r
J
UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
A COMPARISON OF SEQUENTIAL TESTS FOR
THEl POiSSON
P~ER
by
Kozo Ful';.l.l.shima
September, 1961
This research ~~s supported by the Office of
Nava.l Research under Contract No. Nonr-855(09)
for research in probability and statistics at
Chapel Hill. Reproduction in whole or in part
is permitted for any purpose of the United
States Government.
Institute of st~~cs
Mimeograph Series No. 307
ACKnm·7LEDGEr·lEHT
By deepest thanks are due to tJY adviser, Dr. ll. J. Hall, for
his enthusiastic
encouraGe~ent
course of this research.
and invaluable GUidance durinG the
In addition, I au very thanlcful to Hr.
S. Anster for Goirl[5 through sections of the mnuscript and for
his criticisLJ and useful sUGGestions, and thankful for the cooperation of the Research
Cor~utation
Center, the University of North
Carolina.
I an extrenely crateful to the Departnent of Statistics, the
University of North Carolina, and the Office of Naval Research for
.e
their financial aid, without which this work would not have been
possible.
Also, I wish to express
11rs. H. Plass, who gave ne
pY
~y
deep gratitude to Mr. and
first chance to study in the United
States, for their continuous help and encourageDent throughout tJY
studies.
Finally, I thank Miss llIartha Jordan and Mrs. Doris Gardner
for their careful typing of the Danuscript and help at various
ti~es.
iii
TABLE OF CONTENTS
CIlilPTER
I
PAGE
ACKNOHLEDGEMENTS
i1
INTRODUJTION --
1
SEQUENTIAL PROBABILITY RATIO TESTS AND ITS APPLICATION TO THE POISSON DISTRlBUTION- - - - - - - -
5
1.1 Wald's Sequential Probability Ratio Test
(SPRT) for Testing a Simple l~othesis
against a Simple Alternative - - - - -
5
1.2
The SPRT for the Poisson Distribution-
1.3 OC and ASN Functions for the Poisson
SPRT - - - - - - - -
.e
II
....
---
7
8
1.4 The Truncated SPRT - - - - - - - - - - - -
12
MINDiUM PROBABILITY RATIO TEST AND ITS APPLICATION TO THE POISSON DISTRIBUTION- - - - - - - -
14
2.1 The Minimum Probability Ratio Test (MPRT)
for Testing a Simple HYPothesis against a
Simple Alternative - - - - - -
III
14
2.2 Maj or l'roperty of the MPRT- - - - - -
15
2.3 The MPRT for the Poisson Distribution
17
2.4 A Choice for 9o - - - - - - - - - - - -
18
FIXED SAMPLE SIZE TEST .AND HOEFFDING'S LmoJER
BOUND FOR THE EXPECTED SAMPLE SIZE- - - - - - - - - -
3.1 Most
P~1erful ~est
22
for the Poisson Distri-
bution - - - - - - - - - - - - - - - - - - -
22
3.2 Normal Approximation for the Poisson FSST-
23
,.3 Hoeffding's Lower Bound (HLB)- - - - - - -
24
3.4 Hoeffding's Lower Bound for the Poisson
Distribution - - - - - - - - - - - - - - - - - -
26
iv
TABLE OF CONTENTS
PAGE
CHAPTER
IV
THE PROOO/lMS FOR THE
MPRT
AJ:ilD THE
SPRT 0
- 28
FOR THE POISSON DISTRIBUTION .. - - - .. .. - - 4.1
Brief Explanation of the Program for the
MPR~ - - - - ~ - - - - - - ~ - - ~ - - ..
- .. ;2
... _-_ .. _ - - - - - - - _ . _ - -
- .. ;5
4.2 Variable Assignments for
V
~y
.. - 28
t1MPRT .. A"
APPENDIX
I
NOTATION
A.l
.e
II
Notation- - .. - .. - ........ - - - - - - .... - .. - ;7
SAMPLING PLAN
A. 2.
III
A.;.1.
A.3.2
A.3.3
A.;.4
11.:A =
H2~ A
a: =
A.4.2
- - ;9
_...... - - .. - - .. - - -
- - 40
Characteristics of the Poisson MPRT
and SPRT Approximation .. - - .. - - - -
- .. 40
11.: A = .1 against H2 : A = .3 TABLE II, 11.: A = .5 against H : A = .8
2
TABLE III, 11.: A = .5 against H : A = 1.
2
TABLE I,
TABLE IV, HI: A
= 5.
against H :A
2
- .. - 40
= 8.
- 42
- 45
- 47
.......... - - - .. - - - - - .. .. .. - - - 49
Characteristics of the Poisson MPRT
(A
A.4.l
.5 against
.01 .. - - -
= 1.0 "lith
TABLES V - VI
A.4.
-.... - .. - - - - .. .. - .. .. - - - - - -"
sampling Plan for
TABLES I - IV
A.;.
IV
-.... - - - .. - .. .. - .. - - .. .. .. .. .. - .. - ;7
O
= D and
s) and
TABLE V, HI: A
TABLE VI,
H1:
A
SPRT Approximation..
= .5
49
= .1
against H : A
2
= 1.
against H : A • 2. - .... - 53
2
.. -
49
v
TABm OF CONTENTS
PAGE
V
TABLE VII - - - - - - - - - - - - - - - - A.5.
Characteristics of the Poisson MPRT
(A. = D) J SPRT , SPRT Approximation
o
0
and FSST for H2: A. = .01 against
57
H : A = rl - - - - - - - - 2
.e
VI
A.5.1 TABLE VII - 1,
a =.1 - - - - - -
57
A.5.2 TABLE VII • 2,
a
=
58
A.5.; TABLE VII -;,
a
= .01
.05 - - - - - - - - • -
- - - -
59
A.5.4 TABLE VII - 4, a = .001 - - - - - - - -
61
A·5.5 Figure 2,
62
ASN
curves
----
TABLE VIII - - - - - - - - - - -
A.6
Characteristics of the Poisson MPRT
(A. = D), SPRT , SPRT Approlimation
o
0
and FSST for ~: A. = 2. against
H:
A. = 4. - - - - - 2
A.6.1 TABLE VIII - I, a =.1 --
= .05 ;, a = .01 4, a = .001
A.6.; TABLE VIII -
A.6.4 TABLE VIII A.6.5 Figure;,
-
ASN curves
"K. ll'UrdSID'MA"
6;
6;
65
66
- - - - - -
68
69
COMPILER PROGRAM
A. 7.1
6;
64
A.6.2 TABLE VIII - 2, a
VII
57
MPRT - A" -
BIBLIOGRAPHY - - - - - - - - - - -
69
ERRATA PAQ'l:J
for
A COMPARISON OF
Line number fran
the top
7
SE~
TEarS FOR THE POISSW PARAMEn'ER
by
Kozo Fukushima.
Correction
Misprint
7
7
17
17
m Xi
tt A.
1=1
0
17
20
Wich :implies
19
16
Leo
20
.16
f'2<X)
17
f'2m ~
=~
f'lm
Wich implies
=A.
e Ijxg
g
The time for camputaticm.
36
16
'the MPRr is
53
5
(n ;: 11)
os
lm
'"'A.
Leo f'2(X)
rg
A.~ e 2jxt
The time for ecm,pilaticn
'tb.e SDN of the MPRI! is
56
000071
f
o
20
17
<
A. +
33
unif'~
f'2m
00000071
unifo~
INTRODUCTION
This investigation treats a truncated sequential test for testing
a simple hypothesis against a silll:Ple alternative.
Attention is con-
fined to the case of sampling from a Poisson population.
Although
VTald 's £l]}l sequential probability ratio test (SPRT) will terminate with probability one at some stage of an experiment, there is
no guaranteed upper bound for the sample size of this test.
However,
for cases in which time and cost of sampling are involved we often
have to face the situation of making a definite decision within a
given number of trials.
.e
\'lith this in mind, several sequential
tests, such as Anderson's £".!:.7, Armitage's
£g,7
and Donnelly's £2.7,
have been proposed which are truncated within some fixed stage. However, none of these are directly applicable to Poisson sampling.
Here, as a variation of the SPRT, we treat Hall's £!!.7 minimum
probability ratio test (MPRT).
llliereas for the SPRT the rejection
and acceptance lines are parallel, for the
~~RT
these two lines con-
verge so that the test always terminates with a definite decision by
a predetenlined stage.
More specifically, we present and compare several test procedures
for testing whether a Poisson parameter has value
A or A (specil
2
fied numbers) based on a sequence of independent observations fram
a Poisson population.
The test procedures considered here are:
~he numbers in square brackets refer to the bibliography listed at
the end.
2
i)
the sequential probability ratio test, both
~Ultruncated
(SPRT) and truncated (SPRT)
(Chapter I),
o
ii)
iii)
the minimum probability ratio test (NPRT) (Chapter II), and
the most powerful fixed sample size test (FSST) (Chapter III).
A diagram for carrying out these tests appears in Figure I 2 (A.2).
A requirement in all of these test procedures is that each error
probability should not exceed a common specified level a.
The bases
for comparison of the test procedures are the operating characteristic
(OC) function, the expected sample size (ASN) function, and the standard deviation of the sample size (SDN) function.
bound
127
Hoeffding's lower
(IffiB) on the ASN at an intermediate value is also presented.
(Chapter III)
.e
Major attention is given to the 11PRT since the other procedures
are quite well known.
An extensive discussion of this test pro-
cedure appears in Chapter II.
Actually, the MPRT can only guarantee
an upper bound on an average of the two error probabilities. Achieving
equal error probabilities hinges on the choice of an intermediate
'"o value.
'"0
is here chosen to make equal the divergence (a con-
"'0
cept from information theory) between "'1 and
and "'2'
This value of
also given to
"'0 = s,
"'0
is denoted D.
and between
"'0
Some consideration is
the slope of the SPRT acceptance lines.
The
calculations indicate that either of these choices is quite successfUl,
s
D being slightly better for moderate or large a values and
being slightly better for small a values
(~
tables and Chapter V.)
2A11 figures and tables appear in the Appendix.
.01). (See the
For the HPRT, calculation of seven points on the OC curve, ASN
curve and SDU curve were carried out,
Ct
= .1,
.05, ,01 and .001, for
the following pairs of hypotheses:
7-.2
"'1
Table
I
,1
II
·5
III
·5
.'
V
bound at D and
s, and
8.
D
D and s
D and s
2.
Also presented in Table I and VI are the
.e
D
.5
1-
FSST, the maxiD1tUU sample size
D
1.
.1
VI
D
,8
5,
IV
"'0
11
0
for the
F
of the MPRT, Hoeffding I slower
sar~le
size n
s, and the ASN of the SPRT at four values,
"'2' calculated by Wald IS approximation 11];7.
0,7-. ,
1
The exact cal-
culations were performed on the UNIVAC 1105' by a program described ill
Chapter IV and A.7.
For purposes of comparing the three test procedures, tbe ca1culation of 16 points on the OC curve, ASN curve, and SDn curve were
carried out for all tests, for
Ct
= .1,
.05, .01, and .001, and for
the following pairs of hypotheses:
Table
VII
VIII
Figure
2
.01
2.
'In this thesis the computer refers to UNIVAC 1105.
.1
4.
4
Exact calculatUDs were done for the OC, ASH and SDN ftUlctions of the
HPRT (>-'o
= D) and the SPRT truncated at no' the maximum sample size
of the IJ1PRT, and Hoeffding's lower round a.t
the computer.
>-.
III
S
a.nd D by
Also calcula.ted vTere three values of Hald' s approxima-
tions to the OC and ASN functions for the untruncated SPRT.
The
FSST calculation are based on Poisson tabl63 f§l to the extent available and otherwise on the normal approximation.
A discussion of the
results of the calculations appears in Chapter V.
One use for tests concerning a Poisson population is in quality
control worIt where the defects in a unit are counted. and the quality
of a lot or process is judged by the average nUtwer of defects per
unit.
,e
This differs from the test where each unit is placed into
a "defective" or "non-defective" category and the quality of a lot
or process is determined by the total number of defectives.
The defects
per unit analysis is useful under the following
conditions.
i)
If almost every unit contains at least one defect, a fraction
defective plan ..• "defective" or "noi1..def'ective" classifice..tiol1 ...
obviously not feasible for such a case.
ii)
For products which are expensive to produce or inspect and
products customarily inspected in small lots, it is too costly to obtain samples large enough to assure high discrinrlnation by a fraction
defective plan.
However, if the number of samples observed is suffi-
ciently large and the quality of the product is high, the result will
not differ greatly between these two tests.
iii)
" test to apply exactly, the defects must be randomly
For this
and independently distributed.
CHAPTER I
SEQUENTIAL PROBABILITY RATIO TEST
AND ITS APPLICATION TO THE POISSON DISTRIBUTION
1.1.
.~!a~Cl'.~.~~~~~t~lPr~?a.bili.~::r Rati~st JS?RT) __~.~.
Testing a Simple Hypothesis against a Simple Alternative
[ill.
Suppose xl' x2 ' •••• is a sequence of independent observations with a common density function f(x:9) and let
~
= (xl'
x2 ' ••• xm), m = 1, 2, "', be a sample of size m.
Let 01 and 02 be, respectively, the desired probability of accept-
.e
ing the alternative hypothesis H :9 = e 2 when the true parameter
2
is 91 , and of accepting the null hypothesis Hl :9 = 9 1 when the
true parameter is e 2 , and call (aI' 02) the strength of the test.
We also denote by di the decision to accept Hi(i = 1. 2).
For any positive integer m, the joint density function f im
of a sample of size m under Hi (i = 1. 2) is
(1.1.1)
f.1.m
= f(xi:e.)
1.
• f(x-:e
) ••• f(xm:9.)
c i
J.
Then, the SPRT is carried out as follows:
For suitably chosen A and B (0 < B < 1 < A), at the m- th
stage (m
= 1.
2 ••• ) of the experiment,
(i) stop sampling and accept H if
l
B<f2 / flj < A (j=1.2 .•m-l) and f 2m!flm ~ B
(i1) stop sampling and accept H2 if
(1.1.2)
6
(iii) otherwise, continue sampling until f 2m/fJ:m falls into
either category (i) or (ii).
The calculation of A and B to obtain the desired strength
(a , C¥2) is very laborious. Therefore, in practice. Wald suggested
l
putting
(1.1.4)
as substitutes for A and B, respectively.
Denoting the resulting error probabilities byai and a 2 we
can easily see that
and therefore, at least one of the inequalities, ai:5 C¥l and
.e
a :5 a 2 , must be satisfied. Moreover, if C¥l
inequalities are almost achieved.
2
= C¥2
= a, both
The most important characteristics of the SPRT are the operating characteristic- (OC) and the Average Sample Number (ASN) functions.
The OC function, L(e), is defined as the probability of
accepting the null hypothesis H when the true parameter is 9.
l
An apprOXimation [il-.7 is given by
(1.1.6)
L(6) ...
where h . satisfies
Ee
=
1
f(x:6 2 ) /f(X:e )
l
the right hand side of
wi th 21 = log
(1.1.7),
hz
If h = 0 is the only solution of
(1.1.6) is evaluated by taking the
7
limit using l'Hospital's Rule.
The ASN function, Ee(n), is the expectation of the sample
size when e is the true parameter and is approximately givenLJ1_7
by
(1.1.8)
and
(1.1.9)
1.2.
Ee(n)'" {L(e)(lOg 13)2+fi-L(e17(log A)2} / Ee(Z) if Ee(z)=O
The SPRT for the Poisson Distribution.
Suppose xl' x2 ••• is a sequence of independent observations
from a Poisson distribution with mean A. We want to test the null
hypothesis Hl:A=~l against the alternative H2:~=~2(>~1) with strength
(CX ,
1
.e
CX
2
)·
For any positive integer m, the joint density function of X
m
under Hi is
m
(1.2.1)
f
im
=
·~lxi
A~-
i
e
-m~.
~
m
/ ~ x.
i=l ~
Hence,
(1.2.2)
and
Using Wald's values (1.1.4) for A and B and taking logarithms
of (1.1.2) and (1.1.3), the acceptance rules are given as follows:
At the m-th stage of the experiment,
(1) stop sampling and accept H if
l
8
m
~
(1.2.4)
i=l
ii)
xi
log (~2/~1)· m(~2-~1) < log B'
stop sampling and accept H2
m
.~
J.=l
iii)
xi
log (~2/~1) -
if
m(~2 - ~1)
> log A'
otherwise, continue experimentation.
(1.2.4) and (1.2.5) are conveniently written as
(1.2.6)
m
1:: x
<
i=l
.-
m
(1.2.7)
x >
1=1
~
-
log
log
{a
2
~2
/ (l-al )}_
- log
~l
== cl
+ s m ==
log {(1-a2 )/al }_
log
~2
(~2-~1)m
+
- log
-
c
~l
log
+
~2
&m
- log
(say)
(~2 • ~l)
log
~2
~l
tl
• log
~l
s m == b (say) .
2 +
m
Thus, it is seen that the acceptance lines are parallel straight
lines ~th the same slope
s
= (A2
- ~l)/(log ~2 - log ~l)
independent of the desired error probabilities.
which is
However,
cl and c2
are determined by the hypotheses values and the error probabilities.
If a l = a 2 = a, from (1. 2 .6) and (1. 2 .7) it follows immediately that
c2 = -c l = c (say).
1.3
OC and ASN Functions for the Poisson SPRT.
i)
To obtain the OC function of the Poisson SPRT we have to find
that value of h(~)
(see (1.1.7) for which
9
co
e -A./ x!
(1.3.1)
= 1.
Summing the series and taking logarithms, (1.3.1) leads to
(1.3.2)
Then, from (1.1. 6), the OC
function L( A.) is approximately given
by
(1.3.3)
L(A.)'" { (
1 - 0:
2
0:
1
However, it is impossible to explicitly solve (1.3.2) for h.
statistical Research Group, Columbia University
~1~7,
The
gave an in-
direct method of obtaining a solution, but even this method does not
.e
yield the OC
function directly for a given value of A..
For practical purposes, we can find the approximate values of
the OC
function for the following particular values of A. to give
a rough picture of the OC
A.
L(A.)
0
1
(1.3.5)
A.l
1 - a1
(1.3.6)
s
c 2 /(c 2 - cl ) = 1/2 if a l =0:2=a
(1.3.7)
A.2
0:
(1.3.8)
co
0
(1.3.4)
e
function of the test.
Since for A.
=s
2
we have h(A.)
=0
as the only solution of (1.3.2.),
10
(by L'Hospital's rule) (1.3.6) is obtained by taking the limit value
of (1.1.6).
1'le sha.ll introduce a graphical solution for (1. 3 .2) •
By
(1. 3.2)
To solve (1.3.9) for h, mwte the following tra.nsformations:
(1.3.10)
Since 10g(A2/Al ) and (A - Al)/A are fixed numbers for any given
2
A, Y1 is an exponential curve, and Y2 is a straight line. Then
the solution of (1.3.2) can be found as the intercept of the straight
line Yl
and the exponential curve Y2 as shown below •
(A ::; 0)
.e
(11 > 0)
.-l _
-==========
I
I
(A
= co)
I
--1
1
.-.l.- - - - - - - . . . , . . 11
h
The accuracy of the graphical solution can be improved, if desired,
by a Newton itenltive procedure.
From the h· found by this method we can obtain the approximation of
ii)
L(~)
by substituting h
into
(1.1.6).
In order to obtain the ASH function from (1.1.8) or (1.1.9)
11
we have to J:..now the
OC
:f'unction,
in the preceding parts, the
ruately.
OC
However, as we have seen
L(A).
function can only be found approxi-
For the following particular values, the follovTing approxi-
mations to the
ASN function may be found from (1.1.8), (1.1.9) and
(1.3.4) to (1.3.8):
.l\(n)
o
-c1 /s
/(Al -
{ (1 - 01) c l + O1c 2}
= (1
- cl c2 /s
s
.e
(1.3.14 )
s)
- 20)c/(s - Al ) if 01
=
c
2
/s
= 02 = °
if 01 = 02 =
°
{ (1 - 02) c2 + a 2cl } /(A 2 - s)
(1. 3 .15)
00
e
(1.3.13) is obtained from (1.1.9) since Es(Z)
five values of
= O.
These particular
A will serve sufficiently well for most practical
purposes.
Although it is known that
~(tJ,2) exists, neither exact nor simple
approximate formulae to obtain the variance of N have yet been found,
and very few empirical studies have been undertaken.
HOvTever, from
Tables VII and VIII it will be seen that the standard deviation of N
for a truncated SPRT tends to be very large when
and A
2
and
a is very small.
A is between A
l
12
1. 4.
The 'J)runcated SPRT
Since the SPRT has no upper bound for the sample size, for prac.
tical purposes we may have to force the test to terminate at some
definite stage, say no' of the experiment.
't'lald [1J:} gave these
rules for truncation.
If the test does not terminate by the no·th stage, at the no·th
stage,
stop sampling and accept Hl
i)
no· 1
log B < E
i=l
(1.4.1)
ii)
if
Z1 < log A and log B < E
stop sampling and accept
~
if
no·l
.e
(1.4.2)
zi < 0
1=1
no
log B < E zi < log A and 0 < E
i=l
i=l
Zi < log
A
In this thesis we use the following rules for truncating the
SPRT.
Let no be the maximum. sample size of the IJ1PRT (see Chapter
II) under the same hypothesis and strength, then the terminal decision
at the no·th stage will be the following:
i)
stop sampling and accept
~
2
ii)
Stop sampling and accept H if
2
2
if
If the SPRT
is truncated at a sufficiently large no' it is
to be expected that the error probabilities will not be greatly
affected.
However, truncation '\'1111 reduce the average sample size
somewhat, especially for intermediate A values and it will have
a similar effect on the SDN function .
.e
•
CHAPTER II
MIHIUUM PROBABILITY RATIO TEST AND ITS APPLICATION
TO THE POISSON DISTRIBUTION
The expected value of the sample size (ASN) of a seQ.uential
test depends on both the parameter point
sequential test .
0
and the particular
Ideally, we would lUte to find a seQ.uential test
which minimizes the
AS~T
function for all values of
"uniformly most economical" test exists.
lower bound on the ASU at inten:nediate
Hoeffding
", but no such
£27
has given a
"--values (Chapter III) for
any seQ.uential test meeting specified bounds on the error probabili-
.e
ties.
In order to come close to achieving this lower bound under cer-
tain conditions, Hall
£'17
bility ratio test (NPRT).
introduced the seQ.uential minimum proba..
In this chapter we give a general dis-
cuss ion of this test and then consider the special case of the
Poisson distribution.
2.1.
The Minimum Probabilitl Ratio Test (MPRT) for Testing a Sip.;ple
Hypothesis
against a Simple Alternative.
Suppose
..• is a seQ.uence of independent observations
2
'With a common density function f(x:O) ana. consider testing the null
hypothesis
be a
Xl' x
Hl :
9 = 01
against the alternative
parameter point between
Denote by
\t
"1 and 02
and
H : 9 = 02'
2
f
i
Let
= f(X:Oi )(i=O.1.2).
= (Xl' x 2 ••. xm) a sample of size m and f im the
joint density function of
X.
m
00
15
where k + k = 1,
2
l
2
for f
and f , respectively, to obtain a specified weighted average,
l
2
a, of the error probabilities (see section 2.2).
We introduce the weight functions
k
and k
l
The HPRT procedure is described as follows:
at the m-th stage
of the experiment,
i)
ii)
if
~
stop sampling and accept
iii)
.e
~
stop sampling and accept
if
otherwise, continue sampling •
Since the acceptance lines generally converge at some stage of
the experiment, say no' the test has the upper round no
to the
na.ximum number of trials, and therefore, either category i) or 11)
will occur for some m < n .
-
2.2.
Major Property of the MPRT
= ai'
Let l' . ( d .)
~
dj(j
0
= 1,2)
(2.2.1)
J
be the probability of making the decis ion
when Hi(i
kl
ai = k l
= 1,2 and
i ~ j) is true.
00
1'1 (d2 )
=
k
1;
l
m.=l
Then,
J
f
Im
m
00
(2.2.2)
--
k a
2
2
=
k p (d )
2 2 l
=
k
2
1;
m=l
,f
8
1
m
f
2m
16
i
.
where 8m (i = 1.2) is the subsets of the sample space of Xm for
making decision di , by (2.1.1) and (2.1.2), respectively.
vie show one of the important properties of the MPRT by the
following lemma.:
Lemma
1
f!!.7:
For the NPRT with specified Qo' the specified weighted average
of the true error probabilities is not greater than the preassigned
value a.
Proof:
.e
I;)
Il1
Sit is obvious that
ill
that is,
where Po(di ) is the probability of di
parameter value. Similarly, in 8 1
m
k'2 a'2 -< a0
when 00 is the true
Po(dl )
Hence, it follows immediately that
assuming the acceptance lines converge.
exi
For the case k l = k2 = 1/2, from (2.2.3) to (2.2.5) we have
~ 2 a Po(dl ), a ~ 2 a Po (d ) and a + a ~ 2 ex. If 00 can
2
2
1 2
17
be so chosen that PO(dl ) = P0(d2 ) = 1/2 , then aJ. ~ 0: and
In any case, at least one of the inequalities must hold since
1
0:
+
2~
0:
2~ 0:.
0:
2 0:.
No method of choosing
to achieve specified individual error
o
probabilities is known, except for a limited result in the case of
symmetry
./!t7.
Q
Such symmetry does not obtain in the Poisson case
considered here.
We shall attempt to achieve equal error probabilities
by a method described in section 2.4.
2·3.
The MPRT for the Poisson Distribution
Let
Xl' x ••• be a sequence of independent observations from
2
a Poisson distribution with mean ')I.. We wish to test the null hypoHl : ')I. = ')1.1 against the alternative H : ')I. = ')1.2 ( > ')1.1)' and
2
assume weight functions k = k = 1/2 and 0: < 1/2 .
l
2
The ~~RT is carried out as follows:
thesis
.e
at the m-th stage of the experiment,
1)
stop sampling and accept
~
if
<
i.e.
which implies
ii)
f
2m
~
f
lm
stop sampling and accept
~
if
2 0:
18
which iL1plies
iii)
f
> f lm
2m
otherwise, continue sa:crpling.
(2.,.1) and (2.,.2) v~ll define the acceptance lines of the test.
These two lines meet at m:: n' where
o
o =
n'
and hence, this is the upper bound for the saraple size.
maximum sample number -n
nI
o
.e
The actual
will be an integer slightly smaller than
o
•
An illustrative diagram for carrying out this test with a comparison of the truncated SPRT and the
FSST appears in Figure 1
in the Appendix.
2.4.
A Choice for
00
In the MPRT it is possible to choose
°o
at any parameter
point other than 01 and 02' depending on our emphasis on the error
probabilities
D1
and a
2
and the weight ftulctions
and k ·
l
2
in an attempt to obtain approxi-
In this thesis, we will choose 00
mately equal error probabilities at
this point of view, we take
vergence"
r77
-
-
°o
P o(d )
l
between 01 and 0, and between
0
= f 2(X);
= p0(d2 ).
From
as the value which yields equal tld:!_
Suppose the null hy:pothesis is
native H2 : f(x)
00' l.e.
k
I1.:
f(x)
°
0
= flex)
and 02 .
and the alter-
then for a given x, the mean information
19
for discrimination per observation in favor of H
l
defined 117 as
.J
I (1:2) =
against H is
2
dx
f l (x) log
K
~
where
r (2:1) is similarly defined.
is the entire sample space.
Under the assumption that the probability measures for
f (x) and
l
"absolutely continuous", the integrals al'WaYs exist.
f 2 (X) are
The divergence D(l,2) between Hl
(2.4.1)
D(l,2)
= I(1:2)
=~f
+ I(2:l)
and H is defined 117 as
2
flex)
!flex) - fn(X) 7log ----- d x.
i....
c;
f (X)
2
D(l,2) is a measure of the difficulty of discriminating between
.e
hypotheses H and H , and thus will be a reasonable choice to deterl
2
mine Q for the MPRT. Further detailed discussion concerning ino
formation and divergence "Till be found in Kullback 117.
Example:
Poisson case
Suppose H :
l
A-
= A-l
i.e.
x -Aflex) = A-1 e l/x~
i.e.
f (x)
and
H2 :
Then,
by
A- = A-
2
definition
00
I(1:2)
=
L.
2:=0
x
A1
xl
00
•
2
=
L.
x=o
x!
e
-A-l
= A-xl
x e -A-l/x!
1
x -Ae 2/x !
A2
A-
log
-A-
e l/x!
20
(2.4.2)
Sililila.rly,
Hence, from
(2.4.1)
For some
~o'
to
(2.4.3),
"There :\1 <
nating between hypotheses
equal.
That is,
:\0
D (1,0)
= D(O,2);
that is
:\1
~o
< ~2' the difficulty of discrhli..
and
:\0
and betvreen
:\0
and :\2 is
is obtained from
.e
=
0.
Newton's approximation method 'Was used to solve this equation
numerically, starting with the initial value:\ 0.0
That is,
(i
= 0.1.
... )
with the criterion for stopping the iteration
I ~o .J.'+1
.. :\O. .Il -< 10- 5
=(:\1
+
:\2)/2.
21
and
i < 200.
By this method, divergence values in the Appendix were computer
calculated.
This A.
o
value is there denoted by D.
A few examples were considered using
A.
o
=s
instead of
a comparison of these choices is discussed in Chapter V.
that
•
D < s.
A.
0
= Dj
It 'Was found
CHAPTER III
FIXED SAHPLE SIZE TEST .AND
HOEFFDING'S
L01'lER BOUND FOR THE EXPECTED SAUPLE SIZE
Most Powerful Test
for --~.~===::.::,;;;;..:,:::=:=:,;;;;;;.:;.:.
th..~1 POiSS.0h~~D1stJ?1bU.tion.
.
3·1.
-
To make a sample size comparison between the HPRT and a fixed
sample size test (FSST), 't-re need to find the minimum sample size for
the Iilost powerful FSST with specified bounds on the error probabilities.
The most powerful FSST is obtained by the Neyman-Pearson
lemma.
For the Poisson case with sample
.e
size
m, the test is carried
out as follows:
i)
accept
H2
if
m
f
(3.1.1)
1!
i1)
2
i=l
m
21;1
-=
f
i\:lc1
i\xi
1
i=l
lIil
1!
otherwise, accept
e
-i\
2/x1
>
-i\
e
l/X
k
(>
0)
i
H1 .
Taking logaritbIils on both s ides of (3.1. 1), it follows that
m
1:1 xi (log i\2- log i\1) - n(i\2 - i\1) ~ k'
and
m
E
1=1
x
i
>
-
It-II
Iil
e
,f
He know that the random variable
y
= r.
1=1
tribut10n with mean
mi\, i.e.
xi
has
the Poisson dis-
k"
If the Type I error a and the size of sample m are given,
l
is obtained as the smallest integer for which
1 - Fw.. (kit - 1)
1
~
~
where FA,(X) is the Poisson distribution function a.t
x.
If the
Type I error a and the Type II error a are preassigned, the
l
2
sample size can be obtained by finding It" and the mininJal m which
will satisf'y,
(It II
1 - F
rnA.
1)
-
< (Xl
l
and
(:;.1.6)
:;.2.
F
rnA.2
(k" - 1) <
-
Normal Approximation for the Poisson FSST
If A,
is large, the Poisson distribution can be approximated by
the Hormal distribution.
(:;.2.1)
where
FA,(x)
=
That is,
(x - A + 1/2 )
~
t
~
(t)
= .[ (1/
-co
Then, by (:;.1.5) and (3.1.6)
~
/2ic
)
e
_x2/2
dx
are the required inequalities.
If the Poisson mean is not large, the normal approximation may
not be adequate so that the following rule
viaS
used.
If
(m >"1)1/2
< 4.0, several integer values were chosen in the neighborhood of
t~e
m which
obtained from the Normal approximation method.
'WaS
These were substituted into (:;.1.5) and (:;.1.6), with suitably chosen
It" s
from the Poisson tables, until (:;.1.5) and (:;.1.6) were satis-
fied.
The minimum such m is the sample size of the most powerful
test.
In the tables in Appendix those values obtained by the Normal
approximation appear with
:;.:;.
Xl' x
2
.••
Also,
let
H: Q
l
f(x:Q)
is to accept
and
FSST *
£'27
be a sequence of independent observations with
a common density function
null hypothesis
n*
F
such as
Hoeffding's Lower Bound (RIB)
Let
di
*,
f(x:Q), and suppose 1,e 'Want to test the
= 91
= fi(X:Q)
against the alternative
under Hi(i
= 1.2),
H : Q
2
= 92 ,
and the decision
Hi'
As one of the optimum properties of the SPRT, it has been proved
["1E,7
Q
l
that the
and 02
SPRT minimizes the expected sample size at the points
subject to specified bounds on the error probabilities.
In general, its expected sample size is largest when
Q
l
and O
2
but not ahre.ys (see Figure 2).
Q is between
However, there exist
tests which have a smaller expected sample size at intermediate
values than the SPRT.
Kiefer and \'Teiss
£§J
0
have discussed impor-
25
tant qualitative properties of such tests.
by comparing, at any parameter :point
Q,
These tests may be judged
the expected sample size of
the test with the smallest expected sample size attainable by any
test having the same error probabilities at
Q
l
and
Q2'
The Hoeffding Lower Bound on the expected sample size at any
Q
o
is given by
assuming that the following integrals exist
and
Also assume that
and that
If
E (r:
0'1
J=
T
=
log { f (X)/f (X) }
2
l
j
is satisfied where Yj
2
=
y)2
Eo(N)
Hoeffding :proved ~27 that for all
bound among all strength
go' (3.3.1) gives a
l~rer
(OJ.. •cx2 ) tests.
For the Normal density function with variance one and mean
where
Q
o
A
0, gl
= -8
=8
and Q
2
Q
i
> 0, he compared the numerical
values obtained by his 10't'rer bound with those of the fixed sample
size test, one of Anderson I s tests ~J::.7 and the SPRT of the same
strength
(OJ..'
cx2 )·
For
OJ..
=
CX
2
= cx < 1/2,
the results indicate that
26
equality in (;.;.1) is nearly obtainable with a FSST if a is very
small and with the SPRT if a is sufficiently large.
For other a
values, those commonly used in practice, the MPRT (which coincides
1'7ith one ot Anderson's tests) nearly attains this bound.
;.4.
Hoettding's Lower Bound for the Poisson Distribution
Suppose f(x:~)
and we wish to test Hl : ~ :: ~l
against H2 : ~ = ~2 (> ~l) with strength of the test OJ. = ct2 = ct
< 1/2. Then, by definition (;.;.2)
DO
~i =
= ~x
~x
E
x=o
0
e-~/x!
e
-~
~x
0
log
xr
e
-~
0
~x
i
o/x~
-~
e i/x !
-~
DO
-A
~o eO/x!) x log(~o/~i)-{~O·Ai) E ~~ eO/x!
x=o
x=o
DO
=( E
,e
Therefore,
Also by definition (;.;.;)
2
,. =
E
x=o
~ oe
x!
-A
°
~x
2
e
..
~
0
27
co
;:
I;
x
x=o
00
=
I;
x=o
For any A
O
we can obtain Hoeffding's lower bound by substituting
(3.4.2) or (3.4.3) and (3.4.4) into (3.3.1).
From (3.4.2) and (3.4.3) it is easily seen that the relation
tl
.e
=
t 2 ;:
~
is obtained at
and this value of;\
o
is equal to s, the common slope of the accept-
ance lines of the SPRT for the Poisson distribution under the same
For Ao = s, Hoeffding's lower bound is given by
;\2 2
s
11/ 2 1/2 A2 2
- ~ log
[{16 (log ;\1) - (s log A - s+;\l)log 2aJ
l
Es(N) ~ ---------~--------s
s log ~l - s - ;\1
hypothesis.
1
X-J
(3.4.5)
The Es (N) of the SPRT is given approximately by (1.3.13), therefore,
we can com,pare Hoeffding's 101/er bound with the approximation of the
SPRT ASN at ;\0
= s.
The HLB's in the tables in the Appendix were computer ca,lculated.
CHAPTER IV
THE PROGRAMS FOR THE NPRT AIID THE SPRT 0
FOR THE POISSON DISTRIBUTION
In this Chapter we discuss the programs by which the exact
ee,
ASN and SDN functions of the MPRT and SPRTo were obtained for
the Poisson distribution. The programs (for the HPRT and the SPRT o )
were written in the "IT" language 27 and stored as "K. FukushiLJa,
L
NPRT w A" and "K. KukushiLJa., SPRT w A", respectively, in the library
of the Research Computation Center, the consolidated University of
North Carolina, Chapel Hill, North Carolina, for future use.
The
detailed compiler program for the MPRT is shown in A.7.
4.1. Brief Explanation of the Program for the HPRT
Let
E.\
(integer):
ill1 (integer):
the acceptance boundary for the
iwth trial.
the rejection boundary for the i-th trial.
P(~i):
the probability of acceptance at the iwth trial.
p(mi ):
the probability of rejection at the i-th trial.
p(mi .):
the probability of
•J
j
defects (j
=~
x) at the i-th trial .
n t:
o
the point at which the acceptance and rejection lines intersect.
n :
o
the
maxinlUlll
iiii
w
!'}i
possible number of trials; the least 1 such that
~ 1
29
~:
the number of Poisson probabilities to be calculated.
n1
= c2
a 1 , a 2:
5
- c1 +
the ordinates (E x) of the acceptance and rejection lines
at n'o .
A 1, A 2, .•• A 11 (input):
The parameter points for which the OC,
ASN and SDN functions are to be calculated.
By
(1. 2.6) and (1. 2 .7), after we determine the bounaaries for
acceptance and rejection and no' we perform the following ca1culations for
.e
(i < n )
-
0
=
p(rIl .)
i .J
is the Poisson probability of x with mean A.
where PA(x)
otherwise
and
=
ID1 _1 -1
P(IDi)
=
E
{ p(mi _1.1t)
E P A(x) }
x=l1li -k
The OC, ASN and SDN functions are given by
k~i_l+1
n
L(A)
e
=
0
n
~(N)
=
P(~i)
E
i=l
O
E i {
1=1
P(~i)
+ p(rn1 )
}
. 1/2
SDN
=
2
1 { P(lE1) + P(iii1 ) } •
{l\(N)r J
The overall flow diagram is shown on the following page .
.e
30
31
In rut : -1, ,-1 0
AI ,
"2
I'
A.2,
,
tJ(,
All
•
a.c,D
~(X) ,
F
,e
Dc, ASN,
5DM
H
I
J -o
-
P(t11I~j)
•
12.
.,.;:;:r
•
,
(1I~ ....
'I1L+J " . 11ft,
), :: .."
A-I
p(~:.;J
No
p (..",,, oj .J
• J' :c,;;;:i. +1 .. tJ'n,A·-1
... ,
4.
J
The steps A, B •••
A.7
correspondingly.
j
=:.m:,.. I, ... ~.....
I
14 appear in the compiler program in
L
32
4.2.
Variable As s ignments for "l-WRT - A II
Y 1 (input)
= Al '
Y 4 (input)
= 0,
N1
=~,
N 10 -
N 905 -
zo-
N 900
N 1050:
Z 2000 -
= A 1,
A2, .•• A 11
N
-
904 :
m
i
Alpl1a11Ulleric
Y 1999: PA(j)
Z999:
= A2
= nl
N 902
N 500
Z 2400:
,
Y 2000
Y2400: P(E:i)
-
j
r:
P (x),
A
x=j
.e
= no'
~i'
N 500:
Y 1000 -
Y 3 (input)
Y 28, Y 29 •.• Y 38 (input)
= j,
N2
= AO
Y 2 (input)
Z 1000 -
r:
P,,- (x)
x=o
P(iiii) where
4.3. The Program for "SPRT -
Z 1999:
j
= 0,
1, 2, •••
nl
A"
The storage spaces for the variable assignments are very similar to the program for the NPRT, except for the fqllO'Wing changes.
For input,
i)
ii)
iii)
Y 1
= Al
,
Y 2
= A2
,
Y
3
the truncation point (N 900
= ~ , Y 4 = O2
= no ) must be given
the number of parameter points for which the
OC, ASN and
SDN ftu1ctions are to be calculated is 10(Y29, ••• Y38) instead of
11 in ":r-.n?RT _ AII
•
outputs and Capacities of the Programs,
'~/jpRT
- A" and "SPRT-A".
Unconditional outputs for these programs are as follows:
"'1'
and ",1, ",2
"'2' ex
"'0'
f
c l ' r l , c 2 ' r 2 , no and
cl ' s
a
l
and
and a
c
2
2
(where
1=1
a
n
P"'(~i) ,
ASN,
(for the HPRT - A)
=
l
a )
2
;>.. 11
0
E
0
(for the SPRT - A)
For ;>.. =;>..1, ;>"2,
n
n
.•• ;>..11
0
E
P;>..(iiii)
i=l
the variance of IiI, and
SDN
For conditional outputs of the programs, the following outputs are
added:
.e
m. and
1 ' -~
m (i
1
11
P;>..(j) ,
j
E
x=o
p(rn . j )
i
for the
P(rn )
-n
o
and
By "HPRT_A Il
= 0,
1, 2, ••• no)
CQ
P",(x)
and E P'" (x)
x=j
i-th trial (i
p(iii
n
)
(for
= 1.2
...
n
••• no)
SPRT - A)
and lISl?RT - A", we can take any hypothesis values
do not exceed
400
ex <
~ as long as no and
and 500, respectively.
Also, the core
storage used in the computer is 7695 for IlMPRT - A"
"SPRT - Aft.
1, 2,
o
"'1 and "'2 and the error probabilities
E x
j :;: 0,
where
and 7752 for
However, one can change the program according to the
requirements of each particular problem and a computer capacity.
The time for computation 'Was about three minutes, and the calculation of the
00, ASN and SDN functions at one parameter point for
l
each of eight sets of hypotheses with four error levels; that is,
one A value for each of thirty-two test situations, took about
fourteen minutes.
.e
Similar time is required for "SPRT-A" •
CHAPTER V
SUMMARY
In this chapter we discuss the characteristics of the HPRT
froD the data obtained, with comparison of other tests, the SPRT o'
the SPRT and the FSST.
By the use of diverGence to obtain
"0'
and k l = k 2 = 1/2, the
Po( d ) = Po(d ) when a is not very small,
l
2
and "1 and "2 are not very distinct. As already mentioned in
test nearly achieves
section ;.2, the sun of the exact error probabilities,
is smaller than the preassigned level 2a.
ai+ a2 '
Moreover, even though
the test for the Poisson distribution is not syooetric, we observe
.e
in general.
As mentioned in the introduction, the use of
Ao
=D
Gives sliGhtly better results to achieve Po(d ) = Po(d2 ) for
l
larGe a values and " o = S Gives slightly better results for
snall a.
functions of the MPRT with the SPRTo and
FSST the following points may be found:
ComparinG the OC
i)
the SPRT
has generally higher discriLlination than the MPRT
o
between A and A for small a, the OC function of the ~lPRT
l
2
tends to be close to the SPRT ' However,the difference between
o
the OC functions is not sufficiently larGe to be of particular i1:1porlance; and
ii)
the
~(jpRT
has Generally higher discrioination for snall a than
the FSST except near
A
l
and
A •
2
But if
a is larGe, the FSST
tends to have uniforT.uy higher discrimination.
FrOtl the tables and Graphs sho'Wn in the Appendix we see the
following characteristics of the ASN function of the MPRT.
1)
The srL1aller the l)..value, the closer to
is the rJaxil:lUID value
of the ASN of the
~-1PRT.
i1 )
the ASH of the l:1PRT is smaller than both the SPRT 0
For sr.lall
Ct,
and SPRT for sone values of
A between A and A , but the rJaxiour.1
l
2
of the ASH for the MPRT and SPRT
iii)
iV)
o
are not very different.
The ASN of the MPRT is uniforr.uy smaller than for the FSST ex-
cept for extremely small
.e
s
The HLB at
values of
A
a values.
= D (and'" = a)
is nearly attained for larGe
a by the :MPRT. For smaller values of a, even though
the ASH of the r,1PRT is not close to the HLB, it is closer than the
ASH of the SPRT or the FSST.
vIe observe that for any
than for the SPRT
o
a values the MPRT is unifornly smller
which is presUllJably smller than for the SPRT.
Therefore, the lvlPRT under these conditions appears to be advantaGeous as compared with these other tests when the paraneter
point lies near the average of
is s!lall.
a
Al and A , and particularly when
2
Even if' the ASN of the r!lPRT is sliGhtly larger than of the
SPRT, the use of the MPRT may be recotlOended because of its sLJaller
SDH.
and k
Further studies of the NPRT for various weiGht functions
2
should yield useful results.
k
l
,7
APPEHDIX I
A.I.
Notation
T11e followinG notation is
used in the tables and ficures
in the Awendix:
oc
the operating characteristic function
ASH
the average sample
SDU
the standard deviation of the satlple size H
FSST
the lllOst powerful fixed satlple size test
HLB
Hoeffdinc;'s lower bound for the ASN
MPRT
the minimum probability ratio test
~ber
function
(;>"0
= D)
the nunbers in parentheses in the tables V and VI were obtained
by the 11PRT
(;>"0
= 8)
SPRT
the sequential probability ratio test
SPRT
the sequential probability ratio test
the numbers which have
*
in the SPRT
in the tables VII and
o
VIII were obtained by Uald' s approxirJation for the SPRT
SPRTo :
the sequential probability ratio test truncated at no
a
the specified bound on each error probability
D
;>"0
value for which the divergence between
the divergence between
s
;>"0
and
;>"1
and
;>"0
equals
;>"2
the slope of the SPRT acceptance lines
the sample size of the FSST obtained
fr~l
the Poisson distri-
bution ("11th linear interpolation in the tables)
;8
the sample size of the FSST obtained frou Nonal approximation
the tla:>.'imUtl sample size of the MPRT ("'0 = D)
the maximum sample size of the MPRT (i\.
the HLB at
i\.
=D
the HLB at
i\.
=s
o
= s)
e
e
e
APPENDIX"
SAMPLING PLAN
A.2. SAMPLING PLAN
Ix
Diagram for the MPRT, SPRT0
and FSST for testing HI: A:. 5
ogainst H~'" =1. with «=.01.
boundary for accepting HI
1
--- boundary for accepti,ng H2 :
~//
60,," Sampling is continued until the I c::,~~ ./ ~
sample paths cross
.,...; ./
-or----.
;,,-;t'
/
//
/~¢'
40
~<~~
c.~~"'"
.//
~
~
FSST
~~~
/c.~ Xfb~~
,C./4',)("
-I
>",,'-'-
40
60~*
F
SO "0 100
120
~m
Figure I
lJJ
\0
40
APPENDIX III
A.3.
TABLES I - IV:
Characteristics of the Poisson MPRT and
SPRT Approximation.
TABLES I-I, 2, 3, 4.
HI: A = .1 against H2 : A ; .3
s = .18205
D = .18652
TABIE I - I
OC
SDN of
ASN
HLB
A
MPRT
.e
0
1.
.1
·9203
.17
.5854
SPRT*
1.
.90
MPRT
SPRT*
15·
11.
O.
22.83
19.75
7.88
24.63
24.28
s
·5175
D
.4929
24.11
.25
.2112
20·31
·3
.0932
·50
.10
MPRT
16.90
10·37
21.97
20.40
10.62
19.18
10.71
10.97
1}.73
10.13
41
TABIE I - 2
OC
ASN
i\.
RLB
SPRT*'
MPRT
0.
1.
.1
.9594
.17
·5971
1.
·95
21.
15.
34 .15
29.48
D
.4805
38.98
.1497
31.32
·3
.e
11.07
15·09
.5116
~
O.
'9.81
IS
.25
SPRT*
MPRT
39.28
·50
.0455
39.46
34 .37
15·55
32.17
15.71
15.98
24.79
.05
SDN of
MPRT
20·51
13·99
TABIE I - 3
0:
= .01,
no
oc
= 140,
n
F
SDN of
ASN
i\.
RLB
MPRT
0
= 102
1.
.1
.9916
.17
.6237
SPRT*
1.
·99
MPRT
SPRT*
35.
23·
59.68
49.96
MPRT
O.
Iffi.65
79. 49
78.59
s
·5054
D
.4621
77.88
.25
.0706
56.85
·3
.0087
·50
.01
41.79
~·35
96.10
71.46
26.47
66.48
26.87
26.75
34 .76
20.79
42
TABLE I - 4
a = .001, no
0
n~
= 180
ASN
OC
A
= 223,
HLB
MPRT
1.
.1
·9991
.17
.6570
s
.5028
D
.4459
.25
.0256
·3
.00088
SPRT*
1.
·999
MPRT
SPRT*
55.
35·
95.04
76.47
o.
22.04
142.18
.50
141.1
38.03
217.1
139· 7
130.8
40.22
121.1
41.13
92.25
.001
65.01
SDN of
MPRT
39·04
53·19
27.11
.e
- - - - - - -- - - - -- - - - - - - - - - - - - - - -- - -- - - - - - TABLES II - 1. 2, 3. 4.
B
= .63829
D = .64122
TABLE II - 1
= .1,
a
no
= 87
*
~
= 47
SDN
ASN
OC
HLB
A.
0
1I1PRT
SPRT*
MPRT
SPRT*
MPRT
1.
1.
11.
8.
0
.5
·9127
.6
.6448
;1.14
·90
.5064
D
.4958
;6.0;
.2991
;;.84
.e
15.40
;6.09
8
·7
.8
1;.22
27.;8
.50
;6.07
.10
.092;
27.29
;4.24
;1.56
15.80
;0.85
15.8;
15·99
2;.42
14.5;
TABLE II - 2
a
= .05,
no
=126
SDN
ASN
OC
HLB
A.
0
·5
.6
MPRT
SPRT*
llPRT
SPRT*
MPRT
1.
1.
15·
10.
0
46.87
40.88
.9559
·95
58.5;
.677;
s
.5042
D
.4908
58.85
.2491
54.14
·7
.8
.0458
18.41
.50
.05
58.9;
40.66
22·51
61.49
5;.15
2;.2;
51.82
23.28
23.64
;4·96
20.32
44
:>
TABLE II a
r..
0
= .01,
no
oc
= 155
SDN
ASN
HLB
MPRT
SPRT*
MPRT
SPRT*
MPRT
l.
l.
25·
16.
0
82.02
69.29
·5
·9911
.6
.7368
·99
.5018
D
.4831
119·1
·7
.1693
104.8
.8
.0090
38.15
.50
119.3
.01
149.75
a
70.18
= .001,
no
110.4
39·50
107.3
39.62
40.82
59.26
30.58
4
TABLE II -
= 350
oc
r..
27.89
116·5
s
0
n*
F
= 217
SDN
ASN
HLB
HPRT
SPRT*
MPRT
SPRT*
MPRT
l.
l.
40.
24.
0
130.4
106.1
.5
·9991
.6
.7973
.999
36.81
58.43
205.5
.50
s
·5007
D
.4760
214·9
.7
.1025
179·3
.8
.00089
.001
215·3
110·5
338.3
201.9
60.08
195.8
60.36
63.20
90.69
40.10
TABLES III - 1. 2. 3. 4.
HI:
s
~
=
·5 against H2:
= ·72135
~
= 1.
D = .72850
- -- - - - - - - - - - - - -- --- -- - - --- - TABLE III - 1
oc
~
0
.e
ASN
NPRT
SPRT*
1,
1.
,5
.9185
.65
.6683
·90
l1PRT
SPRT*
6.0
5.
13.81
.5088
D
.4929
15.58
.2553
14.11
1.
.0877
.50
.10
15.62
11.47
~
0
6.45
13·93
6.74
12.41
6.76
6.86
6.22
9·21
= 32
SDN
ASN
HLB
SPRT*
l1PRT
SPRT*
NPRT
1.
1.
9.
6.
0
20.33
17·32
.9594
.65
.7054
.95
·5084
D
.4884
24.95
.2026
21.95
.0442
'7·49
24.89
s
1.
12.87
}:IPRT
.5
.85
5.40
11.60
TABLE III - 2
a = .05, no = 52
nF
OC
SDN
lvIPRT
0
15·70
s
.85
HLB
·50
.05
25·02
16.77
9·29
25.02
21.67
9.76
20.84
9.80
10.00
13.76
8.66
46
TABLE III - ;
= .01,
Ct
-
n
o
88
:::
SDN
ASN
OC
HLB
A.
0
l'·'lPRT
SPRT*
l:lPRT
SPRT*
lIPRT
1-
1-
15.
10.
0
;5.42
29.;5
.5
.9915
.65
.7655
·99
48.69
a
.5014
D
.47;6
49.66
.1185
40.82
.85
1-
11.19
49.85
·50
.0084
.01
28.22
15·51
60.9;
45.01
16.48
4;.11
16.59
17.04
12.81
2;.;2
.e
TABLE III - 4
Cl
= .001,
n o ==
142
SDN
ASN
OC
HLB
A.
0
l..IPRT
SPRT*
r.IPRT
SPRT*
MPRT
1-
1-
2;.
14,
0
56.07
44.9;
.5
.9991
.65
.8289
.999
84.77
s
.5006
D
.4640
88.87
.0594
67.96
.85
1-
.00082
14.71
.50
.001
89.25
44.07
2;·5;
1;7.64
82.;4
24.92
78.61
25.17
25.76
;5.69
16.71
47
A. .3.4.
TABIES IV - 1. 2. .3. 4
HI: A = 5· against H2 : A = 8.
e = 6.3829
D = 6.4122
TABLE IV - 1
oc
ASN
SDN
A
HLB
0
5·
,
MPRT
SPRT*
MPRT
SPRT*
MPRT
1.
1.
2.
1.
0
.3.80
2.74
1.59
.90
·9.305
6.
.6551
15
.5290
D
.4919
4.57
7.
.2793
4.31
8.
.0728
1.88
4.54
4.80
.50
.10
3.42
3.16
1.88
3.08
1.91
1.91
3.48
2.34
1.69
TABLE IV - 2
= .05,
a
n
o
=
14 ,
-
ASN
OC
A
0
HLB
SDN
HPRT
l>1PRT
SPRT*
HPRT
SPRT*
1.
1.
2.
1.
0
5.39
4.09
2.04
5.
.9649
6.
.6900
.95
6.87
8
.5427
D
.4908
6.91
7·
.2374
6·37
8.
.0367
·50
.05
7.43
4.78
2·52
6.15
5.31
2.52
5.18
2.61
2.64
3·50
2.22
48
TABLE IV - ;
OC
SDN
ASN
HLB
1\
0
HPRT
SPRT*
NPRT
SPRT*
NPRT
l.
l.
;.
2.
0
8.92
6.9;
2.94
·9929
·99
5·
6.
.7447
s
·5070
D
.4806
13.11
7·
.1605
11·56
8.
.0070
4.0;
12.82
·50
.01
.e
14.98
13.28
11.04
4.09
10.73
4.19
4.;1=
7.76
3.18
5.9;
TABLE IV - 4
oc
SDN
ASN
HLB
A
0
l1PRT
SPRT*
MPRT
SPRT*
l'TI?RT
1-
l.
4.
3.
0
1;.74
5.
.9993
6.
.8010
s
.4989
D
.47;5
22.75
7-
.0969
19·00
8.
.00069
·999
10.60
3.77
6.05
21·70
·50
.001
22.80
11.81
,
33.8;
20.19
6.18
19·58
6.21
6.50
9·07
4.09
APPEIJDIX IV
A.4.
TABLES V - VI:
Characteristics of the Poisson
(~
A.4.1.
= D and
TABLES V - 1. 2.
H1:
~
a) and SPRT Approxiraatiol1
3. 4
=.1 against H2 : ~ = ·5
= .24853
D = .26129
8
---- -
o
HPRT
~
-- - ------
~
-
- - - - - - - -- - - - - - - - - - - -
TABLE V-I
OC
SDN
ASH
A
HLB
NPRT
.e
0
1.
(1. )
.1
.9378
( .9392)
.2
.6856
( .6933)
SPRT*
1.
·90
HPRT
llPRT
SPRT*
7·
(7.)
6.
9.08
(8.92)
7.44
O.
(0. )
2.80
(2.56)
3.66
(3.43)
9.75
(9.51)
.50
9.50
(9.28)
s
·5391
(.5496)
D
.5022
( .5131)
9.39
(9.18)
.4
.1988
(.2094)
7·70
(7.59)
·5
.0909
(.0980)
.10
6.45
(6.40)
7·50
7·03
3·92
(3.71)
6.37
3.98
(3.76)
4.09
(3·95)
4.40
3.78
(3.69)
50
TABLE V .. 2
a:
= .05,
no = 26
(nos
00
= 26)
nF
SDN
ASN
HLB
A
MPRT
0
.e
.e
= 19
1.
(1. )
.1
.9646
(.9663 )
.2
.6975
(.7059)
SPRT*
1.
·95
HPRT
10.
(10. )
13.43
(12.84)
NPRT
SPRT*
8.
O.
(0. )
11.12
3·72
(3.53 )
15.08
(14.60)
.50
14.64
(14.30)
s
·5167
(.5278)
D
.4711
( .4826)
14.43
(14.13)
.4
.1302
( .1397)
11.00
(11.04)
·5
.0417
( .0472)
.05
8.72
(8.89)
5·21
(5.07)
13.47
11.85
5·73
(5.51)
10.68
5.84
(5. 60)
5·95
(5.65)
6·57
5·20
(4.95)
51
TABLE V - :;
oc
SDN
AS1'1
HLB
'}...
MPRT
0
1.
SPRT*
1.
(1. )
.e
.1
.9924
( .99:;5)
.2
.7473
(.7606)
.99
MPRT
MPRT
SPRT*
17.
(16. )
12.
2;.06
(22.16)
18.84
O.
(0. )
5·25
(5.4:; )
8.:;4
(8.45)
28.7:;
(28.41)
.50
28.21
(28.:;0)
s
·5059
(.5218)
D
.44:;5
(.4591)
27·70
(27.88)
.4
.0567
(.0616)
18.94
(19.49)
·5
.0080
( .0092)
.01
14.05
(14.54)
32.80
24.69
9·52
(9.42)
22.07
9.80
(9.67)
9.58
(9.57)
11.13
7.48
(7.56)
52
4
TABLE V 0:
= .001,
no :: 78
(nos
oc
= 78)
11-
= 63
SDN
ASN
HLB
'JI.
MPRT
0
.I!'
1.
(1. )
.1
·9993
(.9993)
.2
.8084
(.8151)
SPRT*
1.
.999
MPRT
HPRT
SPRT*
27.
(25. )
18.
;6.42
(34.96)
28.83
O.
(0. )
7.05
(7.11)
12.45
(12.45)
50.13
(49.02)
s
.5066
(.5175)
D
.4247
(.4358)
49.21
(49.00)
.4
.0185
(.0205)
29.98
(30.84)
·5
.00076
( .00092;
·50
.001
50.26
(49.85)
21.31
(22.14)
74.10
45.24
14.58
(14.05)
40.22
15·20
(14.57)
13.90
(13.60)
17.03
9.80
(9.76)
53
A.4.2.
TABLES VI -
1.
2.
3. 4.
Hl : A = 1. against H : A = 2.
2
s = 1.4427
D = 1.4570
TABLE VI - 1
a
= .1,
no
= 17,
(nos
0;
11)" nF = 11
ASU
OC
SDN
HLB
A
MPRT
.e
0
1.
(1. )
SPRT*
1.
lIPRT
SPRT*
;.
;.
(3. )
1.
.9236
( .9299)
1.35
.6150
(.6275)
s
·5095
( .5205)
D
.4916
(.5040)
8.;6
(8.44)
1.75
.2104
(.219; )
7.35
(7.52)
2.
.0817
( .0867)
·90
7·;3
(7.24)
lIPRT
O•
(0. )
2.86
(2.83)
5.80
8.44
(8.47)
.50
.10
8.41
(8.45)
6.16
(6.36)
;.46
(3.44)
6·97
6.4;
3.55
(;.51)
6.21
;.56
(;.52)
3.58
(;.53)
4.61
3.27
(3.24)
TABLE VI - 2
a
= .05,
n
o
= 26,
(008 = 26), n*~.
OC
'"
0
.e
=16
SDN
ASN
HLB
NPRT
1.
(1. )
SPRT*
1.
NPRT
5(5. )
1.
.9620
( .9635)
1.;5
.6406
(.6462)
8
·5100
(.5140)
D
.4879
( .49;6)
1;.09
(1;.10)
1.75
.157;
( .1607)
11.07
(11.15)
2.
.041;
(.0427)
·95
10.58
(10.46)
NPRT
SPRT*
;.
o.
CO. )
8.66
;.87
(;.86)
13.16
(13.1; )
.50
.05
1;.17
(1;.1;)
8.80
(8.91)
4.90
(4.9; )
12.51
10.8;
5.04
(5.06)
10.42
5.07
(5.08)
5.12
(5.08)
6.88
4.47
(4.41)
'55
TABLE VI -
a
= .01,
no
= 44,
(n.O• S
44),
n-*
.lf
oc
SDN
RLB
MPRT
,e
='2
.AS!'T
A-
0
3
1.
(1. )
1.
·9922
(.9926)
1.35
.6824
(.6875)
s
.5066
( .5083)
SPRT*
},1PRT
SPRT*
1.
8.
(8. )
5.
18.13
(17.78)
14.68
.99
MPRT
O.
(0. )
5.67
(5.70)
8.05
(8.08)
25·57
(25.39)
·50
25.85
(25.60)
D
.4743
(.4802)
25.58
(25.52)
1.75
.0792
( .0820)
19.83
(20.00)
2.
.0079
( .0084)
.01
14.58
(14.76)
30.46
22·51
8.30
(8.34)
21.56
8.43
(8.39)
8.44
(8.37)
11.66
6.50
(6.50)
TABLE VI -
a = .001, no
4
~
= 71, (n DS -=.71),
n* = 57
F
SDN
ASN
OC
HIJ3
A
0
.e
MPRT
SPRT*
NPRT
SPRT*
1.
(1. )
1.
12.
(12, )
28.44
(27.90)
7·
1.
·9992
( .9993)
1.35
.7315
(.7358)
s
.5016
(.5069)
D
.4648
(.4701)
45.24
(45.18)
1.75
.0;20
(.0;;3)
;2.06
(;2.40)
2.
.00076 .001
(.OOO8;)
22.50
(22.84)
·999
MPRT
O.
(0.)
7.40
(7.42)
22.46
12.07
(12.15)
44.73
(44.41)
.50
45·4;
(45.3;)
68.82
41.17
12.58
(12.52)
;9·;1
12.71
(12.63)
12.34
(12.24)
17.84
8.42
(8.39)
57
APPENDIX V
TABLE VII:
A.5.
Characteristics of the Poisson HPRT
(A. = D), SPRT ' SPRT approxiLJation
o
o
and FSST
TP.BLES VII - 1, 2, 3, 4
H1 :
A. =
.01 against H2 :
= .03909
s
D
A. = .1
= .04299
- - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - . - - TABLE VII - 1
a
.e
= .1,
no
= 58,
= 39,
nF
ns
OC
= 22.20,
~
= 19.08
SDN
ASN
A.
.0
l1PRT
SPRT o
FSST
l.1J?RT
SPRTo
l:1PRT
SPRTo
1.
1.
1.
29-
25·
o.
o.
.0025
.9958
·9955 1'-
29.98
26.36
3.84
6.03
.006
·9776
.9820
.98
31.04
27·98
5.60
8·90
.01
.9432
·94
31.84
10.89
.8858
.88
32.36
29.44
26.57*
30.72
6.86
.015
.9566
.90*
·9116
8.03
12.53
.02
.8187
.8553
.81
32.43
31.45
8.98
13.68
.025
.7469
.7916
·75
32.16
31·73
9.80
14·52
.03
.6743
.7240
.67
31.64
31.63
10·50
15·17
.035
.6033
.6555
.60
30.94
31.24
11.08
15.66
s
·5479
.55
30.26
15.56
.4976
.50
29·56
30.74
23·30*
30.16
11.48
D
.6005
·50*
.5495
11.79
16.17
TABLE VII - 1 (continued)
oc
SDN
ASN
A.
MPRT
SPRTo
FSST
MPRT
SPRTO
HPRT
SPRTo
.055
.3628
.4090
.37
27.21
27·93
12.37
16.43
.07
.2369
.2732
.26
24.50
24.81
12.44
16.06
.08
.1759
.2061
.18
22.41
22.78
12.22
15·52
.1
.0950
.10
19.18
14.08
.0694
.07
17.81
19·17
12.69*
17.63
11.40
.11
.1157
.10*
.0865
10.91
13.29
.e
A·5·2.
TABLE VII - 2
'.
.0
-e
SDN
ASN
OC
A.
MPRT
SPRTo
FSST
NPRT
SPRT o
1,1PRT
SPRTo
1.
1.
1.
41.
O.
o.
-
33·
42.52
35.18
5·00
7.86
.0025
.9986
.9989 1.
.006
.9896
·9923
.99
44.50
38.23
7.94
12.73
.01
.9664
·97
46.39
16.14
.9175
.93
48.03
41.42
39.68*
44.68
10·32
.015
·9750
·95*
.9363
12.61
19.22
.02
.8501
.8792
.86
48.83
46·96
14·50
21·32
59
TABLE VII - 2 (continued)
a
= .05,
= 85,
no
nF
= 63,
oc
A.
= 32.03
~
SDl~
ASH
. SPRT
l.1PRT
= 37·53,
ns
o
FSST
HPRT
SPRT o
l·jpRT
SPRTo
.025
·7703
.8076
·79
48.85
48.22
16.13
22.83
.03
.6844
.7268
.71
48.23
48.57
17~54
23·95
.035
.5978
.6423
.62
47.08
48.15
18.74
24.81
s
·5292
.55
45;86
25·32
.4672
.49
44·52
47.34
41.84*
46.26
19.54
D
.5738
·50*
.5106
20.15
25.67
·55
.3054
.3418
.33
39.82
41.81
21.12
25.89
.07
.1681
.1944
.19
33.86
35·51
20.80
24.65
.08
.1097
.1307
.11
30.27
31.57
19·97
23.20
.1
.0448
.05
24.33
19·79
.0282
.03
21.96
25.04
18.95*
22.46
17.68
.11
.0581
.05*
.0388
16.45
18.09
.e
TABLE VII - 3
= .01,
a
no
= 143,
nF
= 117,
OC
n s = 78.39,
~
= 66.26
SDN
ASH
A.
•0
MPRT
SPRTo
FSST
llPRT
SPRTo
MPRT
SPRT
1.
1.
1.
67 .
52.
o.
o.
.0025
.9999
·9999 1'-
71.69
55.47
6.59
9·97
.006
·9991
·9991 1.-
75·75
61.06
10.96
17·30
.01
·9931
.9938
80.62
68.19
14.95
24.19
.99
60
TABLE VII - 3
MPRT
.01
.e
SDN
ASN
OC
'"
(continued)
SPRTo
FSST
MPRT
SPRTo
MPRT
SPRT
0
67.25*
·99*
.015
.9681
.9717
.97
86.39
77.28
18.90
30.68
.02
.9146
·9223
.91
90.92
85.29
22.02
34.97
.025
.8315
.8428
.83
93·55
91.18
24.78
37·73
.03
.7264
.7397
·72
94.05
94.43
27·50
39·75
.035
.6107
.6245
.61
92·59
95·02
30.11
41.49
s
·5164
.52
90.19
42.78
.4311
.44
87.14
93.77
101.89*
91·39
32.00
D
.5296
.50*
.4435
33.48
43.82
.055
.2233
.2327
.2;
75·39
79.66
35.47
44.99
.07
.0832
.0900
.09
60.69
62·90
33.36
41.61
.08
.0401
.0456
.05
52·52
53·27
30.55
37.56
.1
.0084
.0115
.01
40.46
39.21
24.66
28.99
22.15
25·31
32.11*
.01*
.11
.0037
.0059
.05
36.10
34.31
61
TABlE VII - 4
a
= .001,
no
= 243, nF = 212,
OC
ns
= 14;.97,
~
= 120.84
SDH
ASH
A.
.0
11PRT
SPRTo
FSST
l:IPRT
SPRT o
NPRT
SPRTo
1.
1.
l-
110.
77.
0
0
.0025
.9999
·9999 1:
114.0
82.33
7·92
12.50
.006
·9999
·9999 1:
120.3
91.07
13.55
22.34
.01
·9993
·9994 1:
128.5
19.41
33.76
102.9*
·999*
.e
10;.3
.015
·9917
·9937
.99
140.0
121·9
26.1;
47.39
.02
.9610
.9685
·97
151.3
142.1
;1.53
57.45
.025
.8903
·9050
.91
160.2
160.2
35·75
62.96
.03
.7763
.7958
.81
164.8
172.7
39·95
65·63
.035
.6331
.6529
.67
164.0
177·8
44.78
67.97
.5102
·5272
.55
159·9
176.7
48.78
69.61
s
230.2*
·50*
·;992
.4124
.44
153.5
171.1
52.05
73.07
.055
.1526
.1564
.18
127·;
140.9
55·50
76.19
.07
.0327
.0338
.04
96.64
100.8
48.71
65.68
.08
.0101
.0110
.01
81.41
81.01
42.26
55·22
.1
.00077
.0012
.001
61.05
56.20
31.68
;7·96
27·91
32.07
D
.11
.00020
.001*
49.15*
.00044 .00025 54.16
48.51
I
62
A.S.S. ASN CURVES
ASN
220
-----------
-----
- - - - - - - - - - - - - CIt
=.001
- - MPRT
-'-- SPRTO
--SPRT*
180
---- FSST
e-ns
x-no
140
.e
- --------« =.01
100
80
60
....
---
(!)
x
Figure 2. Poisson ASN
H I: A=.01 against H2: A 11.1
5 .039087
0 .042994
o .01 .02 .03 5 0 .05
.07
II
·.1
II
.09 .I
.11
63
APPEIIDIX VI
A.6.
TABLE VIII:
Characteristics of the Poisson lIPRT
(71.0 = D), SPRTo ' SPRT approY..il:Jation and
FSST
TABLES VIII - 1, 2,
~: A.
s
= 2.
against
= 4.
D = 2.9140
TABLE VIII - 1
oc
.e
H : A.
2
= 2.8854·
A.6.1.
3, 4
ASN
SDN
A.
.0
MPRT
SPRT
1.
1.
o
FSST
l1PRT
SPRT
MPRT
SPRT
1.
2.
2.
o.
o.
0
o
1.5
·9925
·9932
·99
3·27
3.08
1.05
1.34
1.75
.9738
.9738
.98
3.62
3.56
1.27
1.76
2.0
.9298
.94
4.00
2.19
.8491
.86
4.33
4.12
2·90*
4.68
1.48
2.25
.9410
.90*
.8658
1.64
2.54
2·5
.7307
.7462
.75
4·56
5·11
1.75
2.78
2.65
.6463
.6572
.67
4.63
5·27
1.80
2.87
2.75
.5871
.5938
.61
4.64
5.32
1.83
2·91
s
.5062
.54
4.62
2·95
.4893
.52
4.61
5·32
3.48*
5·31
1.86
D
.5065
·50*
.4883
1.86
2·95
3·25
.3070
.2938
.34
4.37
4.98
1.89
2·93
64
A.6.1.
TABLE VIII - 1 (continued)
oc
SDn
ASH
A.
HPRT
.e
SPRT
O
FSST
11PRT
SPRT
o
UPRT
SPRT
o
:;·5
.2016
.1858
.2:;
4.08
4.56
1.86
2.80
:;.75
.1254
.1116
.14
:;·77
4.10
1.79
2.60
4.
.0746
.09
:;.45
2.:;4
.0240
.0:;
2.90
:;.64
2.:;0*
2·90
1.69
4·5
.0649
.1*
.0211
1.45
1.82
A.6.2.
TABLE VIII - 2
OC
ASH
SDN
A.
.0
l'1lPRT
SPRT
1.
1.
o
FSST?~
I>1PRT
SPRT
1-
:;.
MPRT
SPRT
2.
O.
o.
o
1.5
.9984
.9984
·99
4.:;5
:;.76
1.:;8
1.65
1.75
·9912
·9925
·99
4.96
4.48
1.72
2.28
2.0
·9647
·95
5.65
2·95
.897:;
.86
6.:;:;
5.42
4.:;:;*
6.45
2.04
2.25
·9711
.95*
.9118
2.29
:;.50
2·5
.7740
.79:;4
.72
6.85
7.:;4
2.46
:;.82
2.65
.6761
.6944
.62
7.02
7.68
2·53
:;.91
2.75
.6047
.6206
.55
7·06
7·80
2.58
3.95
s
·5052
·5166
·50*
.46
7·03
7.83
6.25*
2.6:;
3·99
o
65
A.6.2.
TABLE VIII - 2
oc
A'
SDH
ASH
-.
MPRT
FSST* NPRT
SPRTo
SPRTo
MPRT
SPRT o
D
.4843
.4947
.44
7.01
7·82
2.65
3·99
3.25
.2648
.2644
.25
6.51
7·20
2.73
3·94
3·5
.1496
.1466
.15
5·93
6.42
2.68
3·75
3·75
.0772
.0752
.08
5.30
5.59
2.54
3.42
4.
.0370
.05
4.72
3·00
.0073
.01
3.78
4.83
3.44*
3·70
2.33
4.5
.0369
.05*
.0088
1.87
2.20
.e
A.6.3.
TABLE VIII - 3
a
= .01,
nO
= 22·, nF*
= 16,
.0
•
DV
= 10.78
ASH
OC
A '
= 11.25,
n8
SDN
SPRTo
MPRT
SPRTo
4.
3.
o.
o.
MPRT
SPRTo
FSST* NPRT
1.
1.
1.
1.5
.9999
·9999 1.;'
7.05
5.61
1.67
2.11
1.75
·9993
.9993
1:
8.06
6·79
2.23
2.96
2.
.9928
9.39
4.20
.9567
.94 11.00
8.51
7.33*
10.84
2·91
2.25
.9941
·99*
.9641
3.57
5.49
2.5
.8442
.8601
.79 12.54
13·31
3·97
6.27
.99
66
A.6.3.
TABLE VIII· 3 (continued)
oc
SDH
ASN
A
HPRT
.e
SPRTe
FSST* MPRT
SPRTe
l'·1PRT
SPRTo
2.65
.7285
.7458
.66 13.16 .
14.41
4.10
6.43
2.75
.6356
.6511
.57 13·37
14.84
4.17
6.49
•
.5008
·55 13·35
6.56
.4723
.42 13.30
14.98
15.23*
14.95
4.30
D
.5112
.50*
.4814
4.32
6.58
3.25
.1908
.1880
.17 11.88
13.11
4.52
6.65
3.5
.0757
.0723
.07 10.32
10.93
4.31
6.18
3.75
.0250
.0237
.03
8.84
8.89
3.84
5·30
4.
.0071
7.29
5.83*
5.27
3.29
4.32
2.39
2.84
4.5
.0072
.01*
.00042 . ,00075
A.6.4.
.003 7.62
.001 5.93
TABLE VIII . 4
OC
A
.0
SDN
ASH
MPRT
SPRTe
FSST* HPRT
SPRTe
lvIPRT
SPRTo
1.
1.
1.
4'.•
0
0
6.
1.5
.9999
.9999 1
10.83
7.86
2.10
2.49
1.75
·9999
·9999
.
.
1
12.42
9.58
2·79
3.65
2.
·9993
·9994 1-
14.55
12.25
11.23*
3.75
5·53
·999* ..
"!'
67 .
TABLE VIII - 4 ( continued)
A.6.4.
OC
A
.e
-
. - --.• ---r--'
ASN
,....~_ ..
_-_.--'
---~.-
-_ ..
-
-
SDN
MPRT
SPRT 0
16.51
4.97
8.1,
20.85
22.15
5.89
10.01
.79
22.51
25.15
6.08
10.22
.678,
.68
2,.16
26.46
6.16
10.15
.47
2,.26
10.18
.44
2,.16
26.99
,4.41*
26.91
6.,8
.46,4
.496,
.50*
.4574
6.44
10.21
,.25
.1269
.114,
.14
19·79
22.17
6.88
10.64
,·5
.0,07
.0255
.04
16.4,
17·10
6.24
9.41
,.75
.005'
.0044
.01
1,.64
1,.09
5.19
7,'7
4.
.00069
5·57
.00071
10.,8
8.92*
7·;2
4.24
4.5
.00074 .001 11.57
.001*
.000,4 .0002' 8.89
2.99
,.46
MPRT
SPRT 0
FSST* MPRT
2.25
.9871
.9898
.99
17.45
2.5
.9060
.9151
.91
2.65
.7856
.794,
2.75
.67,8
s
.5005
D
I
SPRT 0
68
A.6.5. ASN CURVES
ASN
35
30
MPRT
---SPRTQ
- - SPRr
----- FSST
x- - - (1)-
"0
"5- ..... -------
_________ ell
=.001
25
20
-----------0( -.01
15
10
5
APPENDIX
69
VII
COHPILER PROGRAM
A.7.
If 1200
K. FUkushima, MPRT-A
Y 2400
n905=$these $
n906=$are va$
n907=$lues 0$
n908=$f c on$
n909=$e, d 0$
n91 O=$ne, c $
n911 =$two an$
n912=$d d
n913=$0
n914=$this i$
n915=$s the $
n916=$upper $
n917=$value $
n918=$for po$
n919=$isson $
n920=$th1s i$
n921 =$s the $
n922=$end of$
n923=$trial $
n924=$these $
n925=$are th$
n926=$e poss$
n927=$1ble v$
n928=$alues $,
n929=$of ma.x$
n930=$ defec$
n931 =$tives $
n932=$theseJ
n933=$two v
n934=$lues m$
n935:::$ust be$
n936=$equal $
n937=$these $
n938=$are th$'
n939=$e uppe$
n94O=$r
n941=$10wer
n942=$boun
n943=$ries $
n944=$these $
n945=$are th$
n946=$e acce$
n947=$ptu.ce$
n948=$ proba$
lI1949=$b1lit1$
n950=$es for$
t1
.e
anU
z2400 S 0150 W 0200
t
t
f
f
f
t
f
f
f
f
t
t
f
f
f
t
f
f
f
f
f
f
t
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
t
f
f
f
f
f
f
t
B
COMPILER mOGRAM (cant.)
n951 =$ each $
f
n952=$tr1al $ f
n953=$these $
f
n954=$are th$
f
~~j~tf'.,;ef
f
f
f
f
. tf
n965=$of acc$
n966=$eptanc$
n961=$e
$
n968=$th1s 1$
n969=$s the $
n910=$prob. $
n911 =$of rej$
n912=$ect10n$
n913=$this 1$
n914=$s the $
n915=$averag$
n916=$e numb$
n911=$er of $
n918=$sample$
n919=$s
$
n980=$th1s i$
n981 =$s the $
n982=$var1an$
n983=$ce of $
n984=$number$
n985=$ of s8$
n986=$mples $
n981=$th1s 1$
n988=$s the $
n989=$end of$
11990=$ a Par$
n991 =$t of c$
n992=$alcula$
n993=$t10n $
n.994=$these
n995=$are po
n996=!1SS0n
11991= prob.
n998= th.eseJ
n999=$are t
n1 OOO=$e cUJDU$
111001 =$lat.1ve$
n1 002=$ poiss$
f
f
f
f
:§~~::;r:~ l
.e
:
n951=$probab$
n.958=j111t1ej
n959= s for
n960= each t
n961 ;"$ri-al $
n962=$th1s 1$
i
~
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
70
=
COMPILER PROGRAM (cont.)
e
.-
t
n
n1oo,=to
t
nl00
= b.
t
nl 005=~ zero:
nl006= these
t
t
nl 007=$are th$
t
nl008=$e cumu$
t
n1OO9=$lative:
t
nl 01 0=: poiss
t
nl 011 = on pro!
t
n1 01 2=$b. up
t
nl 013=$to int$
t
nl 01 4=$inity $
t
nl015=$this i$
t
nl016=$s .the $
t
n1 017=$end ot$
t
nl 01 8=$ tri~
t
nl 019=$ prac
t
nl020=$ically$
t
nl 021 =tthiS it
t
n1022= S the
t
nl023=$s.d. 0$
fnum
f
nl
t
n1025=
er of
t
nl 026= sample
nlO27=$s.
f
$
y1, ••• ,y4
y28, ••• , y38 tukushima input
tyl ty2 ty3 ty4
f
t
ty28 ty29 tY30
f
tY31 tY32 tY33 ty34
t
tY35 ty36 tY37 tY38
t
zl=(*01s,Y3* - *01sjy1*)
t
z2=(*01 s,Y3* -*01 Sjy2*)
t
z3=(*01s,y2* - *01 s,yl*)
z4=(*01s,2xy4*)
t
Y5=z4/z2 t
t
y6=~ y3-y2)!z2
f
y7= (-1 )xz4)/z3
t
y8=(y2-yl )/z3
f
tY5 ty6 ty7 ty8
t
atn905 atn913
f
n902=y7-Y5+5
t
tn902
t
atn914 atn919
Y9=(Y1-Y5)/(y6-y8) t
f
tY9
t
atn920 atn923
f
yl O=y5-f'y6xy9
f
y1' =y7+y8xy9
f
tylO ty11
t
atn924 atn931
f
atn932 atn936
t
nl=O
t
y(1000tn1 )=y5i'Y6xnl
t 1
0001
0116
0130
0117
0118
0080
0079
0018
0017
0076
0075
0014
0002
71
72
COMPILER PROGRAM (cant.)
e
t
zn1 =y7+y8xn1
f
Y(100O+n1) v 0
y(100Ofn1 )=( ...1 .)
f
t
0003 n(lOfnl )=y(l OOOtnl)
n( 500fnl )=zn1
f
t
g4 if znl-n~500hll) u 0
t
n(50O+nl)=n 5C>O+t1l )+1
0004 yO=yO
f
t
tn( 50Ofn1) tn1 ta(1ot-n1 )
n1=nl+1
f
f
g2if n( 500+nl ...1 )-n(l Ofnl -1) v 1
t
n900=n1-1
f
atn937 atn943
f
0072 tn900
The end of Part
f
0071 atnl 015 atn1020
f
y39=y2
n90l=O ( f
y2=y28
f
g40 if 12 u 0
t
~ if
g5
~
.e
0041
f
f
n901=1
y2=y29
f
g41 if y2 u 0
f
g5
f
n901=2
t
y2=Y30
g120 it y2 u 0 .
g5
G122 n90l=5
t.
f
f
f
8120 n901=3
y2=Y31
g121 it y2 u 0
f
g5
t
0121 n901=4
f
y2=Y32
g122 it y2 u 0
g5
f
f
f
f
f
~:=Y33
.. ~
g123 if y2 u 0
g5
f
f
8123 1190l =6
t
12=y34
g124 if y2 u 0
g5
0124
0125
f
f
f
f
n90l=7
y2=Y35
g125 if y2 u 0
f
g5
f
n901=8
y2=y36
t
g126 if y2 u 0
g5
f
f
f
f
A.
COMPILER PROGRAM (cont.)
n901=9
f
y2=Y37
f
g127 if y2 u 0
g5
g5
f
f
f
f
n901=lO
y2=y38
g42 ify2 u 0
0127
73
f
f
0005 yl 000=( *028, ( -1 lxy2*)
.
0006
G>OO7
0008
0009
.e
0050
0051
0011
0012
0013
0014
0015
0017
f
tyl000
f
7,n2, 1,1 ,n902,
f
y12=n2
f
y(1 000+n2)=(Y(1 000+n2-1 )xy2)!y12
n2=y12
f
f
ty(1000k12) tn2
yO=yO
f
The end of
atn994 atn997
f
z1 OOO=yl 000
f
tzl000 t(O.)
f
9,n2,1,1,n902,
f
z(1000+n2)=z(1000+n2-1)+y(1000+n2)
tZ(100O+n2) t n 2 f
yO=yO
f
The end of
f
atn998 atnl005
zO=1
f
t(1.) t(O.)
f
51,n2,1 ,1 ,n902,
f
zn2=1 -z(1 000+n2-1 ) f
tzn2 tn2 f
yO=yO
f
The end of
atn1 006 atn1 014
f
n1=1
f
nl 050=n501 -1
f
1 2,n2,nll ,1 ,nl 050,f
f
y(100+n2)=y(1000+n2)
g13 if nll u (-1)
f
y2001 =y(1 00+nll )
f
ty2001 t(1.)
f
g14
f
y2001=O
f
ty2001 t(1.)
f
z2001 =zn501
f
The end of
t(O.) t(1.) tz2001 f
Part F
n1 =n1+1
f
g21 ifn(1 O+nl) v· 0
f
Part
y(200Ofnl )=0
t(O.) tn1
nl 051 =n(50Offil )-1
. n1 094=n( 500tnl -1 )
nl 095=n(500+nl -1 )-1
20,n2,O..1 .. nl 095,
f
f
f
f
f
f
f
Part B.
f
Part C.
Part D.
Part E.
G
74
COMPILER PROGRAM ,(cont.)
f
0018 y(500tn2)~O
f
0019 l6,n',O,l,n2,
0016 Y(500+n2)~(500+n2)+Y(100+n')XY(l0OO+D2-D') .
ty( 50O+n2) 'linl tn2 t
f
0020 yO=yO
g28 if n(500+nl -1) u n(500kl1)
t
94,n2,n1 094,1 ,nl 051,
t
0091 Y(500+n2)~0
0092 9',n',O,l ,nl095,
f
f
.
Y(50G+n2)=Y(5@0+n2)+Y(l~')XY(1000+n2-1l')
009' ty(500+n2) 'linl tn2
0094 yO=yO
g28
-
f
f
g25
0022
002'
0025
0026
0027
0098
OC85
oc86
0087
f
f
The end of Part H.
g22 if n(1O+nl )-n(l O+nl -1) v 0
y( 2000tnl )=0
t(O.) tnl
nl 054=n(1 O+nl )+1
nl 055=n( 500+nl) -1
t
t
f
t
f
f
f
y(2~l)=0
f
nl 052=n(1 O+nl -1 )+1 f
n105'=n(1O+nl)
f
nl 054=n(1 O+nl )+1
f
nl 055=n(50O+nl )-1
f
23,n4,nl 052,l,nl 05',
f
y( ~000+n1 )=y( 200O+n1 )+Y(1 00+114)xz(l 000+D(1 Q+nl )-n4)
ty( 200O+nl) tnl
f
atn944 atn952
f
nl 051 =n( 500+nl )-1
f
nl 094=n( 500fnl -1 )
f
nl 095=n( 50O+nl-1 ) -1
f
nl 052=n(1 Ofnl-l )+1 f
98,n2,n' 05 4 ,' ,n' 095,
f
y(50O+n2)=0
f
27,n3,n' 052,' ,n2,
f
Y(500+n2)~Y(500+n2)+Y(1 OO+n,)xy(l OOO+n2-n,)
ty(500+n2) tnl055 tn2 tn, f
yO=yO
f
g28 if n(500+nl -1) u n(50O+nl)
t
90,n2,nl 094,1 ,nl 0'1,
t
y( 500k121=0
f
87,n3,nl052,l,nl095,
f
y(500kl2)=y(500f1l2 )+y(100tn3)xy(1000+ft2-n,)
ty(500+n2) tnl tn2
f
f'
.
c090 yO=yO
n2=n(500kll).
f
The
0028
nl 057=n(' O+nl -1 )+1 f
n' 058=n( 50O+nl -1 )-1
y15=O
t
29,n5,n1 057,1 ,nl 058,
0029 Yl5=y15+y(10O+n51xz(n2-n5)
z(200()f-nl )=y15
t
.
end· of Part 1.
f
t
f
f
t
t
COMPILER PROGRAM (cant.)
e
0031
0032
0033
0034
0105
0106
0107
0108
0109
0110
.e
0111
0112
0113
0114
0115
0042
(g043
r
'tn1 'tz( 200Otn1)
The end of
f
atn953 atn961
f
n1 061 =n(l otn1 )+1
n1 062=n(500+n1 )-1
f
f
31 ,n2,n1 061,l,n1 062,
f
Y(100+n2)=Y(500+n2)
Part K
f
g32 if n1 w n900
f
g15
f
y20=O
f
y2l=O
f
y22=0
y23=O
f
f
34,nl ,1 ,l,n900,
y20=y2oty(2000+nl) f
y21=y2'l+z(200O+n1 ) f
y22=y22+n1x(y(200(}ffi1)+z(200O+n1»
y23=y23+n1 xn1 x( y( 2000m1 )+z( 2000+n1 ) )
ty20 tn900 ty2 ty4 f ·
..
atn962 atn967
f
ty21 tn900 ty2 ty4 f
atn968 atn972
~
ty22 ty2 ty4
f
f
atn973 atn979
y24=y23-(y22xy22)
f
ty24· ty2 ty4
f
f
atn980 atn986
f
y25=(*06s,y24*)
f
ty25
f
atn1021 atn1027
f
atn987 atn993
The end of
f
g40 if n901 u 0
f
g41 if n901 u1
u
2
f
g120 if n901
f
g121 if n901 u 3
u4
f
g122 if n901
f
g123 if n901 u 5
g124 if n901 u6
f
f
g125 if n901 u 7
f
g126 if n901 u8
f
g127 if n901 u9
f
g42 if n901 u 10
The end of
f
y2=Y39
g1
ff
75
Part J.
f
f
Part L.
Par M.
,e
77
~127
statistical Research Group, Sequential Analysis of Statistical
~,
Columbia University Press, New York,
1945•
.Ll}} Hald, Abrahatl, Sequential Analysis, John Hiley and Sons,
HevT Yorlt,
["197
1959.
Hald, Abraham and vlolf'owitz, J., "Optit1ULl character of the
sequential probability ratio test", Annals of Mathe:matical Statistics,
.e
19, (1948), 326-339.