Gertruu.~
ON A CLASS OF DECISION PROCEDURES FOR RANKING MEANS
by
Kiron Cha.ndra Sea,l
Specia.l report to THE UNITED STATES AIR FORCE
under Contra.ct AF 18(600)-83 monitored by the
Office of Scientific Research.
Institute of Sta,tisticB
Mimeograph Series No. 109
June, 1954
M.
COX
UNCLASSIFIED
1,
?
3.
4.
Security Information
Bibliographical Control Sheet
O.A.: Institute of Statistics, North Carolina State College of the university
of North Carolina
M.A.: Office of Scientific Research of the Air Research and Development Commend
o .A . : CIT Report No. 13
M.A.: 0SR-TN-54-234
ON A CLASS OF DECISION PROCEDURES FOR RANKING MEANS
Kiron Chandra Seal
5. June, 1954
6. 109
7.
None
8.
AF 18(600)-83
9.
670-088
10. UNCLASSIFIED
None
12. Suppose there are n+l normal populations N(~i,01)' i = 0,1,2, ... ,n, with unkn~wn
means and a cODlllon but unknown variance and that k random observations from each
of the n+l normal populations are given. It is desired to choose a group of
populations from the above n+l populations, with the "help of some decision rule
which ensures that the least upper bound of the probability of not including in
the group the population with the largest mean is a(O<o:<l), whatever may be
the unknown ~ils. Subject to this fundamental requirement the rule may be reqUired to possess other desirable properties such as, (a) the property of unbiasedness, i.e., the probability of rejecting any population not having the
largest mean is not less than the probability of rejecting the population having
the largest mean; (b) the property of gradation, i.e., corresponding to any
a(o<a< 1), there exists a constant ~oa such that the chance of retaining the
population with mean ~ in the group is greater or less than a, according as ~O
is grester or less than ~oa. An infinite class of decision rules satisfying
the fundamental requirement, together with the properties (a) and (b) has been
considered. Certain interesting properties of this class has been derived.
The question of choosing one member from this infinite class haVing further desirable properties has also been studied.
11
ii
ACKNOWLEDGMENT
I wish to express my deep gratitude to Professor
R. C. Bose for his constant encouragement and inspiring guidance in the course of my investigation and to
Professors S. N. Roy andWassily Hoeffding for their
many helpful suggestions and criticism.
My thanks are due to the United states .Air Force,
the Institute of Statistics, the United States Educational Foundation in India and the Institute of International Education for their generous financial assistance without which this work would have been impossible.
Finally, I wish to thank Mrs. H. Kattsoff for her
careful typing of the manuscript.
iii
TABLE OF CONTENTS
PAGE
CHAPTER
....
ii
• • • • • • • • • • • • • • • • •
v
FORMULATION OF THE PROBLEl'1 • • • • • • • • • • •
1
1.1
Introduction ••••.•• • • • • • • • • • •
1
1.2
Statement of the problem • • • • • • • •
·.
1
1~3
Class
decision procedures • • • • • • •
3
1.4 A relation
among t a (c , ••• ,cn ) • • • • • • •
.
1
7
ACKNOWLEDGEMENT • • • • • • • • • • • • •
IN'l'ROD UCT ION •
I
1.5
II
'e of
Derivation of joint frequency function of
X(l), ••• ,X(n) • • • • • • • • • • • • • • •
SOME PROPERTIES OF JOINT
g(X(l)'···'X(n»
12
FUNCTION
FEEQ~~CY
••••••••••••••••
·. ..•
18
• •
18
2.2
An inequality related to location parameters
20
2.3
An inequality related to scale parameters
27
2.4
Monotonicity of expectation
• • • •
30
Property of symmetry • • • • • • • • • • • •
41
Introduction • • • • • • • • •
• •
• •
46
III
Introduction •
..•
• • • •
• • • • • • • •
46
46
iv
PAGli:
CHt\PTER
IV
3.3
Property of unbiasedness • • • • • • • • • • •
3.4
Property of gradation
58
. ...
64
SOME PROPERTIES OF LINEAR FUNCTIONS OF ORD~R
STATISTICS • • .' • • • • •• • • • • • • • • • • •
71
4.1
71
Introduction
. ....
•
..
• • • • •
n
4.2 Asymptotic normality of E ciY(i)- YO • • • •
71
4.3
Asymptotic moments of Y(i) • • • • • • • • • •
73
4.4
Relation among
tics
i=l
~lls
and
~21S
of order statis. • • • • • • • • •
74
4.5 Approximate distribution of Y(n) - YO • • • •
78
A model sampling experiment
4.7
v
• • • • • • • • •
n
n
81
ci-l).
84
• • • • • • • • • • •
90
5.1 Introduction. • • • • • • • • • • • • • • • •
90
5.2
Selection among D(C1, ••• ,cn ) • • • • • • • • •
90
5.3
The case when n population means are close
Minimum variance among
SEIEC'fION OF AN OPTIMUM RULE
1;
i=l
ciY(i)'
(1;
i-I
together • • . • • • • • • • • • • • • • •
~xpected
number of observations retained •
97
·.
98
APPENDIX: RESULTS OF A MODEL SAMPLING EXPERIMENT
100
BIBLIOGRAPHY
101
• • • .. • • • • • • '. • • • • • • • •
v
INTHODUC'l'ION
In recent years it has been recognized l
[5_7, ['18_7, ["19_7,
<L-l_7, ~3_7,
~4_7,
['20_7) that the conventional test of homo-
geneity, such as the F-test in the analysis of variance for testing
the equality of several population means, does not supply all the
information that the experimenter seeks.
In many practical aitua-
tions it is unrealistic to assume that the population means of
several essentially different populations (which, for example, may
correspond to different treatments used in agricultural problems)
will be equal.
A sufficiently large sample will thus enable the
experimenter to detect this difference at any preassigned level.
In most cases what the experimenter actually wants is a decision
procedure which would tell him which population or populations
possess a desired characteristic.
For example, the experimenter
may be interested in determining the population with the largest
mean, from a set of normal populations.
Alternatively he may desire
to select from a given number of populations a group containing the
population with the largest mean.
In ~18_7 Paulson considers a multiple decision problem for
dividing a set of population means into a "superior" and an lllin-
lThe numbers in aque-ire brackets refer to the bibliography
listed at the end.
vi
ferior" group; in
£19_7 he
considers the k decision problem of
finding out the Itbest" of k populations when k-l experimental
populations are compared with a control population; in ~2o_7 he
suggests an optimum solution to a k + 1 decision "slippage" problem considered earlier by Mosteller ~11_7. Bahadur ~1_7 considers the problem of constructing .a
suitablE~
decision rule to
select one or more populations, which minimizes the expected value
of a random distribution function.
In.L
3_7
Bechhofer considers
a single-sample method of designing experiments to determine the
ranking of k normal populations with known variances.
- -7
In,4
Bachhofer, Dunnett and Sobel give 'a two sample decision procedure
for ranking means of several normal populations with unknown variances.
In
L-,-7 Bechhofer
and Sobel consider a seq~ntial multi-
ple decision procedure for ranking means of llormal populations with
known variances.
The problem of determining the population having the largest
mean has a certain connection with the old problem of testing outlying observations considered by many statistiicians
L13J,
~11J,
£21_7, ['22_7,
~24J).
(["8_7, ["9_7,
Various statistios have
been suggested for this problem mostly on intuitive grounds.
After
being detected, the l"outliers It are to be rejected from the sample
to make the retained group of observations appear to represent a
random sample from, say, a single normal population.
To br+ng out
the connection between tihis problem and the problem of ranking means
vii
let us consider only II()ne-sided outliers· i.e., when the outlying
observations are known to have come from a population having 8ignificantly larger mean.
Then acoording to the problem of ranking
means these ·one-sided outliers. are the most desirable observations to choose for further
experiment~ion, whereas
in the prob-
lem of testing outlying observations these outliers are forthwith
rejeoted.
The general problem of ranking means is too complicated for
an explicit solution to be obtained by applying Waldls
eral deoision theory.
L-26_7 gen-
In this dissertation an infinite class of
multiple decision procedures is suggested for ranking means of
(n + 1) normal populations with a common bllt unknown variance cr2 •
l
Our object is to select a group of populations (which may in
particular consist of a single member) from (n + 1) populations,
on the basis of samples of equal size k one from each population
suoh that the least upper bound of the probability of incorrect
choice (i.e., not retaining the population with the largest mean
in the selected group) is a preassigned number
whatever be theunknoWn population means.
a(O < a < 1),
For this purpose we have
considered in Chapter I an infinite class of decision procedures
having the above property.
Certain interesting properties of this
class have been investigated in Chapter II.
In Chapter III it is
shown that the rules of this class possess a property analogous
to the property of unbiasedness in the theory of testing of
viii
hypothesis.
In Chapter IV some results connected to the distri-
bution of a linear function of order statistics are obtained.
These results are used in Chapter V to select one member from this
class, which is an optimum rule in a certain sense.
ClWTER
I
FORMULATION OF THE PROBLEM
CHAPl'ER I
FOWiULATION OF
1.1.
nIE
PROBLEM
Introduction.
In this chapter the problem to be studied in the next few
chapters is explained, and a class of related decision procedures
is considered.
The derivation of the joint probability density
function of n order statistics connected to these decision procedures (though known) is given for future reference.
1.2.
Statement of the problem.
Suppose there are (n + 1) normal populations N(~i,a~),
(i • 0, 1, 2, ••. , n} wi th unknown me ans and a common but unknown
variance and that k random observations xij(i
j •
a
0, 1, ••• ,
n;
1, 2, ••. , k) from each of the (n + 1) normal popUlations are
given, where xij is one of the k observations from the i-th population.
k
Under our assumptions the (n + 1) sample means xi •
2
0'1
2: Xij/k will obey N(~i' k),(i • 0, 1, 2, ... , n} and an estij-l
mate
2
of a
2
oan be obtained which is independent of the sample means
1
xi(i - 0, 1, ••• , n).
We may, therefore, assume for mathematioa1
oonvenience that just one random observation xi from eaoh of (n + 1)
normal populations
2
N(~i,a
)(i • 0, 1 , •.. , n ) i s g i ven, where a 2 •
2
a /k is estimated by
1
and is aS5UIIB d to be known.
Clearly this estimate 52 of a 2 is
stochastioally independent of the given observations xi(i - 0, 1,
••• , n).
It is desired to choose a group of populations from the
above (n + 1) populations, with the help of some decision rule
which ensures that the least upper bound of the probability of
not including in the group the population with the largest mean
is a(O < a < 1), whatever may be the unknown I-Li IS.
Subject to
this fundamental requirement we would like the rule to possess other
desirable properties such as
(a)
The property of unbiasedness, i.e., the probability of
rejecting any population not having the largest mean is not less
than the probability of rejecting the population having the
·largest mean.
3
(b)
The property of gradation, Le., c:orrespondi.ng to any
~Oa
a(O < a < 1), there exists a constant
such that the chance of
reta-Lning the population with mean !-La in the group is greater or
less than a, according as !-La is greater or less than -\Jo Oa •
The
constant !-LOa will in general depend on the decision rule as well
as the unknown means of the remaining n populations, and the
2
common variance cr "
An infinite class of decision rules satisfying the
fundamental requirement, together with the properties (a) and
(b) is given below"
This will be called class ~ of decision rule s.
The question of choosing one member from this infinite class
having further desirable properties has been studied in Chapter
v.
1.3
Class ~ of decision procedures.
Let y.(i
~
= 0, 1, ••• , n) been
+
1) random observations from
2
N(O,cr ) and let y(l) < Y(2) < ••• < Yen) be n ranked observat ions
among Y1' ••• , Y'n"
The Y(i)1 8 will then define another set of
random variables Y(i)(i - 1, ."., n).
i
f
j since the set of points
(yo' Y'1'
It is ,assumed Y
i
0 •• '
'I
Y,
j
Yn ) in (n + l)-dimen-
4
f
sional Euclidean space where 1i
probability 1.
Y"j' i rfj will be obtained with
Let t a(cl , •.• , Cn
) (01 >
-
n
'
c = 1) denote the uPEer a
i=l i
~
10
0
0, i • 1, ••• ,
n,
point in the probability density
1'unctiop ot
(1.3.1)
•
The class ~ of decision rules D{C1 , •.•• on)' (C1 ~ 0, i - I , ••• ,
n
n,
Z 01 •
1) is defined as followst
i-I
~eject
any observation Xo from the given observations
(i.3. 2 )
and accept otherwiseJ where x(l) < %(2) < ••• < x(n) are n ordered
observations among
(X1' %2' ••• , Xn ). (The (n
+ 1) observations
xi(i ·0, 1, ••• , n) taken from normal populations are again
5
assumed to be distinct).
Proceed as above for each of (n + 1) ob-
servations separately, so that, each of (n + 1) observations in due
turn takes the place of X and the remaining l::>rdered observations
o
play the part of X(l), ••• ,x(n)." Thus in the above procedure we
may start with the largest observation among xi(i
as
X
o and
work downwards.
a
0, 1, ••• , n)
If any particular c::>bservation is rejected,
all other observations smaller than this observation are automatically rejected.
For the sake of convenience, we shall denote the decision
rule DCcl' .•• , en) when (i) c i '"" ~(i
=:
1, •..., n) by D and when
Another class ~ * of decision procedures could be constructed
as given below.
It will, however, be shown that it is identical with
'e.
lISt '(1,,(1;a 1, ... , n) and YO have simil&" meaning as before.
Pef~ne. ~:(el"'"
0n)(c ~ 0,
. . .1
i • 1, •.• , n,
n
% 01 • 1) to be the
~l
6
lower a
point in the probability density function (p.d.f.) of
%
YO-II-
(1.33)
t (c l ,···, c n
The class
--e-ll-
)
=
n
~
c. Y(.)
i=l ~ ~
s
of decision rules D*(c , •.• , cn)' (c .:: 0, i
l
i
III
1,
n
••• , n,
~
i=l
c
i
... 1) is then defined in the following way:
~eject
any observation X from x ( i ... 0, 1, ••• , n) if
i
o
xo -
and accept otherwise; where X(i)(i ... 1, ••• , n) are n ranked observations among xi(i ... 1, ••• , n).
This procedure is repeated
for each of (n + 1) observations separately.n
It is now shown that both the decision rules D(c , ••• , cn)
1
a~d n*(c , •.• ,c n ) are identical.
1
From the definition of t*(c , ••• , c ) we get
a 'l
n
*
7
<
t (c ' ••. , c )
a 1
n-
>
-t*(c , ••• , c )
a l
n-
7
1
t a (Cl' ••. ,C n) 7
>
by the definition of t (Cl' ••• ,C).
a
,
Hence it follows that
n
t a (Cl' ••• ,C n )
a
-t*
a(~l'
. ••• 'Cn )
But as explained earlier we know that
for the rule D(Cl, ••• ,cn )
.
any observation
X
o
among the given (n+l) observations xi(i=o,l, ••• ,n),
is rejected if
n
i:1CiX(i) - Xo > sta(cl,···,c n ),
that is, if
xo -
This implies from the above relation (1.3.5) that X is rejected
o
only when
xo -
<
st*(Cl,···,c
),
a
n
which is identical with the inequalities (1.3.4).
and
Hence the class
'e * of decision rules are identical. We shall henceforth study
the properties of class ~ only.
In this section an inte:resting inequality between
ta(C l , ••• , c n ) with different sets of c.ls, where 0i > 0 (i '" 1,
].
-
n
••• , n),
~
i"'l
0i
a
1 is derived.
r
(1.4.1)
~
i-l
.
,
Theorem 1.4•.1
t a (c1 ' ••• , cn
) > t a
(c1 ' •••
, c ),
n
ci
~
r,
Ec
i.1 i
for all r · 1, ••• , n - 1,
.
8
if
and
,
(c1 ' •.• , cn) and (C1 , •.• , Cn) are tHo different sets.
Let
Prooft
(;le ar1y,
an. O.
•
From (1.4.1) we get &r
~ 0,
for all r • 1,
••• , n, where the inequality holds for at least one r.
Consider
the linear function
,
(1.4.2)
where 1'(1) < ... < Yen)
are ranked observations among 7i (1· 1,
••• , n) taken at random from N(O, g2).
It is obvious that (1.4.2) is positive whatever JIlay be the
values of y{i) '. (i • 1, ... , n).
Now (1.4.2) is 8<Jlal to
9
n
•
~Y(l) + r~ (ar - ar _1 )Y(r)
-
,
t
ar - a r-1. cr - c r , (r - 2, ••• , n) •
Henee 1 t follows that
,
whatever may be the values of Y(i) (i • 1, •.• , n).
n
Henee
I
1:1CiY( i) - YO
s
>
t a (ol' c 2' ••• , cn )
,
. implies
s
•
10
Therefore
n
(1.4.4)
p
r
-
Z CtI(i) - YO
> t ( 01' 02' •.• , c ) . 7
s a n -
i-I
n
,
% 0iY(i) - YO
>
PL~r i-I
where P stands for
t~e
> t (
01' •••)
, e
s a n
probability.
,
Also .from the definition of
t a (01' ••• , cn ) it follows that
• «, 0 < a < 1
J
•
Hence from (1.4.4) and (1.4.. 5) we get
11
- y
o
>
t (c , c , ..• , c )
a
2
1
7.
n-
Hence
In the above proof s is assumed to be positive which is true except for a set of probability measure zero.
In particular, we get
from the above theorem,
t (r) > t (r-1), r
a
a
= 2,
••• , n
•
It is obvious from the relation (1.3.,) that under the condition
(1.4.1),
*
* ,
,
t (c ' •.. , c ) < t (clt .•. , c)
a l
nan
,
12
and in partioular,
t*(r)
< t*(r-1),
r • 2, ••• , n •
a;
a
1.,. Derivation of joint frequency function of X(l)' ••• , X(n) •
Consider n Dormal populations
wi1;h
N(~i' ~~h i - 1, 2, ••• , n,
2
unlmown mean P.i and varianoe ai for the i-th population.
9uppose one random observation xi (i • 1, ••. , n) from eaoh of
these
11
populations is given.
The joint frequency funotion of
Xi(i • 1, ••• , n) w111 then be given by
,
(1.5.1)
- 00
< xi <
00,
(i • 1, ••• , n)
•
Let x(l) ~ x(2) ~ ... ~ X(n) be the ordered values of xi(i • 1,
••• , n).
The joint frequency function of XCi) (i • 1, ••• , n) is
now derived.
13
Let R be the n-dimensional
n
subset of Rn where xi
I
~uclidean
xj if i
r j.
peW) - 1
space of points x andW the
-
We can therefore write
•
Let
tet weE) be a subset of W
be a permutation of (1, 2, ••• , n).
such that
(1.5.3)
w( n).
.a.
Sxa x
t - PJ.
< x
P2
<. •• < x
~
Pn
5
The set W is then the union of' nl disjoint sets
(1.5.4)
,
W(E)
•
so that
where ~ denotes summation over n! permutations (PI' ••• , Pn ).
E
Let
•
Hence
•
Let .l be a measurable. subset ot Woe
Hence
(1.5.6)
P(!O e A)., Z P ['!O e: A, !
&
W(F.>-7
E
But A CWO.
Hence (X , ••• , Xp ) &A implies x <... < x •
~
n
~
~
Hence (1.5.6) reduces to
15
J·.... f
•z
E
Let
,
xi •
(i • 1, ••• , n)
Xp ,
i
•
Let
! ·
t
• i, (i • 1, ••• , n)J that is, (~, ••• , t ) is the inverse
Pi
(t , ••• , t >' be the permutation of (1, ••• , n) defined b7
1
n
-~
permutation of (PJ,' ••• , Pn ).
n
From (1.5.8) we then get
,
xi • xt ' (i • 1, ••• , n)
i
•
Hence from (1.5.7) and (1.5.9) we have
f····J
f(x t ,.,.,xt )dx1 •
1
n
LC", ··.,xn )eAJ
• dxn
•
16
If g(x(l)' ..• , X(n»
denote the p.d.L of xO , then from
(1.5.10) we get
(1.5.11) g(x(l)'···'x(
n
»
= 2:f(x(t ),x(t )
t
-
~ 00 <
If ai's (i
= 1,
1
x
2 , ••• , (t)
x(l) < ••• < x(n) <
)dx
n
00
(1)'"
dx
(n)
•
2, ... , n) are all equal to a, we get from (1.5.11),
-
00 <
x(l) < ••• < x(n) <
00
•
17
If', furthermore, all 1J.1'S (i
1, ••• , n) are equal to IJ.' the joint
frequency function of X(I)'
, X(n) from
N(IJ.,
0
2
) reduces to the
familiar form
-
00
< x(1) < • • • < x(n) < 00
•
CHAPl'ER
n
SCME PROPERTIES OF JOINT FREQUENCY FUNCTION g(X(l)' ... , X(n»
2.1.
Introduction.
Let
.0- (
""1' fJ.2' ... , fJ. n )
denote the class of normal
a l , "2' ••• , an
populations
to
({J
N(l-1 ,
i
c{),
i .. 1, 2, ••• , n.
When all ai's are equal
this class will be written as ...Q.. (~, 1-1 2 , ... , ""n J alJ if,
moreover, the common ais equal. to 1 we shall write 11.(~, ""2'
••• , fJ. n ) tor this class.
Let X(l) < ••• < X(n) be n order statistics from
""1' ••• , ""n
.Q.. (
a1 , ••• , an
)
when one random observation from each of these
n populations is taken.
where
It will be shown in section 2.2 that
c ~ 0 (i • 1, ••• , n),
i
- -
c ' > 0 for at least one
i
i(l < i < n) and k is an arbitrary constant, is an increasing
19
function of each of the population means
~i(i. 1, ••• , n).
result will be derived as a particular oase of Theorem
in the next section.
section 2.3.
this
2.~.~·g1ven
A similar general the orem will be given in
Consequences of this theorem will not be studied in
detail since these are not related to the problem of this dissertation.
In section
2.4
a simple expression for the value of partial
derivative with respect to ~i(i· 1, •••, n) of expectation of
any linear function of X(l)' ••• , X(n) with non-negative coefficients will be given after direct evaluation.
In section 2.5 a property of symmetry possessed by the
joint frequency function (1.5.11) for some special values of unknown population
means
and standard deviations will be indicated.
The joint frequency function of X(l)' ••• , X(n) from
JJL
(~,
••• ,
~n;
a) is given by (1.5.12).
Define Y(i) - X(i)/a and
6i • ~i/a, i • 1, ••• , n.
The
joint frequency function of Y(l) < ••• < Yen) will then reduce to
20
-
00
< Y( 1) < ... < Y(n) <
00 •
Hence without any loss of generality we may assume a. 1 in
sidered in later chapters in place of ..0. (1J.1' ••• , FLnJ u) •
2.2.
An inequality related to location parameters.
... ,
Theorem 2.2.1.
is the cumu-
lative distribution function (cdf) of n random variablesXi (1 • 1,
•• • J
n) and T(ul , ••• , un) is a real-valued function of Ui(i • 1,
••• J
n) such that
(2.2.1)
where (al' ••• , an) is an arbitrary set of real numbers and
-
00
< u
i <
00
(i· 1, ••• , n).
If, for an arbitrary constant k,
••• , 6n
21
T(Xl ,
.0.' Xn)
n)
th
denotes the probability of
... , xa-6
n
n
'
> k when 1 1 , ••• , In
en
.0.'
P -;-T{~J
.0., I n ) > k
-~
••• ,
••• , 6
n
Proof a
Let!· (~,
~uclidean
and
.0.'
5n+an
an
J.
X ) be any point in the n-dil'llensional
space in and let
n
22
From (2.2.1) we get
T(x.. +11 , ... , x +11 ) > T(x , ••• , x.)
1
~
.1
n n n
w*
This implies
•
w•
:>
Therefore,
... ,
•
f· .. S
- s·.. )
W*
where
(2.2.6)
... , x
n
... ,
(1
n
,
n
(i • 1, •• _, n)
~
J...w f
+11
n n
-6 -a
n n )
y -6
n n)
(1
6
,
23
••• , 6
= P -;-T(Xl ,
n
••• , X ) > k
n
••• , C1
J.
n
This completes the proof of the theorem.
Corollary 2.2.1
If (2.2.1) is satisfied for all a ~ 0
i
(i • 1, ••• , n), then P ~T(~, ••• , In) > k
_7
ls a nondecreasing
function of each of 6i (1 • 1, ••• , n) •
The proof is obvious.
Corollary 2.2.2
when a i
~
If
0 and a i > 0 for at least one
l(i. 1, ••• , n), and 1£ tre cd!
of T(Xl' ••• , In) assigns a positive measure to every non-degenerate interval in R1 , then peT( ~, ••• , Xn ) > k
J , where
arbitrary constant, is an increasing function of
ea~h
k is an
of 6 (i • 1,
i
••• , n).
Proof:
Under the above conditions it is found from (2.2.3) and
24
(2.2.4) that W is a proper subset of W* and so strict inequality
holds in (2.2.6).
Definition:
~
The corollary follows readily.
(F) is called a strictly monotonic functional of
F (x) such that Fl
2
~
F2 for all x, and F1 > F2 for some x.
m (F) is called a monotonic functional of F(x) if
m(F ) ~
l
m(F ) holds in the above defini.tion.
2
If F(x) is the cd! of a random variable X, then expectation E(X), if it
F.
ex~sts,
is a strictly monotonic functional of
This is readily verified by considering the nondecreasing in-
verse function (cf.
1:10_7
pp. 152-3, ['23_7 p.189) x(F) of the
cdf F(x) which is also known to be continuous on the right.
Assuming (2.2.1) to hold for a i ~ 0 (i - 1, ••• ,
n), any monotonic functional of the cumulative distribution function
corollary 2.2.3
of T(Xl , ••• , In) is a nondecreasing function of each of 6 (i
i
...,
= 1,
n) •
The proof of Corollary 2.2.3 follows immediately from Corollary 2.2.1 and the definition of monotonic functional of a cdf.
Corollary 2.2.4
Any strictly monotonic functional of cdf of
T(Xl , ••• , Xn)' which satisfies the conditions of Corollary 2.2.2,
is an increasing function of each of o.(i
1.
a
1, ••• , n).
25
The proof is obvious.
l~xample
1.
where c
~
--
i
Take
0 and c
i
>
0 for at least one 1(i • 1, ••• , n),
X(i)(1 - 1, ••• , n) are the n order statistics corresponding to
the ranked observations x(l)
~
•••
~
X(1)
~
•••
~
x(n)
from
~-61
F( a
' •.•
1
sgn(u)
-
-1 if u <: 0
{
Oifu-O
•
1 if u > 0
1he conditions of 1heorem 2.2.1 are easily seen to be satisfied
for a i
that
~
0 (1 • 1, ••• , n).
Hence it follows from Corollary 2.2.1
26
is a nondecreasing function of each of 5i (i • 1, ••• , n).
Example 2.
Take
... ,
x -5
n
n n).. J( (a
an
i-I i
/'Tn)
-1
e
-( X -5 )
i
i
2/2a 2
i
and
where
r" 0, 1, 2, •••
and c i ~ 0
(i - 1, ••• , n) and c i > 0
for at least one i.
Here the conditions of Oorollary 2.2.2 are easily verified
and it follows from Corollary 2.3.4 that
n
E( Z CiI(i»
i-I
2r+l
,
r . 0, 1, 2, •••
i8 an increasing function of each of 8 i (1 - 1, ••• , n).
section
2.4
In
the explicit expression tor the partial derivative of
27
n
E(
t ciX(i» with respect to 6i (i
i-I
a
1, ••• , n) will be given.
n
This will again show that E(i:lc~(i» is an increasing function
... , n).
of each of
From Corollary 2.2.2 it follows also that
is an increasing function of each of
n
P(.~
J....l
ciX(i) > k)
6.(i ... 1, •.• J n).
J.
A
particular case of this result will be used in Chapter III in
proving the property of unbiasedness for the class of decision
rules D(c , ••• , cn) as defined in section 1.3.
l
An inequality related.to scale
2.3
Theorem 2.3.1.
Suppose that
F(~,
0'1
variables
Xl, ...
J
p~rameters.
x
... , .2l)n
0'
is the cdf of n random
Xn and T(ul , ••• J un) is a real-valued function
of u. (i ... 1, ••• , n) such that
J.
... , au)
> T(u ,
l
n n -
(2.3.1)
where
-
00
•.• , u )
n
,
(al , ••. , an) is an arbitrary set of positive real numbers and
< u
i <-
00
(i
= 1, ••• , n). If, for an arbitrary constant k,
28
denotes the probability of T(X1 , ••• , Xn) > k when the edf of
~
Xl' ••• , Xn is F(--,
••• ,
(11
v
then
n
... , x )
(2.3.2)
Proof:
xn
~),
n
Consider x t R and let
-
n
and
Then from (2.3.1) we get
> k
29
T(alxl , ••• , a nxn ) -> T(x-l , •.• , xn )
This implies
•
w*::> w.
Now
..
- f· .. J
Yn
••• , (7")
n
w*
where
>
J..- J
w
Yol
d ~c. ( _ .
01
, .. -0'
Yn
)
-0
n
,
30
The proof of the above theorem is
]'lOW
complete.
The similarity of the Theorem 2.3.1 with Theorem 2.2.1 is
obvious and various results concerning scale parameters sinii.lar
to those obtained for location parameters in the previous section
can be derived.
Since, however, these results will not be needed
in the problem under consideration, these will not be discussed
here.
2.4.
Monotonieity of expectation.
Let
51'
... ,
0'1'
... , 0'
5
n
_7
n
denote the expected value of X{P) (~ • 1, 2, ••• , n) when XCI)'
.••• , XCn) arise from
n (6l'.
• •• ,
6
n)
• The probability density
0'1' ••• , an
function ot 1(1)' ••• , X(n) will be denoted by
)1
)
It is then readily verified from (l.S.ll) that
51'
a!'
... ,
6 ,
1
... ,
( 2.4.1)
•
n
t (a
i-l a
J'i)
-1
2
e
-(x(i)-a) /20'
a
... ,
5
n
)
CT
n
2
a
•
a!'
••• J
stands for the probabilit:r density function of
-a'
ria'
... , a
n
)
••• , an
X(l}' ••• ,
32
0 ,
1
... ,
0"1'
... ,
(
0
... ,
a+l' ... ,
0
a-I' ~a' a+l'
O"a_l'
sla ,
a
0
n
0"
) J
n
here (n-l) ordered observations (X(l)' ••• , x(n»"X(i)' 1 ~ i ~ n,
are regarded as (n-l) consecutive ranked observations from (n - 1)
normal populations ;-N(ol'otl ), •••,N(O ,02) 7-N(0 ,02>, 1 < a < n.
n na 0;
--
An interesting expression for
0 ,
1
0"1'
(0; -
1, ••• , n, ~
w.
1, ••• , n) will be developed with the help
of two lemmas given below.
Lemma 2.4.1
!..
... , 0n
_7 ,
... , an
JX(i+l)
do
x (i-l)
33
• a
where
C
x
)...6
~ ({1+1 ) 't'
~
(]
j
x
(x) _.
-00
Proof:
x
)-6
<P {(1-1
.) J
a
...t2/2
e
dt
•
34
(x(i_l)-6 ) 2/2~2 -(X(i+l)-6) 2/2~2
r
+u_ e
-e
~
Lemma 2.4.2
(2.4.3)
If i
~ j,
7
35
Proof I
_t 2/2
e
dt
In Lemma 2.4.1 and 2.4.2 X will be interpreted as -00
o
and X(n+1) as +00.
It then follows that
ep (-00)
• 0,
<P(+oo) -
/2n
,
36
Using Lemma 2.4.1 and 2.4.2 we get from (2.4.1),
•
1~1
i,t~
+
J... J x(~)
1//2a~ ...
[";<x( i-i) -6
... ,
aI'
... , a
5
n
n
2
-(X(1+1 )-6a)
_7
/2a~ ]
wi
fu.J
6 , ••• , ~ , 0
1
a
•• ,
5
n
• gn_1{X(1),···,t(~), ••• ,X(n)
)dx(l)·O~(~) •• dx(n)
a1 ,··· '''a'··· ,an
+
51'
s... ~
31
0 ,.,.,; , ••• ,0
1
a
n
CJ
l
, •. . ,P
a
,•..• ,CJn
where
(i = 1, ••. , n) ,
- ~ + A2 + A
3
Now if in the first set
X(l)
Al
(suppose)
we put
• Y(l)
,
•
•
•
x(i_l)
x(i+l)
8'
..
•
1'(i"l) ,
Y( i)
,
•
X(n)
..•
Y(n..l) ,
•
38
we get
J... f
I··· S
Y{~>le
2/ 2 {
2 2
-eYe i_1)-6 ) 2a - Y(i)-6 ) /2a
a
a-e
a
a j
61 , ••• ,-a , ••• ,6n
) dY(l) .. dY(n_l)
al'···'~a,···,an
.. S... f
wn '
39
5 , ••• ,-a , ••• ,5n
1
-..
~.
Henoe
(2.4.4)
•
J... J
aa
L <f! (X( f!+; )-5· a).. <p (-x( ~-; )-5a)_7
a
51' ••• ''!a , ... ,6n
Therefore,
a
40
(2.4.5)
•••, -a'···' 5n
••• , ria' ••• '
an
_7
where
J.
x
<'P(X) ..
-00
When 6i
a
(cf. ~27
1:.
n
6, 0i -
7,
Thus in
0,
(i • 1, ••• ,
n), it is easily verified
p. 13) that right hand side of (2.4.5) is equal to
this case
which agrees with a known result.
From
(2.4.5)
it is clear that E(X(i»'
when
••• , 5n
)
is considered, is an increasing function of each
n
of 5i (i • 1, ••• , n).
Hence E(i:lCiX(i»' where 01 ~ 0 and
c i > 0 for at least one 1 (i • 1, ••• , n), is also an increasing
function of ~ach of 5i (i • 1, •••, n).
This result, as we have
noticed earlier, follows from the theorem of section 2.2.
2.5.
ProEert;y of symmetrl.
Consider the joint frequency function of X(l) < X(2) < X(3)
1J.-5,
IJ.,
taken from.o..( a , a '
O
1J....5
a
).
Let Yell < Y(2) < YO) be the new
variables when the origin is transferred to
sample space.
(IJ., \:l, IJ.) in the
Moments of X(l)' 1(2)' X(3) except the first will
clearly be unaffected by this transformation.
The joint frequency
-5,
function of Y(l)' Y(2)' Y()
then be given by
0,
arising from ~ (a , a '
O
6
a) will
42
t
(2n)
-3/2
fo
+ e
~§ffl: (~t~~t)
+ e
tt t§ll@w~ that - 'f(J) ~ • Y(2) < • 1(1) have the
same joint frequen9Y function as given in (2.,.1) and hence
43
1, 2, •.•
E
r +l)
E (y2(2)
• 0 , r · 1 , 2, ...
Thus the moments (both rgw and corrected) of pair of order statistics symmetrically placed about median are identical for even
order moments and are of opposite signs for odd order moments.
Similar properties for covariances of Y(l)' Y(2)' Y(3) can also
be easily derived.
The above result can be generalized as follows I
Let X(l) < ••• < X(n) arise from
)
(2.5.3)
if n· 2k + 1 is odd ,
~-5k'
or
... , ~-ol' ~+ol' ••. , ~+8k
(ii)Q(
)
cr k
' •.. ,
cr
l
'
cr
l , ... ,
crk
,
44
if
n
= 2k
is even
Changing the origin to
Y(l) < ••• < Yen)
•
(IJ., IJ., ••• , IJ.) as before the new variables
may be considered to arise from either
... ,
... ,
(2.5.4)
-6 , 0, 51'
1
CJ'1
, 0"0'
... ,
5
k
)
,
0'1'···' O"k
if n - 2k+1 is odd ,
0.(
or,
if
... ,
)
n. 2k is even
•
All moments except those of the first order of
Yell' ••• ,
Yen)
will be identioal with the corresponding moments of X(l)' ••• ,
Xen ).
It can be readily verified that
-Y(2) < -Yell
-Yen) < -Y(n-l) < ••• <
will have the same joint frequenoy function which
45
corresponds to
(2.5.4).
Hence raw and corrected moments of pairs
of order statistics I(v_i) and I(v+i)
(i - 1, 2, •••
~ ~ _7 )
which are symmetrically placed about the median I(v)' (where
v • ~ n~l
_7 _ the
greatest integer contained in (n+l)/2 ) will
be identical tor even order moments and will be of opposite signs
tor odd order moments.
Also E(~)l) • 0, r • 1, 2, ••••
This
property oan also be easily derived for covariances between pairs
It is interesting to compare this property with the wellknown (cf. ~25-l, p. 313) similar property for n order statistics
from N(O,
2
C1 )
•
CHAPTER III
SOME PROPERTIES OF THE CLASS D(c 1 , ••• , on).
3.1.
Introduction.
In this chapter some desirable properties of the class of
decision prooedures suggested in section 1.3 are derived.
section 3.2 the partial derivative of
.
n
P~ % ciX(i) >
i~l
k_7
In
with
respect to any 5a (a - 1, ••• , n) is eva1uated'when X(l) < •••
51' ••• , 5n
< X(n) are assumed to have come frClmfi(
).
••• ,
(J
In
n
sections 3.3 and 3.4 it is shown that the class ~ of rules
D(c , ••• , on) (oi ~ 0, i
1
= 1, ••• , nand 0i
> 0 for at least
one i) possess two desirable properties which may be designated
as uProperty of Unbiasedness· t and "Property of Gradation It respectively.
3.2
Exact value of
In this section we shall derive the exact value of
(a • 1, ••. , n) ,
47
where X(l) < ••• < X(n) are n order statistics from
• •• J
... ,
6
n
), ci'S are nonnegative at least one of which is
CJ
n
posi tive and k is an arbitrary constant.
Was followed in section
2.4
t>imi1ar notations as
will be used here.
We prove first
the following:
Lemma J.2.1!.
[
a
-(a-5) 2/20 2 - ( b-5) 2/2a 2
-e
• e
where a and b are two arbitrary constants independent of S.
Proof:
b
J
a
e
-
( u-S) 2 /2a 2
du
48
b-o
..
c rd-
do
f
a
_t2/2
~ e
dt
C1
2
- ( a-o) 2/2C1 2
•
- ( b-5) /2a
•
- e
e
2
Clearly (J.2.l) is equal to
(a
a
fin)
-1 n
t
i-l
~s:.
vU
a
f·.. f
2
2
-(x(i)-5a ) /2cra
e
w
51'··· '~a'··· ,5n
n
) it"
al' ••• '~a , ••• ,an i-l
where
.... rX.RI
l-
n
<h(i) ,
49
The expression (3. :'.3) can be ~1I'ltten as
(cr
J2R)
a
...1
,
rThere
Ki •
..:L
"00
a
]...f
w
Two cases will now be considered separately aocording as
I.
or,
II.
all c i l s (i • 1, ••• , n) are positive,
at least one c is zero.
i
Case I.
(i • 1, ... , n)
It in
then
•
<3.2.5) we di.fferentiate under the integration sign
50
or, (iii) the range of xCi) is a null set if x(1+l) < k1 - Here
x(O) and X(n+l) will be interpreted as -00 and +00 respeotively;
00 and 0n+l will be oonsidered to be zeros md 0cr(O) • 0n+l%(n+l)·O
will be asSum9 d.
(3.2.6) K i
Hence by Lemma 3. 2.1 we get
.r... ~
wi
~.-(X(i_l)-5a)2/2a~ _.-(X(i.l)-5a)2/2a~-7
,1
where
and
-00
< X(l)<•• '< X(i_l)< x(i+l)<"'< x(n)<
00
(3.2.8)w~. !&Rn_l:ClX(l)+••• +(Ci_l+Ci)X(i_l)+Ci+lX(i+l)+•••+OnX(n)
< k < C1x( 1)+. u+Oi_1X(i_l)+( °i+oi+1 )X(i+l)+'''+Cnin) •
Note that w1 is a null set.
(3.2.9)
Now it we put
Y(l)
..,
X(I)
,
•
X(i-l)
Y(i-l)
a
Y(i)
• X(i+l)
Y(n-.l)
•
•
..,
X(n)
,
52
*
*
in (3.2.6), then wi and wi will oorrespond to vi and vi respeotive-
11, where
and
-
00
< 1(1) < ... < Y(n-l) <
00
Here 1(0) and 1(n) should be interpreted as x(O) and X(n+l) respeotive11.
Vn+1
Also note that
•
{lO
R _ '
n 1
Also (3.2.6) now becomes
-00
< Y(1) < ••• < Y(n-1) < 00 }
•
53
(3. ? .l?) K. =
1
+
j ... f
L-e
.
2
-(y( .. 1)-5 ) /2cr
1-
a
2
2/ 2
-(Y(i)-5 20
a -e
a
a_7
J... f
vi*
cr , •.• ,1. , ... ,Cf
1
a
n
where
54
Case II.
Let
c i ...· 0, for sc:)me. 1:, 1 '" 1 .c:. n.
In this case the relation
imposes no restriation on XCi)'
Hence differentiating (3.2.5)
under the integr ation si gn we get
J...J
2 2
(
2 2
-(Xti_l)-5 a ) /20
- X(i+l)-5 ) /20
a
a
~e
a._e
where
It may be noted that we hav'(3 used the same symbol w. in (3.2.7)
1
7
55
and (3.2.15); this notation is per~tssible since wi in (3.2.1)
is identical with the set wi in (3.2.15) when 0i • O.
It is clear
also that the set w: defined in (3.2.8) is null when 0i • O.
Hence
formula (3.2.6) of Case I can be rega.rded as still valid for case
II if the term k in (3.2.6) is interpreted as infinity (so that
i
-(k -5 )2/2a2
e
i
a.
a.
is zero) for thi:s second case.
Making the Salle
transformation on (3.2.14) as in (3.2.9) we find that formula
(3.2.12) is still applicable if
WIE!
interpret hi(as defined in
(3.2.13) ) as infinity-and remember also that the set v~ in (3.2.11)
is null in this case.
Thus for both Cases I and II (3.2.1) is equal to (3.2.4)
where K is given by (3.2.12).
i
Consider now the contribution of
the term involving
in the formula (3.2.4).
The region of integration over which
(3.2.15) is negative is clearly given by the union of two sets
Vi and Vi* and the region over which thts term is positive is the
56
From (3. 2.10) and (3.2.11) it is readily verified that
set v.I.
l.+
for i = 1, ••• , n,
where
n,
sets.
Hence the term (3.2.15) contributes nothing to (3.2.1).
U represent respectively intersection and union of
Therefore by (3.2.1) . and (3.2.12) we get
J...* J
2
e
- ( h -5 ) /2a
i
a
2
a
vi
a , ••• ,'; , ••• ,a
1
a
n
57
f··J
*
-
\
2 2
-(K -6 ) /2a
i a
e
a.
wi
where
w~ :i,e giv~n by (3.2.8) and gn-1( ... ) has. the same meaning as in
seotion
2.4 •
When 01 •
a
(1 ~ i ~ n)
we have seen that iD this case
wi* is null 8I).d k! may be interpreted as infinit;n hence the i-tb .
term in the summation of (3.2.16) vanishes.
Thus (3.2.16) may be
n
FrOJ1l
this result it follows also that P ( E 0iI(i) > k) is an in..
.
.
i-I
creasing fUnotion of each of 8 (i • 1, ~ •• , n) which was derived
i
befor~
.Ia a
p~rtiou~ar
case ot Oorol1ary 2.2.2.
58
3.3.
property of unbiasedness.
Let X(l) < ••• < X(n) be
~
n order statistics from
(°1 , ••• J an) when one random observation from each of these
n normal populations with cornmon variance equal to unity is taken.
Let X be another independent variable obeying
o
N(OO' 1).
Accord,-
ing to our decision rule D(ol' ••• , c n ) as defined in section 1.3
the probability of rejecting
X
o will
then be given by
where
-00
<
X(1)
< ••• <
XCn)
<
00
n
1~la1X(1) > Xo • at~(ol'
88.'
On)
59
gn(x(l)' •.• , X(n)lol' •.• , on) represents the probability density
function of X(l)' •.• , X(n) and
p( s) _ _....vv/2
:o-__
v.. 2
_vs2/2 v-I
e
s , i.e.,
22f'(~)
the pdf' of sample standard deviation s based on v - (k-1)(n+l)
(cf. section 1.2) degrees of freedom.
~ubstituting
o-
YO for X
8 it is easily seen that right
0
hand side of (3.3.1) can be written as
00 fOO
1
(3.3.4)
ds
0+
where
,
dyo
-00
6i - 6i .. 50 (i • 1, ••• , n) and A stands for the set of
60
points Z in Rn under restrictions similar to (3.3.?).
Hence (3.3.4) can be written as
(3.3.5)
f
oo JOo Pc
0+
-00
f
0
1'·'"
f
(6l ,.·.,6n )p(s)e
-Y~/2
dy.O ds
n
,
t
t
(6 , ... ,6 ) can be interpreted as the conditional
cl, ••• ,cn 1
n
where
P
probability of
when YO and s are assumed to be held constant.
From. (3.3.1) and
(3.3.,) it is obvious that monotonic behavior of P
,
01"·"
,
0
(6 ,
n 1
I
••• , 6n ) with regard to 6i (i • 1, .. .,n) will imply similar
n
behavior for P ~i:lciI(i) -
16 > sta(cl, ••• ,cn ) _7.
~ut it
has bean shown in Example 2 of section 2.3, (as well as in section
61
n
3.2) that
p( l cjY'i) > k) for an arbitrary constant'k
is an
i=l
increasing function of each of
6~ (i
=
1, ••• , n).
Hence we get
the following:
Theorem 3.3.1.
t
is an increasing fuulction of each of 6i • 6i - 50' (i • 1, ••• , n).
In other words Theorem 3.3.1 states that the probability of
rejecting any population N(50 , 1) under our decision rule
D(C l , ••• , On) is an increasing function of each of 5i - 6 (i • 1,
0
• • • J
n).
From Theorem 3.3.1 an interesting property of D(C , ••• , c )
n
l
follows.
Corollaq 3.3.1.
The probability of rejecting any undesirable
population (i.e.,any population which has not the largest mean) is
never less than the probab:llity of rejecting the desirable population (i.e., that population having the largest mean).
62
ProofS
For the present proof let 0(0)
~
•••
~
Ben) denote means
of the given (n+l) normal populations with common variance
By our decision rule
C1 •
1.
DCc , .. _, c ) the probability of rejecting
l
n
the desirable population N(a(O)' 1) will depend on
.
,
The probability of rejecting any undesirable population N(6(i),1)
(i • 1, 2, ••• , n) will, on the other hand, involve
Comparing the arguments of P
cl,o •• ,en
we notice that
in l3.3.6) and (3.3.7)
63
where
i • 1, 2, ••• ,
n,
j -
1, 2, ••• , nJ
j
11 •
Thus we can make a one to one correspondence between the n argu-
no argument of p
°l,···,on
ing argument of P
ol,···,c n
in (3.3.7) i8 less than the oorrespond..
in (3.3.6).
Henae from the monotonia
behavior of P
(8 , ••. ,8 ) with regard to 81 (1 • 1, ••• , n)·
cl, ••• ,on 1
n
.
as stated before it tollows that the probability of rejeoting anT
undesirable population N(6(1)' 1) (i - 1, ••• , n) is never les8
than the probabllity of rejeoting the desirable population
N(6(O)' 1). This property may be denoted by the eroeerty !! ~
bi.sedness whioh is therefore possessed by our decision rule.
D(a l , ••• , 0n)(C1
~
n
0, Z 01 • 1). It mar also be note4 that
i-I
all the arguments of Po
1"
to'
c ln (3.3.6) are nonposl1;ive anet
n
so (3.3.6) wlll not exaeed Po
c (0, ••• ,
1'"'' n
0). This implies
64
that the probability of rejecting the desirable population N(8(0),1)
will not exoeed the desired signifioanoe level a(O < a < 1).
Henoe
a(O < a < 1) will be the least upper bound of the probability of
inoorrect choice (i.e." not inoluding the population with the
largest mean in the selected group), whatever may be the population
means.
Thus any rule D(c l " ••• , on) satisfies the fundamental re-
quirement as stated in section 1.2.
3.4 Property of graqation.
From Theorem 3.3.1 it follows that
i~
a decreasing (in the striot sense) function of 6 J thul when
0
80 - -
00 ,
the value of (3.4.1) is equal to 1 and when 80 • +
the same value is equal to O.
00,
It is easily seen from 0,).1) that
(3.4.1) is a oontinuous function of 60 , Henoe oorresponding to
6,
all'1' assigned value y(O < Y < 1) ot (3.4.1) there exist• • particular value lor ot &0 tor which (3.4.1) is exactl'1' equal to y,
value 6
0r
~,
The
will clearly in general depend upon 6 , ... , 6 and
1
n
••• , on besides the assigned value y and if 61 , 62 , ."., 6n
inorease by a given oon~tant A, then 60y will also
by the same constant.
In this 8i tuation we
shal.'~
P$ 1ncrea,.d
theretor. lind
that
6
0
<
..
>
•
This property will be designated as the Ilroperty; of· gradation.
shall now study the nature of the unknown constant
is taken to be equal to
6
0a
a(O < a <
for the decis~on rule
D (of.
1).
6
0a
when
We
r
It ,will be shown that
Section 1.3) is very simple in
form, but for other decision rules of class
~
no such simple
ax~
66
plicit expression for 50« can be given.
Let
-Xn
•
•
order statistics derived from a random .ample of size n from
8Q
that
~O·
obeys N(O, 1).
n
Z 6
C1 ear1 Y' th e di s t r ibuti on
0f
-Xn" i-l
n i
18 identical -..&th
".... that
n
Zy )
of
i ...1 (i
Yn - - -n- -
•
-
Under our decisielln rule D the probabilitY' ell! rejeoting
o in
X
a single rejection ill equal to
n
.. p£(Yn -
% 6i
YO) + (i-l - 60 > >
n
at«-7
67
>
•< pry
YO
>
It
7·
L
11
CI-
CI
,
n
<
according as
&0;
Z&
i-1 i
n
•
Thus tor D we have the
special property tha; the probability ot rejecting
whoae
113
a~Y'
population
an il greater than the average at the remaining
tion means is less than
CI
D
popua-
and the probability of rejecting any
population whose mean i8 not greater than the sverage ot the remaining n population means is at least equal to a..
It is now shown that 6 « for the general deoision rule
0
n
D(C1 , ••• , an) i8 not in general equal to E~!ICiX(i»' although
ve have just Ihown that this is true tor D.
The existenoe of 8 (Which is a £unotion of 6 ,' ••• ,
1
0a
°
6n J 1 , ... ,
0Il
besides
4)
for whioh the property of gradation
holds tor the general decision rule has already been shown.
.•. implies that
~
n
« - P~i~lCiI(i) - Xo > sta (ol' ••• , c n )
when
_7
I
This
68
n
It is shown that the assumption 60 ... E( 2: 0iX(i»
a
i-1
(which implies
that BOa is independent of a) leads to a contradiction for the
general case.
The right hand side of
(3.4.3) can be written as
n
0.4.4) p Li~loiX(i) -
BOa -
(Xo/ 60a ) > sta (ol,··.,0n)
_7 ·
.'
But Yo • Xo - BOa obeys N(O, 1). hence by
the definition of t a (01' ••• ,
°n ) we
(3.4.3), (3.4.4) and
get
n
p £i:10iY (1) - YO > sta(Cl'H .,on>-7 • a
n
= P L-1:1c1X(i) - BOa - YO
> sta(ol'··· ,on>-7
,
where 1(1) < ••• < Yen) are n order statistios obtained
from a sample of size n from N(O, 1).
Henoe 1t follows that,
when BOa is assumed to be independent of a, the
69
and
distributions of
identical.
As a necessary condition for this we then get
Hence if _
assume that the unknown oonstant,
60"
a
11 I( Z 0 11(1»
i-I
then 1twill tollow that
°
tor an arbitrary set ot 1 ' I fltlCh that
°1 :: () and
n
Z 01 - 1.
1-1
The equation (3.4.7) does not·, :however, hold i.n leneral and henae
]0
n
is not in general equal to E( Z 0iX(i».
i-l
derive the value of 6
0Ct
for D(ol' •• .,
C
We can, however, easily
n ) when 61 ., 62 -= •••
- 6 • It is shown that in such a situation 60a must aLso be
n
equal to 6 •
1
will oorrespond to the n order statistics from N(O, 1).
above relation
betwee~
Using the
Y(i) and XCi) we get from the definition of
t (0 , ••• ,0 ) the equation
a 1
n
0.4.8)
n
since Z 0i • 1.
1"1
Comparing (3 .. 4.3) with
C3.4.8) it follows that
Xo and YO + 61 have the same distribution N(61 , 1).
that 6
0a
is independent of
fl
and is equal to 6 •
1
This implies
CHAPrER IV
·SOME
4.1.
PROP~RTIES
OF LINEAR FUNC1'IONS OF ORDER S'fATISTICS
Introduction.
In this chapter some results are derived which will be use-
fu1 in the selection of an optimum decision rule from our Class
~
to be discussed in the next ohapter.
Throughout this
chapter Y(i) (i • 1, ••• , n) will denote the i-th order statistic from N(O, 1).
~t
is shown that
n
n
Z ciY(i)·Yo,where E 0i-l,
i-l
i-l
has an asymptotio normal distribution and it appears that this
distribution is approximately normal for all n.
4.2.
n
Asymptotia normality of
!Peorem
4.2.1.
Z 0iY(i) - YO.
i-l
Suppose Y(l) < ••• < Yen)
are n order statis-
tics from a normal population N(O, 1) and that
pendent random variable obeying N(o, 1).
bution (as n --->
. where
00)
YO is an inde-
The asymptotic distri-
of
n
Z I ai' < M, an arbitrary constant, is then
i-l
-
N(O, 1).
72
Prooft
(4.2.1)
n
2
n
- Z &i var(Y(i»· Z
i-I
n
Z aiaj cov(Y(i)' Y(j»
i-I j ..l
•
i,tj
frOM
Hence
1:7J,
pp. 374-377, it is known that for
by (4.2.1) it follows that
n
sutficientl;y lara_
73
n
where
k is some constant.
by ~7_7, pp.
Thus
lim Var( Z a Y(i»
n - > 0 0 i=l i
or:
0 and
253-5, it follows that the limiting distribution of
n
Z aiY(i)
+ YO will be the same as that of YO' being N(O,
i=l
1).
This completes the proof of the theorem.
Coro11arl 4.2.1.
lhe random variable
n
where
Ec - 1
i-I i
has the limiting distribution N(O, 1).
Proof of this corollary is quite obvious.
4.3.
As~ptotic
moments of Y(i).
Let U(l) < ••• < U(n)
be the n order statistics from a
rectangular distribution over the interval (0, 1).
~rm(r =
2, 3, 4) denote the r-th moment of U(m)(m
Let
a
Then it is well-known that
(4.3.1)
1J.2"
m
~q
(p+q) (p+q+l)
,
1, ••• , n).
,
74
2pq(q-p)
(p+q) (p+q+l)(p+q+2)
j
~)m.
~ 3Pqt=2(p+q)2+ pq(E+Q-6)M7_
,
,
(p+q)4(p+q+l)(P+q+2)(p+q+)
where p. m and q - n - m + 1.
~~
of U(m)
i8
Also the first uncorrected moment
n::r".
When Ii (i - 1, ••• , n)
obeys normal distribution N(O, 1),
dt
then hau rectangular distribution over (0, 1). Since F(u) il
an increasing function of its argument it follows that the r-th
oorreoted moment of F(Y(m» will be given by ~rm(r· 2, ),
Let the r-th
correct~d
4).
moment and the first uncorrected moment ot
I(m) be denoted by tLrtm (r .,2, 3, 4)
and ...~, m.
respeotively.
7S
From Mean Value lheorem we get
Now 8.asume tha.t min is held fixed as n
I
Since F (u)
--->~.
00.
is a continuous function of u it follows from large
sample approximation that
m
~lm - n+l
I
•
J
"V
-00
where rv represents asymptotic equality.
,
• E(Y(m) - ~llm)
r
_t 2/2
~l,m
dt
.;;;,e
jli
Hence expectation of
for sufficiently large sample.
an asymptotic formula for
~r'm
,
We thus obtain
expressed in terms of
~rm
follows:
....rm
(r ., 2, J, 4)
,
as
76
where
I
""lIm
satisfies the equation (4.3.3).
(4.3.4) was also given in
111_7 for
A formula similar to
r • 2.
From (4.3.4) it is clear that the
P1
and
P2
coeffioients of
m-th order statistio from a N(O, 1) and a rectangular distribution
over (0, 1) are asymptotically equal.
~t
~11ll
P2m
and
PI
denote the
speotively (m. 1, ••• , n)~
and
P2
of U(m)
re-
By (4.3.1)we then obtaLn atter a
little simplification.
.
~1JIl
..
4(n+2) (n_2m+l}2
(n+3)
2 ...............-.e.( n-m+l).
,
and
.,......=""~~0010
+
2
(n+2)
1
(n+3)(n+4). (n-JIl+1).
6. (n+l)
where
•
l;;m< • •
•
'
77
Hence
r
2
4(n+l) (n+2)
•
(n+3)2
-
•
Therefore~l,m+l-~lm
m(n-m+l)-(m+l)(n-m) 7
m{m+1Jrn-m)(n-m+l) -
4(n+1)2(n+2)
2
• (2m-n)
(n+) m(m+l)(n-m)(n-m+l)
>
<
0
according as
m
>
-<
n/2 •
By (4.4.1) we get also
(
)
4.4.3
~2,m+l-~2m
...
6(n+l)2(n~.?J r
{n+JJ\n+4r L
1
1
"('n-m)(m+l) - (n-m+1).
J
78
• (2m-n)
- (Tn~+~3~)(~n+4Jm(m+l)(n-m)(n-m+l)
>
- 0
<
Hence
according as
•
>
m -< n/2.
Thus it follows that for a rectangular distribution both
skewness and kurtosis increase
mono~onioa1ly when
farther away from the median are considered.
order statistics
For median skewness
is zero, for any order statistic below the median it is positive
and for any order statistic above the meqian it is negative - a
well-known result.
Similar argument applies for kurtosis.
Again as we have remarked in section
4.3,
coefficients of I(m)' which may be denoted by
respectively, are asymptotically equal to
it follows that for
~lm
the
~llm
and
~l
and
~2m.
and
~2
~2tm
~ence
N(O, 1) also the skewness and kurtosis of
extreme order statistics are maximum at least for very large
samples.
Intuitively it seems that this result is true tor any
sample size but this could not be demonstrated by any simple method.
4.5.
APproximate distribution of Y(n)-YO•
Here we shall denote the r-th corrected moment ot Yen)
Y(n)-YO by ~rln and ~r(n) (r • 2, 3,
4) respeotively. Also
and
79
~l
and
as
~iln
~2
coefficients of Yen)
and Y(n)-Yo will be written
and ~i(n)(i. 1, 2) respeotive1y.
moment (or expectation) of Yen)
Clearly the first
and Y(n)-YO are identical.
It can be easily shown that
~2(n) - 1 + ~2'n
and
~3(n)
•
~3ln
~4(n)
•
~4\n + ~21n t )
Let
and
By (4.5.1) we obtain after a little simplification
•
80
(4.5. 2 )
and
Tippett
1:25J has
g~ven values of
ranging between 2 and 1000.
Pl\n and ~21n for n
Henee by (4.5.2) the following table
can be oonstructed.
Table 4.5.1
Showing the values of ~l(n) and p2(n)
1'1
~l(n)
~2(n)
2
.00127
.00272
.00282
.00253
.00189
.00162
.00131
.00100
.00082
3.010
3.019
3.022
3. 0 22
3.020
3. 019
3. 017
3.015
3.013
5
10
20
60
100
200
500
1000
81
From the above Table
Y( n) - YO
4.2.1
for 2
~
n
4.5.1
~
it follows that the distribution of
1000 is approximately normal.
By Oorollary
Y(n) - YO has a distribution which is asymptotically normal,
also it is clear that for n • 1 the distribution is exactly normal.
From Table
4.5.1
it may be noticed that greatest departure from
normality occurs between n - 5 and n- 20.
Thus although the distribution of Y(n) departs more and
more from normality, the distribution of Y(n) - YO becomes more
and more normal beyond n. 20.
The reason for this is seen from
the dominating factor whereas for small n
although the effect of
Yo is negligible, the approximate normality is ensured by the
approximate normal distribution of Y(n)
in such a case ~6-7.
4.6. A. model sampling experiment.
From the results of seotion 4.4 it appears that for every n
the greatest departure from normality among all linear funotions
n
1: c -l, will happen for
i~ i
Y(.)"
n
But the mathe-
matieal proof of this conjecture could not be derived.
Assuming
this conjecture to be true it will then follow from the results of
section
4.5
that all' linear functiQns
82
will have approxlmately a normal distribution - the order of
approximation to normality will be closer than that of Yen) - YO.
To verify this result experimentally, the followi,ng model sampling
experiment was performed.
A thousand random samples of size 4 were drawn from
N(O, 1)
wi th the help of Tables provided by Mahalanobis, Bose, Ray and
Banerji ~16_7.
servation of
To each sample another independently drawn ob-
YO was also observed.
Four observations were ranked
in an increasing order of magnitude and frequency distributions of
the following six statistics were derived:
, (a)
,
(c)
,
(d)
,
\ (e)
D
83
(f)
The
~l
and
~2
•
coefficients for the above six frequency functions
are given in Table 4.6.1.
Table 4.6.1
Showing ~l
of six chosen statistics
and ~2
Linear Function
~l
~2
( a)
.25{Y(1)+Y(2)+Y(3)+Y(4)~YO
.011
).067
(b)
•25!(2)+·2 5Y(3)+·50Y(4)-YO
.00005
3.116
(c)
.25Y(3)+·15Y(4)-lQ
.00202
3.101
(d)
Y(4) - YO
.0318
3.285
(e)
.50Y(3)+·5or(4)-YO
.00613
3.465
(f)
.50Y(1)+·5Oy(~)-Yo
.0200
3.114
lhe exact values of
PI
and
P2
for the linear function (a) are
o and 3 respectively and for (d) it appears from Table 4.5.1. that
84
~1
lies between .00121 and .00272 and that
and 3.019.
~2
lies between 3.010
Hence comparing these known values with the correspond-
ing values of Table 4.6.1 an idea of the magnitude of sampling
fluctuation is obtained.
Thus it appears that our conjecture about
n
approximate normality of 2: ciY(i)-Y may be taken to be correct
O
i=l
for n - 4 from the above results.
To bring out more clearly the
departure from normality for each of the above six linear functions
the observed cumulated frequencies and expected cumulated frequencies on the assumption of normal distribution with the same
first and second moment as obtained by sampling experiment are
given in Appendix for each of the functions (a) to (f).
n
4.7 Minimum variance among
n
~ ciY(i) , ( 2: c
- 1) •
i-l i
i-1
Theorem 4.7.1.
!n
n
2: Y
= 2: Yi/n
i=l (i)
i.1
n
among all
2: ciY(i) such that
i~
n
=Y
n
has minimum variance
n
2: c - 1.
i-l i
Proof:
,
8,
where Vij denotes the covariance between Y(i) and Y(j).
Let the
variance-covariance matrix of 'Y(i) and Y(j) (i - 1, ••• , nJ
j ..
1, ••• , n) be denoted by t(n x n).
To minimize (4.7.1) subject to the condition
n
z: c
i=l i
(4.7.2)
..
1
,
we get the following n equations
n
Z CjVij
.. 'A (i-I, ... , n)
,
j-1
where 2'A is used as Lagrange's multiplier.
In matrix notation
equations (4.7.3) can be written as
(4.7.3a)
,
%0
,
1t)'1
,
where ~ (1 x n) and! (1 x n) denote the row vectors (01' ••• , on)
86
and (1, 1, ••• , 1) respectively.
Since t
is non-singular
we get from (4.7.3a)
(4.7.4)
(4.7.5)
Hence Z!·! whioh implies
~-l ! .,! •
(4.7.6)
By
(4.7.2), (4.7.4) and (4.7.6) it follows that
c
i
,..,
1
Ii
(i ., 1, •.• , n)
.
•
This completes the proof of the theorem.
It follows from the above theorem that
Yn
- YO has minimum
n
variance amollD.g
E ciY(i) - yO.
In the following table the ratio
i=l
,of variances of Yn - yo to that of Yen) - YO
is given using the
Table 4.1(.1
'Ihe ratio of variance of
Yn
- yo to that of Y(n) - YO
n
a
Ratio ( /0)
2
89.2
5
82.9
10
81.8
20
82.3
60
84.3
100
84.5
200
86.2
500
88.0
1000
89.0
, Thus it is seen that the ratio of variances is pretty high for
2 < n < 1000.
88-
theorem 4.~.~.
Let X(l) < X(2)
be two order statistics from
JJL (81 , 82 ) (~f. Section 2.1). Among all linear functions
,
(4.8.1)
X(I) + 1(2)
2
has the minimum variance.
ProofS
will be two order statistics from
..n. (-8,
It then follows from seotion 2.$ that
(4. 8 .2)
Therefore
8 +8
8), where 5 • 5 -. 12 2 •
1
89.
(4.8.3)
where
f
12
represents the correlation between Z(l) and Z(2) and
(4.8.4)
By
•
(4.~.1)
1 - 2(1 to (4.8.1).
and (4.8.4), minimization of
f12 )
ci+c~+2elc2
f 12
c c is equivalent to maximization of
1 2
Hence
Var (olX(l) + 02X(2»
n > 20
0
1 2
subject
is minimum for 01 • 02
The proof of the the:> rem i3 now complete.
not be generalized for
c
•
a}.
'ibis theorem could
CHAP'l'ER V
SELECTION OF AN OPTIMUM RUlli
5.1
Introduction.
In this chapter we shall assume that the number of degrees
of freedom (n+l)(k-l) of s (cf. section 1.2) is so large that cr
may be considered to
b~
known.
Under this restriction the rule
D(Cl, ••• ,cn ) as described in section 1.3 requires the obvious modification that s should be replaced throughout by the population
standard deviation cr.
An optimum rule (TI) among the infinite
class of decision rules D(cl, ..• ,cn)(c i ~ 0,
n
~ c
i=l i
= 1) is
derived below for which the probabilities
(i) of retaining the 'best' population in the selected group,
and
(ii) of rejecting the 'worst' population from the selected group
are approximately maximum.
The meaning of 'best' and 'worst' popu-
lations are explained in the next section.
In section
5.4
the ex-
pee ted number of observations retained in the group under the above
optimum rule is just indicated.
It has been shown that the class
~
of decision rules
satisfy the fundamental requirement, i.e., the least upper bound
of the probability of rejecting the population having the largest
mean from the selected group is a (0 < a < 1), whatever may be
the means of (n+l) given normal populations.
If among the (n+l)
population means, all means except one are equal then obviously
it would be desirable to select that rule from the class
~which
(i) maximizes the probability of retaining in the selected
group the population with the unequal mean if this is
larger than the common mean of the other n populations;
and
(ii) maximizes the probability of not retaining the population
with the unequal mean if this is smaller than the mean of
the other n populations.
In case (i) the population with the largest mean will be designated
as the 'best' population, and in case (ii) the population with the
smallest mean will be called the 'worst' population.
Thus if
2
X(l) < X(2) < ••• < X(n) are assumed to have come from N(O, cr )
2
and X from N(o, cr ), then our desirable rule should ensure great-
o
est probability (i) for retaining X in this selected group if
o
o<
0 <
00,
or, (ii) for rej 2cting
l
X
o from
the group if
-00
< 0 < O.
92
From what we have observed in section 3.3 it is clear that the
above rule will be optimum when X(l) < ••• < X(n) are assumed
N(~, a 2 ) and Xo from N(t
to arise from
+
5, a 2 ), (-00
We shall now show that among the class
~
<
~
<
00).
of decision rules
n
D(C , ••• ,
l
C )
n
(c
>0,
i -
E c. • 1) the rule D maximizes (approxi-
. 1 1.
1.-
mately) the probability of retaini.ng the 'best' population in the
selected group.
In an exactly analogous way it can be shown that
D maximizes also the probability of rejecting the 'worst' population from the group •.
~he
probability of retaining
o arising
X
from the 'best'
population when D(c l , ••• , cn) is followed will clearly be given by
(5.2.1)
n!
n+l
ff
•••
2
e
2.
(2n)
where
B
2 n
2
2
Xc
/20
i""l
~
-(xO-S) /2a - Z
1.
1\
iao()
.
dx(i)'
93
-
00
<
x(l) < ••• < x(n) <
00
B
•
-
00
< x
o<
00
We shall show that the expression in (5.2.1) is(approximately)
maximum for
o under
X
'D. This will imply that the probability of retaining
the present model is maximum (approximately) for
D which
may therefore be considered the best rule to follow in this situation.
Using the same notations as in section 1.3 we know that
n
from the previous chapter Z ciY(i) - Yo - u(C l ' ••• , On) (say)
i"l .
may be assumed to be normally distributed for all practical
purposes whatever may be the value of n.
Let the (approximate)
normal distribution of u(c ' ••• , c ) be denoted by N(~ , a ).
l
2
° °
n
Henceforth we shall consider this distribution to be exactly
normal and so the result which is derived below is correct only
approximately.
For the special case when all ci's are equal, i.e.,
ci·~ (i" 1, ••• , n), we shall write
u for
u(*, •.• ,~) and
) distribution of -u.
and N(-~, -2
a ) for the (
exact
!trom section 4. 7
94
we know that
a is
n
the mininnun among all 0c( I: ci '* 1).
- i=1
In the given
s~tuation
n
I: c I(i) - X + 5,will have the
i-I
o
1
2
(approximate) normal distribution N(1;c' 0c)' where mean 1;c
variance 0
2
c
are independent of 5.
Hence
will have standard normal distribution NCO, 1).
Hence the expression in equation (5.2.1) can be
written as
•
and
ot a (ol""'c n ) - ~ 0 + 0
0'
-
o
1
e
-v
2
/2
dv
fin
-00
Also from the definition of t (0 , ••• ,0 ) (section 1.)
a 1
n
it is
now evident that
00
(5.2.4)
1
e
2
-v /2 dv
...
a
ffn
ota (0 1 , •.. ,0n )-~ 0
0'
o
From (5.2.4) it follows that
O'ta (01, ••• ,0n )~ 0
cit - ~
a
-'"'-0'
.
From equation (5.2 . 3) it is easily seen that the prob-
ability of retaining
X
o
in the selected group under the preaent
situation is an increasing function of
particular case of Corollary 2.2.2.
5
~
a result which is a
Now for any arbitrary 5 > 0
the term in (5.2.3) will be maximum (when e i
I IS
are varied subj ect
n
0i> 0, X 0i • 1) when
. i-l
to the conditions
~t (ol, ••• ,e )-( +6
a
n.5!
llI';t
•
a
(0 , ... ,0 )-(
1
n
~
t:f
o
is maximum.
But
> (j (for all oilS subject to the above rea -
t:f
striations) implies that
6
>
for any 6 > 0
•
a
Hence by (5.2.5) and (5.2.6) it follows that (5.2.6) is maximum
for 15.
Thus we have shown thllt the probability of rejecting the
'best' population is maximum under the decision rule
D for
any 6
97
lying between (0, 00).
5.3. The case when n population means are close together.
We shall assume now that n population means among the
(n + 1) given populations are close together and that the remaining
population mean is either greater or less than all of these n
means.
In such situations it is shown that
n would
be a good rule
to follow.
Let
ill
... ,
X
come from N(O,
o
d~) and X(l)
< ••• <
(-51' ••• , -5n ), 8i > 0, (i - 1, ••• , n)J where
x(n)
from
6i's
(i • 1,
n) are assumed to be quite close to each other •
For the present section let 6(1)
population means
of magnitude.
~
•••
~
6(n) denote the
5i (i • 1, ••• , n) arranged in a decreasing order
From 1heorem 3.3.1 the probability P (suppose) of
retaining Xo is an increasing function of each of 51 (i • 1, ••• ,
n)J hence
(5.3.1)
where
n
p>P>P
,
98
i5
c.
Probability of retaining
X
o
(ari.sing from N( 0, cl) )
and
~
o (arising from N(O,
= Probability of retaining X
0
2
) )
Since oils (i = 1, ... , n) are assumed to be quite close to each
other, it follows that 0(1)' and o(n) will not differ very much •
. But P is a cQntinuous function of OilS (i = 1, •.• , n).
P and
~
Henoe
will not be wide apart and the correct value of P will
be approximately equal to the probability of retaining X ooming
o
from N(O,
0
2
) when x(l) < ••• < X(n) arise from!l(-o, ••• ,o;o),
n
where 0" E 0i/n •
i"7l
In the above situation we have seen in section
5.2
that
D
may be considered to be the best rule among all D(Ol""'On)'
5.4
Expeoted number of observations retained.
In this section we shall
st~dy
the expected number of ob-
servations retained in the selected group when
D is
used in the
general case, i.e., when (n + 1) observations are assumed to have
99
come from N(~i' ~2) (i - 0, 1, ••• , n).
The expected number of
observations retained when a is assumed to be known can be easily
shown to be equal to
and
Ty
n
1: . 1' ~..
i-O
I"i
(n + 1) -
i. defined by y -
j
where
00
dt..
(0 < y < 1) •
,",y
When a is not assumed to be known similar result could be derived
with the help of noncentral-t distribution but the result is quite
complicated to be expressed in a simple form and so is not given
here.
APPENDIX
RESULTS OF A MODEL SAMPLING EXPERIMENT
In TablesA.1 to 1.6,
the cdf obtained from experi-
mental sampling along with the cdr based on the normal distribution (with the same first two moments) are given for a sample of
size 1000 for linear functions indicated in the corresponding
tables.
101
TABLE
A.1
~df of .2 5(Y(1)+Y(2)+Y(3)+Y(4)-YO
Class-interval Observed Normal
Below - 3.4
1
1.3
CJa Bs-interva1 Observed Normal
Below 1.0
845
828.2
"
"
1.2
814
810.1
1.4
904
904.6
"
- 3.2
1
2.3
It
- 3.0
3
4.1
It
- 2.8
6
6.8
It
1.6
927
931.4
,~
- 2.6
11
11.1
It
1.8
950
952.2
'"
- 2.4
14
11.,
.
2.0
963
961.6
tt;
- 2.2
19
26.9
"
2.2
912
918.6
It
- 2.0
32
40.1
11/
2.4
980
986.3
- 1.8
55
58.3
"
2.6
988
991.5
"
- 1.6
83
82.4
It
2.8
993
994.8
It
- 1.4
122
113.1
It
3.0
994
991.0
It
- 1.2
162
151.5
II
3.2
997
998.3
tt
- 1.0
202
197.6
3.4
991
999.0
It
- 0.8
249
251.3
3.6
999
999.,
tt
- 0.6
302
311.9
3.8
999
999.1
It
- 0.4
368
371.6
4.0
999
999.9
It
- 0.2
440
447.8
fit
4.2
999
999.9
0
523
519 •.3
II
00
1000
III
0.2
59.3
590.3
It
0.4
615
6~;8 •.3
It
0.6
139
121.6
It
0.8
190
118.6
,.
..
..
It
.
.
1000
Observed standard deviation
(s.d.) ., 1.113
10'
TABLE Ao 3
Cdr of .25Y(3)+.75Y(4)-YO
Class-intervals Observed Normal
class-intervals Observed Normal
Below 2.2
886
882.2
It
2.4
915
912.3
'"
.
2.6
936
936.2
2.8
951
954.8
14.1
It
3.0
964
968.7
17
21.4
It
3.2
978
978.9
- 1.4
2.5
31.7
It
3.4
983
986.1
lfl
- 1.2
44
4.5.8
III
3.6
988
991.0
It
- 1.0
69
64.5
"
3.8
990
994.4
"
- 0.8
96
88.7
It
4. 0
996
996.6
It
-0.6
124
119.0
at
4.2
998
998.0
It
- 0.4
155
1.56.0
"
4.4
998
998.8
'"
- 0.2
196
200.0
4.6
998
999.3
4.8
998
999.6
1
2.1
- 2.4
2
3.5
- 2.2
4
5.7
- 2.0
6
9.1
It
- 1.8
10
It
- 1.6
It
Below" - 2.6
.It
.
.
..
0
244
250.6
"
0.2
306
307.2
.-
.5.0
999
999.8
11
0.4
366
368.9
tt
5.2
..
999
999.9
0.6
435
434.2
III
5.4
999
999.9
It
0.8
520
501.3
5.6
999
1000.0
It
1.0
579
568.4
"u,
5.8
999
1000.0
It
1.2
638
633.6
"
00
1000
1.4
1.6
689
695.1
759
751 ..5
1.8
2.0
808
853
801.9
845.9
't
"
It
III
II
Observed s.d.
•
1000
1.183
104
TABLE
A.4
Cdf of Y(4) - YO
Class-interval Observed Normal
Cl ass-interval
Observed Normal
Below 2.4
881
875.1
It
2.6
910
905.6
4.8
It
2.8
930
930.0
4
7.6
It
3.0
945
949.4
- 1.8
6
11.8
It
3.2
959
964.1
Itt
- 1.6
14
17.9
3.4
967
975.2
It
- 1.4
17
26.3
3.6
980
983.3
If
- 1.2
33
37.9
3.B
987
989.0
IA'
- 1.0
53
53.4
4.0
991
992.9
fill
- 0.8
74
73.5
It
4.2
995
995.5
- 0.6
105
98.9
It
4.4
996
997.3
It
.. 0.4
133
130.3
It
997
998.4
It
- 0.2
169
167.9
4.6
4.8
997
999.0
5.0
998
999.4
5.2
998
999.7
5.4
999
999 .8
Below - 2.6
1
1.9
lit
.. 2.4
1
3.0
It
- 2.2
2
R
- 2.0
'11
,.
..
It
..
..
.
..
.
.
...
0
208
211.9
..
0.2
257
262.0
..
0.4
319
317.5
It
0.6
371.5
tt
5.6
999
999.9
It
0.8
388
450
440.5
It
5.8
999
999.9
III
1.0
518
505.2
It
6.0
999
1000.0
Jt
1.2
569.7
It
6.2
999
1000.0
tt
1.4
1.6
1.8
2.0
2.2
511
636
632.4
tt
6.4
1000.0
693
756
802
839
691.7
746.4
79,.6
838.6
999
1000
fit
If
It
,.
.
00
Observed s.d.
1000
• 1.230
10'5
TABLE A.5
Cdf of .50 Y(3)+.50Y(4)-YO
Class-interval Observed Normal
Cl ass-interval Observed Normal
Below 2.2
912
914.4
It
2.4
936
938.3
7.5
It
2.6
960
956.7
7
11.9
It
2.8
967
970.4
- 1.8
13
18.4
HI
3. 0
976
980.3
It
- 1.6
20
27.7
It
3.2
984
987.2
"
- 1.4
37
40.8
JIr
3.4
987
991.9
"
- 1.2
56
55.1
It
3.6
991
995.0
&It
- 1.0
89
81.2
ltt
3.8
995
997.0
Mr
- 0.8
116
110.5
III
4.0
998
998.3
- 0.6
151
146.6
4.2
998
999. 0
1ft
- 0.4
193
189.8
..
..
998
999.5
tt
- 0.2
236
240.2
.
4.4
4.6
999
999.7
"
4.8
999
999.8
tfj
5. 0
999
999.9
1000.0
Below - 2.6
1
2.7
It
- 2.4
3
4.6
It
- 2.2
5
1ft
- 2.0
It
..
.-
0
292
297.1
'"
0.2
350
359.4
at
0.4
431
425.7
..
5.2
999
"
0.6
502
494.2
,.
00
1000
0.8
582
562.9
1.0
636
629.7
1.2
698
692.8
It
1.4
758
750.7
"'
1.6
809
802.2
1.8
2.0
853
880
846.6
884.0
.
It
..
"
It
Observed S.d.
1000
• 1.157
107
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