ON THE RELATION BETWEEN ESTIMATING EFFICIENCr
AND THE POWER OF TESTS
by
R. M. Sundrum
University of Rangoon
Institute of Statistics, University of North Carolina
Institute of Statistics
Series No. 90
January 1954
M~eograph
1 This research was supported by the United States Air Force,
through the Office of Scientific Research of the Air Research
and Development Command.
UNCLASSIFIED
Security Information
Bibliographical Control Sheet
1. O.A.:
2.
.,.
4.
Institute of Statistics, North Carolina State College of the
University of North Carolina
M.A.:
Office of Scientific Research of the Air Research and Development Command
OoA.:
M.A.:
CIT Report No.4
OSR Technical Note
ON THE RELATION BETWEEN ESTIMATING EFFICIENCY AND THE POWER OF TESTS
(UNCLASSIFIED )
Sundrum, R. M.
5. January, 1954
6.
6
7. None
8. AF 18(600).458
9· ROO No. R.'54.20.8
10. UNCLASSIFIED
11- None
12.
In this paper a condition is derived for the validity of the assumption
that a statistic which has a high efficiency in est~ting an unknown
parameter also gives a powerful test of hypotheses about that parameter.
An example from genetics is given of a failure ot this assumption. A
presumption in tavour ot using the more efficient estimator is estab.
lished by showing that the above condition tails only in situations
where the power of the resulting test is less than i.
On the Relation Between Esttmating Efficiency
and the Power of Tests
by
R. M. Sundrum
University of Rangoon and
Institute of Statistics, University of North Carolina
It appears to be generally assumed that a statistic which has a
high efficiency in estimating an unknown parameter also gives a powertul test of hypotheses about that parameter.
For example, a common
criterion tor comparing distribution-tree tests with corresponding
distributional tests and with other distribution-free tests is the
asymptotic relative efficiency first proposed by Pitman (1948). Recently, Alan Stuart (1954) has shown that this is equivalent to using
the estimating efficiency of these statistics to compare their performance when used in tests ot significance.
In fact, it was by this
argument that this measure of efficiency was earlier arrived at by
Hotelling and Pabst (1936) in their study of the Spearman rank correlation coefficient and by Cochran (1937) in his study of two tests of
the mean and of the correlation coefficient of normally distributed
variables.
In this note, we derive a condition for the validity of
this assumption in the case when the statistics have normal distributiona, a case tor which the idea ot estimating efficiency has most
relevance.
Let t
l
andt be two statistics which are normally distributed,
2
unbiased estimators of a population parameter Q with variances
oi(Qo) and ~(Qo) respectively under the null hypothesis HO: Q = Qoj
and oi(Ql) and a~(Ql) respectively under the alternative hypothesis
- 2 -
holding at least once, so thet t
l
m~y
be called the more efficient
=
1
estimator.
If A. is defined by
a
~(A. )
a
{2it
the one-sided critical region of size a based on t
l
is given by
(l)
and for t
2
by
The power of the critical region (1) is then
and of the critical region (2 ) is
(4)
As ~(x) is a monotonically decreasing function of x, the test based on
t l (the more efficient estimator) is more powerful if
- 3 -
Qo - Ql + >-(10'1 (QO>
0'1 (Ql)
i.e., 1f
(i=1,2).
(5)
If Vo ~ V , .then the right-hand side of (5) is less than or
l
equal to
° and the inequality is satisfied.
But if V1
> V0 ? 1, so
that the relative efficiency of t 1 isGe:e:t;: )under the alternative
than under the null hypothesis, then
> O. As a, the
.
V1 - 1
size of the critical region is always taken less than! in practical
applications, >-a is always positive and can be chosen so large that
l -Va)
"(V
,..~
V
1
0'1(Qo>
> Q1
- Qo
1 -
and
~he
inequality
(5) is reversed.
An example from genetics is given by Fisher ~1950, pp. 314-5
concerning a test for linkage in inheritance of two factors.
Given the
frequencies of four combinations in a sample as a, b, c, and d with
a+b+c+d=n
and the corresponding probabilities of occurrence as
-7
(6)
- 41 1 1
'4
'4
(2 + 0),
4 (1 -
(1 - 0),
0),
1
4
0
the problem is one of estimating Q, when;-g- is the recombination
ratio.
The maximum likelihood method gives a statistic t l as the
positive solution of the equation
2
'
nO - (a - 2b - 2c - d) 0 - 2d
=0
(7)
with a sampling variance
~(t )
= 20(1
- 0)(2 + 0)
(1 + 20) n
1
Another statistic t
2
(8)
may be defined by
t
:2
=a-b-c+5d
2n
(9)
with expectation 0 and sampling variance
(10)
It is easily verified that ~(tl) ~ CJ2(t 2 ).
Now consider a test of HO: 0 = ~
o=
01
1
> '4.
We find V0
=1
(no linkage) against
so that, assuming n large enough
for a normal approxtmation to hold satisfactorily for the distributions
of t l and t 2, and for the bias of the maximum likelihood estimator to
be negligible, the inequality (5) is reversed for all 01 satisfying
1
(°1 - 4)
<
~a:3
4/n
(11)
- 5However, this possibility is not very important in practice.
For if Aa is chosen so as to reverse the inequality (5), the power of
the test will be reduced very much. In fact under the special assumptions of the above argument, the situation in which the more efficient
estimator gives a less powerful test cannot arise, if the level of significance is chosen so as to make the power of that test greater than
~.
This can be seen as follows:
( 6)
From the condition Vl
> Vo ~
1, we have
,
(7)
From (6) and (7) the inequality (5) follows.
.'
.. 6 ..
References
1:1_7 Cochran l W. G. (1937)
J:2J
Jour. R. Stats. Soc." !QQ, 69.
Fisher" R. A. (1950) statistical Methods for Research Workers,
11th ed." Edinburgh" Oliver and Boyd.
L3J
Hotelling" H. and Pabst, M. (1936) Ann. Math. Stats, 1, 29.
f4J Pitman" E. J. G. (1948) "Lectures on Non-Parametric Inference"
University of North Carolina
1:5_7
Stuart" Alan (1954)
Jour. Amer. Stats. Assn"
!t2, ..•