k\1S 197IJ Subkct
Cla~3ifirations
Key Words and Phrases
Pri'-B.ry 62F07 Secondary 62Q05
Panking and Selection; Indifference zone; Subset . .
selection: ApproxiJ!lations.
* This research was supported in part by the 0ffice of Naval Research ContracL
N00014-67-A-0226-00014 and by the National science Foundation Grant 563080-13995, both at Purdue University.
SOME
APPROXI~~TIONS
IN SELECTION THEORY
by
Raymond J. Carro11*
Department of Statistics
University of North Carolina at ~aapeZ Hill
Institute of Statistics Mineo Series #957
October, 1974
SOME
AP~KOX!~~TIO~S
1M
SEL~CTION
THEORY
by
Raymond J. Carroll *
Department of Statistics
University of Nort;71, Carolina
SUMf-1ARY
Simple approximations are given for the following:
of correctly selecting the largest mean from
(1) the probability
K normal populations with
unknown variance using a subset selection approach
cow~o~
(2) the same probabilities
for selecting the largest or smallest normal variance using either the subset
selection or indifference zone approaches.
One class of approximations
invclve~
computing tail probabilities of the beta distribution, while another (less
efficient) class may be
co~puted
by hand.
The bounds are evaluated and shown
to compare quite nicely with known exact results.
AMB 1970 Subject Classifications
Key Words and Phrases
Primary 62F07 Secondary 62QOS
Ranking and Selection; Indifference zone; Subset se1ectic!)
Approximations.
* This research was supported in part by the Office of Tlaval Research Contract
1"lOOOl4-67-A-0226-000l4 a.l1d by the National Science Foundation Grant 563')-8013995, both at Purdue University.
-2-
1.
INTRODUCTION
Dudewicz (1969), Dudewicz and Zaino (1971), and Ramberg (1972) considered
the problem of approximatiTIg the probability of correctly selecting the largest
mean when one is considering
K normal populations with common known variance
using Bechhofer's (1954) indifference zone approach; they then used their inequalities to derive explicit approximations (depending on the inverse of the
standard nomal c.d.f.) to find the sample size necessary to guarantee a probability requirement outside the indifference zone (see also McDonald (1°71)).
In this paper, explicit upper bounds on the probability of a correct selection
are obtained for selecting a subset containing the largest or smallest normal
mean when the common variance is unknown using Gupta's (lS65) rule, and for the
probability of correctly selecting the largest or smallest normal variance
using either subset selection or the indifference zone.
only easy
These bounds are not
to obtain, but some of them may be computed by hand.
Using them,
approxinations are given for the various constants necessary in each
and, when compared to explicit tabulated
favorably.
are shown to behave quite
The method of proof is simple and relies on two facts;
If
(1.1)
X
is a r.v. and
If
Xl' X2 , ...
of any
n
1
E
lJ
are exchangeable r.v. 's (i.e., the joint distribution
~),
m of them depends only on
~A
P{X.
r > 0
e -ra ~( e rX) .
P{X > a}
(1.2)
results~
problem~
A.,
1.
then for measurable sets
,
i=l •... ,n}
~
n
IT
i=l
n
P(X.
1.
E
A.)
1.
~
1 -
I
i=l
P(x. ~ A.) .
1.
1.
-3-
(1.2) is given in Dykstra, Hewett, and Thompson (1973).
the beta censity with
(a,S)
degrees of
freedo~
Finally, recall that
is
O<x<l,
and that if VI' V2
parameters
with
(r , r
l
are independent standard
has a beta densit,
Z) degrees of freedom.
(unknown) ordering,
t~e
of a correct selection
M~ANS
PROBLEM
are normal populations with !'leans
III' .•. ,Ilk
a common unknovffi variance
P*
(J
2
If
lJ
[1]
~
(p(eS))
(2.1)
r
(J
2
for
~1ich
the probability
If
one takes a sample of size
Xl' ... ,5[,:l.. , and if
rS
such that
2
r
Thus, one wants to find
~
max
x. -
l~j~k
c
the parameter configuration
J
given
lJ
l , ...
n
frolJ.
is an
has a chi-squared distribution
degrees of freedom, Gupta (lq65) proposed to eliminate
x.1
and
denotes the true
~
~l
lJ 1 , ... ,lJk
is at least as large as a specified constant
each population anc foms the sal'lple neans
independent estimate of
...
goal is to find a rule
for aU 'parameter configurations.
with
random variables with shape
and
2. THE BOUNDS FOR THE NORMAL
Suppose
Ga~~a
II.
1
if
cS r n -1/2
k,r,P * , such that
,lJ
k .
peeS)
~
P* , whatever
-4-
Define
(2.2)
Lemma 2.1.
(2.3)
A(P * )
= 2(P*)1/k-l -
B(a.,S,y)
= Pr{Beta r.v. with
F(a.,(3,y)
= Pr{F r.v. with
(a., S)
d.f.
(a., 13) d.f. ~
~
y}
y} .
Using the rule given by (2.1),
P{CS}
(2.4)
~
{I - ~ (1 _ F(1,r,c 2/2))}k-l
~
{I - 2- 3/2 c(l + c 2/2r)-(r+l)/2 (l + l/r)
~ {1
Hence, the
1.
valt.~e
(r~1)/2}k-l
2
}k-l
- (2- 1/2 (1 + c-/4r))-r/7.
of
c
for
w~lich
inf peeS)
= P*
satisfies
(2.5)
(2.6)
where
Proof:
where
Co
is the value for which
Letting
VI
IS
(2.4) equals
P*
denote an incorrect selection, one finds by using (1.2),
and Vz are independent chi-square r.v. is with
respectively.
Thus,
I
and
r
d.f.
-5-
Now, by using (1.1),
co
= ~.
f P{bV l
~
o
2
c xb/2r}dP{V 2
00
J e-xc2b/2r
o
:S ;
(1 - 2b)-1/2
=1
(1 _ 2b)-1/2 (1
2
x}
:S
+
dP{V
2
:S
x}
c 2b/r)-r/2 .
The last term achieves its minimum at
b
=
(c
2
2
- 2)/[2(c /r)(1
which yields the two bounds on
Note that the value
Co
r)] ,
P{CS} .
is easily cOMputed using a method of bisection if
one has extensive log and exponential tables
calculator.
+
OT
a reasonably sophisticated
Note also that these approximations hold for the dual problem of
selecting the smallest mean.
Remark 2.1. The case where n
=1
independent replication ( and hence
For
ex&~ple,
and
r
s;
is formed by a small amount of
is small) is certainly non-trivial.
in any large scale experiment with interactions where one wants
the largest cell mean, taking more than one observation from each population
may be infeasible economically; thus, the case where
r
is small
(:S
60)
is of interest.
k
is large
(~SO)
but
No tables are available for this problem,
but the approximations (see Tables la and Ib) seem to perform quite well.
-6-
3. THE BOUNDS FOR THE GAMMA SCALE PARAMETER PROBLEM
Suppose that
f '-i -( x)
=
TIl' ... ,TIk are Gamma populations with densities
..
, -xIS .
r1.
I.
( r(r) 13 r)-l
X
e
, x > 0 (i = 1, ... ,k) , where the shape
i
parameter
r
is
kno\~
(if
s.I. = 1
, the density is called a standard Gamma).
The goal is to find subset selection and indifference zone rules for selecting
13 I.. . The best set of tables
either the largest or smallest scale parameter
for selecting the largest
S.1
are due to Gupta (1963), while Gupta and Sobel
(1962) have tables for selecting the smallest
v
"1'
...
v
,A
f ,
follow the densities
k
1
...
It will be assumed that
R••
I.
,.L./:k
C-F
I.~
samples of size n were
available from each population, then nX ,
,nXk would still have Gamma
l
'.J
densities with shape parameter nr , so it suffices to conSI._,er
n = 1 ) . The
problem of selecting the largest
13[1]
$
13[2]
...
~
$
B[k]
Pi
will be considered first.
denote the correct (unknown) ranking of the
I.
the first integer so that
P{CS
I
r
= r O'
13(1]
solved assuming n
and (assuming
n
Since
c
=1
nr
= 1.
= 1 with r known) selects
max
l~j~k
X.
J
0
TI.
I.
0
$
1
if
(0 < c < 1)
in (3.1) corresponds to the indifference zone rule, one can get
answers to both problems by using (3.1) with
~,d
Thus, this problem may be
The subset selection approach assumes that
c
is
n
r O ' where
is as large as
= ... = S[Ic-l] = o8[k]} = P*
x.I. ; .:
(3.1)
where
which gives rise to the largest of
II.
parameter~
and selects the
The indifference zone formulation assumes
population
Let
= 1 simultaneously). Let
c
~
1,
0
~
1
(but not
c
=1
-7-
A (P*)
2
A (P*)
3
(3.2)
lemma 3.1.
=1
_ (p*)l/k-l
= (A 2 (p*)fl/r .
Using the rule (3.1),
?
-r
- {(I + co)-/4co}
k- l
J
,
so that in the subset selection case,
co
(3.5)
(3.6)
~
* )},
B- l (r,r,A (p*))/{1 - B-1 (r, r ,A (P)
2
Z
~
2
*
2 A2 (P*) - 1 - [(2 AZ(P ) .. 1) 2 _ 1J1/
and in the indifference zone formulation
pees I
&) ~ P*
when
(c
= 1)
, the smallest
r
such that
satisfies
(3.7)
(3.8) r 1
= first
number
r
such that
B(r,r.o/(l
(3.9) r Z
= first
number
r
such that
r
+
0)) ::; AZ(P *)
~ - log Az~P*)/ldg (1+0)2/ 40
Proof: Again, the proof is a simple application of (1.1) and
(1. 2) .
From (1. 2), if V1 ,and V_ are i.ndependent withf,ie,nsJties
2
(r (r) r~.xr-1e-x ;, then
-8-
(3.10)
co
!(-l
= [1 - "D(r,r'l+Co)]
.
This gives (3.3).
From (1.1),
00
(3.11)
P{V
2
~
coV }
k
= (1
~ f
e-x/a(l - co/a)-rdP (V
o
(1
+
which gives (3.4).
2
co) /4co
~
a
>
(3.8), do
T~e
req~ire
x)
for
Co
<
a
Co ,
(1 - co/a) (1 + l/a) ,
(3.5) and (3.6) are now immediate.
now follow from (3.3) and (3.4) with
Remark 3. 1.
~
- co/a)-r(l + l/a)-r
A sinple argument shows that for
(3.12)
2
c
(3.7), (3.8) and (3.9)
=1
approximations (3.5) and (3.7), while better
th~,
(3.6) and
the use of a computer routine to calculate the Beta distTi-
bution, while (3.6) and (3.8) may be done by hand.
Note also that the results
permit the use of a mixture of the subset selection and indifference zone
approaches.
The corresponding problem of selecting the SWRllest
rule that selects
I
S[i]/S[l] ~ 0 1 ~ 1
ously).
would use the
if
x.I
(3.13)
with
1T.
S.I
~
c
l
(but
min
v
A.
l~j~k
c 1 = 1.
J
(l
and
c )
1
~
0
1
1 =
may not happen simul tane-
For exact results, these rules are Gifferent and cannot be solved by
-9-
solving the largest parameter case: however, the approximations are symmetric
in that in
Le~~a
3.1, if one sets
°
= 1/° 1
and
c
= l/c l
' one gets appro-
priate bOQ'1ds.
Of course, the proble!!" of selecting norr'a1 variances is covered by these
results.
Remark 3.2. The simple method given here also can be used to yield approxiwations for the indifference zone
soallest)
~ince
B.
1.
proble~
of selecting the
t
largest (or
\'lhen
tables are not generally available for this problem if
Carroll, Gupta, and
r~ang
t~2
(see
(1974) however), the approximations so derived waule
be of interest in practice.
4. TABLES AND EVALUATION OF THE BOUNDS
The reSults of Sections 2 and 3 gave bounds using twa methods; Method I
uses (1.2) and requires evaluation of Beta probabilities, while Method II uses
(1.1) and gives bounds which may be conputed by hand.
uniformly better than the Method II bounds.
The Method I bounds
ar~
Letting Problem 1 be that of
Section 2, Problem 2 that of selecting the largest or sMallest gamma scale
using subset selection, and Problem 3 that of selecting the largest or
gamma scale using the indifference zone, Tables 1
t~rough
smalle[~
3 lead to the
following conclusions:
(4.1)
Method I bounds are very good for all three problems, and get better as
as either
r
or
P*
increase.
It appears hOlllever, that as
k
-10-
.increases, the. ratio of the Nethod' I bounds to· the .exact :~re~n:l1ts also
increases slightly.
Thus, i f
is the largest
covered by
k
existing exact tables, and if the ratio of the bound to the exact
results is
(4.2)
a, one might, for
k
>
ko
' c1.ivide the Methoc I bound by
?
Method II bounds perform uniformly well in Problem 1 and quite well
for Problem 2 i f
conservative.
~
r
10.
For Problem 3, they seem to be somewhat
For Problem 1, the ratio of Method I bounds to the exact
results decreases as
ratio decreases as
r
r
1
or
k, or
P*
P*
increase.
For Problem 2, this
increase, but increases as
if one wants to select the largest gamma scale parameter.
k
increases
In this
latter case, one should use the method suggested in (4.1).
(4.3)
Conclusions about the bounds from Problem 3 are difficult to mm:e,
since this author
for
as
0
~
.50.
0, or
applied.
can derive exact results from existing tables only
It may be that the Method II bounds actually get worse
P* increase, so that the suggestion in (4.1) might be
The Method II bounds seem to get worse as
k
increases if in
Pro1:len 3 one wants the largest parameter, but better otherwise.
(4.4)
Recall that the Hetholl II bounds (2.6), (3.6), and (3.9) Jrlay be compute}.
by hand.
Their good perforBaTIce in these problems will be
quite
helpful.
Tables la and lb compare the bounds given by (2.5) and (2.6) respectivel
with the tables of Gupta and Sobel (1957) for
k :: 2, 10, 18, 30, 50
and
P* = .90, .95, .99
r
= 16(4)24,
36, 40, 60, 120, 3f
The numbers in parentheses
are the ratio of these bounds to the exact results and are most informative.
-11-
Tables 2a and 2b are concerned with the bounds (3.5) and (3.6) respectively for the problem of
selectin~
the largest gamma scale parameter.
bounds are conpared with the exact results of Gupta (1963) for
4(2)10, 10(10)50,
k = 4(2)10
and
P* = .90, .95, .99.
v
These
= 2r =
The perform2nce of
the bounds of this paper for selecting the smallest gamma scale
para~eter,
while not included here, turns out to be even better.
Tables 3a and 3b give the performance of the bounds derived from (3.8)
and (3.9) for the indifferenc0 zone problem of selecting the largest gamma
scale parameter (again, these bounds will perform slightly better if one wishes
to select the smallest scale parameter).
The exact results were derived from
Gupta (1963) and Gupta and Sobel (1962); direct comparisons are fairly difficult since the range of
0
values one can obtain using the above tables does
not include anything for which
TI1US,
the bOillld given in (3.8)
and any
0 > .50 , which is the most interesting case.
seel~s
very practical for use where
8 > .5
k , and use of the suggestion given in (4.1) should improve the per-
formance of (3.9).
Acknowledgement.
I wish to thank Professor S. S. Gupta for his support
during the writing of my doctoral dissertation, where
Method II was studied.
-12-
REFERENCES
[1]
Bechhofer, R. E.
(1954).
A single-sample multiple decision procedure
for ranking means of normal populations with known variances.
Statist. 25, 16-39.
Ann. Math.
[2]
Carroll, P.. J., Gupta, S. S. and Huang, D. Y. (1974). On selection procedures for the t best populations and some related problems. ~meo­
graph Series #369~ Purdue University.
[3]
Dudewicz, E. J.
problems.
An approximation to the sample size in selection
Ann. Math. Statist. ~O, 492-497.
(1969).
[4]
Dudewicz, E. J. and Zaino, N. A. Jr. (1971). Sample size for selection.
In Statistical Decision Theory and Related Topics. (edited by S. S. Gupta
and J. Yackel), 347-362, Academic Press, IJew York.
[5]
n. A. Jr. (1973). Events
which are almost independent. Ann. Statist. !_, 674-681.
[6]
Gupta, S. S.
populations.
(1963). en a selection and ranking procedure for gamma
Ann. Inst. Statist. Math. 14, 199-216.
[7]
Gupta, S. S.
(1965).
Dykstra, R. L., Hewett, John E., and Thompson,
rules.
[8]
On some multiple decision
Teahnometrias.
I, 225-245.
Gupta, S. S. and Sobel, M.
(1957).
selection and ranking problems.
[9]
[10]
(selection and rankin;
On a statistic which arises in
Ann. Math. Statist.
~~,
957-967.
Gupta, S. S. and Sobel, M. (1962). On the smallest of several correlated
F statistics. Biometrika. 49, 509-523.
McDonald, G. C.
(1971).
en approximating constants required to implement
a selection procedure based on ranks. In Statistical Decision Theory and
Related Tonics. (edited by S. S. Gupta and J. Yachel) , 299-312, Academic
Press, New York.
[11]
Rambe~g,
Statist.
J. S.
(1972).
43, 1977-1930.
Selection sample size approximations.
Ann. Math.
-13-
TA8LE la.
Values of the Bound given
~
2
10
(1. 00)
(1.00)
(1. 00)
3.54
4.04
5.14
(1.10)
(1. 07)
(1. 04)
1. 87
2.43
3.57
(1. 00)
(1. 00)
(1. 00)
3.47
3.94
4.95
(1.10)
(1. 07)
(1.03)
5.34
1. 86
2.42
3.52
(1. 00)
(1. 00)
(1. 00)
3.42
3.87
4.84
(1. 09)
(1.06)
(1.03)
36
1.84
2.38
3.44
(1.00)
(1.00)
(1. 00)
3.35
3.77
4.65
40
1.84
2.38
3.42
(1. 00)
(1. 00)
(1. 00)
1.83
2.36
3.38
20
24
60
120
360
a
(2.S)a,b •
18
1. 89
2.46
3.65
16
by
30
(1.13)
(1.09)
(1. 06)
4.34
4.83
5.92
(1. 07)
(1.12)
(1. OS)
(1. 05)
4.22
4.67
5.66
(1.14)
(1.10)
(1.06)
4.99
5.98
3.82
4.25
5.20
(1.11)
(1.08)
(1.13)
(1.10)
(LOS)
4.46
4.87
5.79
(1.11)
(1. 04)
4.14
4.57
5.50
(1. 09)
(1. 05)
(1. 02)
3.72
4.12
4.98
(1.10)
(1.07)
(1. 03)
4.02
4.41
5.24
(1.11)
(1.08)
(1. 04)
4.31
4.68
5.50
(1.13)
(1. 09)
(1. 05)
3.33
3.75
4.61
(1. 08)
(1. 05)
(1. 02)
3.70
4.10
4.94
(1.10)
(1.07)
t.oo
4.38
5.19
(1.11)
(1. 08)
(1. 04)
4.28
4.65
5.45
(1.13)
(1.09)
(1.05)
(1. 00)
(1. 00)
(1. 00)
3.29
3.69
4.51
(1. 08)
(1. 05)
(1.02)
3.64
4.02
4.81
(1. 06)
(1. 02)
3.93
4.29
5.06
(1.11)
(1.07)
(1. 03)
4.19
4.54
5.28
(1.12)
(1.08)
(1.04)
1.31
2.34
3.33
(1. DC)
(1. 00)
(1. 00)
3.24
(1.07)
(1. 04)
(1.02)
3.59
3.95
4.70
(1. 09)
(1.05)
(1.02)
3.36
4.20
(1.11)
4.11
(1.06)
4.92
(1. 03)
4.44
5.14
(1.11)
(1. 07)
(1. 03)
1. 80
2.32
3.31
(1.00)
(1. 00)
(1. 00)
3.22
3.60
(1. 07)
(1. 04)
(1. 02)
3.55
3.91
4.63
(1. 03)
(1. 05)
(1. 02)
3.82
(1. 09)
4.14
4.84-
(1.06)
(1.02)
For each
3.63
4.42
4.36
3.98
4.47
5.56
3.88
4.3L1·
r ~ the first row is for
and the third for
P*
(1.03)
(1. 09)
p*
= .90
(1.15)
(1.11)
50
4.70
5.18
6.28
(1.18)
(1.13)
(1. 09)
4.55
(1.16)
(1.12)
(1.07)
(1.15)
(1.06)
4.05
4.33
5.03
, the second for
(1.10)
(1. 07)
(1. 03)
P*
= .95
,
= .99
b The numbers in parentheses are the ratios of the bound in (2.5) with exact
results given in Gupta and Sobel (1957).
-14-
TABLE lb.
Values of the Bound given by (2.6)a 7 b •
\
\
lc
\"'.
r\
16
20
2
\
10
18
3.74
4.34
5.59
(1. 97)
(1. 75)
(1. 53)
5.48
6.01
7.24
(1.70)
(1. 58)
(1.47)
5.96
6.50
3.69
(1. 97)
(1. 74)
(1.51)
5.30
(1. 68)
(1. 57)
(1. 44)
5.74
6.21
7.30
(1. 66)
5.79
(1.66)
(1. 55)
(1. 42)
4.25
5.41
3.65
6.83
30
6.90
8.13
(1. 69)
(1. 57)
(1. 47)
6.75
7.30
8.55
(1.69)
(1.57)
(1.47)
6.03
6.57
7.64
(1.64)
(1. 56)
(1.43)
6.45
(1.56)
(1. 43)
6.92
8.01
(1.64)
(1.55)
(1. 43)
5.59
6.05
7.03
(1. 63)
(1. 54)
(1. 41)
5.92
6.37
7.35
(1. 62)
(1. 53)
(1.40)
6.25
(1. 61)
6.68
7.66
(1. 52)
(1.40)
5.33
5.76
6.61
(1. 60)
(1.50)
(1. 37)
5.67
(1. 57)
6.Q5
6.83
(1.49)
(1.36)
5.94
6.32
7.15
(1.56)
(1.43)
(1. 36)
5.61
(1.57)
(1.48)
(1. 36)
5.88
6.25
7.04
(1.55)
(1. 47)
(1. 36)
(1.54)
(1.45)
(1.32)
5.72
6.05
6.77
(1. 52)
(1. 44)
(1. 32)
(1.52)
(1. 43)
(1. 32)
5.56
5.87
(1.50)
(1.42)
(1. 31)
5.45
5.74
6.34
(1.48)
(1.40)
(1. 29)
5.39
5.67
6 .2~
(1.47)
(1. 39)
(1. 28)
7.73
(1.70)
(1. 58)
(1.47)
6.36
4.20
5.28
(1. 96)
(1.73)
(1. 50)
5.19
5.65
3.58
4.09
(1. 93)
(1. 71)
5.01
5.08
(1.47)
6.28
(1.63)
(1.51)
(1.38)
3.57
4.07
5.05
(1. 93)
4.98
(1. 62)
(1. 71)
(1. 47)
5.38
6.21
(1.51)
(1. 38)
5.32
5.70
6.54
(1. 59)
(1.49)
(1. 37)
60
3.54
4.01
4.94
(1. 93)
(1.59)
(1. 46)
4.37
5.23
6.01
(1.60)
(1.49)
(1. 36)
5.19
5.56
6.30
(1. 56)
(1.49)
(1. 34)
3.51
3.96
4.35
(1.92)
(1. 69)
4.76
5.12
5.41
(1. 54)
(1.45)
5.63
(1.45)
5.83
(1.58)
(1.47)
(1. 34)
5.07
120
6.08
(1. 32)
6.30
3.47
3.92
4.78
(1. 91)
(1. 68)
(1. 44)
4.70
5.03
5.70
(1. 57)
(1. 46)
4.99
5.23
(1. 33)
5.30
5.96
(1.53)
(1. 43)
(1. 32)
5.52
6.14
(1. SO)
(1. 41)
(1. 30)
3.46
3.90
4.74
(1.91)
(1.67)
(1.44)
4.67
(1.56)
(1. 45)
(1.32)
4.96
5.27
5.88
(1. 52)
(1.43)
(1. 31)
t; .18
5.47
6.06
(1.49)
(1.40)
(1.29)
24
36
40
360
00
6.65
5.41
4.99
5.65
50
5.99
6.79
5.47
5.81
6.54
5.32
6.50
aFor each r , the first row is for p* = . 9D , the second for p* = .95, and
the third for p* = .99.
b fhe numbers in parentheses are'the ratios of (2.6) with the exact results of
Gupta and Sobel (1957).
-15-
TABLE 2a.
Values of the Bound given
'""
v ~"k
4
by
(3.5)a,b •
3
10
4. ::>0000
0.12558 (1.26609)
0.08,388 (1.23980)
0.03513 (1. 22411)
0.09372 (1.440.39)
0.06307 (1.41118)
0.02682 (1. 37937)
0.07818 (1.57333)
0.05269 (1.55640)
0.02272 (1.49637)
0.06806 (1. 70445)
0.04623 (1. 64387)
0.01967 (1. 62714)
5.00000
0.17931 (1.17531)
0.14638 (1.14773)
0.07761 (1.12097)
0.05935 (1. 27389)
0.11882 (1.24558)
0.06417 (1.21547)
0.13873 (1. 3.5518)
0.10434 (1.32259)
0.05703 (1.26244)
0.12558 (1. 41738)
0.09489 (1.38048)
0.05214 (1. 32324)
3.00000
0.25298 (1.13051)
0.17931 (1.11499)
0.11821 (1. 032(3)
0.21148 (1.20581)
0.1672e (1.17764)
0.10137 (1.14431)
0.18897 (1. 26475)
0.15024- (1.22472)
0.09198 (1.18510)
0.17398 (1. 31053)
0.13873 (1.26868)
0.08561 (1.21484)
).00000
0.29743 (1.10613)
0.23933 (1.09052)
0.15413 (1.06405)
0.25452 (1.16691)
0.20719 (1.14387)
0.13557 (1.10644)
0.23114 (1. 20706)
0.18897 (1.13008)
0.12497 (1.13631)
0.21507 (1. 24611)
0.17667 (1. 21128)
O.117EO (1.15647)
:::.00000
0.43367 (1. 06071)
0.37588 (1.04320)
0.28200 (1.03190)
0.39083 (1. 09510)
0.34163 (1. 07426)
0.25992 (1. 05032)
0.36670 (1.11535)
0.32172 (1.09413)
0.24612 (1.06452)
0.34959 (1.13276)
0.30821 (1.10639)
0.23709 0.07555)
0.50866 (1. 04589)
0.45300 (1. 03753)
0.36125 (1.09620)
0.46863 (1.06695)
].00000
0.42074 (1.05290)
0.44430 (1. 03364)
0.40034 (1. J6909)
0.32428 (1.04539)
0.42867 (1. 09408)
0.38706 (1.07734)
0.31493 (1.05103)
0.49762 (1.06707)
0.45506 (1.05480)
0.38052 (1. 03280)
0.48245 (1.07577)
0.33812 (1. 03512)
0.55801 (1. 03762)
0.50644- (1.02678)
0.41583 (1.01964)
0.S198S (1.05607)
:'.00000
0.00000
0.59439 (1.03298)
0.54503 (1.02371)
0.45817 (1. 01491)
0.55801 (1.04837)
0.51423 (1.03650)
0.43669 (1. 02132)
0.47391 (1.04450)
0.39367 (1.02624)
0.53696 (1.05781)
0.49653 (1. 02906)
0.42272 (1.02906)
I
0.44175 (1.06170)
0.37036 (1.0395':)
0.52211 (1. 06491)
0.41338 (1.03171)
0.41388 (1.03171)
a For each r , the first row is for p* = .90 , the second for p* = .95,
and the third for p* = .99 .
b The numbers in parentheses are the ratios of (3.5) to exact results given by
Gupta (1963). v = 2r .
-16-
TABLE 2b.
Values of the Bound given
\.
\
by
(3.6)a)b .
\ k
v\\
10
4
6
8
4.00000
0.05133 (3.09740)
0.03486 (2.93339)
0.01489 (2.88734)
0.03897 (3.46429)
0.02663 (3.34252)
0.01146 (3.22880)
0.03258 (3.77528)
0.02232 (3.67324)
0.00966 (3.52010)
0.02853 (4.06533)
0.019.59 (3.87G08)
0.00850 (3.76508)
6.00000
0.09816 (2.36356)
0.07409 (2.26748)
0.04047 (2.14948)
0.03031 (2.S2780)
0.06105 (2. ~·2415)
0.03370 (2.31441)
0.07057 (2.66409)
0.05386 (2.56193)
0.02991 (2.40744)
0.06416 (2.77438)
0.04910 (2.66'787)
0.02737 (2.52061)
8.00000
0.14005 (2.04217)
0.11143 (1.97425)
0.06866 (1. 86426)
0.11395 (2.14385)
0.09534- (2.06640)
0.05940 (1.95.300)
0.10713 (2.23089)
0.03622 (2.13399)
0.05406 (2.01631)
0.09922 (2.29803)
0.08007 (2.197g6)
0.05042 (2.06272)
0.17648 (0.86421)
0.14501 (1.79983)
0.09604 (1.70757)
0.15335 (1.93671)
0.12691 (1.86746)
0.03499 (1.76498)
0.14021 (1. ge9S'1)
10.00000
0.11650 (1.91410)
0.07851 (1.80859)
0.13130 (2.04107)
0.10941 (1. 95601)
0.07405 (1.83655)
J.OOOOO
0.30324 (1.51696)
0.26690 (1.47621)
0.20543 (1.41657)
0.27673 (1.54666)
0.24500 (1. 4~7(6)
0.19036 (1.43412)
~.261l7 (1. 56606)
0.23201 (1.51718)
0.18126 (1. 44541)
0.25039" (1.53151)
0.22296 (1.529L!·4)
0.17486 (1.45835)
"J. oooeo
0.38037 (1. 39681)
0.34421 (1. 36545)
0.28018 (1. 41337)
0.35420 (1.41163)
0.32172 (1.37698)
0.26482 (1. 32567)
0.33835 (1. 42455)
0.30322 (1.33862)
0.25415 (1.33385)
0.32729 (1.43299)
0.29874 (1.39583)
0.24715 (1. 33028)
-1. 00000
0.43513 (1.33063)
0.39914 (1. 30281)
0.33512 (1.26523)
0.40899 (1.34233)
0.37683 (1. 31360)
0.31868 (1.26771)
0.39334 (1.34996)
0.36335 (1. 32105)
0.30860 (1. 27351)
0.38237 (1.35733)
0.35384 (1.32548)
0.30140 (1. 2773.5)
0.00000
0.47607 (1. 2(971)
0.44100 (1.26531)
0.37785 (1.23066)
0.45063 (1.29819)
0.41911 (1.27175)
0.36146 (1. 23388)
0.43532 (1.30477)
0.40532 (1. 27647.)
0.35136 (1.23304)
0.42456 (1. 30960)
0.39642 (1.28147)
0.34414 (1. 24076)
\
'-'-'
a
For each r , the first row is for
and the third for p* = .99 .
b
The numbers in parentheses are the ratios of (3.6) to exact results given
by Gupta (1963) .
v
= 2r
p* = .90 , the second for
p* = .95 ,
-17-
TABLE 3a.
= 2r
Values of v
.25
.30
.35
.40
.45
by
(3.8)a,b .
10
6
4
',,-
given
8
11
17
(1.00)
(1.10)
(1. 00)
10
13
20
(1. 25)
(1.08)
(1.11)
12
14
21
(1. 20)
(1.16)
(1. 05)
12
15
23
(1. 20)
(1. 07)
(1.15)
11
14
(1.00)
(1. 00)
13
17
(1. 00)
25
15
13
27
(1. 25)
(1. 12)
(1.03)
16
20
23
(1.33)
(1. 25)
22
(1. 08)
(1. 21)
(1. 04)
13
18
17
21
32
(1.21)
(1.05)
(1. 06)
19
24
29
(1. 08)
(1.12)
(1. 03)
35
(1.18)
(1.20)
(1.09)
21
25
36
(1.31)
(1.13)
(1. 05)
17
23
, 37
(1. 06)
(1. 04)
(1. 02)
21
27
(1.16)
(1.12)
(1. 05)
24
30
45
(1.20)
(LIS)
(1. 07)
26
33
47
(1.18)
(1.17)
(1.06)
22
(1.10)
(1. 07)
(1. 04)
(1.16)
(1.12)
31
39
58
(1.19)
(1.14)
34
42
61
(1.21)
(1.16)
30
48
42
28
36
54
-
-
(1.07)
-
a
For each c , the first row is for
the third for p* = .99 .
b
The numbers in parentheses are the ratios of these results to exact results
given by Gupta (1963).
p*
= .90
, the second for
p* - .95 ,
-18-
TABLE 3b.
Values of v = 2r given
IE
IS
(1. 90)
26
(1. 44)
26
31
(2.16)
(1. 93)
(1.57)
57
(2.12)
(1. 86)
(1.58)
.425
33
46
65
(2.11)
(1. 91)
(1. 54)
.45
44
S3
74
58
70
.25
.35
44
34
.40
41
(2.00)
{3.9)a b .
5
8
6
4
by
10
18
21
28
(2.25)
(1.75)
(1. 55)
19
23
30
(1. 90)
(1. 91)
(1.50)
20
24
31
(2.00)
(1.71)
(1. 55)
30
32
LlO
50
(2.00)
(1. 90)
(1. 56)
34
"R:>
'.'
48
(2.14)
(1. 75)
(1. 60)
52
(2.12)
(1.31)
(1.52)
39
46
62
(2.16)
(1.91)
(1. 55)
42
49
65
(2.10)
(1.83)
(1. 54)
44
51
67
(7. .00)
(1. 82)
(1.52)
44
52
70
(2.20)
(1.85)
(1. 52)
48
56
74
(2.18)
(1.83)
(1.54 )
51
59
77
(2.12)
(1. 96)
(1.54)
(2.00)
(1. 89)
(1. 60)
50
60
(2.08)
(1. 87)
SS
(2.11)
(1. 88)
58
67
(2.07)
(1. 86)
(2.23)
(1. 91)
66
78
35
-
.50
-
(2.20)
(1. 85)
-
64
72
154
-
-
(2. 00)
(1.83)
-
76
88
(2.11)
(1. 96)
-
a
For each t'i , the first row is for
and the third for p* = .99 •
b
The numbers in parentheses are the ratios of these results to exact results
given in Gupta (1963).
F*
= .SO
, the second for
p*
= .95
,