
-r
1\
as m -r
0
00
•
Var(a )
mn
By an argument identical to that used for showing that
< Var(~)
"
Var(~*)
for large m, we can show
Var(a*)
----"--'- -r
A
as m -r
0
00
Var(a)
and thus
a'V*-l a ;::: a'V-la for m sufficiently large.
Now suppose the c.d.f. of X is of the form
both unknown.
F(X~~),
~ and a
Here again one can find, in certain situations,
better linear estimators of
~
" and a*
" by choosing
and a than ~*
a modified ranked set sample.
However,
~
and a cannot both be
estimated with linear estimators if the same quantile is chosen
from each of the mn samples, since in that case
r1
a
I1
a
li
a
I
A =
(r:m)
(r:m)
(r:m)
1
has rank 1.
J
Therefore, at least two different quanti1es must be included among
those order statistics which make up the modified ranked set sample.
32
Suppose we include exactly 2 different quantiles in our
sample; i.e., suppose we quantify rnn/2 r-th and mn/2 s-th order
statistics, each from a sample of size m, with r>s; then the gen"'/1
eralized variance of 8
is
.
"'/1
I Var ~
I!,;
2
= (0 2 /mJY [[ 0 2 1
(
l2
O(r:m)
i (s:m)
+
~~s2 :ml ]
0
(s :m)
.] 21~
)s :m) .
(r:m)
2.
20 0
J fa(r:m) +
1
+
(r:m)
_ l)r:m)
2
(s :m)
0
(r:m) (s :m)
mn(a
-a
)
(r:m) (s :m)
If the r-th and s-th order statistics are the
o
< Al < 1, 0 <
1..
2
<
1..
1
- and A -quantiles,
2
e/1l~ is minimized
1, then for large m, IVar
by minimizing
over the rectangle
0 < Al < 1, 0 <
1..
2
From this expression we
< 1.
can see that
"'/1
IVar ~
!,;
/2
1
= 0(-2-)
as
m -+
00
mn
regardless of which pair of quantiles are chosen.
We know from
(2.3) that
"'/1 ~
IVar -8
1
I = 0(---)
mn
as
in any regular estimation situation.
"'/1
IVar 8
00
Consequently we have
!,;2
1
'" !,;
/Var ~I
m -+
2
-+ O. as
m -+
00
•
33
Suppose X has a distribution symmetric around
~
and we wish
optimal pair of quantiles X
X
([mAl ]+1 :m)' ([mA ]+1:m) ,
2
where Al = 1~A2' for estimating ~ and a. If the quantified sample
to find
th~
includes
mn/2 each of the Al - and A -quantile, then
2
AI:" ~
/Var ~
since
/\1:"
cov(~
2 2
I
= a a([mAl]+l:m)1 mna ([m\]+l:m)
/\1:"
2
. a
, a ) = 0, a(r:m) = -a (m-r+l :m)'
(r:m)
= a 2(m-r+l:m)
These optimal quantiles, A* and I-A*, are found for large m by
minimizing
for 0 < A < 1.
--
For symmetric distributions, the BLUE of
e
from a conventional
ranked set sample has generalized variance Urom (2.10))
/Var
~*I~1
1 [rl
-l2
1l
2
= a n
~ a
2
m
~
- r-l
I/a~r:m) ) ( m~ a (~:m)
r-I a
r [m/2]
n
1
~
2
r= [( 012) m] a(r:m)
)J ~
(r:m)
J(
. 2
2
[m/2]
I
a(r:m) )1~
2
r=[(0/2)m] a (r:m)
JJ'
(2.11)
By using the same argument as for the I-parameter case, we can
2
show that each of the sums of (2.11) are O(m ) as m + 00, which leads
us directly to the conclusion that
A
k
IVar ~*12
/Var
~ I~
+
0
as m +
00
•
34
Suppose X ~
N(~,cr).
Then hl(A) is minimized by
A~
= .5.
This indicates that the best linear unbiased estimator of
~,
for
' based on a modified ranked set sample is obtained by
cr 2 k nown,
ctoosing the sample median from each sample.
unknown, but
~
If
~
and cr are both
is the only parameter of interest, it will still be
estimated most accurately by this linear estimator.
can show that h 2 (A) is minimized by
A~
=
.942.
So if
Similarly, one
~
is known,
cr can be best estimated (for large m) with a linear estimator by
choosing the ([.942m]+1)-st order statistic from each sample of size
m.
If
~
and cr are both unknown, but cr is the only parameter of
interest, it can be estimated most accurately by choosing
m~([.942m]+1)-st and m~(m-([.942m]+1))-st order statistics.
~
If neither
nor cr is known, then the optimal symmetric pair of quantiles for
estimating both
~
and cr from linear estimators are the 87.3 and the
13.7 sample percentiles.
So far, we have established that modified ranked set BLUE's
e = [~]
can be found whose generalized variance is smaller than
cr
that of the BLUE based on a random sample, for any distribution for
of
which the Cramer-Rao inequality holds.
Furthermore, we have shown
for this same condition that a BLUE based on the conventional ranked
set sample has smaller variance than that of the random sample when
its distribution is one-parameter or symmetric.
Notice that if
we could show that the BLUE based on a ranked set sample is efficient, we would have proved this fact in greater generality.
Un-
fortunately, I have not been able to prove this, although I believe
that it is true.
Recall that Blom and Jung have shown that BLUE's
based on random samples are efficient, for distributions satisfying
e-
35
certain (fairly restrictive, in Blom's case) conditions.
They did
this, as we have mentioned, by displaying another asymptotically
unbiased linear function of the order statistics which was
efficieTlt.
([ 2],
To arrive at his results, Blom proved and used a theorem
p. 53) which states that, for distributions for which
-1
F(u) = G(p) satisfies certain conditions,
EX
and
r
= G(-) + R
(r:m)
IRr I
<
m+l
M*
m
r
where M* does not depend on r
or m ,
if X(r:m) belongs to a sequence of random variables of the form
{X
)} where n <n 1 and r,,/n + c as V + 00, for all but
(rv:n
V v+
v
V
.
v
possibly a finite number of points of the interval [0,1] (and
another condition for the remaining points)
A second theorem
, ([ 3 ], p. 55) states a similar result for covariance:
Cov(X(r . m),X(s . m)) =
where
IRrs I
M*
<2'
1
m+2
r(m-r+l) G'(--.!..-)G,(2-) + R
(m+l) 2
m+l
m+l
rs
(rS;s)
and M* does not depend on r,s, or m.
m
It may be that a proof of my conjecture could be produced by
mimicking Blom's method.
However, a more direct approach might
be possible, although perhaps not without the introduction of
additional restrictions, since (for example) we see from Blom's
theorem that
r
V.'ar X( r:m ) =
r
m+l(l - m+l)
mf 2 (-!-)
m+l
A
Var(p*) =
+
I
2
n 1 a
/ (r:m)
r
and
00
= nm
2
J
_00
f2 (u)
F(u)[l-F(u)] du + Oem).
36
Unfortunately, Blom's theorem holds only for quantiles, and does
not assure us that his condition on the variance of X(
holds
r:m )
for all r, even in the range [om] < r < [ (l-o)m], o > O.
2.3
s2_ type Estimators of a 2
So far in this chapter, several methods for estimating scale
parameters have been discussed.
But for each of the methods, know-
ledge of the form of the distribution is necessary, as well as the
assumption that perfect ordering is present.
Dell and Clutter's
paper dealt with an estimator of the mean, X, which was improved
(or at least not hurt) by the process of ranked set sampling, regardless of the underlying distribution or of the presence of ranking
errors.
Here our purpose is to determine how ranked set sampling
affects
sN = N-I Li=l (Xi-X)
2
1
\N
- 2
as an estimator of the variance a
2
(when it exists) of any distribution.
Let X[l:m]'" "X[m:m] be a set of judgment ordered random variables, and let
EX
[r:m]
T[r:m] = ~[r:m] - ~.
=
~
2
Var X
= a
[r:m]'
[r:m]
[r:m]'
Then
m
\ E(X r
]_a)k = mE(X-a)k
L
I. r : m
r=l
As particular cases of this relationship, we see that
m
L T[ r.m
. ]
r= 1
2
m
L a[r:m]
= 0
r=l
(2.12)
m 2
L T [r:m]
r=l
2
= rna
A natural variance estimator calculated from the ranked set sample
of size mn = N,
*2
X[l:m],l""'X[m:m],n' 1s
s[m]n
1
mn-
-1
n
m
(X[ . ]. .
. L1 Ll·r:m,l
1=
r=
A
- ~[m]n)
2
e
37
Then
n
E s*i
[m]n
= mn-l
(~2
A2)
r=l EX[r:m] - mE~[ m] n.
L
m
mn
-Z- ((
L T[r:m]
m(mn-l)
s*~m]n
r=l
2
is a biased estimator of o.
Es *2[m]n
~
However,
LO~r:m] <£
2
L.. T [r:m] = mn-l - m(mn-l) - mn-l
m(mn-l) r=l
02 = ~_l,,--:-~
2
0
*2
.
So s[m]n
IS asymptotically unbiased and the bias is no larger than
that which results from estimating
0
2
N-I
2
.
bYT sN ' though i t cannot
be removed in the same way.
Ranked set sampling does not always im rove the precision of
.
2
the estImator
SN; i. e. , it is not always true that
*2
*2
MSE s[m]n
= Var s[m]n
+
(Es 1;]n
)
- o22
~
as was the case for Dell and Clutter's estimator
Var sN2
However,
of~.
if mn = N is sufficiently large, we will show that
MSE
. . S*2
[m]n <- Var sN2
regardless of the underlying distribution (as long as the appropriate
number of moments exist) or of the presence of ranking errors.
To simplify notation in the following calculations, we write
A
x[r:m],i ~ Xri ' ~[r:m] ~ ~r' ~[m]n ~ ~,
T
[r:m] -= Tr'
Then
0
2
[r:m]
2
+ ~ )]
m 2
I
so
~ 2
L O[r:m])
r=l
38
(mn-l)s*
2
n
=
=
m
A
I I [(Xri-]Jr)
i=l r=l
(]J-]J)
+
2
+ I
Ii rL (Xr1.-]J)
I
r
r
i
- 2
? rI (Xri-]Jr) (~-]J)
+
I I
r
-]:1)]
A
(]J-]J)
2
I I
i r
1
- 2
(]J
2
+
2
Ii rI
(]Jr -]J)
2
(Xri~]Jr)(]Jr-]J)
A
(]J-]J)(]Jr-]J)
i r
Let
T = I I (X ri -]Jr)2
i r
+
mn(~-]J)2 - 2(v-]J)I
+2
I I.
r
1
T
r
y2 I y )2
= .
ri
mn.
ri
1 r
1 r
II
I
(Xri-]Jr)
i r
-leI
(X r1.-]Jr )
2 I(I T Y .)
. r r1
r 1
m
m m
y2. - - 2 I I I
.y . - - 2 I I I y .y .
= (mn-l)J..
mn . .
r1
mn
mn r<s i=l j=l n SJ
1 r
r""l i<j n TJ
+
I
Iy
where Y . = X . - ]J.
r1
r1
r
Ii rI
2
+
Then
T
Y • ,
r r1
2
E Y . = 0, Var Y . = 0 , and
r1
r1
r
cov(Y .,Y .) = 0 if r F s or i ~ j, and var(mn-l)s*2 = var T
r1 sJ
2
ET _ (ET)2 .
ET
m
= n(mn-l) I (l
mn 1=
. 1 r
+
4 I I T 2y 2 .
i r r r1
+
4(mn-l)L I T y 3 .
mn.1 r r r1
+
terms having
expectation 0 }
39
where ].1kr
=
k
E(X r -].1 r ) .
Var(mn-l)s*2
So
= n [(mn-l)2\
-ron- L
r
+
].14r
+
4\L Tr2ar2
r
4n 2 L L a 2a 2
(mn) r<s r s
+
+
4(mn-l)\
mn L Tr ].13r
r
2(n-l)-(mn-l)
(mn) 2
2
L a4 ]
r
.
r
(2.13)
Note as a check that if the ranked set sample were indeed a random
sample (i.e., judgment ordering no better than random), then
a
2
r
=
a
2
= ].14 = E(X-].1) 4 ,
'
Var(mn-l)s*
2
m.n-l)2
= n [(-m-nm].1 4
=
(mn-l) 2
+
yielding
4n m(m-l) a4+ 2(n-l)-(mn-l)2
4]
---=-2 2
2
rna
(~)
(~)
(~-3) (mn-l) a4
mn~4 + -
mn
= Var ( N -1) sN2
(2.14)
Var T will be rewritten in a form which contains only expectations of powers of (X r -].1) and Tr by noting:
(a)
(b)
40
(c)
(d)
(2. IS)
Thus from (2.13) we see that
+ (mn-3)(mn+l) L T4
(mn) 2
r r
+ _4~(m_n_--::-l",-)
2
(mn)
Then
MSE (s *2 ) = Var (2)
s*
+
(Es*2 _ a2 )2
+ (mn-3)(mn+l) L T4 +
4
2 2
(mn-l)2 m2n
r r
(mn-l) m n
2
= -l
mn
E(X_1I)4
t-'
L Tr E(Xr _11)3
r
2
+ -2(mn) +6 L T 2E(X _ )2 _ (mn) -2mn+3 L a4
2 2
r
r 11
2 2
r
(mn-l) m n r
(mn-l) m n r
(2.16)
Lemma.
Let X[I:m]' ... 'X[m:m]
be a ranked set sample of judg-
ment order statistics from a continuous distribution whose fourth
moment exists.
(a)
m
Then:
2
2
L T [ r.m
• ] E(X [ • ] -11)
= 0 (m)
r.m
r= 1
(b)
m 4
\' T
- Oem)
L
[r:m]r=1
and
m 4
\' a [r:m]-- Oem)
L
r=l
41
m 2
(c)
( \ T
I...
r=l
[r :m]
)
2
=
m 2
2
2
and ( I o[ .]) = Oem )
r=l r.m
O(m 2)
m
L T[ r.m
. ]E(X[ r.m
. ]-1l)3
(d)
= Oem)
r= 1
Proof:
[E(X _1l)2]2 ~ E(X ~1l)4
(a)
r
r
o ~ Tr2 = E(Xr _1l)2
-
\
r
0
and
2 ~. E(X _1l)2
r
r
22·.
r
22
=> I... E(X r -ll) Tr ~ HE(X r -ll)]
(b)
\ T4
r
~ I...
r
r
L 0:
r
(d)
\ E(X -ll) 2
4
Tr ~2
mE(X-ll)
r
I...
\
r
~ I... E(X -ll)
r
4
= mE(X-ll)
and
~ L[E(Xr -ll)2]2 ~ mE(X-ll)4
r
I T E(X -ll)3 ~ (I T:)1/4{I[E(X _ll)3]4/3}3/4
r
r
r r
r
r
~mE(X-ll)
.
Theorem.
by Holder's inequality
4 3/4
4
4 \
[I... E(X -ll)]
= mE(X-ll)
r
r
Let X[l:m],l""'X[m:m],n
= Oem)
be a ranked set sample
of judgment order statistics from a continuous distribution whose
fourth moment exists.
Proof:
4
Then
a N* s.t. for N = mn
From the lemma and (2.16)
> N*,
42
2
1
·41
MSE s·" = mnE(X-jJ)
+ -2-
L T r4
mn r
1
4
=-=£(X-jJ)
mn
1 L T4
2
r
mn r
+ -
+
2 L T 2E(X -jJ) 2 - -1- L[E(X -jJ) 222
-T ]
2
r
r
2
r
r
mn r
mn r
- -
D(~)
(mn)
1
=-E(X-jJ)
4
mn
1
mn
= -=-E (X-ll)
- - 21 \'L[E(X r -jJ) 2 ] 2
mn r
+ D(
1 2)
(mn)
4
and from (2.14)
Var
s~
=
m~
E(X_jJ)4 -
m~ a4
+ D(
1 2)
(mn)
But
rna
2
.
~>
2 4
ma
=
so
Table 2.1 compares MSE
*2
s(m)l
and var 52
m
for several dis-
tributions to see how large m must be before MSE S(;)n s var
when n
= 1.
s~
44
e
Table 2.3.1 (continued)
Distribution
m
-------*(5)
Gamma
r=3 [6]
2
f(x) = (x e-x) If (3)
=
(6)
o
otherwise
Exponential
f(x) = e
=
* (7)
x~O
-x
x~O
o otherwise
S[~]l
MSE
2
var s
m
2
37.11
27.00
3
18.28
15.00
4
11. 55
10.50
5
8.18
8.10
6
6.19
6.60
7
4.89
5.57
2
6.38
5.00
3
3.56
3.00
4
2.39
2.17
5
1. 76
1. 70
6
1. 37
1.40
7
1.10
1.19
Lognormal [7]
2
1337.4
1253.7
f(x) = _1_ exp{-~ [.R-n x]2}
xl2TI
x~O
3
873.1
828.3
4
645.2
621. 0
5
510.0
494.9
6
421.0
412.1
7
357.7
353.0
8
310.6
308.8
9
270.7
274.4
10
245.4
246.9
=
o otherwise
'"
+
+
e·
+
-
~
*
References for tables of moments of order statistics needed for
performing these calculations are indicated.
e
45
If n
1, an
>
unbiased "s2_ type" estimator can be found.
~2
Consider the estimator s , calculated from a ranked set sample:
~2
s
1
= mn
n
m
[I
I (x[r:m] ' i
i=1 r=1
+
n
1
(n-l)m
2
m
r (x[r:m] ' i
r=1
I
i=1
where
1
A
~
A
~[m]n)
-
A
-
~[r:m])
2
]
n
=-lx
-.
[r:m]
n i=1 [r:m],l
Then
1
~2
Es
[m]n
= -[(mn-l)a
mn
2
1
-;
m 2
\' T
L
[r:m]
r=1
+
1
(n-l)m
m
2
\' ( n - l ) a l
[r:m]
L
r=1
= a2
and
~2
Var s
1
1
__ A
222
(mn) (n-I) m
[m]n
2
1
B
(mn)2 (n-l)m
where
A
2
A = Var[l I (X rl. - ~[r:m]) ]
i r
A
2
A
2
(X . - ~rr:m]) ]
(Xrl. - ~ [m]n) , I
B = cov[l
rl
i r
i r
and
r
r
Using (2.14), one can see that
2
m
~
A
2
(n-l) \'
A = L.Var r L (X .-~[ . ]) ] = n
L~4r r=l
i=1 rl
r.m
r
(n~3)(n-l)
n
Ia4
r
r
Therefore
1
1
1
\'
(n-3)
Ia4
--2
2 2 A = --22- L~4r 2
2 . r
(mn) m n r
(mn) (n-I)nm r·
(mn) (n-I) m .
1
= O( --3) by (2.15 a) and the lemma.
(mn)
By a method
be obtained.
j
dentical to that used for fi nding VaT (JIlIl-!") 5*2, B can
46
1
--2
(r.m)
1
(n-l)m
+
Sn
(mn-l)
\
-mn+3
\a4
3 2
L~4r +
3 2
L r
(mn) m (n-l) r
(mn) m (n-l) r
B =
Var ~2
=
2 2
2
~3
(mn) (n-l)m r r r
IT
Var s*2 + O(1/[(mn)2]).
= 0(
1
3
(mn)
)
by (2.15 a and b) and
the lemma.
Therefore we have shown that for
mn sufficiently large,
Var ~2
s[m]n
2.4
Judgment Ordering with a Concomitant Variable
In this section, we discuss the use of a concomitant variable
to judgment order the X's.
Suppose the observations of X cannot be
easily ordered, out there is a related variable Y which is readily
observable and which can be easily ordered.
We use the related
variable Y to acquire a ranked set sample of X's.
Again, mn samples
of size m are chosen; in the first sample of size m, the X associated with the smallest Y is measured; in the second sample, the X
associated with the second smallest Y is measured, etc.
As before,
this cycle is repeated m times until mn X's have been quantified.
This procedure is similar in theory to stratified sampling.
That is, the "extra information" available from the Y variable is
utilized to divide the population' into m homogeneous strata which are
as different as possible from one another.
Obviously, the benefit derived by using this method should depend on how much information Y yields about X.
pendent, the estimators
precision to
X and
s2.
If X and Yare inde-
0[m]n and s1;]n should be equivalent in
If
Ipl = (Ia xy /)/(a xay ) = 1,
s1;]n should be equivalent in precision to
~(m)n
A
then ~[m]n and
and Sc;)n' esti-
mators which are calculated from a ranked set sample of actual order
statistics of X.
47
Let X[l..m ] , 1'·· .,X[ m.m
. ] ,n
\'"
be a ranked set sample selected
on the basis of an ordered concomitant variable Y.
Furthermore,
X-]J
assume that the regression of X on Y is linear, and that _,_x
a·
x
Y-]J
and ~
both have c.d.f. F.
a
CThese properties are characteris-
y
tic, for example, of both the bivariate normal and bivariate Pareto
distributions.)
Then
ECX IY) = ]Jx
pa
+ -
x
a
y
CY-]J )
y
and
2
2
VarCXIY) = ax (l-p )
Thus
pa
+ ~CE
E X
= E[ECX/Y : m))] = ]Jx
[r:m]
Cr
= ]J x
where
a Cr : m) = E UCr : m)'
y
YCr:m) - ]JyJ
,
pa a
x Cr:m)
+
Y-]J
=--Y
a
U
a
has c.J.f. F, and
y
Var X[r:m] = E[VarCX/YCr:m))]
Var[ECxIY Cr : m))]
+
2 2
P ax
= axCl-p ) + ~ Var Y
2
2
Cr : m)
y
=
2
2
a Cl-p )
+
x
2
where aU
222
p axau
Cr:m)
Then
= Var UCr : m)
Cr:m)
m
2
1
2
-'[rna - L CEX : ) - ]Jx) ]
2
x
Var ]JCm)n
Cr m
r=l
mn
=
m
1
' 2
2
Var ]J[m]n
-[rna
L
CEX[r:m] - ]Jx) ]
2
x
r=l
mn
A
A
2
ax
mn
a
2
2
T L T~ Cr:m)
= __ m n r
1 -
......:::.-.:.:.-=-_~..::...:::.~
p2 a 2
EY
__
x \ C
2
mn
L
r
Cr: m)
a
y
-]J
~ L T Cr :m)
= __ r
2
~ \
_
~--'C-
y)
1 -
2
m L T : m)
Cr
r
48
where
= EU(r:m) - EU = a(r:m)'
TU
(r: m)
Likewise, we can compare
.,
.
A
the precision of ~[m]n with
X by
examining
A
A
A
Table 2.4.1 displays some values of Val' ~(m)n/Var ~[m]n '
a measure of the loss of precision obtained as a result of ranking
by a concomitant variable V rather than by ordering the X's themselVes, when (X;V) has a bivariate normal distribution.
_
Table 2.4.2 displays some values of Val' X/Val'
I
A
~[]
mn
,a measure
of the gain in precision obtained as a result of ranking by a con·
comitant variable V over that obtained by random sampling, when
(X,V) has a bivariate normal distribution.
Simila:t1y, we can examine the effects of ranked set sampling
with a
MSE
~oncomitant variable on s1;]n' an estimator of o~.
2
*
s. (m.).. n
2
. ) 4 - --2-1 L\[ Val' X : ) +(EX "' )
ron1ECX-~
Cr m
Cr. m
m. n. I'
.
_ i'~ )2]2
X
=
1
. --------------'-..-...:......::;....-----..........----------------------- + 0 C )
MSE S*['m ]'n
1· C·)4 - ---2'"
1 \["
X .. ]
--'
mn E X-~
. ' L Yar[ r.m
m
n r
Likewise, we see that
Yar
Val'
+ (E' x[r.••. m.]
_ lIX)2]2
~
urn
49
Table 2.4.3 displays some values of (var
s(~)n)/(var S[~]n)'
a measure of the loss in precision obtained as a result of ranking
by a concomitant variable Y rather than by ordering the X's themselves, when (X,Y) has a bivariate normal distribution and n is
large. Table 2.4.4 displays some values of (var s* )!(var s*[2] ),
mn
mn
a measure of the gain in precision obtained as a result of ranking
by a concomitant variable Y over that obtained by random sampling,
when (X,Y) has a bivariate normal distribution and n is large.
We see from examining the tables that the advantage of using
ranked set sampling for estimating the variance is almost negligible if
Ip/ < .75.
Gains in precision of the estimator of~,
however, are substantial for
Ip I
as small as .5.
losses in accuracy of estimators which
resul~
Similarly, the
from ordering the
-e
concomitant variable Y rather than X itself are considerably more
.
marked .in the estimation of mean than in estimation of variance .
Table 2.4.1
A
A
Var U(m)n!Var ~[m]n
m
p=±.25
p=±.5
P=±.75
P=±.9
2
.70
.74
.83
.92
4
.44
.50
.63
.80
6
.33
.38
.51
.71
8
.26
.31
.43
.64
10
.22
.26
.38
.58
50
Table 2.4.2
.
."..
Var X!Var ~[m]n
m
p=±.25
2
1. 02
4
p=±.75
p=±.9
1. 09
1.22
1. 35
1. 04
1.17
1.48
1.87
6
1. 05
.121
1.63
2.25
8
1. 05
1.23
1. 73
2.55
10
1.05
1.25
1.80
2.79
p=±.5
Table 2.4.3
Var S(;)n!Var s *2
[m]n
p=±.25
p=±.5
p=±.75
p=±.9
2
1.00
1. 00
1.00
1.00
4
.85
.86
.89
.94
6
.73
.74
.79
.89
8
.64
.65
.72
.84
10
.57'
.58
.66
.80
m
\.
Table 2.4.4
Var
2
S'
!Un
*2
!Var s[m]n
m
p=± .25
p=±.5
p:;:±.75
p=±.9
2
1.00
1.00
1.00
1.00
4
1. 00
1. 01
1.05
1.11
6
1.00
1.02
1.10
1.22
8
1. 00
1.02
1.13
1. 31
10
1. 00
1. 03
1.16
1.40
e""
3.
ESTIMATION OF PARAMETERS OF ASSOCIATION
In this chapter, we will examine the process of estimation
of the correlation coefficient of a bivariate normal random vector
(X,Y).
We will show that a modified form of ranked set sampling
can improve the estimator of p over that from a random sample of
comparable size.
Estimation of p with All Parameters Known
3.1.1
The simplest, but most unrealistic, situation is that in which
)Jx'
]J ,
y
(J
2·2'
, and (J
are assumed to be known.
x
y
Suppose t'he pairs
(X,Y) of a sample of size m are ordered on che basis of
of the Y's.
th~
rank
Then the pair (X[r:m]'Y(r:m))' the r-th largest Y
and its concomitant X, have joint p.d.f.
f(r:m)(x,y) = f(r:m) (y)f(x Iy)
m!
r-l
m-r
= (r-l)! (m-r)! F
(y) [l-F(Y)]
f(x,y).
(3.1)
Suppose we choose, in the usual way, a ranked set sample of
Y's of size mn=N, and measure the pairs consisting of those Y's and
their concomitant X's.
(Note the similarity of this procedure
and that discussed in Section 2.4.)
mator of p,
p,
can be found by solving the equation
a
....
A maximum likelihood esti-
').R-n
op
{m n
IT
ITf(
. 1
r= 1 1=
r:m
,r1
) (X ., Y .)
where f(r:m) (x,y) is given in (3.1).
r1
\
J=
0
52
But this is equivalent to solving the equation
arm
n
'" in I II II f(X ., Y .)
ap
-r=l i=l
rl rl
]
=0
(3.2)
where f(x,y)is the p.d.f. of a bivariate normal distribution.
This is because p does not appear in any factor of f(r:m) (x,y)
except f(x,y).
Therefore the MLE of p from a ranked set sample
is calculated in exactly the same way as that from a random
sample.
In fact, even if ranking errors do occur in the ordering
of the V's, the MLE ofp will still be calculated in the same way,
since the joint p.d.f. of Y[r:m] (the r-th judgment ordered Y)
and its concomitant X is
f[r:m](x,y) = f[r:m](y)f(xI Y)
where f[r:m](Y) is the p.d.f. of the r-th judgment order statistic
and f(xIY) is the usual conditional normal p.d.f.
Note here also
that regardless of the distribution of the random vector (X,Y),
the ranked set sampling
~ x , ~ Y, ax , a are known is
.. MLE of. p when
.
Y
always calculated in the same way as the MLE of p from a random
2
2
sample.
Equation (3.2) is equivalent to the cubic equation
n
2 2 + a a (l.+p 2 ).L ~ (X
= mnaX2
aYp(l-p)
X Y
.L
ri-"'x )(Y. ri-"'y
r=l 1=1
fi1
g(p)
11
11)
There must always be a root of this equation between -1 and +1
53
since
m
n
g( -1) =
I .I [cry (Xri -].1x)
r=l 1=1
=
-I . I1 [cryCx r1. -].1 x)
+ cr (Y -].1y)]
x ri
2
~
0
~
0
and
m
g(+l)
n
r=l 1=
I
r
2
>
note that there is a root < 0 in (-1,1) according to
~urthermore,
whether
crx (Y ri -].1y)]
I (X . -].1 ) (Y . -].1 ) ~
. r1 x
r1 y
o.
By adapting Kendall and Stuart's
1
argument ([13], p. 40), one can show that as m +
00
or n +
the
00,
probability of more than one root of the likelihood equation existing will tend to O.
If there are three roots ofg(p) = 0, there
will be two real turning points satisfying the quadratic equation
g'(p) = O.
But this equation has two roots iff
(3.3)
Using -Chebyshev's inequality, we see that
1
2
E(U l U2)
Prfl-IIu
.U
·-pl>E:]~
2
mn
. l jr1,
2 r1
r 1
mnE:
2
+0
asm+
or as n
+
oo
00
,
since
l
E -!
I
Iu
.u
.
=
~
E
I
U
U
=
-m
mEU l U2
mn
. 1,r1 2,r1
mn
l,r 2,r
r
2
=P
r
1
and
l'i''i'
)
n'i'
22
,2
Var(-2 L[EU I r U2r - (EU I r EU 2r ) ]
mn L L. Ul ,r1.U 2 ,rl. = ----2
rIm n r
2 2
1 ['i' Eu lr
u 2r ] = .-!.
~ --2-- L
mn E(U 1U2) 2
mn r
Similarly,
Pr [I-l
mn
I I.
r
1
EU
2
.. -1/ > E] :::; -- 3 2 + 0 for k= 1 ,2
k , r:J
mnE
as m +
00
or n +
00
54
since
E -l
mn
2
LL
. Uk,ri
r
1
= lm E
2
2
Uk,r = lm mEUk = 1,
l
r
k=1,2,
and
Var(-l
mn
lL
U2 .)
. k , rl
r
1
= _n
l[EU 4
_ (EU 2 )2]
2 2
k, r
k, r
mn
r
3
EU4
k,r =-mn
mn r
< _1_ \
-
2
{,
Therefore, the left hand side of (3.3) converges in probability to
p
2
-
.
2
3(1+1-1) = P - 3 < O.
So for m or n sUfficiently large, there cannot be three real roots
of g(p) = 0, so there is a unique root to the likelihood equation.
Even though the estimates from a ranked set and from a random
sample are calculated in the same way, it is not necessarily true
that they have the same asymptotic variance.
To simplify calcu-
lations of Fisher Information, IN (p), we assume that
~
y
= 0,
a
2
2
= 1, a
x
y
= 1.
~
x
= 0,
In this case
and
mn
2
(l-p )
-
2mnp2
+
Cl_ p 2) 2
\ \
2
r i
rl
+
4p
2
{, L (X. -2pX . Y • +Y .) [
rl rl
rl
l r XTl.Yrl.
2 2 r i
(1-p)
1
2
(l-p)
If the N observations are simply a random sample, then
(3.4)
55
A
This tells us that the MLE PN from a random sample of size mn = N,
where
~"
x
~
2,
y
2
, ax , and ay are all known, has asymptotic variance
2
[(1_p )2]/ N(1+p2).
set sample.
Now suppose the mn observations form a ranked
Notice that
00
r
EX1r:m]Y(r:mJ = J xy f(r:m) (y)f(x/y)dy dx
-00
00
=
f
00
Y
f (r:m) (y)
_00
Jx
f(xjy)dx dy
_00
00
2
= Jy f(r:m) (y)py dy = PEy (r :m)
00
and
00·00
2
EX
=
[r:m]
J IX2f(r:mJ(y)f(X/ y)dY dx .
_00_00
2
2
=1 - P (l-EY(r:m))
Then
mn
This tells us that the MLE
mn = N, where
~x' ~y'
[ (1 _p2) 2]/mn(1+p2) .
2
p*(m)n
from a'ranked set sample of size
2
ax' and ay are known, has asymptotic variance
S
' varIance
. .IS concerne d ,
o ar as f
asymptotIc
the process of ranked set sampling does not improve the MLE of p.
Suppose now, however, that we modify the ranked set sampling
procedure so that instead of choosing n first order statistics,
n second order statistics, etc., where all mn are independent, we
include in our sample only extreme order statistics.
That is,
from each group of m judgment ordered random variables, we choose
56
for quantification Y : m) and its concomitant X and include N
Cm
vectors in our sample. CNote that we could just as appropriately
choose all smallest V's, YCI:m),or eYen choose YCI;m) and Y : m)
Cm
.
t
h
.
h
.
2
d 2
1t
t
a ema e 1y. Th·e reason IS
at In t IS case, 1.I,
x 1.I y , a·,
x an
.... ay
are known, so the probability structure of CY. m;m )-1.I y )/ay is
C
known and is the same as that of -CYCI:m)-l.Iy)/ay. Later we will
consider the case where 1.I and a2 are unknown. Then we will see
x
x
.
that we are better served to alternate Y : m) andY : ))'
CI
Cm m
When we sample in this way Cwhich is another form of modified
"'11
ranked set sampling), the MLE of P, PCm)N' is still a root of
gCp)
= 0, but the asymptotic variance of the estimator is the
reciprocal· of
1I1 )N CP )
Cm
=
2
N 2 IEu : )
cm m
Cl-p )
2
+
2p ];:: I Cp)
N
CI-p2)
where
)/a y , since EU 2~:~ ;:: 1.
. . U = CY-ll
. y.
C3.5)
We note here that the choice
of the extreme order statistics far inclusion in the modified ranked
set sample was made because it maximizes the Fisher Information
OVer all possible choices of order
statistic~,
Table 3.1 gives some values of the ratio
fOr particular values of m and p.
These values are a measure of the
amount of improvement an estimator based on a modified ranked set
sample allows over that from a random sample.
([3], p. 376) one can see that
as m -+
po.
2
EY Cm : )
m
= OC~n
From Cramel'
m).
Consoquontly,
57
Table 3.1
Comparison of asymptotic variances of pA and p"I:i
I:i
I (m)N(P)/IN(p)
m
p
P =±.5
= 0
P ;:±.75
2
1
1
1
4
1.55
1.15
6
2.02
1. 33
1.65
1.29
8
2.40
1. 84
1.39
10
2.71
2.03
1.48
3.1. 2 Confidence Intervals for p with All Parameters Known
Suppose a confidence interval for p is desired rather than a
point estimate.
Large sample theory allows us to find an approxi-
mate confidence interval based on the asymptotic normality of maximum likelihood estimators.
sample size N from which
(Theorem B, Appendix).
P~m)N of Section
So if the
3.1.1 is calculated is
"sufficient ly large," a 100 (1-0.) % confidence interval for p would
be
=
(3.6)
58
where U ~ N(O,l).
But what can be done to obtain an exact confi-
dence interval?
We choose a modified ranked set sample as required for P~m)N
above.
Note that the random variable X[ m:m ] IY(m:m'
), has p.d.f.
f (m:m) (x,y)
where f(m:m) (x,y) is given by (3.1) and
f(m:m)(Y)
=
1
I l _p 2
f(m:m)(Y) is the p.d.f. of Y(m:m)
¢(I ~)
2
where ¢(u)
Cl/I27f)e- u /2
=
l -P
assuming once again that ~,x
.- X[ m:m ]'.
.•
,1 - p Y( m:m ),1
2
= ~Y = 0 ,
2
a x = a = 1.
Y
Let
T [m:m], i
Then
T[ m.m,1
. ] . I Y( m.m,1
• ) . ~
N(0 , 1-P
2
)
e-
.
Since this distribution does not depend on Y(m:m)' it follows that
T[
.
m:m ],1
~ N(O,
2
l-p ) ,
and since the T[ m:m ] ,1.'5 are independent, we have
N
I T[ ] .
. 1 m:m,1
1=
2
~ N(O,N(l-p ))
Thus we can use the statistic
tN
'
Li=l(X[m:mJ,i - pY{m:m),i)
.T(m)N
=
IN(1~p2)
to construct a confidence interval for p.
~
NCO,l)
(3.7)
59
Consider the event
From (3.7), we know Pr(A) = I-a, and A is equivalent to
From this form of the confidence interval, we see that it cannot
contain values of p s.t.
ipl
> 1.
This, in turn, may be rewritten
as
(3.8)
where
Xm = L\'~1 = IX[ m:m,l
] ./N
-Ym = IN1=
. lY (m:m,l
) . /N.
and
side of (3.8) is a parabola in p
a
= Nz 2l-a/2
+
.
of the form
ap
2
The left hand
2bp + c.
Since
N2y 2 > 0 ,
m
we know that the only turning point of this parabola is a minimum,
2- -
2
and it occurs at P = b/a = N XmYm/(Nzl_a/2
value
ac-b
a
2
+
N2y2), with minimum
. m
2
But if ac-b
a
> 0, or equivalently, in our case,
ac-b 2 > 0, there will be no values of p which satisfy (3.8).
The
probability of this occurring is
Pr[ac-b
2
>
0]
=
2 2 _ N2
) _ N2-X2V2 > 0]
Pr [ (N. 2-y2m + Nz 2l-a/2 ) (N -Xm
zl-a/2
mm
=
2.2
-2
-2
2
Pr[N zl_a/2(NX m - NYm - zl_a/2) > 0]
=
Pr[Xm - Ym > zl -a /2/ N]
-2
--22
We now show that this probability converges to 0 with increasing N
or m.
(Assume that p
2
< 1, since for p
2
= 1,
~~
Pr[Xm-Ym = 0]
=:
1,
60
-2 -2
2
m m > 2 1 -ex. /')/N]
..
so Pr[X -¥
x
=0
V'N,m.)
222
- Y = zl-ex./2 /N
has as asymptotes the lines x
-2
-2
Pr[Xm - Ym
= Pr[(im,Y)
m
$
E
AI]
1 - Pr[(i. ,Y.)
E
m m
=1 -
Pr[-¥
<
First note that the h)'IJerbola
+ Pr[(i
=y
and x = -y.
2
>z 1
,Ym)
m
Therefore
N
-ex. /2/ ]
E
A2. ]
B1 ]
X < Ym and Ym > 0]
mm
. _ - - - - - , , . . . - ' - - - - - - _._----_.
Figure 3.1
=1
- {Pr [- Y
m<
+
X < Y and Y - > 0 II Y
m
m
m
m
Pr [- Y < X < Y and
m
m
m
1 - Pr [- Y < X <
m
m
Ym and Ym > 0 II Ym
$
1 - Pr [-l.! +E < i
< l.! -E] • Pr [
m
m
= 1 - Pr [- j.l (l +p) + £ <i
m
m
11
~m
I. < e:]
• Pr [ IYm - l.!mI < e:]
Ym > 0 II ym - l.!mI > E] • Pr [ IYm - l.!mI > E]}
$
m
-
- pj.l
IYm -
- l.!
l.!
I
m
I
m
< E] • Pr [ IV
m
<. £ ]
where
£
< j.l (l-p) - £] • Pr £I Y
mm
m
- l.!mI < E]
< l.!
m
- lJl
<
m
£]
61
$
{II - Pr[/Xm - p~ml < ~m(I+p) - E] " Pr[/Ym - ~ml < E]
- Pr[/xm - p~ml < ~m(I-P) - E] " Pr[/Ym - ~ml < E]
~
$1-1-
va.r X
(~
(m:m)
J ,-
m"I
(l±p) -E) 2J _ -
Var Y ]
E2
.
if p
$
0
if p > 0
$
m
if p > 0
by Chebyshev's inequality
=1 since [3]
Jr1 a~m:m)l
2 2
1+p (a (m.m
. ) -1)
[1 N(~
(1±p) - E) 2
(m:m)
~(m:m)
L
= EU (m:m) = O(/in
NE
m)
2
J
0 as
m~oo
or as
N~oo
~
2
and a(m:m)
= Var
U(m:m)
If there are values of p which satisfy (3.8) (i.e., if ac-b
= o( in 1·
n/
2
~ 0),
then the IOO(I-a)% confidence interval is
b+~J
=
l(NXmYm-Z,I-a.
/2fN(Ym2-X;)+z'~_a/2
2
-2
zI-a/2 + NYm
NXmYm + Zl-aI2 YN (V;-X;)+zi-a/2)J
' ------'=2:-·
-2
zl_a/2 + NYm
(3.9)
The interval of (3.9) is an exact 100 (l-a)% confidence interval
for p for any sample size, but (3.6) gives an interval which is a
valid 100(l-a)% confidence interval only when N is large.
Conse-
quently, it is reasonable to compare the lengths of the two confidence
intervals only for large N.
The length of the confidence interval
derived from the properties of the MLE is, from (3.6),
62
2
where ]JCm:m) = EU Cm : m)' 0Cm:m) = Var UCm : m)' U ~ N(O,I).
length of the exact confidence interval is, from C3.9),
The
We know that (under certain regularity conditions which
are fulfilled here),
a MLE converges a.s. to the parameter i t
estimates, and that L~=IXi/N a~s. EX
SLLN.
when Xl' ... ,XN are iid by
Therefore
Ab,
a.s.
PCm)N
P
+
V a.s.
EY
= ]J(m:m)
m +
(m:m)
X a.s.
EX
= PJ..\rn:m) ,
m +
[m:m]
and
a.s.
+
a.s.
+
all as N +
1f::(7/ ]J(m:m) ,
2Z I _a /2
00
•
Let the ratio of these limits
2
]J(m:m)
be denoted by R(m,p).
of m and p.
+ 02
(m:m)
2p2
+--.
2
l-p
~
It is clear that R(m,p) < I for any values
But since
02(m:m) =
and
.
we know that
lim R(m,p) = I
rn+oo
63
So the limiting length of the exact confidence interval approaches
that of the MLE-based confidence interval as m increases.
values of R(m,p) are tabulated in Table 3.2.
Some
This table shows
that R(m,p) approaches I fairly rapidly (with m), especially for
small values of
Ipl.
If errors in ranking of the Y's occur, this method will still
produce an exact IOO(I-a)% confidence interval since the p.d.f. of
x[ m:m ]·/Y[ m:m.] will still be
f[m:m] (x,y) = f(xly)f[m:m](y)
f
f [m:m] (y)
[m:m] (y)
where f[m:m](Y) is the p.d.f. of the largest judgment order
statistic.
vN L2
-+
00
However, if the ordering is no better than random,
as N -+
since J.l[
00
.] = Ell
m:m
= O.
In order to use the
MLE-based confidence interval for the case where ranking errors
. te
h Y' s occur, one must f
"estlmate
·
2
ln
lrst
J.l[2
m:m ] + o[ m:m ]
. h
Wlt
an estimator such as
1.
N
Ly2
.
N i=l [m:m]'l
a.s.
-+
2
2
J.l[m:m] + O[m:m]·
Then the interval will also be a valid confidence interval for
",fj.
large samples, since P(m)N will still be the MLE of P, as noted
in Section 3.1.1.
In this case, if ordering is no better than random,
, IN L2
-+
~
-2
l-p
2 zl _a / 2 ----2 '
l+p
64
so this method will still give us a useful confidence interval.
Table 3.2
R(m,p)
p-±.25
m
p=O
2
.564
.530
.437
.299
4
.826
.793
.691
.507
6
.891
.863
'.773
.591
8
.919
.894
.813
.639
10
.934
.912
.837
.669
-
p=±.5
p=±.75
There is another approach to finding an exact confidence
interval for p when
~
x
,
~
y
,
2
0 ,
x
and
0
2
y
are all known.
It too,
is based on the fact that E[XIV] is a linear function of V for
the bivariate normal distribution.
Suppose here again that our
data is a modified ranked set sample of size N.
Consider the
random variable n(m)N where n(m)N is the number of the N observations which fall in the region X[m:m] - PV(m:m)
2 = 0 2 = 1.)
we assume that 11t-'x = 11t-'y = 0 , 0 X
Y
= py
/Figure 3.2
~
O.
(Here again,
65
Then n(m)N ~ Bin(N,~) , since
Pr[X[m:m] - PY(m:m)~ 0]
py
00
Jf(m:m)(y) JIl-p1 2 l11--py)2J. dx dy
(x
=
ct>
_00
p
_00
= <p(0) =
~
(Note that this does not hold true for p=±l, since in that case
Pr[X[m:m] - PY(m:m) = 0] = 1.)
So by randomizing, one can find a 100(1-a)% confidence interval
[n* < n(m)N < n*]
based on the binomial distribution of n; or if N is moderately
large, the normal approximation to the binomial can be employed
to yield
n*
N
=2
+
~
zl_a/2 vN / 2 •
The implementation of this confidence interval procedure can
be handled graphically by simply counting the number of observations
to the left of the line x
p = +1.
= pyas
its slope changes from p
= -1
It can also be carried out analytically by calculating
for each pair (X[ m.m,
. ] Y( m.m
. )) the interval of p's satisfying
X.
- PY(
)
[m:m]
m:m
~
O.
These intervals are:
X
e
P
~
p
<
~
[m:m]
Y
(m:m)
for Y(m:m) > 0
X
[m:m]
Y
(m:m)
for Y
(m:m)
all p's if Y(m:m) = 0 and X[m:m]
< 0
~
0
no p's if Y(m:m) = 0 and X[m:m] > 0
to
66
Then the IOO(I-a)% confidence interval I _ is the set of p's for
l a
which n(m)N' the number of intervals covering P, lies between n*
and n*.
Now suppose (X,Y) '" N2
[Q, (~
(a)
Pr[P I
E
II_a]
+
0
as m +
00
(b)
Pr[P I
E
II_a]
+
0
as N +
00
n
iJJ and PI 1 p. We will show:
Proof:
(a)
But
where
By Chebyshev's inequality, we know that
P [IY
r
(m:m) -
So if PI < P
+
j.l (m:m)
I
0
> e:]s;
2
(m~m)
e:
67
:;; l'Pr[/Y(m:m)-~(m:m) I>E] + <I>
2
a(m:m)
<
-
2
+
E
(
(~(m:m)-€)(P1-P))
11-p 2
~l(~(m:m)-E)(PI-P)'
'*'
J -+ 0
as m-+
11-p 2
Pr[ly( . )-~( . ) I:;;E]
m.m
m.m
00
and if PI > P
r
PI
J
2
Iy
(m:m)
-~
<I>
(Y(P1~P)J
2
I<E
(m:m)<
>
- pr[ly( m.m
. )-~(. m.m
. ) I-E] <I>
.
2
(1 -
2
a(Em2:m)]
-
<I>
f(m:m) (y)dy
11-P
l(~ (m:m) -E) (P1-P),
J
/1 -P 2
y
((~(m:m)-E)(P1-P)'
__
J -+
11-p2
1
as m -+
00
•
Thus lim Pr[P EI _ ]
1 1 a
m~
=0
(b)
.
Considering PI
=f
00
-00
we see that
f ( . ) (y)<I>
m.m
(y(P 1 -P))
.
dy
11-p2
as a function of PI
68
QO
=
Y(Pl-P))
. 2 J
Io l
<!>
Il-p
Y
2(f(m:m)(Y) - f(m:m) (-y))dy
..
Il-p
00
=
Io
(y(Pl-P})
<!>l.·
Il-p 2
-L. m<!>(y)[<P-(y)
m-l
II-P 2
- ~(-y)
m-l
if m>l,
]dy>O
Therefore PI is a strictly increasing function of PI'
This tells
us that
Thus
as N -+
ranking occur,
if m > 1.
However, unless the ranking is somewhat better than
random, we cannot be assured that lim Pr[P El 1_a ]
l
N-+oo
3.2
00
= O.
Estimation of P with 11 y
, y
(i Known;. 11x , ax2 Unknown.
Now suppose we are interested in the estimation of P when only
2
2
11y and ay are known, with
11x , ax un~nown.
.
2
The MLE's of 11x' ax , and
P from a random sample of size N are [11]
(3.10)
•
69
_e
1
where s2 = N
x
N
- 2
I (X.-X)
. 1 1
,
s
2
1 N
N
I
=
i=l
y
1=
- 2
(y.-y) ,
rs s
x y
1
1 N
= N
I
. 1
1=
(x.-x) (Y.-'Y).
1
1
If we choose as our bivariate sample a ranked set sample of y's and
2
their concomitant X's, then the MLE's for p, a , and ~ will be
x
x
calculated in the same way. This is true for any modified ranked
set
sample as described in Section 3.1, or, in fact, for any ranked
set sampling scheme where the Y's are judgment order statistics.
The reason for this, of course, is that the p.d.f. of (X[ r:m ],Y[ r:m ])
is
where f[r:m](y) is not a function of p,
a; .
~x' or
In order to find the asymptotic variance of the MLE of p, we
must determine the Fisher Information matrix
I~l(~,a
-;'1
X
x
,p).
After
calculations similar to those of the previous section,we find that,
once again, a random sample
a~d
a ranked set sample produce identical
Fisher Information matrices', which are both equivalent to
1
0
0
(2_ p 2)
P
a2
x
a
a2
x
~(~
=
x ,ax ,p)
N
0
(l-pl
0
a
Thus
r a2
-1
~ (~
e
x ,ax ,p)
=
(l_p2)
N
x
l+p 2
--2
l-p
0
0
P
a 2 (l+p 2)
0
x
2
pa (l_p2)
0
x
x
2
2
pa (l-p )
x
2
2
2
(l-p ) (2-p )
2
(3.11)
70
Now suppose the observations in our sample are chosen according to
a modified ranked set sampling scheme; i. e., choose for quantification N/2 each of Y : m) and Y : m)' along with their concomitant
Cm
CI
X's. Then the upper triangle of the Fisher Information matrix
for this sample will be
o
o
222
2(l-p )+p EU m.m
.)
C
po C2-EU 2 ))(I_p2)
x
Cm:m
2
2EU cm : m)
EU
2
(m:m)
2p2
+--
l-p
2
Thus
o
o
2
2
POxC2-EUCm:m))CI-P)
o
2
2EU m:m )
C
=
o
C3.12)
where U - N(O,I).
Since
2
2 2]
2
[2EU (m:m) - p EU Cm:m)
2NEU Cm : m)
(l-p 2)
71
2
=
(l;P ) [2EU 2
Cl 2)
Cm:m) -p
2NEU Cm : m)
2
(l-p )
2
2NEU Cm : m)
[2Cl -p 2)
+
2 2
]
P EU Cm:m)
we know that the asymptotic variance of '"
P is always larger than that
N
''''f'..
of the MLE from the modified ranked set sample, PCm)N'
that the asymptotic variance of
Note also
...:.1::.
NPCm)N
~
"'I::.
Here again, we can use the fact that PCm)N is asymptotically
normally distributed to find an approximate lOOCl-a)% confidence
interval for p.
Unfortunately, the methods of Section 3.1 cannot
be used to find exact confidence intervals, since estimates of V
x
--
2
and a have to be introduced.
x
2
If V and a are replaced by their
x
x
maximum likelihodd estimators in TCm)N or nCm)N' those statistics
could still be used to find approximate confidence intervals for P
when N is large.
They are asymptotically normally distributed and
2
will have the same asymptotic properties as when Vx ' ax were known
because of the consistency of the MLE's.
However, I have not
demonstrated that a confidence interval procedure based on,TCm)N
,
"'f'..
or nCm)N is in any way preferable to that based on PCm)N' although
no investigation of the speed with which they reach their asymptotic
distribution has been undertaken.
3.3
Estimation of P with All Parameters Unknown.
2
2
Now suppose that V , V , a , and a are all unknown.
x y
x
Y
The MLE's
of all five parameters are well known ·if the sample Xl"",X N is a
random sample.
But suppose our sample is a modified ranked set
,
72
sample of the type which includes N/2 largest and N/2 smallest Y
observations and their concomitant X's; i.e .• our bivariate sample
4It
! •.
is (X[1:m].1'Y(1:m),1).· .. '(X[1:m].N/2. Y(l:m),N/2). (X[m:m].l'
Y(m:m).1).···'(X[m:m],N/2. Y(m:m).N/2)·
Then (writing Ymi for
Y(m.m
. ') .1.• Xl'·1 for X[1..m ] .1.• etc.) the new likelihood
function is
.
N/2
m-1
N/2
m-1
L = IT m F (Y .)f(X .• Y .) IT mel-Fey .)]
f(X.Y.)
. 1
m1
m1 m1 1=
. '1
m1
m1 m1
1=
2
where f is a j oint normal p. d. f. with parameters j1 x , j1y •(i.
0
and
x y
p; and F is a normal c.d.f. with parameters j1y and 0 y2 . Then the
five likelihood equations are:
dQ,n L
=
dj1
x
dJin
dj1
L
=
y
V-V}
{ ___
X-V x p ---l.
N
o
0
2
o (l-p )
y
x
x
N
ay C1-p 2)
Y-j1 , _p __
X-j1x }
{ -J.
o
a
y
x
Y .-j1
Y -j1
f( 11 y)
f( mi y
a
ay
N/2
l
+ (m-1)
Y .-j1
Ymi -j1y)
ay
i=l 1-F( 11 y)
F( a
ay
y
N/2 (Y ·+Y .)
N/2(X ·+X .)
11 m1
11 m1
and Y =
X=
N
. 1
N
1=
i=l
r
where
= 0
r
r
+p
=
(3.13)
=0
e~
73
+ p
\·N/2
L· l[(Xl·-~ )(Yl·-~
1=
Y)+(Xm1.-~ X)(Ym1.-~Y)]
a a
x Y
1.
x·
N/2-f[(Y .-~ )/(a)]
I
(m-l)
2a
. 1
2
Y
-
f[(Yl.-~
1
Y
Y
.-~ )/(a)]
m1
{ i=l F[(Y
Y
m1
)/(a )]
Y
Y
(
Y
Yl · -~ )]}
Y)
~
=0
Y
atn L
ap
=
0 .
An exact solution to these equations would be difficult to find
numerically.
One finds, using Tiku's approximation which was intro-
duced in Section 2.1.3, that an approximate solution to (3.13) is
(see Appendix, p. 107) .
"
~
x
=X
~
Y
= y
But this approximation does not produce an explicit solution to the
remaining three equations.
One intuitive approach to estimating all parameters with only
the data at hand is to first estimate
~
Y
and a with their BLUE's.
Y
These estimates are, for our modified ranked set sample of Y's,
74
~y =
where
N/2
L. l(Ym1.-Y l 1,)
= 1=
~
J and
A
o
a N
m
y
U ~ N(O,l).
am = EU(m:m)'
with their BLUE 's.
Then replace
~y
and 0y in (3.10)
The remaining estimators wi 11 then be
A
A
~x
=X
~2
.2
2 2~2 2
= s (l-r +r 0 /s )
o
x
x
~
y
y
2
~
2-k
2~2
P = (roy/s ) (l-r +r 0 /s)
Y
L~/2 reX
where
1=1
2
Y Y
, -X) 2+ (X • -X) 2]
11
m1
N
2
s
Y
'\N/2
- 2
- 2
+(Yml.-Y) ]
1= l[(Yl'-Y)
1
= ----"'-'''':-:----=--L'
N
'\N/2
-.
-..
L' l[(Xl,-X)(Yl,-Y)+(X i-X)(Y .-Y)]
1=
1
1
m
m1
ssr=.....=.-=.-...::::.::-----=-----=---=--
x
y
N
A simple Monte Carlo study was undertaken in order to gain some
"'-
information about the behavior of the estimator
A
A
P.
The mean and var-
.
iance of p were calculated on the basis of 200 samples, which were
chosen by the modified ranked set sampling procedure described in this
section.
The results of this experiment are given in Table A.l
of the Appendix (p. 109) for m = 5, 7, 10, P = 0, .5, .75, and
N = 10, 20, 30.
For comparison purposes, we give in Table A.2 the
ratio of the Crarner-Rao lower bound of the estimators of p from a
random sample ((l_p2)2/N) to the variance of
(1_p2)2/N is the approximate.variance of
$.
For large N,
r, the Pearson product-
75
moment correlation coefficient.
Examination of this table shows
that in all cases, these ratios are larger than 1, indicating that
ranked set sampling substantially improvffithe precision of the
estimator of p over that from a random sample for an m even as
small as 5.
~
A drawback of the estimator p is that it is not valid when
A
errors in ranking occur, since
A
estimator of a and
y
~
y
& will
y
no longer be an unbiased
will not be an unbiased estimator of p
y
unless the ranking errors one makes are "symmetrical" (i. e., unless
E(Y[m:m] - p y )
=
-E(Y[l:m] - Py)), which is unlikely.
If errors
in the ranking of the Y's does occur, then a two stage estimation
process could be employed; i.e., independent estimates of p
ay
can be introduced into (3.9).
y
and
Presumably, this process will be
relatively inexpensive, since, according to
heframework of the pro-
blem, the observations of Yare more accessible than those of X.
.
.e
4.
HYPOTHESIS TESTING AND RANKED SET SAMPLING
In this chapter, test statistics calculated from ranked set
samples are compared with appropriate conventional tests, and, in
other cases, with each other.
4.1
Tests of Location
Since the first step in the ranked set sampling procedure is an
ordering process, it is natural in some instances to compare a test
based on a ranked set statistic with a nonparametric rank test.
XCI'.m ) , 1""'X Cm.m
.) ,n
Let
and YCI .m
' ) , 1""'Y Cm.m
.) ,n be ranked set samples
Y
from each of two populations, with n + n = n. Suppose we wish to
x
x
Y
test the hypothesis that the two populations are identical against
the alternative that they are of the same form, but differ in location;
i. e.,
against
Consider the test statistic
A
- l1 yCr ))
2
77
n
~y(r) =(l/ny )'
y Y(r:m),i' and likewise for X.
Li=l
For large values of n
x
and
,. n y ,
m 2
m 2
+ (n -1) II s
r~l y(r:m)
y
r~l x(r:m)
nx + ny - 2
( n -1) II s
x
+
m 2
II a
L
r=l
as n x ' n
Y
+
(r:m)
00
Thorefore,
when nand
n y are large, T (m)n ,n is approximately
'.
x
x y
normally distributed with
]JT
(8)
(m)n ,n
x y
for
sampl~sfrom
=
e
_I,
211
a T
(8) =
+
(m)n
,n
m
x
, x y
Z(n nI )
2
lLr a(r:lJl)
y
distributions satisfying certain weak conditions.
A
test based on T(m) will be compared with the Mann-Whitney test, a
linear rank test for detecting location differences.
Recall that because of the ranked set sampling process, there
were originally mn x samples of X observations and IJlny samples of Y
observations, each of size m.
But
T
(m)n ,n
x
is calculated only from
y
the quantified observations of which there are mn x X's and mn y V's,
a total of mn.
Computation of theMann-Whitney statisttc, however,
requires only that the order of the observations within a combined
sample be known, and no exact measurements are needed.
So if the data
is the type where ranked set sampling is appropriate, i.e., where
quantifying is much more difficult than ordering, a Mann-Whitney test
would seem a reasonable choice.
Notice, however, that it is not
always possible to eye-order the combined sample, since the sample of
X's m<iY not become available at the same time or at the same place as
the sample of V's, thus eliminating the opportunity to directly com10
(4.1)
78
pare the members
of the two samples,
Two situations will be considered.
In the first, T
(m)n ,n
x
y
is compared with the Mann-Whitney statistic U
which is
(m)n ,n '
x y
based qn one Mann-Whitney test applied to all the observations.
That is, assume the ranking of all
m2n observations to be established,
and calculate U( mn
)
n from<this information. This is usually an
x' y .
unrealistic approach, of course, because our concern all along has
been to keep the number to be ordered in each sample reasonably small
so that the ordering may be done accurately.
tion, this "advantage" given to
let n x
U (m)
In the second situa-
is partially eliminated.
= ny = N/2.. Then suppose we combine one sample of
X' s
Here
with
one sample of Y' s , both of size m, and that these 2m observations
are ranked among themselves and a Mann-Whitney test performed.
This
process is repeated mn/2 times (since there are mn/2 pairs of samples),
and the resulting mn/2 test statistics are combined to produce a
"Mann-Whitney type" statistic
.
(m)n
U*
=
mnf2
U·
.
mi'
1=1
.
where U . is the Mann-Whitney statistic from the i-th combined samples.
m1
For large values of n, both U( m) n ,n (because it is a linear rank
x
y
statistic) and UCm)n (by theCLT) have approximately normal distributions,
and
I
00
=
= m4n~ny
2
2
(m n ) (m n )pr[Y>X]
y
x
_00
2
22
(8) = m n m n (m (n +n )+1)/12 x
y
x
y
-
F(z+8)f(z)dz
4
m n n (mn+l)
. x y
12
•
}
(t1.2)
79
lJ
U*
(8) =
m~m2 Pr[Y>X]
m~ (~;l) m3n~im+l)
(m)n
2
(J U*
(8) "
m2
(m)n
"
I
(4.3)
}
Theorem C, of the Appendix enables us to calculate in a simple
manner the Pitman efficiency of T wrt
U and U*.
Following the
notation of the theorem, each of our statistics is based on mn
observations.
2
(Note that we get m n - mn
"free" in the nonparametric tests.)
d
d8 lJ T
= mn(m-l)
From (4.1), we have
1
(0) =
. . 1I r=l a~r:m)
,
m
(m)n ,n
x y
and
From (4.2) and (4.3),
00
d
d8 lJ U
(m)n ,n
x y
4
(0) = m n xn y
r
J
f
2
_00
and
au
2Jn n (mn+1)
(0) =
(m)n ,n
x
m,,-x--'l':--,2:O----
y
00
d
d8 11U*
(m)n
and
aU*
(m)
•
(0)
3
(0) = m2 n
r
J
f
_00
=.. . /~3n (2m+ 1)
,.
24
observations
2
(z)dz .
(z)dz
80
n
n
-rn ~
Thus if ~ ~ p and
n
= lim
(l-p) as n ~
1
•
m
n~f
mn
00
,
m~nx:y
2
"a
L
(r:m)
r=l
c
U(m)
00
=
J f 2 (z)dz
m 112p(1-p)
_00
=
J;:;A
3
v24 mn
lim
n-~
2
f (z)dz
2~(mn)m3n(2m+l)
___
... /
00
[00
6m
00
2
r
Jf
=, 2m+l
2
(z)dz .
_00
Therefore, the Pitman efficiency of
U (m)
relative to T(m) ,
""Joo 2
2
12m p(l-p)
[-00
2"m
2
f (z)dz] Lr=lO(r:m)
mp(l-p)
00
=
l2mo 2 [
f
f
2 (z)dzJ.2 (Lr=lO
"m
2(r:m/O 2 )
_00
A
=
var p(m)n
2
m e
u,t var X
mn
where u, t are the conventional Mann-Whitney and t-tests.
(
.
cUem) , T (m)
=
l
c U*
2
(m)
c
T (m)
]
Similarly,
81
A
=
2m 2 e
var ~(m)n
2m+l u,t
X
var
mn
var
We noted in Chapter 1 that
~(m)n
2
--~""-- 2 ~(-m-+~l):-
, with equality holding
var X
mn
only for the uniform distribution.
e
V (m)' T (m)
=
2m 2
(m+l)
So fUr the uniform distribution
= D(ln)
-+
var
For many other distributions,
as m -+
2
~(m~n
00
•
is close to its lower bound
var X
mn
of 2/(m+I).
For these distributions then,
2
2m
'" - - - e
m+l u,t
~
~
4m
(2m+l}(m+l)
~
From these results, we see
iency is concerned.
ranking of all mn
2
Table 4.1 displays
form distributions.
2
e
t~at V(m)
u, t
.
is better as far as Pitman effic-
However, as we pointed out earlier, the accurate
observations will quite possibly be very difficult.
e *
V
T
for m = 2, ... ,5 for the normal and uni-
(m)' (m)
Here again, the nonparametric test would be pre-
ferred over that based on the statistic T(m) on the basis of Pitman
efficiency.
We must keep in mind, however, that:
82
(1) direct comparison of the two samples may not be feasible
(2) T(m) requires the ordering of m observations while U )
em
requires ordering of 2m
(3) if ranking errors occur, T(m) will still produce a valid
size ex test for large samples, while UCm) (or U(m)) will not.
Table 4.1
4.2
-m2
Normal
Uniform
1. 04
1.07
3
1.28
1. 29
4
1.45
1.42
5
1.57
1. 51
Tests of Correlation
Suppose that (X,Y) has a bivariate normal distribution wIth
corr(X,Y) = p.
In Chapter 3, two procedures for producing confidence
intervals for p were developed.
In this
sec~ion,
these procedures will
be used to construct tests of
against
First suppose
HO:
P = Po
HI:
p::f Po
~x' ~y'
cussions we assume that
~x
2
2
crx ' cry
=
~y
are known.
= 0,
ix
=
cr
2
Y
For the following dis=
1.
Recall from Chapter
3 that
=
T(m)N
if 11 0 is true.
I~=l (X[m:mJ,i
- POY (m:m) ,i)
/ N (l-P~)
The power function of this test is
~
N(O,l)
83
BT
(p) = prp[the test rejects HOl
(m)
N
$
i~l (X[m:ml,i
- PY(m:m),i)
-
(
1 - E¢
(
N
(PO-p) i~lY(m:m),i
+
zl-a/2 VN(l-P~)
2
'VN(l-p )
L
N
r (PO-p) i~lY(m:m),i
- ¢
=
- zl-a/2 VN(l-P~)
)
e·
~N(l-P~)
if
Ipi
where ¢ is the c.d.f. of the unit normal distribution.
< 1 ,
This function
is a difficult one to evaluate, so we introduce an approximation which
is valid when N is large enough that
~(1-P~) TN = I~=l (X[m:m],i
P Y
o (m:m),i ) will have an approximately normal distribution when
corr(X,Y) = P f PO· We know that
ET(m)N =
(N/jN(l-P~))(PlJ(m:m)
= IN(P-P O)lJ(m:m)//1-P6
- POlJ(m:m))
84
and
var TCm)N = E[var[TCm)N1YCm:m),i
+
where
f.l( m:m )
var[E[T m)N1YC m.m
.' ) ,1.
C
2
J IiL
N
(X[m:m],l -
l
by
where
we have
T
(p) ~·8+
(m)
i"l,
,N]]
,N]]
= EU( m:m) ,a(m:m ) = var U( m:m)' U '" N(O,l) .
Thus, if we approximate
8
i=l,
(p)
(m)
';~(l-P~)
85
=I
-
o
-
Cm:m)
<P
Now we calculate the power function of the test statistic n
from Chapter 3 where
Cm
)
nCm)N = number of observations CX[m:ml,vCm:m))
in a sample of size N such that X[m:m]
~
POY Cm : m) '
If HO is true,
nCm)N ~ Bin(N,~), and if N is large enough, nCm)N ~ NCN/2,N/4).
The
power function of this test is
B
Cp) = Prp[the test rejects HOI
n Cm)
(4,5)
where
00
P
= Prplrx [m:m]
~
r
J
p0 Y (m:m) ] --
f
[(P o-p) Y1
(y)<P
Cm:m)"
If:P7 J dy
-00
Thus
B
Cp)
n. Cm)
,To
=
I -
1-<pl~C~-P)
_.
+
~ZI-a/2J
Ip Cl-p)
test the one-sided hypothesis
HO: p
against
~
Po
HI: p < Po '
-
<p(INC~-P) - ~Zl-a/2Jl
Ip (l-p)
-
.
86
we may still use either test statistic T(m) or n(m)'
We would re-
ject HO if T(m)N is too small or if n(m) is too large; i.e., reject
at level a if:
T (m)N
or i f
<
-z
I-a
n (m)N > N/2
where <1>(zl -a )
=
IN zl_al2 ,
+
1 - a.
Since once again all conditions of Theorem C are fulfilled, the
Pitman efficiency of T(m) with respect to n(m)' e T
• can be
(m)' n(m)
calculated by observing that
d
dp].lT
(m)
aT
(PO)
= 1N]l
(po)
=1
(m)
=
(m:m)
-N
].l (m:m)
~
o
dr JOO
l(Po-P)Y) 1
n dp
f(m:m) (y) <1> ~ . JdYJ _
II-P
c
=
T(m)
and
..e
.
P-Po
cj>(O) ,
Thus
c
U
•
--00
=
/11-p2
=
n(m)
lim
1N]l
N~
(m:m)
IN(l-p2)
I'
-Ncj>(O)].l(m:m)
1m
N~
=
0
N/l-p2/2
0
].l(m:m)
1(l_p2)
0
-2cj>(O)P(m:m)
=
Il-p Z
0
87
Therefore
e.
1.57
= --2
l-p
o
There are, of course, infinitely many test statistics similar
to n(m) which could be used to test the hypothesis p
= PO.
For
n~m)N be the number of observations in a sample of size
example, let
N such that
x
-
p
y
[m:m]. 0 (m:m)
Il-p2
~-"",,==::;;;:=-..>------"-
::; k .
o
We know that if HO is true
n~m)N ~ Bin(N,~(k))
.
So we construct a size a test by rejecting H if
O
n(m)k - N~(k)
(N~(k) (l-~(k))~
I
> zl-a/2 .
The power function of this test is
=
(3 k
n
'I
1· _
(p)
[-"'tlN(~(k)-P)
+
'!'
v'j)Oq
(m)
where
p
= Ey
r l(Po-P)Y +
(m: m)
zl_a/2(~(k)(l~~(k)))\
J
'L'~
k/1=P'6).,J
Ii"="P7
88
The form of this function makes it difficult to compare with the
power function of
cular p.
S
n (m)
or to find an optimal value of k for a parti-
Suppose, however, that we restrict our definition of optimality
to be that value of k which maximizes the efficacy of n~m) for a given
Po and m.
Then
= _d
d
P
=N
N Ey
cj>(k)·
[
and
/- ((Po-P) y + k/~JJ
!P
(m:m) II_p 2
).l
(m:m) -
2
II-p a
1
kp O
2
(l-p)
-
a
= y'N!P(k) (l-!p(k))
So
2
cj>
=
--2
2
(k) [().l(m:m) I /l-P'~ - (kPO / (I-PO))]
!p(k)(l - !p(k))
c k
can be maximized for any values of m and PO'
n(m)
Table 4.2 shows the
optimal value of k for various values of Po and m (which we write as K),
and the Pitman efficiency of T(m) with respect to n(K ) , e
K
m
T (m) , n (m)
Note that for Po
= 0,
•
the optimal value of k regardless of m -is
K = 0, yielding, as we have seen, e
K
T(m)' n(m)
= 1. 57. We see from
this table that inasmuch as Pitman efficiency is concerned, T(m) is a
better test statistic than nZm) when /pol is small. As IPol increases,
however, the situation reverses. Note also that for small values of m,
nZm)'s advantage is more marked.
Graphs of the approximate power functions of T(m)N (from (4.4))
-
and the power functions of n(m)N (from (4.5)), with N = 25, for testing
rn
5
Po
K
0
0
e
7
K
T Crn)' n Crn )
e
K
10
K
T Crn) , n Crn)
K
e
K
TCrn),n Crn )
1.57
0
1.57
0
1.57
.1
-.23
1.54
-.20
1.55
-.18
1.55
.3
-.62
1. 33
-.56
1.38
-.50
1.42
.5
-.91
1.01
-.84
1.10
-.78
1.18
.7
-1.12
.665
-1.07
.764
-1. 01
.851
.9
-1.34
.272
-1.30
.334
-1.27
.395
Note:
For negative P, the signs of
K
are reversed.
90
,.,
e
against
at level a
HO: P
=
0
HO: P
~
0
.05 are illustrated in Figures 4.1 and 4.2 respectively.
=
The four solid lines of each figure are the power functions for
m = 2, 5, 7, 10.
The dotted line of Figure 4.2 shows, for purposes
of comparison, the position of ST
Cp).
Values of the power func-
CIO)
tions of T ) Cp) and nCm)Cp) CN = 25, m = 5, 7, 10) for some particuCm
lar values of P are given in Table 4.3, along with values of the power
function of r, Pearson's product moment correlation coefficient, the
conventional test statistic from a random sample.
as a test statistic when
2
are unknown, while T ) and n ) reCm
Cm
quire knowledge of these parameters, although they can be estimated from
]J
the data, as we have seen.
and
CNote that r is used
0
Hence, judging r l'y the standards of n
)
Cm
and T ) is not entirely appropriate, but is given here, with that in
Cm
mind, for purposes of a rough comparison.) The values of the power
function of r are interpolated according to David's method from her
tables [4].
Graphs of the approximate power functions of TCm)N and the power
.
K
functions nCm)N' N
= 25~
for testing
HO: P
against
~
~
.5
HI: P < .5
and for testing
H: P
O
against
~
.9
HI: P < .9
all at level a
=
respectively.
CNote that the optimal value of k for each particular
m and
.05, are shown in Figures 4.3, 4.4, 4.5 and 4.6,
K
Po is used for nCm)N')
Tables 4.4 and 4.5 give some particular
91
Power Function of T
Cm
)25 for
Po
=
0
e
1.0
.67
.33
o. - \ - - - - - - t - - - - - - r - - - - - - r - - - - - - - 1 P
-1
-.5
o
.5
-+
1
Figure 4.1
Power Function of n~m)25 for
Po = 0
1.0
.67
.33
/
J . - - - - - - - t - - - - - - ; - - - - - - - - t - - - - - " 1 P -+
-1
-.5
o
Figure 4.2
.5
1
,-.
,.
e
92
Power Function of T(m)25 for Po = .5
1.0
.67
.33
0
p-+
-1
-.5
0
.5
1
Figure 4.3
'~e
\'
K
Power Function of n(m)25 for Po = .5
1.0
.67-
.33
o J.-----+-------r------;~---___l P
-1
.5
1
-.5
o
Figure 4.4
-+
93
Power Function of T )25 for Po
Cm
=
.9
1.0
.67
o ~-----;--------r-----r-----""'" P
-1
-.5
o
.5
-+
1
. Figure 4.5
K
Power Function of n
Cm )25
=
Po
=
.9
1. 0 -.--------------=:::::.:::::-~~;;::---......,
.67
.33
o --.Jl~-----;I-- - - - - r l----11"-----., P
-1
-.5
0
Figure 4.6
.5
1
-+
values of these functions, along with those of the power functions of
e"
r for these tests.
We see from the graphs and tables that the power of TCm)N appears
to be larger for small values of IPo I and smaller for large values of
than that of n~m)N' over all values of p.
Ipol
This conclusion is
the same as that suggested by Pitman efficiency for the power of the
two test statistics close to the null hypothesis; i. e., that T(m) is
a better test statistic for Ipol small and n~m) is better for Ipol
large.
k
The use of any n (m) besides n (m)
(i. e., letting k
= 0)
for the
construction of a confidence interval is questionable, however.
For
some small val ues of m and an unfortunate choice of k, there wi 11 exi.st
at least 2 values of p which satisfy the equation
.'
96
pep)
One solution to this equation is PO' the true value of the parameter.
If k is such that
_P_o_
= ]J(m:m)
I _ 2
l p
k
o
then there will exist another value p which satisfies pcp)
=
<I>(K).
This is because, for such a k,
= <1>(10.j-kP 0
_(l_p 2 )
o
and
_
]J (m:m) ]
=0
Il-p2
.
0
This tells us, that the function pcp) has either a maximum or a minimum
at P
= PO.
But
lim pcp)
p+l
=0
lim pcp)
p+-l
=1
.
Since pcp) is continuous over the interval -1 < P < 1, we know that
pcp) must intersect the line y
= <I>(k)
at least once more, say at
pcp)
<I>(k)
e
I
J
-1
I
Po
I
PI
... :1
97
k
So the test statistic n(m) cannot distinguish between Po and P
l
j
since
n~m)
,....
Bin(N,~(k))
if P = Po Or if P = PI •
This problem does not develop when we use the test statistic n(rn)
(i.e., k=O) though, because in that case
~
dp
pep) = </>(0) \- -lJ(m:m)
0
_ I _p 2
l
o
J
< 0 for m > I .
5.
COST CONSIDERATIONS
The benefit derived from ranked set sampling depends, of course,
not only on the amount of increase in precision of the estimator, but
also on the relative cost of the two methods of sampling.
Dell and
Clutter define:
c
s
- the cost of stratification associated with each quantified observation.
This is the cost of drawing m·l
elements and judgment ordering the m sample elements.
c
q
- the cost of drawing and quantifying 1 element (without
classifica~ion).
Then the cost of a random sample of size mn
ranked set sample is mn(c +c ).
q
s
=
N is Nc , and of a
q
They then consider an 'expression
for the relative efficiency (RE) of the two procedures, given as
the ratio of the variance of the estimate from a random sample and
that from a ranked set sample, when the total cost C is held constant for the two procedures.
They show that
= cq 'a 21
/ -m
m
2
\' (c q +c s )a[ r:m ]
L.
r=l
1 ~ 2
-1
cq
= (1 - ~ L. T[r:m])
-c-+~cma r=l
q s
where T[r:m] = ~[r:m] - ~.
if
So
1 m 2
\' T
- 2 L. [r:m]
ma r=l
~
c
_-,-s_
c +c
q s
Since the cost of stratifying a random sample involves choosing
99
m-l observations and ordering m, I have modified Dell and Clutter's
cost function to reflect the fact that c s should be dependent on m.
Let c
s
= (m-l)c*.
Then the cost of choosing a ranked set sample of
s
size mn is mn(c +c ) = mn[(m-l)c*+c ].
s
s
q
In that case,
q
1 m 2
-1
c
= (l - - 2
T [r :m] )
"'Cm---=-l')~~*;:-+-c
rna r=l
s q
r
and
c*
if 2.
c
< _1_
- m-l
q
2
Consider the "s -type" estimators of variance discussed in
Section 2.3.
We know that
2
1
1
Var sN = mn ~4 - mn a
. and
4
1
+ OC(mn):2)' where ~4 = R(X-~)
4
2
sN is the sample variance from a random sample and
1
~
1
2
221
= mn ~4 - --2-- L (a[r:m] + T[r:m])
m n r=l
+ O(
(mn)
2)
Thus for large samples
1
m
t
m r~l
c
= _---...;q:L.-__
C
q
+(m-l)c*
s
*2
sN' s [m]n
) > 1
-
2
[r:m]
+T
2
[r:m]
1 rm
a2
+T 2[r:m]) 2
~4 - -m r=l ([r:m]
Therefore
RE( 2
(a
if
c*
a 4 _ 1.. t m (a 2
+T 2
)2
s < _1_ [ _ _:::.m_L..::.r_==..1--!..[~r.::..;::m~]-....J[l...::r-:..: m;;:;;.],,-- ]
c - m-l
4
q
~4 - a
)
2
100
,e
and if n is sufficiently large.
If l.l , l.l ,
x
°x2 , °y2
Y
are known, we saw in Section 3.1 that we could calculate
an MLE of p from either a modified ranked set or a random sample.
Then from (3.4) and (3.5), we see that for large samples
=
c O_p2);. [(m-I) c*+c ] (1_p2)
q
s q
l+P
2
2
EU[m:m]
c
=
where U ~ N(O,l).
2
2
9
(l-P 2) EU 2
(m-l) c*+c
[m:m]
s q l+p
(5.1)
Thus for large N,
If only l.ly and 0y
are known, then the MLE's can still be computed.
(See Section 3.2.)
From (3.11) and (3.12), we see that for large
values of N,
=
cq (2- P2) /
2
[2(1-p 2)+p 2EU[m:m]l[(m-l)c;+C
g]
2
c
= -:----:-::.....:g~-
(m-l)c*+c
s q
2EU2
[m:m]
EU 2 . (2- 2)
[m:m]
P
2(1-p
2
)+p
2
2
(5.2)
EU[m:m]
Thus for large N,
c*
.
2 O_p2) (EU 2 . -1)
if 2.. < _ 1 _ .
[m:m]
c - m-l 2(1- 2)+ 2EU 2
q
. p
p
[m:m]
If the form of the distribution is known
~nd
if perfect ranking
101
is possible, one can determine if ranked set sampling is beneficial
as far as relative efficiencies for equal cost are concerned.
Tables 5.1 and 5.2 display RE(XN,G(m)n) and
RE(S~,S(~)n) respec\
tively for several values of m and c */c for the normal and uniform
s
q
distributions.
'" "'/1
Tables 5.3 and 5.4 display RE(PN,P(m)N)
(all
~
~/1
parameters known) and RE(PN,P(m)N) (only
2
~y,ay
'"
known).
~
"'/1
~/1
One can observe from the tables that RE(PN,P(m)N) and RE(PN,P(m)N)
begin to decline with m for P = ±.5 and P = ±.75 respectively when
cs*/c
q
= .1.
This is true not only for these particular values of
P and c */c , but for all P, c */c
s
s
q
= O(JI,n
q
m)
~
0, since, from (5.1),
-+
0
Oem)
as m -+
00,
and from (5.2)
~
~/1
as'm
RE (PN' P (m) N) =
since
EU~m:m)
-+ 00 ,
= O(JI,n m )
However, if the major expense in the stratification step is the
ranking process itself, rather than choosing the (m-l) observations,
it is quite likely that
C
s
is not a linear function in m of c;, but
rather increases more slowly with m.
If this is the case, the rela-
tive efficiencies may not decrease to O.
A
"'/1
then RE (PN' P (m) N)
~
~/1
and RE (PN' P (m) N)
-+ 00
In fact, if c s Ic s * = o(JI,n m),
-+00
as m -+
00
If the form of the distribution is not known, or if the samples
are judgment ordered, then a rough idea of the benefit derived from
ranked set sampling may be obtained in some cases by estimating the
relative efficiencies.
X = 1:.. t n X
r
n Li=l [r:m],i
tm
- '"
2
For example, let A = Lr=l(Xr-~[m]n) where
and let
B = t n t m (X. -X ) 2 .
Li=lLr=l rl r
Then
102
EA = m-1
nm
So A -
m
\ (0 2
[r:m]
r=l
l
+ T2
)
[r:m]
IT!
2
L o[ r.m
. ]
and EB = (n-1)
.
r- 1
(m-1) B is an unbiased estimator of
nm(n-1)
\m
2
lr=l T[r:m]'
is not known, it too must be estimated, and the, two estimates can
~ 1. Similarly, if P1m)N
be used to get an idea if RE(XN,G[m]n)
~nd
P1m)N-
are based on judgment order statistics, then
(O~[r:m]
in (5.1) and (5.2) is replaced by
.
f
.
th e- expresslon
orI
reta 'lve e ff"lClenCles.
is an unbiased estimator of
O~m:m]
+
+
T~[r:m])/O~
Then
T~m:m]'
EU~m:m)
in
_lIN (Y
_)2
~
N i=l [m:m],i y
allowing us to estimate
the relative efficiencies.
Table 5.1
A
RE(~'~(m)n)
r'e
Normal
Uniform
m
c s* Ic q= .01
2
1. 33
1.59
1.49
1.37
3
1.45
1.88
1.96
1.67
4
2.28
1. 81
2.43
1.93
5
2.66
1.98
2.89
2.14
7
3.40
2.25
3.79
2.50
10
4.40
2.52
5.05
2.89
c Ic = .1
,s* q
c
s*
'Ic-~
q
.01
'C 5* Ic q= .1
--1
Table 5.2
2
2RE(sN'SCm)n)
Normal
"
""
e
Uniform
c s* Ic q= .1
C5* Ic q= .01
c 5* Ic q= .1
m
c s* Ic q= .01
2
.90
.91
.99
.91
3
1.06
.90
1.09
.93
4
1.15
.91
1. 21
.96
5
1.23
.91
1.35
1.00
7
10
1. 37
1.62
.91
.93
1.62
2.02
1.07
1.16
103
e.
Table 5.3
"
,.
AI1
RE (PN' p(m) N)
P
m c s */c
. q=.01
=0
P =±.75
P =±.5
c s */c q=.1
c s */cq=.01
c s */c q=.1
c s */c q=.01
c s */c q=.1
2
.99
.91
.99
.91
.99
.91
3
1. 25
1. 07
1.15
.97
1. 06
.90
4
1. 51
1.19
1.29
1.02
1.12
.89
5
1. 73
1. 29
1.42
1.06
1.17
.87
7
2.09
1.39
1.63
1.08
1.26
.84
10
2.49
1.43
1.86
1.07
1.36
.78
Table 5.4
m
c s */cq=.01
c s */c
.. q=.1
c s */cq=.01
c s */c q=.1
c s */c q=.01
c s */c q=.1
2
.99
.91
.99
.91
.99
.91
3
1. 25
1.07
1. 20
1.02
1.13
.96
4
1. 51
1.19
1.40
1.11
1.24
.98
5
1. 73
1.29
1.55
1.15
1.32
.98
7
2.09
1.39
1. 78
1.18
1.42
.94
10
2.49
1.43
2.00
1.15
1.49
.85
APPENDIX
Theorem A
[3]
Let XI"",X n be a random sample of iid random
variables, each with p.d.f.
fe(x), SEe.
Consider a transforma-
tion from the set of variables XI"",X n to a new set of variables
~l""'~n-l,e.
Let J be the Jacobian of the transformation.
Let
g,h be probability density functions s.t.
n
'i~lfs(Xi)IJI = gs(e)h(~I""'~n_Ile,s)
Suppose that for almost all values of x,e'~l""'~n_l' the partial
derivatives
I o~. Sf/
where b(S)
< FO(x) ,
I'
= E(S-S).
I o~he I
l:
I'e)
< HO(l:sl"",sn_l;
The sign of equality holds here, for every SEe,
when and only when the following two conditions are satisfied whenever
ge(8) > 0
A)
h(~I""'~n_Ile,S) is independent of e.
B). We have d ~~ g = k(e-e), where k is independent of e, but
may depend on
e.
For a rigorous proof of this last statement, sec Wij sman [23].
105
Let Xl"",Xn be a sample of size n where fa(x) is the
Theorem B [3].
p.d.f. of X..
1
(1)
Suppose that the following conditions are satisfied:
f a 2 tn f
For almost all x, the derivatives a tn
aa ' aa 2
exist va
(2)
a 3 tn f
aa 3
belonging to a non-degenerate interval 0.
For every aE0, we have
l~al
< Fl(X),
Ia
2
f, < F (x), and
2
aa 2
the functions F and F being integrable over (_00,00),
l
2
while fooH(x) fa (x) dx < M, where M is independent of
_00
(3)
.
For every
eE-0 /00 l(a
_00
(A. I)
e.
tn f 2f dx IS
. f"Inlte an d posItIve.
..
aa
J
Then there exists a solution to the likelihood equation, and it converges in probability to the true value of
e
as n
-+
00.
This solution
is an asymptotically normal and asymptotically efficient estimator of
e.
LeGam [14] has replaced the conditions of (A.I) with a set of
less stringent conditions which do not involve or imply the existence
of derivatives higher than the first.
Theorem
C ([ 13],
p. 276 and 285).
Let T1 = {TIN}
and
T2 = {T2N} be
two sequences of test statistics for testing HO: a = eo against
HI: a
~
eo' each based on N observations, where N
~(TiNla) = ET
0(T
iN
le)
= 1,2.
a,
00.
Let
when the parameter is a ,
= standard
is
for i
iN
-+
deviation of TiN when the parameter
106
~e
Suppose that for all k > 0, i=1,2:
lim
l.
a~ ~(TiNI8=80 + kilN)
a
ae(T iN ' 8=8 0)
N~
•
2.
lim
a
a8 ~(TiNI8=80
lim
N~
4.
kilN)
= c.1
IN aCT iNI 8=8 0)
N~
3.
+
a(T iN I8=8 0
+
= 1
kilN)
= 1
a(T iN I8=8 0)
lim Pr[
T'N-~(T.NI8=80 +
1
N~
kilN)
1
a(T
iN
/8 0
kilN)
+
~ z18=8
0
+
kilN] = G(z)
for a fixed continuous distribution function G.
Then
1.
Show that under appropriate regularity conditions
IN*(d) = N E(ff')2 + N(m-I) E( f2 )
F (I-F)
a2
a2
Proof:
IN(a)
= E fa J/,n L*.) 2
~ a8
where
J/,n L* = -nm J/,n a
n
+
m
L I
i=I r=l
n
+
m
L I
n
+
I
m
I
X..-~
J/,n f( (r:m),l
i=l r=I
a
x-~
(r-I)J/,n F( (r:m)
+
)
a
x-~
(m-r)J/,n[1 - F( (r:m)
i=l r=l
•
J
terms not involving a.
a
J]
107
So
d
dO .l/,n L*
f' (U .)( - U .)
f (U .) (- U .)
r1
r1 + \ \
r1
r1
feU .)
L L (r-I)
F(U .)
i r
r1
i r
r1
= -nm
+ \ \
L L
o
•
feU .)U .
\ \
r1 r1
+ ~ L(m-r)I_F(U .)
1 r
r1
Suppose~L*
is such that
Then we have
d
2
f' (U )
L \ f "(Un .)U2.
n I L \ 1ri
J2
f(U ri ) - 0 2 i ; L f(Uri ) Uri
.l/,n L* = nm
d0 2
02 i ;
2
f' (U .)
+ 02
?1 rL f(Ur~)Uri
r1
2
+ --.
feU .)U .
LL
r1 rl
.
F(U .)
o
2
1
r
2
1
+
"2
0
?
2
LL
-"2. (m-r)
rl
0
1 r
_ --!. L L(m-r)
0
2
2
rtJ2. f' (U .)
Url.f (Url.)]
L(r-I) I-r~(u .)'1
F 2 (U .)
1 r
r1
r1
i r
feU .)U .
rl r1
I-F(U .)
rl
-u 2 • f' (U
In
1-
.)
rl
1 ... F (U .)
rl
2
U £2(U )
\' ri
ri
L
.
r I-F (U
1
2J
-.)]
r1
So
d2 .l/,n L*
-E~---
d0 2
'U 2
f
= mn + mn E(-)
02
02
f
.
fllU 2
_ mn E(--)
02
f
U2f 2
nm(m-l)
2nm(m-l)
E(--)
13 (Uf)
+
2
2
F
0
0
22
2nm(m-I)
nm(m-I)
2
nm(m-l) E'(~)
+
E(Uf) +
E
(f'
U
)
2
+
2
I-F
2
0
o
0
e
...
107
{
e
nm[l
2
=(i
2
(1) [ U f 2 ]
f "U
E (_ _)] + nm mf
02
E pel-F)
(f') 2
+EfU
(A.2)
2
f"U
If we assume that two moments exist, we can show that E(--2--)
00
E
f"
r
2
U =
f
J
2.
00
f" (u)
r
2
feu) u f(u)du
Ju
=
2
f"(u)du
_00
-00
00
=
u 2 f'(u)
1+
00
_00
-
f f'(u)u du .
2
(A.3)
_00
Consider the first term of (A.3)
00
00
2
2
u f'(u)du = u f(u)
f
I:: - 2 f uf(u)du
_00
00
_00
since first and second moments
lim
··e
<
2
u f' (u)
=
exi~t.
Therefore,
2
lim
u f' eu) =
o .
u+- oo
u,"*+oo
Now consider the second term of (A.3)
00
00
f f' (u)du = uf(u) 1+
r
00
_00
_00
J
f(u)du
=o-
I
_00
since the first moment exists.
Therefore, we have
E~
f
U
2
= -2 (-1) = 2
and from (A.2).
2.
We now show that Tiku's approximation allows an explicit approxi-
mate solution to equations (3.13) to be found when m is large enough.
Recall that this approach is based on Tiku I s linear approximati on of
the function
fez)
= ¢ (z)
<I>
,.,
where ¢,
<I>
(z)
are the p.d.f. and c.d.f. of the unit normal distribution;
108
e
i.e., we have
g (z)
RJ
a + Bz
)
B = g(h~=f (t) , a :: g (h) -hB,
where
Note that (Y
Q.nd (t,h) is an interval containing z.
.-~
ml
y
)Iay and
- (Y 1, -~ y )/aY have
the same distribution;
that of Z('m:m )' where
,
"
I
Z ~ N(O,I).
Note also that(ep(z))/(I:"'CP(z)) = g(-z).
Now if m is fairly large, then Var Z(m:m) is fairly small (for
m=IO, say, Var ZOO:IO) = .344).
Then (EZm-KlVarZ m, EZ m + K/Var 2 m)
will have a high probability of covering (Y m-~y) lay and
- (Y i -p y ) lay
if K is large enough '(but still small enough to keep the linear approximation valid).
Then we see that (3.13) simplifies to:
1\ = p
xa
x
y - ~
_ _,,-Y
a
(A.4)
y
and
Y-~
-----:L
ay
=p
{N/2
L
X-~
2
__
x + (m-l) (l-p ) ,
[(a -(3
ax
N
'1
mm
1=
. ~11
y
- (exm+ (3m . ' m~ ,y)]
y
= P
X-~
a
2
x
+ (m-I) (l-p )
Na
x
y
YI · -~
1
y)
ay
}
.
B
N/2
L [2~
, m'·
1= 1
y
~ (Y . + Y .)]
I1
ml
(A. S)
After substituting (A.4) into (A.S), we have
Y(l-(m-l)B) = ~, (I-(m~I)Bm) ,
m
y
yielding the only solution to (A.4) and (A.S) as
'"
~
Y
=
Y,
~x :::
X
1
J
109
'1
e
Table A.l
A
A
Means and Variances of p
•
.0
N
E
A
A
20
A
A
Var p
P
A
A
/\
EP
m
A
A
A
A
Var p
Ep
Var p
p = 0
5
-.024
.056
-.006
.027
-.007
.019
7
-.003
.053
-.004
.025
.003
.015
10
-.004
.049
-.001
.019
.005
.013
P
=
.5
5
.497
.034
.492
.016
.498
. OIl
7
.513
.031
.511
.016
.509
.010
10
.520
.027
.509
.012
.508
.008
p
·e
,J
30
A
= .75
5
.750
.014
.742
.006
.750
.004
7
.760
.013
.760
.006
.755
.004
10
.765
. OIl
.756
.005
.754
.003
Table A.2
A
Ratio of Cramer-Rao lower bound of Var p from a
A
random sample to Var
N
20
10
5
1. 78
1.84
1.76
7
1. 89
2.02
2.20
10
2.06
2.61
2.55
P
,e·
30
p = 0
m
.,
p
=
±.5
5
1. 63
1. 80
1.66
7
1. 80
1. 81
1.86
10
2.06
2.43
2.25
110
Table A.2 - Cont'd
N
10
20
30
e
'},
P = ±.75
ill
5
1.35
1. 52
1.53
7
1. 47
1.65
1.60
. 10
1. 68
1. 96
1. 84
t
e
•
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•
j