Research partly supported by an NSF Grant No. GP-19568.
and Institut de Statistique math~matique, Universi~ de
Gen~ve.
CoNFIDENCE SET FOR 1liE RATIO OF MEN-Js OF Two [\bRMAL
DISTRIBUTIONS WHEN 1liE RATIO OF VARIANCES IS UNKNOWN"
I. M. Chakravart; ....
Department of Statistics
University of North Caro'Zina at Chapel HilZ
Institute of Statistics Mimeo Series No. 765
July, 1971
CoNFIDENCE SET FOR THE RATIO OF MEP.NS OF Two rhRrW.
DISTRIBUTIONS WHEN THE P\ATIO OF VARIANCES IS UNKNOiJN*
I. M. Chakravarti**
Department of Statistics
University of North Caro'Zina at Chapel, BiU
I.
••. , xl
dom variables.
INTRODUCTION
independent normal ran-
;
nl
We assume,
=
for
u
= 1,2, ••• , n i '
We assume
J.1
2 =I 0
A,
dence interval for
ances
o 2
1
and
i
(1.1)
0 2
i
= 1,2.
and set
A = J.1 1 /J.1 2 •
The problem is to derive a confi-
without any assumption about the ratio of the two vari-
O 2•
2
Let
S 2
s
i
2
i
i
=
= 1,2.
(1.2)
u=l
We give a confidence set for
teed to exceed or equal
than
*
**
A with a confidence coefficient which is guaran-
(I-a),
when
a
is a preassigned positive number less
1.
Research partl,y supported by an NSF Grant No. GP-19568.
and Institut de Statistique
math~matique~ Universit~
de
Gen~ve.
2
2.
Let
t a (m)
CoNFIDENCE SET FOR THE RATIO
A
be a constant chosen to satisfy
(2.1)
where
t(m)
dom and
a
follows the Student's
t
distribution with
is a preassigned positive number less than
this article, we denote
ta(ni-l)
by
t ,
i
i
m degrees of free1.
For simplicity, in
= 1,2.
We note that the random variable
y
•
(2.2)
crf/n l + A2cr~/n2
follows chi-square distribution with
1
degree of freedom and the probability
density of this distribution is
fey)
=
(2n) -~ exp(-y/2) y -~ ,
(2.3)
Consider the integral
g(v)
Noting that
=
f: (2n)-~
g'(v) > 0
and
exp(-y/2)
g"(v)
<
0
y-~
(2.4)
dy.
it follows that
(2.5)
for
0 < p < 1
and
0 < VI'
v2 <
00.
That is,
g(v)
is a concave function.
Let
v =
=
(2.6)
w
1 (n -1)0'2
1
1
(n -1)0'2
2
2
3
where
w
2
=
Note that
...
Let
(n -1)
i
~ - S2/
2
i ai'
~i
and
0 S wi S 1,
i ="
1 2
t h en
1
and
Now
V
t
with
(n -1)
i
i ... 1, 2 •
(n -1)
i
(2.7)
h as a chi -square distrib ution wi t h
Zi
T~ = (n i -l) Y/Z i
degrees of freedom and
of the Student's
with
1
is distributed as the square
degrees of freedom (or equivalently as an
degrees of freedom),
F
i ... 1,2.
can be rewritten as
V
...
2
2
t 2 Z2
t 1 Zl
wI (n -1) + w2 (n -1)
2
1
P
=
Prob
...
Prob {Y :::;
V}
=
Prob
ti ZI
Z2 }
wI (n -1) + w2 (n -1) •
.
(2.8)
Let
{
Y
S
t~
1
(2.9)
2
We then prove the following
THEOREM
P ~ I-a.
PROOF:
Y, Zl
(n -1)
1
and
and
(n 2-1)
Z2
are independently distributed as chi-square's with
degrees of freedom respectively.
note the density of the distribution of
n -1) degrees of freedom)
i
as the triple integral
i'" 1,2.
Zi
Let
h(zi; n -1)
i
1,
de-
(chi-square distribution with
Then the probability
P
can be written
4
co
p
...
co
rr w1v1+wZvZ
fo fo l!o
. J: J:
J
(2")-~ exP(-Y/2)Y-~d~h(Zl;nl-l)h(Z2;nl-l)
g(w1v1+w 2v 2) h(zl;n1-1) h(zZ;n 2-1) dZ 1 dz Z
where
(2.10)
But from (2.5)
and
(Z.ll)
Further
i •
(2.12)
1,Z.
Thus, from (2.11) and (2.12) we have
A confidence set for
_ _ 2
(X -AX 2)
1
$
A is thus, given by
2 2
2 Z
t 1 sl
2 t 2 s2
+ A
n
n
1
1
(2.13)
or
A2
[iZ
Z
_
t~ s~]
n
Z
_
2 A
XX
1 Z
+
[xz1 _ tin
si]
$
0
(2.14)
1
with a confidence coefficient greater than or equal to
(l-a).
Consider the quadratic equation
A2 [X;
-
t~n:;J -2 A X1X2 +
= o.
(2.15)
't'lriting
c
...
(2.16)
5
(2.15) becomes
aA 2
= o.
2Ab + c
-
(2.17)
The two roots of (2.17) are given by
(2.18)
Let
l3
=
2
b -ac.
3, CoNFIDENCE INTERVAL, LIMITATIONS
If
a
0
>
terval for
and
A,
8
>
0,
D
the confidence set (2.14) becomes a confidence in-
namely,
(3.1)
If
(a
<
0,
8
0),
>
(2.14) implies that
A is outside of the interval
(A-, H).
It is easy to show that the case
cases
a
=0
and
8
=0
(a > 0,
8
<
0)
is impossible.
The
are ignored, because the probabilities are zero for
these events.
The case
(a < 0, 8 < 0)
may, however, occur with positive probability.
In this case (2.14) will imply that
-00
S AS
00.
The confidence set for
A,
therefore, as constructed here suffers from the same limitation as the one constructed by PieZZer (1940) for a bivariate sample from a bivariate normal
distribution.
Recently,
Sahefie
(1970) has constructed a confidence set for Fieller's
problem, which is free from this limitation.
Using
Scheff~'s
ideas, it is
6
possible to derive a modified confidence set for out problem, which will not be
improper in Scheffe'e sense.
This will be reported in a subsequent communi-
cation.
4 ILLUSTRATI ON
D
Using the formula (2.14) of this paper, confidence sets were calculated
for the ratio of mean caries increment in two groups, experimental and control,
from data compiled by different investigators and reported in Table II of a
paper by MarthaZer (1970).
In every case, the confidence set resulted in a con-
fidence interval of the type
A-
~
A ~ A+.
Here we give one example based on
data by MarthaZer (1970).
n
n
xx
s
s
c
=
number of subjects in the control group
e
=
number of subjects in the experimental group
c
=
mean caries increment in the control group
e
=
mean caries increment in the experimental group
c
=
standard deviation of the individual caries count in the
control group = 8.55
=
standard deviation of the individual caries count in the
experimental group = 7.59
e
=
=
4.17
=
=
4.08.
=
32
=
=
42
15.25
=
11.33
Let
=
mean caries increment in the experimental group
mean caries increment in the control group
The confidence set for
to
0.95,
is
A then, with a confidence coefficient greater or equal
7
0.5518 S A S 0.9976,
and
x-
= x_e
c
=
0.743.
My sincere thanks are due to Professor A. Linder for drawing my attention
to this problem and stimulating discussions.
I wish to thank, also, Dr. Th.
Mal'thater for letting me see his manuscript,
"Statistical Treatment of Percent-
age Inhibitions of Human Dental Caries Incremental Data".
REFERB'lCES
Pietter, E.C., 1940: The biological standardization of insulin.
Supp1. J. Roy. Stat. Soc., 7, 1-64.
Marthater, Th., 1970:
Statistical treatment of percentage inhibitions of human
dental caries incremental data. (To be published.)
Saheff~,
H., 1970:
Multiple testing versus multiple estimation.
fidence sets. Estimation of directions and ratios.
Ann. Math. Stat., 41, 1-29.
Improper con-
8
SU~f'1ARY
In this article, a confidence set for the patio of the means of two normal
distributions, has been derived without making any assumption on the Patio of
the two vaPianaes.
The construction of the confidence set uses the sample
means and sample variances of two independent random samples from the two distributions.
mation.
The solution is exact and does not use any large sample approxi-
The confidence coefficient, however, is guaranteed to be greater than
or equal to
(I-a)
rather than be exactly equal to
assigned positive number less than
(I-a),
where
a
is a pre-
1.
ZuSAf'IfIENFASSUNG
Ein Konfidenzbereich fUr das VephaZtnis deP
Mitte~epte
von zwei NopmaZ-
vepteiZungen wird abgeleitet, ohne Annanmen Ubep das VephaZtnis dep heiden
VaPianzen zu treffen.
Zur Berechnung des Konfidenzbereiches werden die
Mittelwerte und die Varianzen von zwei
beiden Verteilungen benUtzt.
Ann~herung
unabh~ngigen
Zufallsstichproben aus den
Die Losung ist genau und stellt keine
mittels grosser Stichproben dar.
Vom Konfidenzkoeffizienten kann
jedoch nur behauptet werden, dass er grosser oder gleich, jedoch nicht genau
gleich
als
1.
(I-a)
ist. Hierbei ist
a eine vorgegebene positive Zabl kleiner