The Nsearoh dssOPibed in this !'eport ruas supported by the National.
Scnence Foundation unde:r- G:r-ant GU-2059 and the U.S.A. F. under Contmot
AFOSR-68-1415.
•
AMORE PRECISE ESTIMATE OF THE COST OF NOT KNOWING THE
VARIANCE WHEN MAKING A FIXED-WIDTH
CONFIDENCE INTERVAL FOR THE MEAN
by
J.K. Osborn
and
G.D. Simons
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 677
Ap4U. 1970
t<·
,
...
~
,.,
"
", , ...!'
"
>.....
..
.'
*
r
ANDre 'recue I8t:l._te of the CoIIt of Bot IIaovlDa
the Van.ce vbea HaUoa a rlxed-w1dtb
e·
Coafi4ena Intena1 for the Me••
J .IC•. Osborn
Let '1
1
ad G.D. Sillou
*
X2 • ••• be a ••quence of iDdependellt nopal ruclOli
t
nth 1mIcaowo' _an•. lI.
anel
uaJmGWD
CODfiMa. tAten&! for
~
covua.. ,robal:ti1J.ty
(0 <
CI
d
of fixed width
CI
0 2•
vart_ee.
< 1).
In orcler to obtalu a
(d > 0)
ad prucribed
• procedun ol'i.a:l.aally
Starr baa beea • •lopeel by Gordon S:l.mou [3].
a.. Ill. [4 J, (5].) Tid.. pnC8dua ia }). . .d
Y.riab~.
A. .p.ted
by
(POl' further refereru:..,
OIl tba
foUGWiIaa ......tial
acowtaa Mal
e'
(1)
..
n
x... •
*
I
1-1
x
f••
•
Xi
2
Sn
•
n!r
n
I 'CX,.Xn)2
t-l
.u2/2
• ;r;- dv
CI
•
2te.)· 1.
"1'
The reaearch ' ..crib.eli. tid.• report " .. auppol'tecl ., tile ••tlau1
ltiaoe 'oUildatin " r Gnat CU-20" aDCI tile 1.8•••' • •
OoDtract
AlOIW8-1415 •
2
.-'
e
SilllGllieh.. pCOft4 that
the~.ex1.t.
aflaUe iDteaer k .uch that if
aftu lIOII1a&11y .toppag accoTdiDI to (1).
k IIOre oblervAUou are t . . . .
thn
for all
TIlllute il ccmcerae. with •
2.
.4l d.
lavestilatlOD of the value of k.
Jt\rna),
1.
01'.1'
obtaiDelI.
to evaluate
k~
the ,robebl11ty distribution of
N
IlUt
be
However. the probl_ of finel!:q tbi. di.tributiOl1 i8 n1ativel,
iocraat.le , 10 wa IIocI1fy the 8toppi1ll 1'01. N in the £011011101 wayl
Mote that N* ~ N.
v111 ••
DO
k* for (2)
Bence the aoneaponcliDl fillite iataler k* auch Chat
·f....
laqel' than k (e.f. (26) ofI3}). That ta, 1f the value of
u
"bftiti:t'l (2). we
it 1141 UIlderelUaata the value of k for· (1).
set
•
3
At this point. we follow Robbins [2].
e
quence of radom variables
(0,1)
random variables.
N*
(5)
(n· 2,3•••• )
1s the aame
88
thet of
{02(y~+. ":Y~-l )/(n-1)}. where the Y1 arc independent nor-
the 8equence
mal
{Sn 2}
The joint distribution of the se-
2
0
'S
.=:
Thus (4) is equivalent to
smallest integer
2m
" 2
L Y1
s .(2m+1 )d2
•
2
1-1
'
for which
where Y1 .... N(O,l)
r
,/
i · 1•••••2m.
for
(
)
NO.1.
then
(6)
ADd by noting that if y 1
%(y 12+y2)
2
zl·
N* . -
. .llest integer
m
I
... e -z •
s j2m+l >.mr 2
a2
%1
1-1
•
and y 2
are independent
we get. in analogy with (5)
n - 2m+1
where
~
Z1 ...
2mo+1
~
3
e- Z for
for which
i · 1•.•••m.
r • d/o.
Or, the probability that
II
N*· 2mf'1 i8 the 8ame as the probability that
i8 the 8malle8t integer for which (6) i8 8ati8fied.
ins1y constructed an iterative 8che_ for determiniog
Bobbi1l8 haa accorcl-
P{N*-2I*t-l}. ma1d.q
it pos8ible to compute
M
P(r.x)·
i
t(rl2m+1+x)p{N*·2.....
n.
m-mo
(With
....
"
M larae, 2P(d/oJ(*)·1
closely approaillate8 the probability ia (3).
An adequate truncation point for our ranse of computatiou....
C•
a2/r2 •
i.e..
PtN* ~ 2(e] + 21} ~ .0001.).
Ma[CJ+l0.
4
For fixed value of
we let
",
h(r) • smalleat iDteaer 1 .ueb thai: P(... i) a u
an.
k(r)
•
P{r•.x) ~ (d.
fnf{x:
Note thath(r)-l < k(rl s; h(r).
k(r)
imate
by computing the coverage probability at
Ia and Ib show a plot of C
Graphs
r· .2, .5 (.05)
for
Lineal' interpolation was used to .pprox-
k(r),.
VB
l\(r)-l al her).
where we obtained k(r)
andr· 0.5. 1.0 (.10).
Althouah it seems extremely difficult to prove the unimod.lnY of .. k(r).
our computations certainly support this character.
If Chi. is true. then
the
k*.
appropriate1nllllber of additional observations,
cou1cJenel i . listed in the
foll~1ng
mo
2
3
4
5
mo· 1
The moele for
occurs at
(I ..
for the c....
table.
0.95
(I •
0.98
8
10
6
8
1
6
6
5
C". or at some·point outside our com-
putable range.
If
\I
0
2
were known, then the confidence interval
of prescribed coverage probability
a
[Xn-d.
1s aBsured prOVided n
Bence, in orcJer to aeasure the cost due to ianorance of 0 2 t
C
'98
EN· +
It.·
x,.+d]
C. Graphs l1a an4 lIb
~
for
c.
we plotted
Ih_ the plota for the caeea
5
we considered.
rno=4
li~8
Due to the discreteness of
entirely above the plot for
k*. in graph lla the plot for
mo=3.
It i8 interesting to
that the plots in set II" remarkably support one's intuition.
nete
6·
Ia
a • .95
C versus k(r)
17
16
15
. rnO"D 1
14
•
13
12
11
10
9
k(r)
7
•..
6
ID0 •
nn -34
5,
ID
O
4
3
to
nu"
2
1
0
2
10
20
30
40
50
C+
60
70
80
90
5
7
Ib
a • .98
C versus k(r)
Ii\
23 .
22
21
20
19
18
17
16
15
14
13
k(r)
11
10
9
m -2
···0·
8
7
6
5
4
3
2
1
0
10
20
30
40
50
C+
60
70
80
90
8
Tables of values obtained for snpbe
Is and lb.
a ... 95
C
3.84 4.74 6.00 7.84 10.67 15.37 18.97 24.01 31.36 42.68 61.47 96.04
mO·1 0.12 0.72 1.48 2.50 3.91 5.95 7.26 8.78 10.43 12.87
mO·2
0.90 2.23 3.96 4.97 6.02 6.92 7.39 7.02 5.93
- - - - - - - - - - .- -
mO· 3
Il1o.4
m0 ·5
-
1.00 2.80
C
75.88 120.44 170.74
UUa1
14.31 15.93 16.84
3.79
4.79
5.58 5.87 5.35
4.26
1.83 2.92
3.97 4.82
5.15
4.73
3.82
0.84 2.12
3.33 4.29
4.74
4.43
3.66
a • .98
C
5.43 6.70 8.48 11.08 15.08 21.71 26.81 33.93 44.32 60.32 86.86 135.72
UU-1 1.33
UUa2
UU·3
mO=4
m CIS
0
-
2.10 3.17 4.63 6.62
0.50 1.50
- - - -
0.09
-
- -
9.34 11.05 12.95 15.03 17.21 19.57 22.50
2.80
4.50
6.60
1.50
3.20
5.21 6.22
4.24
8.70 9.31 9.19
8.25
6.87
7.00
7.34 6.88 5.69
4.45
5.29
6.07
6.42 5.99
4.95
3.94
1.03 3.41 4.57
5.45
5.86
5.56
4.65
3.81
0.23 2.10
-
7.73
..
l
I
..
9.
IIa
a
= .95
16
15
14
13
12
11
10
9
8
mO=2
EN+k*-C
tna=3
6
tna=4
m -5
0
4
3
2
1
o
10
20
30
Co+-
40
50
."..
10
lIb
(l
= .98
11
10
EN+k*-C
B
7
6
5
4
~
0
.-
tna=5
10
30
20
C+
40
50
...
l
l
..
11
Tables of values obtained for graphs
IIa
and lIb
a ... 95
c
0
5.43 6.70
8.48 11.08 15.08 21.71 26.81 33.93 44.32 60.32 86.86 135.72
mO·2 13 11.13 10.65 10.08 9.44 8.73 8.07 7.85 7.77 7.89 8.21 8.63
mo·3 13 10.31 9.67 8.95 8.18 7.38 6.67 6.42 6.32 6.41 6.68 7.02
8.96
mO·4 15 10.83 9.94 8.96
tna- 5 16 11.63 10.53 9.24
5.76 5.60 5.63 5.85 6.13
6.33
4.80 4.77 4.93 5.17
5.35
7.94 6.92
6.06
7.85 6.52
5.41 5.02
7.22
a ... 98
c
0
3.84 4.74 6.00 7.84 10.67 15.37 18.97 24.01 31.36 42.68 61.47 96.04
mO·2 15 9.87 9.43 8.90 8.27 7.53 6.69 6.29 5.95 5.78 5.86 6.23 6.73
mo.3 15 9.37 8.74 8.01 7.19 6.29 5.34 4.90 i.S4 4.34 4.39 4.70 5.10
m o4 16 11.19 10.39 9.40 8.28 7.07 5.88 5.34 4.90 4.64 4.62
O
tna=5 17 12.16 11.28 10.12 8.67 7.04 5.46 4.76 4.20 3.85 3.76
4.87
5.19
3.95
4.23
...
'
)
....
l2
.
'.
REFERENCES
,e
[1]
Chow. Y.S. and Robbins, H. (1965).
[2]
Robbins, H. (l959). Sequential estimation of the mean of a normal
population. Probability and Statistics -- The HaroZd CI'Cl!1l~1!
VoZume.. 235-245, Almquist and Wiksell, Uppsala, Sweden.
[3]
Simons, G. (1968). On the cost of not knowing the variance when
making a fixed-width confidence interval for the mean.
Ann. of Math. Statist.
1946-1952.
.
On the asymptotic theory of
fixed-width sequential confidence intervals for the mean.
Ann. Math. Statist. 36. 457-462.
39.
(4)
Starr, N. (1965) •. The performance of a sequential procedure for the
fixed-width interval estimation of the mean.
36-50.
Statist.
[S)
37,
Ann. Math.
Stein, C. (1945). A two sample test for a linear hypothesis whoae .
f2"er is inclependeut of the variance. Ann. Math. Statist.
lb, 243-258.