Gertrude M. Cox
SO~lli
USES OF QUASI-RANGES
by
John T. Chu
Department of Statistics
University of North Carolina, Chapel Hill
Special report to the Office of Naval
Research of work at Chapel Hill under
Contract NR 042 031, Project N7··onr284(02), for research in probability
and statistics.
Institute of Statistics
Mimeograph Series No. 124
February 23, 1955
SQIIE USES OF QUASI-RANGES
By
~ohn
1,2
T.Chu, University of North Carolina
1. Summary. Confidence intervals for, and tests of hypotheses about, the difference
of two quantiles ( of the same distribution ) are obtained, usine one or two
properly chosen quasi-ranges.
proved.
Consistency ( of the estimates ano tests ) is
Applications are discussed to the standard deviation of a normal
distribution and the parameter of a binomial distribution.
2. Introduction. Let a population ( not necessarily continuous ) be given with
cdf ( cumulative distribution function) F(x).
~
p
For a given p, 0< p < 1, any
satisfying
(1)
F( ~
P
- 0 ) < P
= <;= F(
~
P
)
is called a quantile of order p ( or p-quantile ) of the given distribution.
Usually one of them is chosen, by one way or another, as the
Let
~q
be the
is drawn and x
q-quantile, where
r
and x
order of magnitude).
s
p < q < 1.
p-quantile.
Suppose that a sample of size n
are the r-th and s-th order statistics ( in ascending
Intuitive17, it seems obvious that if n is sufficiently
large, and r is small compared with n while s is not much less than n, then
X
- x r is likely to be relatively large, consequently the event
~q
-
s
~p
occurs with probability close to 1.
: for given 0
<;
p < q < 1, and 0
<;
a
<;
s - xr
X
~
An interesting question then arises
1, what would be a nice way, assuming
there exists at least one way, for choosing two order statistics x
r
and x
s
from a sample of given size n, such that
(2)
P( x s - xr >= ~ q - ~ p ) ~ 1 - a,
where P(E) denotes the probability of the event E?
One may also ask similar
questions concerning
,.
lSponsored by the Office of Naval Research under Contract NR 042 031.
2Presented at the 1954 Annual meeting of the Institute of Mathematical Statistics.
-2p(
(2 1 )
X
S'
-xr
P( x
-x
Sf
f
<1;
=
q
-1;
P
» l . a , and
=
-1; <x -x ) > 1 - 2a.
q
p = s
r
=
These questions, of course, are rather loosely stated. For example, the meaning
r
f
<1;
=
of the word " nice " is not specified.
Besides, there are at least two different
approaches to the problems, namely: parametric and non-parametric.
in a sample of size n,
Incidentally,
x n .. xl .is known as the range, and xn-r+ 1 - x r , where
1 < r ~ ( n + 1 ) / 2, a quasi-range or the r-th range, e.g., ~4_7.
Here, how-
ever, any x s - xr , s ?_ r, will be called a quasi-range.
In this paper, an attempt is made to obtain some distributionfree methods, optimal in a certain sense, of choosing r,s,r l ,and
(2) and (2 1 ) hold for given p,q,a, and n.
(6),
Sl
such that
In section 3, lower bounds (4), (5),
(14), and (15), are obtained for the probabilities given in (2) and (2 1 ).
Corresponding to the bounds (14) and (15), two integers k and kl are defined by
(19) and they can be found by using
(5.7.
If r,
5,
r l , and
51
are defined by
(8) and (9) in terms of these k and k ' , then the corresponding order statistics
satisfy (2) and (2').
'When the parent distribution meets certain continuity
requirements, we are able to show ( section 3, Theorem ) that both x
- x and
s
r
x s ' - xrl ' chosen in the way just described, are consistent estimates of ~q -
~p
and provide consistent tests with respect to various hypotheses and alternatives
concerning
~
q
-
~
p
•
2
For a normal parent distribution with variance (] , if we
choose q
=1
- p , then
defined by (17).
~
q
-
~
= 200 , where
{(-a)
p.
= p , and I ( x ) is
To obtain, e.g., a 100 ( 1 - 2a ) per cent confidence interval
for (], one proceeds as follows:
Take any ,p
(0 < p < 1 ).
Let
wr -= (xn-r+ 1 - x r ) / 2a,
wr 1 = (xn-r '1
+ - xr , ) / 2a,
where I(-a) = p and r and r' are respectively the largest and smallest integers
(3)
which satisfy (26) with
q'" 1 - p.
Then p( w , < (] <
r
-
=
W
r
)
> 1 - 2ci.
=
Here, a
question follows naturally: p is taken arbitrarily, but, does the value of p
-3effect the efficiencies of the corresponding estimates and tests ?; and if so,
how to choose p to maxlluze the efficiencies? To these questions, some answers
are given in section 5.
The efficiencies of w and .wrt relative to S' of (29),
r
are defined by (30) in the conventional way, i.e., as the ratios of the variance
of 8 '
to the variances of w (j/E(w ) and w
r
r
r
,a/ E(wr ,)
which
are unbiased
estimates of a.
However, exact values of the efficiencies of w and wrf are
r
hard to obtain, because the variances of w and w r usually cannot be found
r
without very laborious computations.
r
For large samples, a method of approximation
is suggested for obtaining the expectations and variances of w and w , and
r
r'
the efficiency of w ( and similarly w,) is approximated by the asymptotic
r.
r
)/2a l , where ~f'
= rnpl 7' + 1,
efficiency of ( x ,-
X I
~
/ n -< pI < r/n, q'
1 - pt, and ~(-at) == pl.
v
wrl
are about
==
.65.
~
-
vI ==
rnqJ
7+ 1,(r
--
Maximum efficiencies of w
- 1 )
and
r
A method is suggested of finding, for given n and a, the
values of p which maximize respectively the efficiencies of w
r
and
Table
2 in section 5 is given for illustration.
For samples of moderate sizes, we made no attempt to solve
the problem of " how to find P ".
There is evidence, hOl1ever, that if p
is
properly chosen, the efficiencies of the corresponding wr and wrl are reasonably high. We find, e.g., for sample sizes around 20 and 30, if p == .25, the
efficiencies of the corresponding w and w are about .70 ( Tables 3 and 4 ).
r
r'
In section 6, we discuss_some applications to the parameter
Q of a binomial distribution f(x)
= Ql
tests are obtained for testing p
Q ~
tives Q < P or Q > q, and
known one
["7,
p.
14_7,
Q
f
~
p.
- x ( 1 _ Q )x, x
q and
Q
For testing
= 0, 1. Consistent
= P with respect to the alternaQ == P , our method improves a
in making unique the choice of critical region and the test
consistent with respect to Q ~ p.
Finally He note that the idea is
by
no means new of using
quasi-ranges, as well as sDuilar statistics, in estimations and testing of hypothese~.
In fact, much work has been done along these lines. For references, see
[1_7 , (4_7, and those cited there.
·43.Consistency.
Let a population be given with cd!
F(x).
O' < P < q < 1 , the corresponding quantUes of orders
are uniquely defined.
~
Let xl
x
~
2
••• ~ x
ponding to a random sample of given size
If
Lemma 1.
(4)
=1
L
- P( x
<;
s
~
)
q
1
r
p( x
<;
(5)
L
=1
- p(
X
s
> l::'q ) - p( x
f
""
p(
<;
=
,
(6)
L + L
- 1
p( x
<;
=
occurrence of the events
L
<;
=
- x
_ sl
Proof.
p( x
> ~
p( x
-
=
f
= r <;= r'
X
t
s
<;
<;
f
<;
=
~
t
q
~p
,
and t q ,
are integers, then
• xr =
> ~ - ~ )
q
p
) + p( x <; ~ ) = U ;
p( x
<;
=
s
r
l::'
)
""p
<;
='
p( x
) + p(
X
x
- x
t
-
<; S <; n
=
=
~:
)
q
and
Then,
<; Sl
> t
= q
s
r
p
p
be the order statistics corres-
n
n.
<;
Suppose that for given
<;
r
p
_ x:
sIr'
t
> ~
)
<; l::"
_
= ""q
l::"
""p
)
1
):: U ;
p
<;
J
U ... U
<N
1.
- ~
) >
rJ = q
p = s
r
=
Let p( A, B ) denote the probability of simultaneous
A and B.
Clearly p( x
s
- x
r
> ~
=
q
p
=
> ~ , x <; ~ ) > p( x > ~ ) ... PC x <; ~ ) - 1. Therefore we have
s=q
r=p'"
.s=q
r=p
p( x - r > ~ - ~ ) . Likewise we obtain the other inequalities.
s
r= q
p
For samples drawn from a binomial distribution, say
Remark.
with pdf ( prohability densit-,r fl.ll1ction)
i f we define
~p
=0
and
~q::
f(x)
= pI
- x ( 1 _ p )x , x ". 0, 1,
1 , then the lower bounds in Lemma 1 are actually
attained.
Lemma 2.
For all integers
k
and
kl
0;; k, k I ~ n - 1,
;
and
c
= q - p ,
choose
r
r
where
t
=~ (
n - k ) p / ( 1 - c
=~ (
n -
,
k ) p /
(
1 • c
[a.? denotes the integral part of a.
).7 + 1,
)_7 + 1,
non-decreasing and non-increasing functions of
c
2
be given such that
0
<;
01
<;
c
<;
c
2
Sl
=r
+k ;
=rl
+ k' ,
Then the corresponding RHS ( right
I
hand sides) of (14) and. (15), 10lfer bounds for
and
s
<;
L and L , are respectively
k
1.
and
If
k1.Further, let c
k = [n0
.7
2
and
k
l
1
-5- [nc1 •..7
'
then
(10)
Lim
L
~
00
=
Lim
LI
n-->
..,.
:t.
00
t:
I
On the other hand, if k :: [" nc1 _7' and k ::
nc 2_7
parent cdf F(x) is continuous at X= l;
and x = ~
p
(11)
Lim U ::
n-> 00
q
,
and, if, in addition, the
, then
I
Lim
U:: O.
n~
00
t
t
Proof. It can be seen that 1 ~ r, r ~ n and 1_~ s, s
;; n , e. g., s ~ n because s < ( n - k ) p / ( 1 - c) .. k + 1 < n + 1. Hence
I
L and L, as well as the RES of (14) and (15), are well defined functions of k
I
and
k.
Now
(12)
p( x s <
t'
~q
)
= 1-
B ( s - 1, F(
n
t'
-
~q
0 ) ) =
< 1- Bn (s - 1,q).
where
n( r, p ) =
(13)
e
B
r
~
i =
°
is, for fixed n and r, a decreasine function of p, 0 < p < 1.
(14)
Hence
Bn (s - 1 , q ) - Bn (r - 1 , P ) ;
I
t
t
L > - B (s - 1 , q ) + B ( r - 1 , p )
(15)
n
n
=
Now r is a non-increasing function of k. But i f k is increased by 1
L>
::
decreased at most
by
1.
.
Hence
quent1y so is the RES of (14).
s
, r
is
is a non-decreasing function of k , conse-
In a silnilar way, we show that the RHS of (15) is
t
a non-increasing function of k.
It is well knm~n, ["2, p. 200_/ and ~3, p. 193_7 that,
as
n tends to
00,
(16)
B ( r, p ) n
uniformly in r
I(
n tends to
is fixed, and
-->
, 0 :; r :;: n, where x:: ( r - np )
(17)
As
I( x)
1.- np
( 1 - p ) _7 - 1/2
2
x) ::
and
-
( 2n ) -1/2 e -t /2 dt •
it can be shmin that if k:: ["nc2_7
and s are defineu by (8),
00,
r
0,
,
where
c2 > c
and < 1
-6(18)
for
( r - 1 - np ) / n
( s - I-nq ) / n
= ( c
-> _ 00,
n-1/2 ( r _ 1 _ np)
2
... ( c - c
1
+ o( n- )
=b
2
b
1 - q ) / ( 1 - c ) > O.
- c ) (
Lim L:: 1.
n-> 00
... o( n-1)
) p / ( 1 - c)
where
->
n-1/2 ( s - 1 - nq)
=(
1 - c
2
and
) p / ( 1 - c)
+ c
2 - q
(16), and (18), we obtain
Combining (14),
Likewise we prove the rest of (10) and (11).
Lemma 3.
k
(k
t
= the
)
Corresponding to given
n
and
a, 0 < a < 1, let
least ( ~reatest ) integer between
t
s
are defined accordingly
~ 1 -
by (8) and (9), the RHS of (14) and (15)
.
( From Lennna 2, such
and
k
For fixed
k
Pi and
t
and n - 1
0
,
such that, when r, s, r , and
ly large ).
00,
-
a.
exist for any a, 0 < a < 1 , i f n is sufficient' i =- 1, 2, where
qi
PI < P < P2 and
ql < q < q2,
define
(20)
= {npi.7
ri
Assume that
l~ge
F(x)
+ 1
is continuous at
x
= (nqi_7 + _1
= ~ P and x = ~ q •
si
,
•
Then, for sufficiently
n,
< s
=
2,
(21)
t
where
r, s, r , and
s'
are defined by (8) and (9) with
Proof.
k ~ ~nc2_7 for any fixed
Choose
c
2
By
~ nP1 + 1 ~ r 1
and
s ~ n [
n
k
I
given by (19).
is sufficiently large.
r ~ n ( 1 - c
c, then
2
( 1 - c 2 ) p / ( 1 - c ) + c2
Similarly following (11), we have
and
(10), if k is the integer defined by (19), then
c 2 > c , provided that
sufficiently close to
k
r ~ r2
) P / ( 1 - c )
_7' ~ nq2
~ s2.
and s ;;; sl.
The fol10lTin[~ lemma is a known fact (2, p. 369_7.
We
state it without a proof.
Lemma
F(x)
and pdf
defined and
of x
=~
p
f(x).
4. Let a continuous population be eiven with cdf
Suppose that for
0 < p < q < 1 , ~
p
and 1;.
q
are uniquely
t
f (x), the derivative, exists and is continuous in some neighborhoods
and
x
=~ •
q
If
I-L =
r- np- 7 + 1
and
\l"
r- nq- 7 + 1,
( we assume that
-7np
ancJ
nq
are not integers ) , anc1
statistics in a sample of size
x
and
!J.
n, then as
x
are the corresponding order
\I
n -->
00,
- x
X
\I
ally normal distrihution with mean
~q
-
~p
!J.
has an asymptotic-
and variance
2p ( 1 .. q )
(22)
f(~p) f(~q)
As a consequence of the previous lemmas, we have
Theorem.
and
a
pdf
satisfy the continuity conditions stated in Lemma
k and
, let
by (8) and (9).
a sample of size
I
If
x
x .. x
s
r
n , then both
Lim
(x
n -> 00
and
X
S1
l ,
and
be defined
Sl
r-th etc. order statistics in
etc. are respectively the
r
cdf
4. For given nand
k be the integers defined by (19), and r, s, r
~q" ~p in the sense that
of
Let a continuous population be given whose
..
xrl are consistent estimates
Lim
(x
.. x
)
s .. x r )::: n-->ooSI
r'
=
~q - ~p
in probability.
Proof.
PI and
p( x
q2' and 51 are properly chosen and
- x >
sr..
~ 6' ) ~ e.
4.
Following Lemmas 3 and
4,
for given 6, e > 0 , i f
n is sufficiently large, then
~
• ~ + 6 ) < p( x
.. x > ~ .. ~ + 6 ) < p( x .. x > ~ - ~
q
p
:::
s2
rI
q
P
:::
s2
r1
q2
PI
In a similar way we easily complete the proof.
Inference.
In section 3 we proved among other things the existence of
k
and
kt ~ defined by (19), for sufficiently large
k
and
k' , there is no difficulty either.
be evaluated by means of ~5..7,
(23)
I
where
P
(r, s ) =
k
As is well known, binomial
J
J
0
Therefore for a given
k' , defined by (19) and the corresponding
by (8) and (9).
cdfs can
through the relationship
0
and
To actually find such
B ( r - I , p ) = I - I ( r , n .. r + 1 ) ,
n
p
00
p
r
r
x - I ( I .. x ) s - ldx
x - I ( I - x ) s - ldxl
is the incomplete beta function.
find
n.
Of course, for a small
a
and
n, we can easily
r, s , r
a, in order that such
k
l ,
and
and
s' defined
kl
exist,
-8~ar_g;p.!.;,
n hos to be
.
', ..
;....
~
~. Confidence ;.n,tervals-.
.
..;'.
_.
When "~'1~'f,'"
cribed, x s '" xr
bounds for
( :rSf
...
and r sf - X;:r'
~q'" ~p
xr !
, X
rnd
arechosen in the way previously des-
Sl
are- respectively confidence upper and lower
with the same confidence coefficient ]. ..
cdf and pdf, then both x
estimates of
and
s - X'r ) is a confidence interval with confidence coefficient 1 - 2a.
4,
If, in addition, the continuity conditions, dtated in Lemma
the parent
em ,
~q - ~p
.. x
s
and x r - x.
s
r
r
I
are satisfied by
are consistent
•
B. Tests of hypotheses.
Let
(24)
Then the tests, using as critical regions:
,•
-x <d •
%
s
r
'
X's - xr < d or X r - X r • > d ,
are respectively: 1. of significance levels a, a, 2a ; and 2. consistent with
s
, respect to the alternatives
~
q .. ~ p < d;
~
q .. ~ p > d ; and ~ q .. ~ p
provided that the continuity conditions in Lemma
4 are
satisfied.
I
d ,
A test, for
testing a given
hypothesis H0 , is said to be consistent with respect to a certain
.
alternative, if its power, when the alternative is true, tends to 1
as
sample
size tends to infinity.
C.
A special case:
q
=1
- p
0
If, in par'c,icular, q = 1 -p, then, fC1110wing (8) and (9),
r
= L"
( n .. k
for which
(25)
) / 2_7 + 1, s = r + k , etc..
If we use only those
k and k'
(n.. k )/2 and (n - k r ) /2 are not integers, then
s
~
n .. r + 1 , and
Sl
=n
.. r
l
+ 1 •
From (14) and (15), it follows that L > 1 .. 2I ( n .. r + 1 , r) and
.
=
q
1
L ~ 2Iq< n .. r t + 1 , r' ) - 1. For a given n, we use [5_7 to find the
largest r
and the smallest integer r'
for which
•
-9..
I q ( n - r + 1 , r ) ~ 0./2
(26)
,
Iq ( n f
Then the corresponding L and Lf ~ 1 - a. •
For example: i f n = ,30
in this way.
largest r
~~~~75 - ~:25 ) ~ .95 •
~ .95 •
~8 - ~ )
t
and q ::
satisfying the first inequality in (26).
P( x 28 - Xj
r
+ 1
r
I
l
)
> 1 - 0./2 •
Table .3 in section 5 is obtained
= .05 ,
a
I
-
.75 ,
then .3
From (25), s
is the
= 28.
.
'!hus
-
Likewise, p( ~7 - x l 4 ~ ~. 75 - ~ .25 ~
For large n, a fairly good approximation for Bn(r,p)
case, the largest integer
1.( ( r
7
mere !.(x) is given by (17) • In this
f for which
and smallest integer
holds are
+ 1/2 - np )/ rnp( 1 - P )
the largest integer r
r
is
1/2 )
(26)
r
and smallest r l
satisfying
r
r
r <
1/2 + np - aa- np ( 1 - P ) ..71/ 2 , r' >= 1/2 + np ... aa- np( 1 - P ) - 71/~
2'
where !( -aa ) ... a/2. Table 2 in section 5 is computed
_ in this way. We see that,
(27)
¥
with the exception of a few cases, r' - 2np ... 1 - r •
5. The standard deviation of a normal distribution.
Suppose that the parent distribution is normal with mean
and variance
(J
2
Then, for each p, 0 < p < 1 , and q
•
(28)
where
a
=p
~(-a)
= ( ~q
=1
~
- p ,
- ~p )/ 2a ,
and I(x) is given by (17).
Obviously, as was pointed out
in
section 2 , confidence intervals for, and tests of hypotheses about a, can be
readily obtained from those concerning
~
q
-
~
p
• A new problem arises, however,
as to how to choose p to maximize the efficiencies
of the corresponding wr
and wrt in (,3), since they, for given n and a, are determined by p. We
shall now consider this problem.
A f~~iliar statistic used to estimate and test
n _ 1 )1/2 r( n...;..l )
2
~
S ,
r( n/2 )
n
2 n
2
S = 1: (xi - i ) / ( n - 1 ) and i = 1:
(J
is
f
S = (
where
i
ciency of w
r
=1
i
xi/n •
The relative effi-
=1
(and similarly w ' ) with respect to Sf is defined to be
r
·10-
()o)
The variance of
[2, p.
484.7.
I
100 var S t
E1: -
var
[w
(J!E(w )
7 •
r is easy to compute and for large n
St
The variances of
wI'
and
r
wI'''
is about
for arbitrary n,
(J
2
I
2n
and
I' ,
are, on the contrary, difficult to obtain, ( Recently general formulas
r'
have
213 7 ), consequently so is the value of e
sample size is large, and ( r . 1 ) I n and rln are fairly close to
been derived by Ruben
1rJhen the
;-6, P.
E(Wr ) ,
each other, we suggest the follOWing method of approximation for obtaining
var w
,etc..
r
In section 3, Lemma
and ~,= ~np'_7
1 and
+
VI
is asymptotically normal.
4,
0 < pI < ql < 1 ,
it was stated that if
= [nq' ..7 +
1 , then
the distribution of x
For large samples, the mean and
V
I
-
X
IJ.
I
variance of the asymp-
totic distribution may be used as approximations to the true mean and variance of
x , - X I . Now if p' and
v
IJ.
q I "" 1 - p' , then x ... x,
the denominator of
()o)
(x
,
.. x
)/2a l
!J.'
~(-a') ... pl.
where
at2
e
is ,follouing
~(a')
If
var S'
Lermna
4,
2
¢(x) ... (2n)-1/2 e- x /2 ,
/ pI ( 1 _ 2pl ) , where
nwnerical investigations, e.g.,
w
I'
and
at
pI .... 07
{4.7 ,
approximately.
is maximized i f p
a:, this value of
to
is replaced by ,i/2n,
which, in fact, is the asymptotic efficiency of
.65
and
and x
1 = X I • Therefore am approximation
n-r+
v
is the variance of the asymptotic distribution of
then an approximation to
(31)
1 ) / n < p r < I' /n
(I' -
IJ.
I'
v'
are such that
qI
(x I - X , )
v
!J.
revealed that (31) has
Therefore, for large
is so chosen that
p
r/n
n
is about
2a l
/
a
Previous
•
ms.."Cimum value
, the efficiency of
.01n
•
For given
can be found approximately by equating to
.01n
n
the
RHS of the first inequality in (27) and then solving a quadratic equation in p
In a similar way we can find the optimal p with respect to wrl • Obviously,
values of
p which maximize the efficiencies
larger than
Houever, as
of the corresponding
.01, and those maxir.lize the efficiencies
sample size increases, both
r/n
and
W
I'
's
are
of the 'tvrl's , smaller.
rl/n
tend to
p
(Lemma 3 ).
-11Therefore eventually the efficiencies of w and wrt are maxilrlized if P'" .01.
r
In the following, Table 1 gives, for different values of p,
the asymptotic efficiencies . (31) of (x\I .. x_~ )/ 2a
\I ...
(nq.7 + 1 , q ... 1 .. P
imum value
J
= p.
and ::I(-a)
.65 occurs at P'" .01.
J
where ~
= ~np - 7 + 1
J
As was mentioned before, the max-
In the neighborhood of P" .01 , the
variation is small of the corresponding efficiency.
In Table 2 are given, fer
different choices of n and a, the values of p, and corresponding to each p,
and r t
the subscripts r
of w
r
and w
r
They are not found, however, by
t.
the method of solving a quadratic equation mentioned in the previous paragraph.
Instead, four different values of p are chosen , then for each p, the corresponding r
ani r
t
are found by (21).
For most combinations of n and a
given in that table, at least one of the four
with an efficiency fairly close to
p( ( x 64 .. x
) / ~.56~;;; (] ) ~
31
4
statistic is about .65.
.65.
:9S, ( p
p's will provide a wr
For example, if n" 500, then
....10 ) , and the efficiency of the
For samples of moderate sizes, Cadwell ~1, p. 609.7
obtained recently the efficiencies e
(0), for
n .. 10, 20, ••• , 60 , and r
:;;z
1, 2, 3 • Using his results, we see ( Tables 3 and 4 ) that if p = .25 ,then
the corresponding w 's have efficiencies around •10 for n
r
Table 1.
Aymptotic efficiency of
= 20,
( x \I ..x )/ 2a
IJ.
P
.01
.02
.03
.04
.05
.06
.01
.AE
.39
.51
.58
.62
.64
.65
.65
p
.08
.09
.10
.ll
.12
.13
.14
.AE
.65
.64
.63
.62
.61
.59
.51
30•
12
Table 2 The Largest r and smallest r'
p
2a
1000
500
.10
.25
n ""'a
= .04, .07, .10, and .13
= 3.50, 2.95, 2.56, and 2.25
.01
.025
.05
.005
33
48
30
51
28
53
26
55
24
57
23
58
61
80
57
84
54
87
52
89
49
92
47
94
89 112
84 117
81 120
79· 122
76 125
73 128
118 143
113 148
109 152
106 155
103 158
100 161
15
26
13
28
11
30
10
31
9
32
8
33
28
43
26
45
24
47
22
49
20
51
19
52
42
59
39
62
37
64
35
66
33
68
31
70
56
75
53
78
50
81
48
83
46
85
44
87
5
12
3
14
2
15
1
16
0
17
10
19
8
21
'7
22
6
23
5
24
4
25
15
26
13
28
12
29
10
31
9
32
8
33
21
32
18
35
17
36
15
38
14
39
13
40
2
7
1
8
0
9
0
9
0
10
0
11
4
11
3
12
2
13
1
14
0
15
0
15
7
14
5
16
4
17
3
18
2
19
2
19
9
18
7
20
6
21
5
22
4
23
3
24
200
100
n
= Sample
size
3 14
a = Level of significance
13
Table 3.
The Largest r and Smallest r'
= .25
p
n~Q
2a
.10
.25
= 1.35
.01
.025
.05
.005
10
1
5
20
3
8
2
9
30
5
11
4
13
3
13
3
14
2
15
2
40
7
14
6
16
5
17
4
17
4
18
3
19
50
9
17
8
19
7
20
6
21
5
22
5
23
Table 4.
2 10
1
1
Efficiencies of the Statistics
(Xn _r+l - xr ) / 2a in Table 3
n~a.
.25
.10
.05
.025
.01
.005
10
.85
*
*
*
*
20
.66
.73
.73
.70
.70
*
*
30
*
*
.70
.70
.70
.70
40
*
*
*
*
*
.69
-146. The parameter of a
binomia~istribution.
We shall..consider a practical example.
To classify lots of, say, factory products, according to the
proportion
e
of defectives contained therein, the following way might be of
It
practical interest.
~
> q , a prescribed quantity, then the lot will be
rejected for containing too many defectives.
If p
~
e
~
q , where p < q is
another given quantity, then the lot will be accepted as of average quality.
if
Q
< p, then the lot will be considereCl as superior ( and probably will be sold
at a higher price).
i.e.,
But
Q < P ;
p <
While the problem, to decide between three alternatives,
= Q =<
q ; and
Q
> q , is actually a
3-c'ecision one, we may ,
perhaps as a preliminary approach , start with testing the hypothesis
p <
< q.
=Q =
H:
(34)
o
Corresponding to a random observation taken from a given lot, let x
=1
is found defective, and x
o,
1 , where
f(x)
if good.
.
=0
and S
~
=q1 .
in a sample of size
Let xr
xs
and
n, then
lity of E given that H
3, a pair of integers
r
=
•
be the
r-th and s-th order statistics
p(x s -xr::
>~ -~ IH )=p(x =l,x =0
q
pas
r
is true.
o
and s
p( E I H
) , where
0
) denotes the probabi-
Following the general method given in section
can be determined, (8) and (19) , for given n
such that
or x => 1
r
Therefore the test using as critical region
(35)
p( x
(36)
xs = 0
where r
x
is true, we may define, following (1) ,
0
I Ho
) = 1 - p( x - 0 or x = 1 I H
s.
r
. a
and a
Then f(x):: Q 1 - x ( 1 _ Q )x ;
if it
denotes the probability of obtaining an observation x
For this distribution, whenever H
~
sp
=0
and s
s
= 0
or x
r
=>
1
H
0
)
< a •
=
,
are given by (35) , is of level of significance
a
for testing
We sho1"J'ed ( section 3, Lemma 3 ) that for arbitrary but
-15-
I1.
fixed
< p
and q2 > q , it r~ ... [nP-r _7 + 1
in (35).
=:
(nq2
_7
+ 1 , then,
and 6 ~ 6 ' where r and s are given
2
1
Using (16) , we easily show that the test defined by (36) is consistent
when n is sufficiently large,
with respect to the alternative
r
~
and s2
Q<
r
P or g > q •
The hypothesis
may be considered as a limitine:: case of (34) as
To test H at a pre1
scribed level of significance a, we use critical region (36) with 6 s r + k ,
and
r = [ ( n . k ) p_7 + 1 , where
(38)
q -> p.
is the least integer for which
k
Bn ( s - 1 , P ) - Bn ( r - 1 , p ) >
1 - a •
=:
If, in particular,
P'" 1/2 , then, from (25) and (26) , s
=n
- r + 1 and r
is
the largest integer for which B ( r - 1 , 1/2 ) ~ a/2. It should be noted that
n
the tests just suggested are not new ["7, p. 14_7. However, for testing HI when
p
f
1/2 , according to the knOlTn method, one chooses a pair of r
and
s which
satisfy (38) and uses the corresponding region (36) as critical region; and if
there are more than one pair of r
be made
and
s
satisfying (38), then selection should
"in accordance with practical consideration"
1:7,
p.l5.7.
removes this 60mewhat ambiguous way of selection, since r , s
uniquely determined for given n, p , and a.
is consistent with respect to the alternative
See
{7.7,
Our method
and k are
Further, the test thus obtained
Q
f
p •
for applications to interval estimations
testing of hypotheses of quantiles of an unknown distribution.
and
-16References
["1_7
Cadwell, J.H .. ,"The distribution of quasi-ranges in samples from a
normal population ", Ann. Math. Stat. , Vol. 24 ( 1953 ),
pp. 603 - 613 •
[2_7
Cr~r, H., Mathematical Methods of Statistics, Princeton University
Press, 1946 ..
["3_7 r£vy, P. ,
[4_7
Calcul des probabilites, Paris •
Mosteller, F., " On some useful" inefficient" statistics ", Ann. Math.
-stat., Vol. 11 ( 1946 ) , pp. 311 - 408 •
[5_7
Pearson, K., Tables of the Incomplete Beta Function, The Biometrika
Office, University College, London, 1934.
[6_7
Ruben, H .,
On
II
the moments of order statistics in samples from normal
populations ",
[7_7
Wilks, S.S.,
II
Biometr!~,
Order statistics
It,
( 1948 ) , pp. 6 - 50.
Vol. 41 ( 1954 ) , pp. 200 - 221.
Bull .. A.rner. Hath. Soc. , Vol. 54