UNIVERSITY OF nORTH CAROLINA
Departnent of Statistics
Cl~pe1 Hill, N. C.
SOHE SEQUElJTIAL MTALOGS OF STEIN'S THO- STAGE TEST
by
Jacl~son- Hall
Hilliau
-_.- .. -
Septenber,
_ 1961
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-
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!
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'I
,-~.~~
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:•
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~_ t;;~.u- ;:;,v~~
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This research 'WaS supl)6Ftedby the Office of
Naval 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 Governcent.
Institute of Statistics
Mimeocraph Series H9. 308
SOME SEQUENTIAL ANALOGS OF STEIN I S TWO- STAGE TEST
by
l
William Jackson Hall
University of North Carolina
This paper presents several sequential analogs of Stein's two-stage test procedure for
testing hypotheses about the mean of a normal
population with unknown variance and with specified error probabilities. When sequential experimentation is feasible, they provide alternatives to the sequential normal test (variance
known) or the sequential t-test. If the
variance is assumed known, the procedures may
still be recommended since the added cost may
be only a very few additional observations on
the average, and the performance of the tests
does not depend on the validity of any assumption about the variance. Moreover, unlike the
t-test, these procedures do not reqUire that
the alternative hypothesis be specified in
standard deviation units.
1.
INTRODUCTION
When sampling from a normal population with unknown mean
unknown variance
~
and
2
rr , one may Wish to test the composite hypotheses
vs.
with pre-assigned strength
It is a well-known fact
(a,
~) (bounds on the error probabilities).
tha~unless
no such non-sequential test exists.
Hl
at least bounds are placed on
rr,
A common solution is to restate
in (unknown) standard deViation units and use the t-~ (non-sequen-
~s research was supported by the Office of Naval Research under
contract No. Nonr.855(09) for research in probability and statistics
at the University of North Carolina, Chapel Hill, N. C. Reproduction in whole or in part is permitted for any purpose of the
United States Government.
tial or sequential), or, equivalently, allow a to be a f\ulction of
the unlmolll1 cr. HeitIler of these reforwulations my be coupletely
satisfactory.
The only know11 solution to the probleo as stated is
Stein's two-stace procedure (Stein l 1945, MoshQan, 1958): a preliDinary sample of fixed size m (> 1) is t~~en in order to esti~te cr2 and a second staGe sat~le, of size dependinG on this firststaGe estinate
s~, is then taken if necessary; since the first-staGe
sample oean and variance are statistically independent, the inforr.~­
tion from the first sample about the Dean ~ can be utilized, toGether wi tIl that from the second sample, in mking the terninal decision. The size of the second sal:J.Ple depends only on s2, so that
2
m
its distribution depends only Oll cr, and not on !.t.
A sequential analOG of Stein's procedure is presented here.
2
A6ain a first staGe sample is used to estimate cr, but samplinG is
then continued, if at all, one observation at a tine rather than in
a non-sequential fashion.
It is otherwise analOGOUS to Stein's pro-
cedure, but, as one would expect,
nOv1 depends on
~
the distribution of the sanple size
2
as well as on cr •
This procedure, test T, may be described as a sequential probability ratio test (SPRT) v1hich is not peI'Llitted to teroinate before
ill
observations and in which cr2 is replaced in the probability
ratio by the estimate
2
so; tl1e usual tenlination bounds
A and B
are oodified by a oethod due to Paulson (1961) in order to achieve
the required strenGth.
An eqUivalent interpretation of the test T,
useful for studyinG its properties, is that it is a conditional
SPRT, Given
sand cr, with ter.oination boundaries dependinG on
m
Its behavior can be studied by averaGinG (takinG expectation) with respect to S. Thus approximtiona to its OC (operatine
m
characteristic) function and ASH (averaGe sample nurmer) function are
obtained by averaGinG the correspondinG approxir.Jations of Hald for the
conditional SPRTj these approxir.J8.tions are valid if the test is not
likely to tertlinate with the first stage.
considered in this light.
Stein's test can also be
Some numerical comparisons of the power
(or OC) and ASN functions, based on these approxirations, for the
test T, for Stein's two-stage test, and for the SPRT
~ize ~
(FSST) assULIing cr known, are presented.
and!~
sample
The approx1IlJa-
tion obtained for the A&f of T sugGests that substantial savings,
compared ,dth Stein's procedure, are possible.
For the case
~
= a,
an alternative sequential test procedure T'
t"'l,"\
is described, usin~mi~itrun probability
stead of the SPRT.
ratio test (Hall, 1961) in-
TW'o-sided test procedures, analogous to T and
T', are also briefly discussed.
2
If' an estimate of cr is available from previous experiments,
the need for the first stage is elininatedj minor modifications of
these procedures would :cake then applicable.
In fact, this is the
context in which Paulson's method 'Was introduced.
In none of these procedures is any of the information about cr,
other than frOD the first-stage sample, utilized, so that the tests
do not depend on a sufficient sequence of statistics.
sequential procedures,
T
n
Alternative
and T', using all available
n
infor.n~tion
about cr but without theoretical justification, are proposed.
Some
empirical evaluation of these procedures is planned.
Illustrative diagrams for carryillg out these sequential tests are
presented.
2.
Let
-~
<
JS.'
X '
2
0 < 0"
j.L <~,
j.L
T.
be independent
0"2) randoLl variables,
Let
<~.
Consider a SPRT of
THE SEQUEIJTIAL TEST
=0
be a specified intecer exceedinc one.
0
va.
j.L
=A
Xn+2' ••• , with ternination bounc1a.ries
placed by
s
o
l'T(j.L,
(0" Imown) based on
A
tl
and B
o
XLl ,
Xm+l'
and with 0"
re-
where
v
=0
1
-
(1)
We refer to this test as test
vnth
A, B
approxir~te
T.
m > A == l/a and
Note that A
equalities instead of inequalities if
are the conservative ternination bounds of Hald
Ll
B
0
< B ==
is larce, and
(1947,
42)
p.
appropriate if 0" vTere known.
Denotinc
(n ~ 0) ,
T
is found to be:
observe
successively and, for each
(Xl' ••• , Xn) and then
n >
-
0,
after observinc
XT.1+l' Xr.1+2'
X,
n
stop saIlpIinc; and nal~e decision
dO (accept H ) if
stop SC1T.1plinC and r.Ja.ke decision
HI) if
continue saoplinc
if b
r.l
<
d (accept
l
r (s ) < a
non
o
(3,
r (s ) < b ;
n n - m
r (s ) > a ;
n
D
-
n
5
3.
For Given
of I.l = 0
VS.
THE STREHGTH OF TEST
T.
(SOl a)1 consider the conditional SPRT I T(Sm' ~),
I.l =
X,
X 1
o tl+ 1
A based on
with tenlina.tion
•• ,
bounds ~ and Br.l where
2,2 =1
xn/A
= as/a
, bo = i n -BI:l = btl
tl
tl rl
-
aI:l
sl:l2}~2
COI:lputin3 the relevant probability ratios l notina that Xllll Xmtll •••
are statistically independent of S, one finds that decisions are
I:l
made accordinG as r (a) < b , r (a) > a I or b < r (a) < a .
n
0
n
0
1:1
n
1:1
Using Waldls conservative bounds on the error probabilities of a
SPRT, we have
(4)
Pr {do usinG T(sI:l,a)lsl:l, a,
1.l~}BC = 6XP
s2/~2) •
om
(b
But r n (a) :: r n (sI:l ) s2/~2)
so tlmt T(sl:l I a) is seen to have
III
precisely the sane dec~sion rule at each stace as does the test T,
with sI:l cotl]?uted froLl the observed values of
ePr {di usinG T(SdO") JSr.l~'Il}
I:
£,
Pr
{d
i
)L,
usinG
:: Pr {di using TJ
and therefore, usinG (4)
(5)
Pr
for all
{~ usinG Tf
~
since
a,
Pr {dO
us1na
X.
I:l
Thus
Tf
~>Il}
.
)
I
O} < £: exp
222
vso/a :: Xv and
(-atl S2/a
2)
l"J
£ exp
= (1
V 2
+ 2a,jvf
/
....
(t Xe) :: (1 - 2tr V/
SiD1la rly I
(6)
••• 1
-~
rl::- exp (b S2/a2 )
DB
2
•
6
~.
for all
SPRT, its
T(S,~)
Since the conditional test
I:1
is a nortnl-oean
function is monotone 111 J..L (Hald, 1947), 1.e.,
OC
Pr f~ di usinG T(a,~)
f B0 '~IJ..LI is oonotone in J..L for every fixed
m
So and ~. This, tOGether vnth (5), (6) and (1), 1r~lies
(7) Pr
{~uSinG
and (7) states that
Tf HO} < a,
T has
Pr {do usinG Tf H1 }- < 13 ,
strenGth
(a, 13).
Note also that since the SPRT T(s
,~)
m
teroinates with certainty
for every fixed So' the test T also tero1nates with certainty.
4. THE OC FUNCTION OF T.
For the conditional test T(stl ,~), Wald's approxioation to the
OC
(8)
function may be used, naoely:
Pr {dol
'\-There h(J..L)
Srl'~'J..L} ~ ~ -
=1
(11 ~ 0)
1)/<A: - B:)
- 2J..L/6 and vThere the "e" implies neGlect of excess
over the bmundaries; this excess should be soall if the test is lilrely
to have a sample nur.lber substantially larcer than
tations vnth respect to SL1 in
Ill.
Talt1nc expec-
(8), usinc (3) and droppinG the sub-
scripts on aIII J band
SU J we have
tl
(h > 0)
which does not depend on
~.
The intecrand on the RHS r.l8.Y be expressed
7
as an alternatinG series 'Hi.th terl:1s of decreasinG r:acnitude so that
successive partial
expectations
SULlS
Give u];lper and lower bounds on it.
te~-by-te~}
Taldna
the RHS thus yields successive upper and
lower bounds on the approxllJation to
p~dol cr,j.L}
.I
1'1hich may be
expressed as
1 - (1 + 2ah/v,-V/2 + (1 + 2 a-b h/v
-
r V/ 2
(1 + 2 2a-b h/v)-V/2 + (1 + 4a:b h/v)-V/2 - •••
- ... .
These are valid fori h
> 0, i.e., for
j.L
< 6/2. For
j.L
> 6/2 (h < 0),
1-1e obtain analoGously
1 _ C
(10)
C
2
(1 + 2bh/vr V/ _ (1 + 2 b-a h/vf V/ 2
=
(1 + 2 2b-a h/vr 2 / V
+
-
(1 + 4 b-a h/vr V/ 2+ •••
(1 + h - 1~-2/v)-V/2 _ (1 + 2h _ ~.2/V
=
-
2 . 2)
1- exp ( - bhS:Lcr
- h82/2
1- exp (
-b-a
/cr )
h
a- 2/ V )-V/2
+ (1 + 3h _ 2h ~-2/V_ ha- 2/ V)-V/2
_ (1 + 4h _ 2~-2/v _ 2hc1,-2/vf v/ 2 + ....
For
~
= Ci,
we have
8
(11)
pr
{do.IO'I~} ~ !:~::O
(_1)1£1 + 1
IhJ(ct 2/ V• 11T V / 2 (~16/2)
uhere acainl as in (9) and (10)1 the partial sums provide succes.sive
upper and lower bounds.
At ~:: 6/2 (h :: 0) 1 we obtain by taltinc 11mts in (8) as
h ---> 0 and usinc l'Hospita1's rule:
irrespective of s, so that
(12)
Pr {dO
I 0'1
~
6/2}
a/(a-b)
=
(a- 2/ V• 1)/(a- 2/ v+ (3-2/V_ 2)
uhich equals 1/2 if (3:: a.
The series (9)·(11) converce reasonably fast except near h:: 0
(~
= 6/2). For exar.lp1e 1
'Wi th
find the followinG values for
to the power function Pr
a
tl~
= (3 = .05
and
Ll::
16 and 31, we
successive approx1r:ations in (11)
{C\ I~ } :: 1 • Pr {do I ~} :
-n
!J./6
11
1/4
1/2
16
.19,
.14,
.16,
.15,
0
1
16
.0501
.044,
.045 1
.045
.1/4
3/2
16
.016,
.015~
.015
-1/2
2
16
.0059" .0056,
1/4
1/2
31
.21"
0
1
31
-1/4
3/2
-1/2
2
.0057,
.0057
.161
.17,
.17
.050,
.046"
.046
31
.014 ,
.0131
.013
31
.0041" .0040 1
.0040
.16,
.151
.15
9
The saue speed of converGence viaS found for ex
= f3
;: .01.
Ca1cula-
tion beyond two siGnificant fiGUres is actually unwarranted since
these formulas ignore the excess.
5.
~ ~
For
~/2
THE ASH FUNCTION OF T.
(h
~
~ rN(~) ~
(13)
and for all
0)
~ ~ Z:~=1
::
(~
Do
Also, droppinG the subscript
£ rIl~)
(14)
~/2)
-
~, we
have, uninG (2):
~/2)/a2
(Xi -
t: !r/el :: - ~
0,
C ~ L ru(a) 1.~7:: ~ (z:i::o C£ rl1(~) IS,di _7
;:
Pr {dilS } ) •
Now, still droppinG the D'S, and icnorinc excess,
c -
dl _7 •
)1
7
0' 1.r r ( a s , dOC
n
e -b
II;
a::: a
=b
6
2, 2 .
la
,
1 2
s 2 la
•
The excess should not be sicrnificant if H tends to be laree relative to m.
(15)
(14) thus leads to
crN(a)
e
E:'fb
;:
a
s2 Pr {dO IS} + a S2 Pr
- (a-b)
G £s2
Pr {dolS
{~IS}_7/~2
}_7i~2 .
Usinc (8) and proceedinc as in the previous section, we obtain for
h>O
£r N( 0-) ~
a - (a-b)
2' ~~ Ll-exp( -abS" /0-2)7 E~=o exp( -1 a-b bS2/0-2 ) }.
10
NotinG that
CLX2 exp(-t 1)/v_7 = (1 +2t)"·1 -V/2
1
\1e find after
takinG expectations tern- by-tero, and eqU£l.ting 'With (13) I that for
11>0
(16) (6/cr )2 ~
~
N
_ 2: _
2{~..b):L(1+2ah/vr1-V/2_(1+2
+ (1 + 2
=
*~-2/V_1
h/vf 1- v/ 2
2a:b h/v)-1-v/2..(1+4 a-b h/v)-1-V/2
_ (a- 2/ V+ f:;-2/V _
a- 2/ V)-1-V/2.
h
a-b
+ (1 - 3h + 2h
(1 _ 2h + h
a- 2/ V +
h
2)
.L(l -
h +
a- 2/ V+ h f:;-2/v)-1-V/2
f:;-2/v) ..1-V/2
2
- (1 - 411 + 211 ct / V+ 211 f:;-2/Vr 1- V/ 2 + •••_7
For
< 0 1 the same formula. holds with h replaced by -11 and with
h
a and
(17)
}.
~
interchanGed.
~ ~
(6/cr)2t: H
Ih'
~
For
(a.- 2/ V
1) {1 + 2 r,f1J_ (-1)i £1 +
i 1
_
-
i
At
JJ.
=0
=al
h
(a- 2/ V ..
1L7- l -V/2}
(h
~
0).
and JJ. = 6 1 llald's conservative bounds (4) on the
error probabilities of the conditional test may be used instead of
(8); thus (15) leads to
{6/cr)2
C £N
e
IJJ. =
27 > -
b .. (a-b)
= v ~~-2/v_
and s1Llilarly for
(6/cr)2~£n
1JJ.::;6_7
e s2exp(_as2/cr2)/cr2
1 _
a(l + f'-2/V a.2/ V)_7
with Ct and f:;
interchanGed.
11
If
~
(18)
= a,
these two relations reduce to
(t:./cr)
2
2 LU fj.1=O
or
e
6.-7
> v (a-
2/
V_I .. 2a).
and
(20)
co f
r
G.
2 e 2 ~ 4 4
n(cr t7 ;:: a c: S /cr
- (a
2
2 ~r 4
- b )C L S Pr
L"7 4
l.(J
do ,SJ-!/cr •
EquatinG (19) and (20) and usinG (12),
(21) (t:./cr)
2ro
C _
11
rJ
J.1
=t:./'S7 e=fa 2..
=
f
~42
2
a(a+bt7c; Xv/v;:; -ab(l + v)
(1 +
~)(a"2/V..
1) (f3- 2/ V
-
1).
Some successive approxim.tions froo (17) appear below (a = f3 ;:: .05)i
evaluations froo (18) and (21) also appear:
J.1/t:.
h
- Ll
1/2
0
16
15.4
frOLl (21)
1/4
1/2
16
10.2,
11.2,
0
1
16
, 6.9,
6.9i
-1/4
3/2
16
4.8,
4.8
-1/2
2
16
3.7,
3.7
1/2
0
31
1/4
1/2
31
8.3,
9.4 ,
9·1,
0
1
31
6.1}
6.1i
10~'Ter
-1/4
3/2
31
4.3,
4.3
-1/2
2
31
3.3,
3.3
10.9,
11.0, 10.9, 11.0,
11.0
lower bound frOLl (18) is 5.9
11.7 frorl (21)
9.2,
9.2
bound frorl (18) is 3.6
12
ConverGence
sliGhtly fa.ster for a::: t3 :::
'WaS
bound frotl (18)
liaS
.01" and the lower
ouch closer to the approxiIJation (17).
6. COUPARISON i'lITH THE OC
AIID
ABlY
FUNCTIOnS OF
STEIN'S THO-STAGE TEST
Let t
~
v"....
be that number which is exceeded by at-statistic
with probability a (v decrees of freedoo).
N in Stein's (one-sided) procedure
and error bounds
where "f
_7"
Ca"
t3) is Given
neans
~dth
The total saople size
initial sample size n::: v+l
by
"larGest inteGer in"" and
(t v,a + t V, ....R)
2
"
the approxioation beinc valid if it loplie8
~ N is sOtlewhat lar13er
(Stein" 1945).
than n
The ter.o1nal decision rule for Stein's test nay be written
~ EN (
~/2)/ 2
i=l Xi sm
ApproxirJatinG 11 by
s; (t~
>
(t 2
<
v"a _
decide d
t 2 R)/2
l
decide dO
V, ....
+ t~)/t?, the OC
function is found
to be
PI'
where F
V
fdol~}
;.
'
L
Fv-rt v,a -
~(t v,a + t V,....R) /~ -7
is the distribution function of a (central)
with v decrees of freedor.l.
t-statistic
The true OC function is presU11Jably
sliGhtly steeper since the true sanple size tends to be larGer than
the approx1I:Jatinc value.
1;
Tables 1 and 2 at the end of this paper present
srn~e coc~ri­
sons of the a:pproxmate pOlTer and ASH functions of the sequential
test T and Stein I s two- staGe test (ex = 13
= .05
and .01),
Ala 0
included in the ta.bles are the correspondinG approxir:ate values for
the SPRT (with A
(FSST) if
about
Cl"
Cl"
= I-a/a = l/B)
and the fixed saople size test
'l'Tere Imown (correctly).
Of course" if the aSsULlption
were incorrect, the power ftu1ctions of the SPRT and FSST
ray be drastically altered.
It ia of interest to note tlmt the power functions of these tests
becooe steeper as one Doves frOD left to riGht in Table Ii thus" the
test T discrioinates best for inter.nediate
discrluinates best for extrene
~-values
and the FSST
~-values.
These calculations sUGGest that substantial savinGs Day be
possible usinc the sequential test T -- at least if one
hypotheses is correct.
of the
The conparison between T and Stein's test
is analOGOUS to the cor.1parison bet",een the SPRT and the FSST of the
saoe strenGth.
Actually 1 the conparison "'ould be in closer analoGY if the SPRT
with conservative boundaries (A
= l/a = l/B)
the test T uses conservative boundaries.
were oodified, by increasinG a and 13
probabilities equal to a, the
~SN
",ere considered, since
If the boundaries of T
in (I), to achieve error
of T would be further reduced.
(Calculations indicate that substitution of a/l-p
for
a and
I'll-a for P in (1) still Gives conservative bounds on the error
probabilities.)
The lack of ImO'l'Tlec1.ce about
few observations
(perl~ps
Cl"
costs only a very
two or three) on tile averaGe, and thus the
14
test T or stein's test may be recauoended even if
~
is thought
to be l010rm (if a one-stage test is not essential).
7. AN AIlI'ERNATIVE SEGJJEIHIAL TEST T'
For the
ratio
~
s~etric
(a = ~)
one-sided case, a niniouo probability
(rIPRT), which has converging straight-line boundaries
(Hall, 1961), can be adapted in the same nanner as
'WaS
the SPRT above.
The MPRT is equivalent to one of Anderson's (1960) tests.
obtain the followinG test T'
"lith decision rule:
61~~=1(Xi - 6/2)j/8; >
stop saDpling as soon as
2
cn - 1162/ 46I:1
and choose
~
He thus
(n
? n)
or dO according as ~(X1 - 6/2) is
> or < 0
where
After m observations have been taken and sn conputed, an upper
2, 2
4cDO
B ,Ii
•
bound on the total sample size is
Ho approximtions to the OC
tained.
PresUIJably T'
conpared idth T would have a smller ASN
in the neighborhood of ~
ASH at (and beyond)
~
or ASH functions have been ob-
= ~/2
= 0 and
at the cost of a slightly larger
~.
The cOI:lparison '\-,ould be analogous
to the cOtlparisons of the SPRT and UI?RT (0" Imown) given by Anderson
(1960).
If a ~ ~ , the test can still be used uith 2a in (24) replaced by a +
~,
but then one can only assert that the stu'J of the two
error probabilities is less tl1an a +
~
(Hall, 1961).
15
8.
~le
fuJ.'lcti~n
THE THO-SIDED CASE
two-sided nomal test, variance knovm, based on the weicht
1:1ethod (Hald, 1947) or the invariance oethod (Hall, 1959),
cannot be adapted to the case of unlmow variance as 'WaS the onesided test,
ty ratios.
since cr
2
does not factor out of the relevant probabili-
However, the Sobel-Hald test procedure (Sobel and Hald,
1949), in which one in effect
is easily adapted.
To test
two T (or T') tests
l:'WLa
two one-sided tests sioultaneously,
IJ;: 0 aGainst IIJI
?
of IJ;: 0 aGainst
6, one can :run
IJ;: 6
and IJ
=0
acainst IJ::: -6 -- smultaneously und continue satlplinG tulti1 both
tests have teroinated.
9·
HEURISTIC TESTS
T
n
and T'n
In discussinG the sequential estimation of IJ (cr mUcnow), Anscaube
(195,) noted that, if the procedure were not allowed to terninate early,
cr would essentially be lcnovm.
TllUS,
if one uses a procedure requirinG
lmowleclce of cr but replaces it by an estitJate, the properties of
the procedure should not be Greatly affected.
The tests
T and T'
are lllce this; in fact, the test boundaries suitable if cr were
are widened to account for the fact that cr
crees of freedao.
However,
cr
is not
IU10WU
is estiLated on 0-1 de-
re-est~ted at
each successive
stace, and the choice of 1:1 seeDS arbitraryj in fact, if cr were
Duch smller than expected, a cor.lpletely sequential procedure my
te~inate
before the first staGe of T is coop1eted.
16
The followinG tlod1fication of T is proposed purely on intuitive Grounds:
re-estiDate ~2 by s2n at each staGe nand
base the test on rn(Sn) (n > 1) 'With boundaries
by (1) 'With v replaced by n-l.
(~
This test,
(an I bn ) Given
Tnl is lilte a SPRT
Imawn) ,,11th ~ replaced by a new estimate at each staael and
,,11th the boundaries 'Widened in an atteopt to account for the lack
of Imowled6e about
~.
The boundaries of T
n
conservative boundaries (-in CX I
in~)
converGe to Hald's
which are appropriate (thoUCh
sliGhtly conservative) if ~2 is ItnO't'lIl.
A diacram for carryinG out
this test is illustrated in FiGure 1 1 tOGether with diacrans for other
sequential tests.
boundaries,
a
= in
(The SPRT i1) the diaGratl uses vlald IS approx1r.a.te
(1 - f3/CX)
The alternative test T '
T'
n
'With the roles of
c
D
and b
= in
(f3/1='"a) .)
can be oodified analocously, obtaininc
and s
D
replaced by c
n
and s.
n
Its
boundaries depend on s 2 and thus cannot be Graphed in advance (in
n
the diaaram, the expected values of the boundaries are Graphed).
Ho theoretical evaluation of these procedures has been possible.
17
~ ~
DiaCratlS for six sequential tests of
I'n
\..L
~A
(0 = 16, ex
0 acainst
= f3 = .05).
13
~T
~ ... ~--
upper boundary:
choose d
lower boundary:
choose do
-
(lL
1
------_
~ L)
(jJ.~0)
..
-h-rrr.,-----r----------------------....,...~n
O
.1
16
6
- .-
b
brr:
-- -
- -
-~....
_.
-
~"f
- - - - - - - - - .-. -- -- ..- - - -- -- --
-...
./
I
I
I'n
=
0
v2
=
10
\
\
2
/ v
i::. ~(xi - li)
2
2
for
s 2
m
2
s
n
SPRT & MPRT (0 known) - - -
for
T
for
T
n
&
T'
&
T
n
, -
--
'.
c
- ,_.
o
-+r-r-T""n-----r--------------------=-'":::=:-=--r~lJ.
--
-c
-c m
/'
/
-10
I
~
.. --~
(boundsr'y
-- -------
~-
-~
---====
f: (boundary (
~
"----- - t
-- ----
I~
~
-
1
2 )
~
)
for
for
T :
n
--.
18
RETh'REHCES
AIIDERSON, T. H. (1960).
A I:lodification of the sequential
probability ratio test to reduce the sanple size.
Ann.
Nath. Stat. ;1, 165..197.
l'-JISCOBBE, F. J. (195;).
Soc.
Sequential estiI:Ja.tion. J. Roy. Stat.
B, 15, 1..21.
HALL, H. J. (1959).
The relationship between sufficiency and
invariance with applications in sequential analysis, I.
to::> appear in Ann. l-1ath. stat. (Institute of Statistics
lIioeo Series, Ho. 228, Chapel Hill.)
HALL, W. J. (1961).
Sone sequential tests for syouetric probleos.
Invited address at Eastern ReGional 11eetincs of Inst.
~mth.
Stat., Ithaca, IT. Y.
MOSHr.1AH, J. (1958).
A nethod for selectinc the si~e of the
initial saople in Stein 1 s t"TO sanple procedure.
Stat.
Ann. 1-1ath.
29, 1271-1275.
Pi\I1L60lJ, E. (1961).
A sequentia.l procedure for comparinc several
experinental cateGories with a control.
Invited address at
Eastern Recional Heetincs of Inst. Math. stat., Ithaca, N. Y.
PEARSOU, E. S. and HARTLEY, H. O. (1954).
Statisticians, 1.
Bionetril;:a Tables for
Caobridce University Press.
SOBEL, M. and WALD, A. (1949).
A sequential decision procedure
for choosinc one of three hypotheses concerninG_ the unl;:novID
nean of a nornal distribution.
STETI1, C. (1945).
Ann. Math. Stat. 20, 502-522.
A two-sample test for a linear hypothesis
whose power is independent of the variance.
Ann. Hath. Stat.
16, 24;-258.
HALD, A. (1947).
Sons.
Sequential Analysis.
New Yorl;:: John Hiley and
19
APPENDIX
Explanation of Tables
Table 1:
Pr
{~,
calculated froLl (11) and (12). : For the SPRT1 it
.
ws calculated
frOtl Wald's approxirlation ( 1 - e -l~)/( eha- e-ha)
where a
J.1}
For the sequential test T, the power function
llaS
= {n
(1 - a/a).
For the other tests1 it ws calculated
froD (2;) which reduces to F (-tt
(t
);:; 1 - a; for
v
v,a ) where Fvv,a
the fixed saLlple size test (FSST)1 v = ~ 1 i.e., F is the standard
v
normal d.f. The Pearson and Hartley (1954) tables of F vrere used.
v
Table 2:
For the test T, the ASH function 'Was calculated from
(17) and (21).
(1::/0')2
t
For the SPRT, Hald's approximtions \Tere used, naLlely
2
If ;:; 2a £1 - 2P(~17/h
if h ~ 0 and = a if h = O.
For the other tests, it
v~6
calculated frOtl (22), and is the same for
all J.1-values.
The teroinal decision rule in every instance is the sane, naLlely
choose dO (J.1
:s 0)
if ~f < 6/2 and choose
'S.
(J.1 ~ 6)
if ~l > 6/2.
The approxiInations to the power function and ABU function icnore
a)
excess over the boundaries in the sequential tests,
b)
the restriction that N wst be inteGral,
c)
the restriction that N > 0 for test T and Stein's test,
and are thus valid if If is larGe relative to
In
with hiGh probability.
The author wishes to acknowlec1Ge the assistance of Hr. K.
FW~ushina
4 and 5.
in preparinG these tables and those presented in sections
20
TABLE
APPROXIHATE
PO~1ER
FUIJCTIONS"
FOR TESTING II ~ 0
1
P~~ JII}"
II ~ 6 (a
AGATIJST
SEQUEHTIAL TESTS
a
ll/6
h
Test T
l'l
·5
0
.25
.1
=
16
IJ
=
OF PROCEDURES
= (3)
ONE- OR TWO-STAGE TESTS
SPRT
31 (0"
kno'Hn)
Stein's Test
tl
= 16
tl ;::
FSST
31 (0"
know )
·5
.5
.5
.5
.5
.5
0.5
.154
.168
.187
.197
.201
.205
0.8
.0727
.0777
.0866
.0906
.0923
.0941
1.0
.0450
.0463
.05
.05
,05
.05
-.1
1.2
.0285
.0278
.0284
.0264
.0253
.0242
- .25
1.5
.0149
.0132
.0119
.0095
.0081
.0068
-·5
2.0
.00565
.00405
.00276
.00159
.00097 .00050
.5
0
·5
·5
·5
·5
.5
·5
.25
0.5
.062
.075
.091
.106
.114
.122
.1
0.8
.0192
.0216
.0247
.0275
.0293
.0313
0
1.0
.0095
.0097
.01
.01
.01
.01
-.1
1.2
.00J.J.96
.00453
.00397
.00349
.00307 .00262
-.25
1.5
.00206
.00154
.00101
.00071
.00043 .00019
.05 0
.01
21
TABLE 2
APPROXIHATE
FUNCTIONS, (t::./0")2
ASH
!.L ~ 0
FOR TESTING
!.L ~
t::. (ex
h
I
Test T
Ll
==
16
ill
==
SPRT
31 (0"
OF PROCEDURES
= l3)
ONE.. OR THO-STAGE TESTS
SEQUENTIAL TESTS
I !.L/t::.
ex
AGAINST
£ {Nh.},
Imown)
Stein's Test
==
tl
·5
0
15.4
11.7
8·7
.25
0.5
11.0
9.2
7.4
I .1
·05 0
I I... 1
-.25
0.8
8.2
7.2
6.1
.
1.0
6.9
6.1
5·3
12.3
1.2
5.9
5.3
4.6
1.5
4.8
4.3
3.8
2.0
3.7
3·3
2·9
I
I
I
1
I
I
I
I!
-.5
I
.01
.5
0
45.8
31.0
21.1
.25
0.5
23.1
18.8
15.0
.1
0.8
15.5
13.0
10.9
0
1.0
12.6
10.6
9.0
-.1
1.2
10.5
8.9
7.6
1.5
8:5
7.2
6.1
... 25
I
I
I
Ll
==
31 (0"
known )
11.5
10.8
24.1
21.6
j
27.1
I
16
FSST