c,"_.. .
UNIVERSITY ()F NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
II!
Mathematical Sciences Directorate
Air Force Office of Scientific Research
Washington 25, D. c.
AFOSR Report No.
THE ASYMPTOTIC POWER OF THE KOIMOGOROV TESTS
OF GOODNESS OF FIT
by
Dana Quade
<~- e
..
"'-:-1
/
'
December, 1959
Contract No. AF 49(638)-261
The asymptotic power of the one-sided and two-sided
Kolmogorov tests of goodness of fit of a hypothesis
distribution H(x) against sequences of alternatives
Gn(x) for 'which s~ JnIH(x)-Gn(x)I tends to a limit
is investigated by application of Doobls "heuristic
procedure"; bounds on the power are found, and some
numerical examples provided.
Qualified requestors may obtain copies of this report from the
ASTIA Document Service Center, Arlington Hall Station, Arlington
12, Virginia. Department of Defense contractors must be established for ASTIA services, or have their "need-to-'know" certified by the cognizant military agency of their project or cnntract.
~'!'1
,I
Institute of Statistics
Mimeograph Series No. 243
1i
•
TABLE OF CONTENTS
iii
ACKNOWLEDGEMENTS
•
I am greatly indebted to my adviser, Professor Wassily Hoeffding,
for his unfailing guidance and encouragement, especially in those
dark days when after interminable whirlings my brain-wheels had
ground out nothing at all; to the Air Force Office of Scientific
Research (which is probably not getting its money's
worth~)
for
financial aid, and also to the National Science Foundation for
-enabling me to attend the 1958 Summer Statistical Institute; to
•
Joan Castle and Helen Jane Wettach of the Instttute of Natural
e
Science for devotion far beyond the call of duty in translating my
abominable typescript into acceptable form; to Martha Jordan,
departmental secretary, for unerringly pointing my way through the
tangled skeins of administrative red tape, past troublesome
deadlines, and around various other pratfalls; and finally, to any
other source of assistance whose name may have slipped from my
mind but shouldn't have.
iv
SUMMARY 1
•
Let there be a random sample of size n from some unknown distribution function F(x), let Fn (x) be the empirical distribution function
of the sample, and let H(x) be the null-hypothesis distribution.
Then
the one-sided Kolmogorov test rejects H(x) for large values of
Dn+
= sUP/n-LF
x
n (x) - H(x)J, and the two-sided Kolmogorov test rejects
H(x) for large values of Dn = s~Pln IFn(X) - H(x)1 • In this paper
we investigate the asymptotic power of these tests against sequences
.
..
of alternatives G (x) for which 11m sup jn- /H(X) - G (x)1 exists.
n
n-tco x .
n
In particular, we extend Donsker's justification of Doob's "heuristic
procedure" as applied to this problem; we find upper and lower bounds
on the asymptotic power; and we provide some illustrative numerical
examples for the case where G (x) consists in a translation of H(x).
n
1. This research was supported by the United States Air Force
through the Air Force Office of Scientific Research of the Air Research
and Development Command, under Contract No. AF 119(638)-261. Reproduction
in whole or in part is permitted for any purpose of the United States
Government.
CHAPTER I
THE KOLMOGOROV TESTS
Let Jrbe the class of all univariate distribution functions F(x)
on the real line, and j* the subclass of all continuous F(x).
given any F and G in
J, define
(1.1)
=
D( F G)
'
-00
I
Then,
I
F( x) - G( x)
<sup
x < co
The functional D(F,G) obeys the relations
.
,,..
(1.2)
e
a)
D(F,F)
=
b)
D(F,G)
= D(G,F)
c)
D(F,G) < D(F,H) + D(H,G)
0
for all F, G, H€ l' and is therefore a true distance between distribution functions.
0 ~ D(F,G) ~ 1 and
Furthermore,
D(F,G)
=
0
if and only if F{X)
We may define also a "directed ll or
1I
= G{x)
0ne- sided" distance
(1.4)
then
And let
(~,x2"."Xn) be
distribution F€
t.
c(x).
a random sample of size n from some
If
{~
x
> 0
x < 0
,
2
then the random distriuution function
1
(1.6)
= -
n
l:
n J.=
. 1
c(x-x.)
J.
is called the empirical distribution function corresponding to the
sample; it is the proportion of the sample which is less than or equal
to x.
As the sample size increases, the empirical distribution
fupction approaches the true distribution function; in particular, the
Glivenko-Cantelli lemma
(1. 7)
U
[12-71 states that with probability
lim D(F F)
-+ 00
n'
=
1
O.
Now suppose that we desire to test a hypothesis
(1.J)
F(x)
"
where
•
the following procedure:
H€f is
_ H(x)
completely specified.
Then it seems reasonable to adopt
Reject HO if and only if Dn >
~
where
(1.10)
and ~
(1.11)
Dn
=- ~(ex)
=
in D(Fn,H)
=
-00
<s~p<
00
jn
I
Fn(X) - H(x)
is determined so that the Type I error is
ex = 1 -
In(~)
=
pr {Dn >
~
1
This test was first proposed by Kolmogorov in
(F = H)
L14J.
Alternatively,
we may
(1.12)
Reject HO if and only if D~ > R
n
n
,
where
..
1
The numbers in square brackets refer to the bibliography.
3
where
and Rn
= Rn(a:) is also determined so that the Type I error is
(1.14)
= 1 -
a:
J+(R)
n n
= pr
f
D+
n
> Rn
1
(F
=
H)
Since the test (1.12) is based on the one-sided distance we call it the
one-sided Kolmogorov test, thus distinguishing it from the two-sided
Kolmogorov test (1.9).
For a fuller
e~ository
treatment of these tests
than can be given in this introductory chapter we refer to the paper
by Darling
{7J which
also includes an extensive bibliography.
The two Kolmogorov tests are distribution-free in
!*;
that is, the
distribution functions I n and I n+ do not depend on H so long as E€
Even if
Et 1* the
J*.
tests are conservative, for
(F = HE
r)
and
(F
= HE 3')
In his original paper Kolmogorov gave recurrence relations suitable for
calculati ng J (y) and proved that
n
y>O
J(Y)
In
lim
= n-+oo
LIUJ Smirnov showed that
o
y~O.
4
where n* is the greatest integer not greater
J~(Y) =
(1.16)
but earlier
than n(l-L), if 0 < Y < in
jn
L17-7 he
o
if
y:s
1
if Y ~ /n
0
;
had found the limiting distribution
1 - e
lim
(1.17)
In+(Y)
n~co
_ 2
2y
y>O
=
o
1/
Tables of these distributions are available; for I
LI9-7,
and for J~ see
determine
~,
1J-7.
n
see
L2-7,
for J see
Thus, given any ex , 0 < ex < 1, we can
R , and also Q and R by the equations
n
(1.18)
where Q
=
lim
n --)- co
~andR
=
lim
n~co
Let an alternative hypothesis to HO be
11.:
where G€ f and D(G,H) > 0
F(x)
==
G(x)
,
Then the power of the two-sided Kolmogorov
test of H against G (in more usual terminology,
"of HO against
is
(1.20 )
Similarly, the power of the one-sided test of H against G is
11." )
5
(1.21)
(F
Furthermore, if
=
G)
fGn1 is any sequence of alternative distributions,
we define the asymptotic power of the two-sided and one-sided tests of
H against
fGn1 to be
(1.22)
and
lim
(1.23)
n~co
respectively, if these limits exist; otherwise the asymptotic power is
not defined.
•
Suppose that Gn
= G, n = 1,2, •••
Then it is not
difficult to see, from (1.7) and (1.2c), that D(Fn ,H)~D(G,H) > 0
with probability 1.
1
and hence pr {Dn > ~ } ~l, or
n
In other words, the two-sided Kolmogorov test
But
Q -+ Q
P(H, ( Gn
,cr:) = 1.
is consistent against every fixed alternative distribution.
Similarly
we can see that the one-sided test is consistent against every alternative stlchthat D:(G,H) >
0."
jnthia paper it will be our purpose to investigate the asymptotic
power against sequences of alternatives which converge to the hypothesis
distribution; specifically, we shall restrict our attention to the case
where (H,
{Gn }
) belongs to the class
H€
1* ;
Gn €
t*,
jand nlim
n D(H,Gn ) =
-,H);,)
n
~
= 1,2,... ;
exists }
However, in order to do this it will be convenient first to re-express
6
these definitions in terms of a new notation which is explained in the
next paragraph.
For any F6 3'* define
F(x)
=
t}
O<t<l
=
lim
-l()
l>t~lF t
t = 1
note that
(1.26)
O<t<l
And let the stochastic process [Zn(t), teT)
(1.27)
Zn(t)
=
In
be defined by
(FnL~l(t)-' - t), teT
where T is the closed interval
jy,lJ.
,
Then all probability statements
about Zn(t) are entirely independent of F, so long as FE
f"*.
In par-
ticular, it is easy to verify that
a)
(1.28 )
b)
c)
Zn (0)
=
E{In(t)J
Zn(l)
=
=
o with
probability 1
0
covL.Zn(t), Zn(t f
)J =
min (t,t') - ttl
Next, if aCt) and bet) are any two functions defined for tET, and
,«
n
is any subset of T, we define the event
Then it is easy to see that
7
...
where A(x)
=....! a{ff(x)J
+ F(x)
in
and B(x)
=....!
In
bLF(x)J + F(x) .
Now if
.
F(X)€.fl but A(x) ~ ~ ~ B(x) is false
~
J
and
yES.
~
~
x > y and YES. 1 ==> x < yl
~-
-
f
then
where x(.) is the ith smallest observation; but since I. is an interval,
-
~
~
the event E /a(t),b(t),n 7 has a well-defined probability which we may
nwrite as
(1. 30)
),..nJ =
(3nLa( t) , b( t
Finally, for any F, 0 e;
pr
f a( t) ~ Zn(t) ~ b( t), t€ n ~ .
J*" define
(1.31)
from (1.26) it follows that
(1.32 )
Note also that
D(F,G)
=
1
In
sup
tET
and
(1.34)
()
D+F,G
sup
= /_1 t€T
L-Sn (tjF,G )J
.
In
Then, starting from the definition (l.lO),we have
8
Dn
=
=
<si< co jn IFn(X) - H(x) I
l
l
~~~fi I F LF (t)J - H{F- (t)J I
-00
n
= ~~~
t j nW.JJ- l (t)J -
t) - jn
l
(HLif- (t)J-t)
I
or
~
and
p;r'f
Dh~}'
Hence, using
r~~~
IZn(t) - Sn(t;H,F) I > ~ 1
pr { I Zn(t) - Sn(t,iH,F) I ~ Qn' teT}
=
pr
=
1 -
=
1 - pr
=
1 - ~
f
Sn(t,iH,F) -
~ ~ Zn(t) ~ Sn(t,iH,F)
/s (t,iH,F) - Qn , Sn (t,iH,F) +
nL:n
(1.32),
we can rewrite
(1.11)
+ Qu, teT}
7
Q ,T
~-
as
(1.36)
the power of the two-sided test becomes
and the asymptotic power, if it exists, is
(1.38)
f Gn 1,a) = I-n~oo~J.Sn(t,iH,Gn)-~,Sn(t;H,Gn)+~,TJ .
since we consider only (H, fGn 1 ) e J, we have from (1.24)
P(H,
Note that,
and (1.33) that
lim
n -to co
sup
tET
~
exis.ts
9
Proceeding in the same manner we obtain also that
(1.40)
we can rewrite (1.14) as
,.
(1.41)
the power of the one-sided test becomes
(1.42)
and the asymptotic power, if it exists, is
=
.
A test of flO:
F(X)
==
lim
1- n""-7OO
L(
)
~ .
n -OD,Sn t;H,Gn + Rn ,T
~
H(x) is called partially ordered if', for
any two alternative distributions A and B such that n+(A,B)
=
0, its
power against A is less than or equal to its power against B; the terminology is due to Chapman, who showed in
[6J that the one-sided
Kolmogorov test (among others) has this property.
In our new notation
this is particularly easy to prove, as follows:
Theorem 1.1.
Proof.
If n+(A,B)
A-l(t) ~ B-l(t), tET.
=0
then A(x) ~ B(x) for all x and hence
But then
and the theorem follows immediately from (1.42).
true for H, A, BE 1') •
(The theorem remains
10
CHAPTER II
THE PROCESS Z( t )
Consider the behavior of the stochastic process Zn (t) as n tends
to infinity.
If, for any k, we have 0 < t
(&1,Z2' ""zk) is
l
< t 2 <.•.< t k < 1, and if
an arbitrary vector, then by the multidimensional
central limit theorem
lim pr [z (t.) < z., 1 < i < k} = pr I Z( t. ) < z., 1 < i < k}
n----tco
n 1 - 1
l
1
1
where Z(t) is a Gaussian stochastic process with the same mean and co( 2 .1 )
variance functions (1.28) as zn(t); that is
(2.2)
Thus if Zi
a)
Z(O) = Z(l)
= 0 with probability 1
b)
CLZ(t)J =
0
c)
cov {Z(t), Z(t')J = min(t,t') - te
= Z( t i ),
1 ~ i ~ k, then (Zl'~' ... ,2..k) have a joint multi-
variate normal distribution with all means zero and with covariance
matrix Ek = (cr .. ) where cr .. = ti(l-t.) for 1 < i < j < k. From this
1J
1J
J
it is easily verified that the joint probability density function of
(z.-z.
1)
1
1-
2
t.-t.
1
1
1-
where
Zo = zk+l = O.
o<
< 1 and all real z
T
In the special case k=l, (2.1) says that for
,
11
where
x
q>(x)
(2.5 )
= -
1
J2rr.
J
u2
e- 2 du
;
-00
or, in other words, that Z(T) is normally distributed with mean 0 and
variance T(l-T), as is Zn(T) asymptotically, which statements we write
(2.6)
and
(2.7)
The process Z(t) specified by (2.2) is related to the classical
Wiener process { W( T), 0
~
T < co} , a Gaussian stochastic process
such that
(2.8)
= 0 with probability 1
a)
W(O)
b)
E{W(
c)
cov LW(T), W(T')-7 = min (T,T')
T)
J =0
by the following transformation of Doob
(2.9 )
Z(t)
= T+11 WeT)
where t
LrOJ
=
11
T+
It may also be verified that a conditional Wiener process, so constrained
that W(l)
=
0, is identical with Z(t).
Now we may and shall assume
that Z(t) and WeT) are separable; then the sample functions of Z(t) are
continuous with probability 1, since this is a well-known property of
a separable process W(T).
so is Z(t) •
Finally, WeT) is a Markov process, and thus
12
In analogy with (1.29), if aCt) and bet) are any two functions
defined for t€T, and n is any subset of T, we may define the event
(2.10)
ELa(t),b(t),fl.J
= fa(t):s Z(t):s bet), ten} •
Now only trivial changes are necessary in order to derive from Donsker's
argument in LBJthe following
Theorem 2.1.
Let A be the space of all real single-valued functions
f(t) which are continuous for teT except for at most a finite number of
finite j.umps, and let F be a functional defined on A which is continuous in the uniform topology at almost all points of =:
continuous
s~le
functions of the Z(t) process.
fFL"'2{t)J :s 1is measurable for every
Q
Q;
, the space of
Then the set
that is, F{Z(t)J is a
(generalized) random variable.
By a "generalized" random variable we mean one which may take on
infinite values with positive probability; that is
and
But this theorem allows us to define, in analogy with (1.30),
(2.11)
t3Li( t ), b( t ), 1l.J
=
pr [ a( t) :s Z( t) :s b( t ), te fL }
in all cases; in fact we have
Corollary 2.2.
If aCt) and bet) are any two functions (possibly
infinite)defined for teT, and.n. is any subset of T, then
13
f3*LQ; a( t), b( t), 1\.J
(2.12)
= f3L,a( t )-Q, b( t )+Q, .11.J
is the distribution function of the (generalized) random variable
(2.13)
Proof.
X
= max [~~~ la(t) -
Z(t)J,
~~~
Lz(t) - b(t)J}
For any function f(t) defined for t€T, let
Fl[f(t)J = :~~ La(t) - f(t)J
,
= :~~ L1(t) - b(t)J
,
F2L1(t)J
FL1( t)]
f
= max FlLf( t )J, FJ1( t )J }
Then certainly the functional F is defined on A.
Now, to say that F
is continuous in the uniform topology at almost points of =: is to say
that whenever we have a sequence of functions {fn(t)
verges uniformly to a function F(t)
lim sup
n-4oo t€T
€
=:, that is
I fn(t) - f(t) I
=0
then with probability 1
So let such a sequence be chosen.
a(t) - f(t)
Then
= a(t) - f n (t)
+ f n (t) - f(t)
and letting n tend to infinity yields
sup la( t) _ f( t)J < lim sup
te:J), L;.
- n...-+ co t€
la( t) - f (t)J
n
L;.
1 in =: which con,
14
or
On the other hand, since
a( t) - f n (t)
,
:;: a( t) - f( t) + f( t) - f n (t)
so that
n~co ~~~
{aCt) - fn(t)J
~ ~~~
{aCt) - f(t)J
+ lim
n~co
sup [r(t) _ f (t) 7
t€.n
n -
or
lim
n~co
Thus
and since it can be shown in a similar manner that
,
we have finally
:;: F{rJ
;
that is, F is continuous in the uniform topology.
But then the corollary
15
follows immediately from the theorem because the random variable
FLZ(t)J
is exactly the X defined by (2.13) and the f3~(Q) defined by
f
X~ Q}
(2.12) is just the probability of the event
Theorem 2.3.
If aCt) and b(t) are any two functions defined for t€T
and 0 < t' < ttl
(2.14)
~
1, define
sLa(t),b(tht',tl',z'
,z"J
= pr {a(t)~z(t~(t),'tA~"~ Z(t')
where it is understood that if t'
z" = o.
= 0
then z'
;;r
a',
z(t Jl )":;:
Z'I}
= 0 and if t" = 1 then
Then
sLa( t ) , b( t ); t' ,t i, ,z t , z"
f
= pr MLT; aJ
J
~
W( T)
~ MLT; bJ,
0
< T < CD}
where
;r:;.
J
(T
2
t " ) f{ Tt "( til.t I) + t' t 11 )
T(t".t l )+ t,,2
(2.16)
MLT;f
Proof.
It is well-known, and easily verified, that :if
(2.17)
= t"" + t".t'
wet)
= !T WeT)
= !T
where t
TZ"
t" Z r
• (~ + t".t') .
,
thEm wet) is a Wiener process; and if it is given that W(x) = y then
the process
~t)
(2.18)
= W(T). Y
is a Wiener process also.
{aCt)
~
Z(t)
~ bet),
where
t
= T·
X
,
So consider the event
t' < t <t"t Z(t')
= z', Z{t"} =
z"} . . "
16
Applying (2.9), we see that this is equivalent to
...
I
z"
1
1
Z"
Now apply (2.13) with x ::: t" -1, Y = t".
1
{
Zll
1
a(----y-) ~ (-1-) Lill(t) + trrJ~
t+:tv
t?
I
1
W(t" -1) = t" ' W(tT -1)
This yields
1
1
b(-r),t"
t"'t"'
-1
1
< t + t""
-1
1
1
z"
z'
c.o( 0 ) ::: 0, W(tr - trr) + t"" = tT
1
< tT -1
1J
Applying (2.17) to this gives
til
~ b(tyT),
1+:t
(~I -~)
w(l
1 1 ) =
tr-:rr
t't"
t"-t' <,.<00
"
.
Next, applying (2.18) agal.n, thlS time Wl.th x:::
we have
I
~ ~
tIt"
W(t""'=t'")
-
=
}
tllz'-tIZ"}
t"-tl
.
tIt II
t " Z I_t IZ II
t"_t,and y::: t" -t' ,
17
+ tt:,t_tll,J
t"I:,-;;;'(t)
, ,,t "/+t
LV
LW\
+,tIlZ'-tIZ"
ttl_ti J+Z
{t + tIt"
ttl_t,J
a(
tit"
) < --------------t
til t't"
+ ttl_t' +
t"+ t + t":tT
f
II tIt"
t Lt +ttl_t l 7
<
b(
t 't It
II
t + tll_'t i +t
or
t't ll
),
t't"
t":tT <t+ttl_t l
rL
a(TtIl(t"-t' )+t't Il2 ) < tll(tll_t' )W(T) +TZ"(tll_t')
T(t"-t' )+t 1l2
T(t"-t') + t,,2
< b(
-
Tt Il (t"_t')+t,t,,2
T(t"-t') +t,,2
) ,
°< T <
00
}
which is equivalent to
f
M{T;aJ
°< T <
~ WeT) ~ M{T;bJ,
,.
00 }
and this yields the theorem.
Theorem 2.3 and the procedure used in proving it were suggested by
LI5J which is essentially a
the proof of Malmquist's Theorem 1 in
special case of it.
a(l) ~
(2.19 )
We also remark that if aCe) ~
sLa(t),b(t);O,l,O,o-l
f(t)
= A(l-t)
then by substitution into
(2.21)
~
b(O) and
° ~ bel) then
= ~La(t),b(t),T-l
Consider now M {:r;fJ when f(t) is linear:
(2.20)
°
M{T; f -l =
+
~t
(2.1~)
say
,.
we obtain
frr Lr(l-t") + ~t" -
til
z"-l + ttl_t' Lr(l-t I )~t I- z 1.1,
18
which is also linear.
in
But the following formulas were derived by Dooo
LrOJ: if we define
(2.22)
then
if
o
Gkl (c,d,y,8)
c,d,y,8 > 0, and
otherwi se ,
where
Lk(c+y)-yJ L'k,(d+8)-8J
:=;
Gk2 (c,d,y,8)
= k2 (c+y)(d+8) + k(yd-c8)
Gk3 (c,d,y,8)
= Lk(c+y)-cJ Lk(d+8)-dJ
,.
2
Gk4 (c,d,y,o) = k (c+y)(d+o) - k(yd-c8)
and if
(2.24)
G+( c,d)
= pr { W( T)
~
+ d, 0 <
CT
T
<
00 }
then
1 - e
-2cd
if c, d > 6
o
otherwise
It now requires only simple algebra to establish, using Theorem 2.3,
that
(2.26)
~Le(l-t)+$t,t..(l-t)-liJ.t;tl,t",zl
=
G(
,zllJ
t..(l-t" )-liJ.tl-zllt"l~·(l-tI )+j..Lt' -z'
t
-
Ii
' . .
e(l-t")+~t"_z"
til
, -
t"_t
'
7
t"/e(l-t l )+~tl_z'
tli_t'
,
7
)
19
and that
+ A.(l-t" )-/1J.t"-z" t"/I(l-t '}fiJ.t '-z'
tit
,
tll_tt
G (
==
l-e
=
Proof.
7
_ 2/I(1-t' )of1.Lt l -z' 7/I(1..t l )-/1J.t"-z"
t"-t'
if zt < A.(l-t' )-/1J.t', Z" < A.(l-t")-/1J.t", and
o
Lemma 2.4.
7)
otherwise
If 0 < 8 < ~ and A. > 0 then
Certainly
pr
f '5B~P
0
'5. 1> Z( t »I.} = prf0 '5.
Si''5. 1> Z( t »1.,
+ pr { 0 '5.
Z(1) »1. }
Si''5. 1> Z( t »1.,
Z( 1> l'5.1.) .
Now using (2.4) we have
pr
[0 <_s~p_<
8 Z(t»A., Z(8»A.}
==
pr {Z(8»A.)
=
~
( - A.
).
/8(1-8 )
But
A.
-L {o '5.8~P'5.
pr { 0 '5."':,P'5. I; Z(t »1., Z(I;)'5.I.} =
pr
1> Z(t )>I.IZ(1) )=z
1
£'1(z )dz
where fl(Z) is the probability density function of Z(8) defined by
(2.3) with k=l and
20
pr
fa ::5':;'S. B z(
t
hI.
I
Z(II) "
z}
= l-sL~oo,~; 0,5,0,z-!
2
e
by (2.27); hence
pr
fa 'S.s~p'S.
- 8' ~(~-z)
~
II z( t)>I.
I
( -g~(~-z)
z(o)<~ } = )e a
-
-00
_~z:;-2"'l':'")
1
e 25(1-5 dz
j2'Jt 5{ 1-8)
Adding these results, we have
(2.28)
pr{o
<_s~p<_ 5 Z(t»~} = ~(
where (2.28) holds if
for x >
°< 5 <
1.
-~
/5{1-5)
Now it is easily established that
°
and the statement of the lemma then follows easily from (2.28) using
these inequalities.
Lemma 2.5.
11. be
any countable subset of the closed interval
°< 5 < ~
Proof.
Let f(t) be an arbitrary :f\mction defined for te:T, and let
L5,1-5J where
. Then for every real V and ~
The lemma is trivially true if V =
~
; so without loss of
21
generality we may assume 0/ > $ , and then certainly
~L~oo,f(t)+o/,1L-l - ~~co,f(t)+$,1L-l ~O
Let
n=
{tl , t , ••• }
and, for each k, let
2
..fl
k
tul (k),u2 (k), ... ,uk(k)}
{t ,t , ... t } =
l 2
k
=
,
where 0 < 8 ~ ul < u 2 <...< uk ~ 1-8 < 1. Let f i = f(u i ), 1 ~ i ~ k,
and assume the f.'s are all finite. Then, using (2.3), we have that
J.
for anys
fl+o
~Eco,f(t)+i,.nk-l
=
S ...
-00
fk+t
f
eke
k+l (z.-z. 1)
_~ E
J.
J.1 1 u.-u. 1
=
J.
2
J.-
-00
Transform the variables of integration so that Xi = fi+e -zi' 1
then
Differentiating with respect to
e
1
we obtain
~
i
~
k;
22
or, transforming back from the x's to the z's,
<
f... 7
Ok
-00
-00
=~
But, from (2.6), Z(u1 ) : NLt),u1 (1-u1 )J and Z(Uk ):
and
covLZ(u1 ),Z(uk )J=
~(1-Uk) by (2.2c)j so
J~Z(U1)
Z(Uk )
t....:.
+ 1U
u1
- k
J:
N{O,..!.
+ 1-1:..u
-uk
1
J
NLO,Uk (1-uk )Jj
23
and thus for all';.
From this it follows that
We have assumed that the f.
I
1
S
are all finite:
but if any f. =
1
-00
then
13L:-oo" f( t )+t,n kJ = 0 and the result above is trivial; whereas if any
f i = +00 then the corresponding u
proceed as before.
Now since
can be ignored and the argument will
n k C .n,
13~oo)f(t)+t,llJ -
And since n
i
13~oo,f{t)+t,llkJ ~ 0
00
=
U.n. k there
must be a k=k( t, 4l) so large that
k=l
Then using this k we obtain
by adding the last three inequalities; and this completes the proof
of the lemma.
Given any function f(t) defined for t€T, let ref) be the set of
all points of discontinuity of f(t) in T, always including the two
24
endpoints 0 and 1 of T.
Here f(t) need not be finite:
= +00 at t = to' then to
~
ref) if and only if f(t)
but if f(t)
= + 00
in some neighborhood of to; and similarly if f(to) = -00.
everywhere
Then we
shall say that f(t) is piecewise-continuous (in T) if and only if ref)
is countable.
Theorem 2.6.
Given any two piecewise-continuous functions aCt) and
bet), let r be any dense countable subset of T which includes rea)
and r( b ), and define
lim
r
sup
()
sup
()
= 5 ~ 0 max 10 < t < 5 a t , 1-5 < t < 1 a t ,
- -
sup
r:)J
sup
o ~ t ~ 5 L-b(t
'1-5 ~ t ~ 1
L--b(t )J 1
J -:
then
(i)
(ii)
(iii)
Proof.
r3*( Q) = 13*LQja(t), b(t), TJ = 13*LQ;a(t), b(t) ,rJ
13*(Q) = 0 for Q < Q*j and
13*(Q) is continuous for Q =1= Q*
Let the stochastic process
X( t)
r
X( t), t€T } , be defined by
f
= max a( t) - Z( t ), Z( t) - b( t) ~
j
then from Corollary 2.2 we have that 13*LQja(t),b(t),TJ is the distribution function of the random variable X = ~~~ X(t) and
13*LQja(t),b(t),rJ is the distribution function of the random variable
~ = ~~t X{t).
Let E be the event that Z{O)
=
Z(l) = 0 and Z{t) is
continuous for all t€Tj then E occurs with probability 1.
Finally,
25
Now since r is dense in T it includes
let T be any point in (T-r) .
a sequence ft n } such that nl~
. . . . . oo t n = T.
lim ( )
n ~700 X t n
lim max
= n,oo
-A
SO, if E occurs,
t()
a tn
- z( t n ), z( t n ) - b( t n
)}
lim Z(t) _ lim b(t ) 1
n -teo
n
n~eo
n J
,
because then Z(t), a(t), and b(t) are all continuous at t
..
= Tj since
Z(t) is necessarily bounded when E occurs the possibility of a(T) and
b{ T) being infinite need not trouble us.
Now certainly
lim X{t) < sup X(t)
n"-'OO
n - t€I'
'
or, that is,
and since this is true for every T€(T-r) we have that in fact
sup
()
sup
()
t€(T-r) X t ~ t€I' X t
But then
X=
~~~
X(t)
=
maxf~~~ X{t), t:(~-I') X(t)} = ~~~ X(t)
Thus we have shown that X
(i) .
=
~
=
XI'.
With probability 1, and hence we have
26
There is no loss in generality in supposing that
*
11111
Q =0 ..... 0
and suppose that
Q
< Q* :
say
Q
sup
()
O<t<oat
= Q* -
,.
where c > O.
CJ
Then there
is a 0o(c) such that for 0 < 00
sup
()
*
c
c
O<t<oat >Q -'2 -Q+'2
But for any 0 < 00
f
t)*(Q) = pr aCt) - Q ~ Z(t)
~
pr
~ pr
f
z( t)
~
a( t) - Q, 0
SUP
[
~
()
0 ~ t ~ 0 Z
~ pr [ 0 ~s~p~ 5
t
Z( t)
2:
0
bet) + Q, teT}
~
t
~
sup
<t <
0 }
()
5 a t
}
- Q
~~}
< 85
-2
c
by Lemma 2.4.
Thus we have immediately statement (i1), and a fortiori
t)* (Q) is continuous for Q < Q* •
NoW if Xl
= ~~~ {aCt) - Z{t)J and X2 = :~~ {Z{t) - b{t)J, then
t)l(Q) = t){a{t) - Q, +
00,
J
(32{Q) = t){=oo, bet )+Q, T
..
TJ
is the distribution function of
X:t
and
is the distribution function of X2; and if
both t)l{Q) and f3 2 {Q) are continuous for Q> Q* then f3 *(Q), which is
the distribution function of X = max (JS..,X ), is certainly continuous
2
for Q > Q* also. A distribution function is by definition continuous
to the right.
We shall show that f32{Q) is continuous to the left for
27
Q > Q*; the proof for
~l(Q)
is similar, and statement (iii) then'
follows.
So suppose that Q > Q*:
5,
0<
< ~, let r 5
5
= [t:
say Q = Q* + c, where c. > O.
ter, 5
~
t
~
For any
Then for any e ,
1-5 }.
c
0<8<3"
J - ~L:: 00, b( t )+Q-6 ,rJ
= pr f EL:: CD, b(t )+Q-c ,r5 J () EL::
~L:: 00, b(t )+Q-c,r5
= pr[
(]I),
b(t )+Q-6, r J
}
Z(t) ~ b(t)+Q-8, ter5 ; 0ut for some ter,
Z(t) > o(t)+Q- 6 }
~ pr { Z(t) > b(t)+Q-c for some t, O·~ t ~
+ pr
f Z(t) > .:>(t )+Q- c. for
5}
some t, 1-5 ~ t ~ l}.
But, from (2.29), there is a 5 (c) such that for 8<5 we have
1
1
b( t) ::: -Q* - ~ for 0 ~ t ~ 5 and 1-5 ~ t ~ 1; and then
b(t) + Q -
e>
(-Q* - ~ ) + Q - ~
=~ ,
and
pr {Z(t »b(t)+Q-£ for some t, 1-5
~
t
~
1}
.r
sup
() c I
~ p~~ 1-8 < t < 1 Z t >j {
l
J
185
<2
c
by Lemma
2.4,
and hence
28
"
13
Now since
E 00, bet )+Q-[ ,raJ - I3L=- CD, b(t)+Q-6 ,r7
rae r
for all
< 36~ •
c
a,
and by Lennna 2.5
Adding these last three inequalities, and applying statement (i), we
obtain
..
Thus for any
c,~
> 0 we can choose a0
(c,~) and then £ (a ,c,~) so
0 0
that for 0 < 6 < €
o
•
that is, 13 (Q) is continuous to the left for Q > Q*.
2
The arguments for statements (ii) and (iii) have tacitly assumed Q* finite, but the changes that are necessary if this is not
so are obvious, and thus the proof of the theorem is complete.
We J;lote that from Theorem 2.6 it follows that 13*(Q) is continuous in Q for all Q if and only if 13*(Q*)
examples:
if aCt) =
-00
and .b(t)
=0
= O. Consider two trivial
for t€T then by (2.19) and
Q>O
Q~O
,
29
which function is continuous for all Q, and 13*(Q{/-) = 13*(0) = 0; but
if aCt) = -
= bel) = 0,
CD
and bet) =
+00
for 0 < t < 1, yet a(O) = b(O) = a(l)
then
13*(Q)
Q<O
=
Q~O
which has a discontinuity at Q = Q*
Theorem
= O.
2.7. Let aCt) and bet) be any two functions defined for
l < t 2 < •.• < t k < 1 be chosen arbitrarily; then 13ta(t),b(t),T~ can be expressed as
t€T, and let k
Proof.
~
1 and 0 < t
Certainly we have
b(t )·
l
13La(t),b(t),TJ =
b(t )
k
f ... f
a(t l )
pr{ aCt)
~
Z(t)
~ bet),
teT
a(t k )
IZ(t1 ) = z1'
1
~ 1 ~ ~ fk(zl"",zk)dzk ••• dz1
where fk(Zl'''''Zk) is the joint probability density function of
Z(t l ), Z(t 2 ), ••• ,Z(t k ) given by (2.3).
Markov process,
But, because Z(t) is a
30
=
k+1
.
I
II pr taCt) < Z(t)<b(t),t i l<t<t. Z(t. l)=z. "Z(t.)=z.
i=l
1..
1.J.-u.
1.
1
by (2.14), and this clearly gives the theorem.
We shall say that a piecewise-continuous function f(t) is
f
piecewi se-1inear with (k+1) segments if r( f) = to' t 1 , • • •,t k+1 .~
where 0 = t@ < t 1 < ••• < t k < t k+1 = 1 and f(t) is linear in each
open interval (ti,t i +l ), 0 ~ i ~ kj f(t i ), 0 ~ i ~ k+l,may be arbitrarily defined. Then (2.24), (2.25), and Theorem 2.7 enable us to
calculate ~*LQ;a(t),b(t),T_7 whenever aCt) and bet) are piecewise-
.
linear.
However, it will be more useful to specialize further to
the case where aCt) and bet) form a coincident piecewise-linear
pair of functions; that is, where rea) = reb) and
aCt) = bet) = Ai6(1-t) + ~.6t
1.
for t.1. < t < t.l+1,0 <
1 < k,
-
but a(t i ) and b(t i ), 0 ~ i ~ k+1, may be arbitrarily defined. The
apparently superflUOUS parameter 6 has been inserted for convenience
in a later chapter.
Note that here
Q*La(t),b(t)J = max ja(o), a(l), -b(O), -bel), IA16 1,
and
Q*L=oo,b(t)J = - min {b(O), bel), A16,
~k+16 }
•
l~k+161
}
31
Now if we define
C(Q , A. ,...it ,t' ,til , Z' , Zll)
= ~8(1-t)
(2.31)
'" pr
+ IJ.t-Q,)"(l-t) + IJ.t + Q; t' ,tll,z',ZIlJ
r
A.(l-t) + IJ.t-Q
~
~
Z(t)
] z( t')
A.(l-t) + IJ.t + Q, t' < t < til
= Zr
• z( til)
=Z
II }
we have, using (2.26) and (2.23) to write out the formula e.x;p1icit1y,
-~Cik
4
k til
ti
E (-1) e -
co
1 +E
1=1 k=l
(2.32)
C(Q,A.,IJ.,t' ,til ,Z' ,Zll)
=
if Q>
maxi tA.(l-t' )-fi,lt'-zll,
IA.(l-t") + IJ.t" o
Ci1 =
•
i(2i-1)Q+8(1-~)+lJ.t'-ZIJ
otherwise, where
1i (2i-1)Q+8(1-tll)+lJ.ttl-zIlJ J
t 2iQ + (IJ.-A)(tll_t l ) (
C = 2iQ
i2
ci3 =
Ci4 =
zlll }, and
)
(ZIl_Z')
3
.11
[(2i-1)Q-Lr(1-tl)+lJ.tl_ZIJ}f (2i-l)Q-8(1-t ll )+lJ.ttl ..z
2iQ
t 2iQ-(IJ.-A.)(t ll -t' )+(ZIl_ZI)}
;
and if we define also
=
~.[.= oo,A(l-t)+lJ.t+R;t' ,t",z' ,ztlJ
= pr { z(t) ~
then from (2.27) we have
A(l-t)+lJ.t+R,t' < t < till z(t' )=z' ,Z(ttl)=Z"
J'
32
(2.34)
C+(R
,r.,..-, t' ,til , ,Zll)
A
Zl
I.
_ 2,0+A.(1-t' )+j..Lt'-~IJ~+A.(l-tU)+j..Ltll_ZllJ
(l-e
t -t
=)
l
o
if R > max [Z'-A.(l-t' )-I-Lt' ,Zl_A.(l_t")_f.l.t"}
otllien1ee.
Thus we can now establish
Corollary 2.8.
If a(t) and b(t) form a coincident piecewise-linear
pair of functions satisfying (2.30), let
a i 6.
= maxf681_1(1-ti)+l-Li_lt1J,a(ti),6L,5:1(1-ti)-/ilt1J J
b16.
= min {6L,"f:1 _1 (l-t1 )+1-1.1 _1t 1J,
b(t i ) ,!:iY:.i (l-t 1 )-/iltiJ }
for 1 ::: 1 ::: k, and define
(2.36)
QoLa(t),b(t)J
= max [Q*LB:(t),b(t)7;~
(a1 -b i ), 1::: 1 ::: k }.
then
~ta(t) - Q, b(t) + Q,
TJ
Lib +Q
k (k+l
~ IT C(Q,&.,4J,.,t11 ,t.,z. 1,z1)]
S
1
t 1=1
6.a -Q
k
=
o
Also,
otherwise.
J.
J.
-
J.
J.-
33
=
if R ~ Q* L-=ro,b(t)J, and
o
Prodf.
that
otherwise.
From the definition
(2.31) and from Theorem 2.7 it follows
~ La(t) - Q,b(t) + Q,T~ can be expressed as
for all Q, and the refinements included in
Q < Qo' or if, for some i, 1 <_ i <_ k,
(2.37) are trivial: if
bJa.. - Q < z.
~
-
~
<
6b. + Q fails
J.
to hold, then from (2.32) we see that
k+l
11 C(Q,.6A.;,4J..,t.
l,t.,z.
l'z.)
= O. And the proof of (2.38) is
. 1
J.
~
J.J.
J.J.
J.=
similar and even simpler.
It will be useful to have in explicit form some of the formulas
for ~ ta(t)-Q,b(t)+Q,T~ and ~ L-::oo,b(t)+R,T~ when a(t) and b(t)
form a coincident piecewise-linear pair with the number of segments
very small.
In each case we wite the formula only for
Q ~ Qota(t),b(t)~ or R ~
Q*L-= oo,b(t)J
•
If k
•
~
~l'
for
=o
(i.e., there is k+l
=1
segment), write A. for A.l and
Then substitution into (2.32) gives
~La(t)-Q,b(t)+Q,T~
=
c; [e-21J2i-l)Q+AAJ[J2i-l)Q+l..LA~
1 _
i=l
_e-4i(~L2iQ - A(A-fJ.)J
(one segment)
+e-2[[2i-l)Q-AAJ1J2i-l)Q-fJ.AJ
_e 4iQ['2iQ
+ A(A-IJ.) J
};
and (2.34) gives
(2.40)
~ L-::oo,b(t)+R,TJ = 1 _ e-2(R+~)(R+!J.A)
(one segment)
If k
= 1 (i,e.,
there are k+l
=2
segments), write
T for t l ,
a for a , and b for b • Then by Corollary 2.8 we can e2q)ress
l
l
~La(t)-Q,b(t)+Q,TJ as
z2
Ab+Q
1
C(~~1'~1,O,T,O,Z)C(Q)~2'~2,T,1,z,O)e
./2TCT(1-T)
_Q
Let
qjl
Pi2
= 2iQ
Pi3 = AlA - (2i-l)Q
then
= -2iQ
= fJ.~ + (2j-l)Q
qj2 = 2jQ
(2.41)
Pi 4
-2T(1-T)
qj3
= ~z6'
qj4
=
..
-2jQ ;
(2j-l)Q
dz.
35
and
and with some computation it can be shown that
~ a(t)-Q,b(t)+Q,T~ = ~ ( jT{1-T
6b + Q)
)
(2.42)
-
La.
(two seSIDents)
00
+ ~
4
k
~ (-1)
1=1 k=1
x
- 2p·kL.Pik-6(~1-1l1)J
~
f ~(&+Q-2pik(1-T)
·l
~
j=1
rl ~
ex>
K=1
(Ab+Q-2 q j
t ) _~ (
jT (1-T )
4
ex>
rl~ (
Furthermore,
~ (6a-Q-2pik(1_T»)~
jT (1-T )
J
i (-1)~ e- 2qjVLqjK-6(~2-1l2)J
6a-Q- 2 'l.j(
jT (1-T)
4
+ ~
~ ~
~ (-1
1=1 j=1 k=1 /=1
X
\ _
jr (1-T » )
+
x
e
)k+ U
~ e
-
)
j
2LrPik-qjll)2_6(~1-1l1~+~~-1l2bI17
6b+Q-2pik(1-T )-2 Qj ,() _
jT(1-T)
\1
~
'Jx·
~(6a-Q-2pik(l_T )-2'l.j(\ 1.
jT{1-T » ) j
36
.~
( 2 • 43 )
(two segments)
J = ~ (i
[: 00, b( t )+R) T
_
R + Ab'\
J"";ry=;(::;:l-="::;) )
e-2~R+"'16)(R+J.L~6) ~(R(2"_1)+b.b-2.(1_")"'l~)
,!,-(l-T J
.
I
- e
-a(R-M~)(R+t-L~)
~
.
.
( R(1-2,. );.6b-2Tt-L~)
/,.(1-,. )
Finally, define
2
x
f
(2.44)
.lY
e- u -2puv+v
2(1 2)
-p
2
dvdu ;
-00 - <D
then some rather tedious calculation Will establish that
- e
-2(R+"'16)(R+t-L16 )
x
F
R(2t l -l)+6bl -2"'16(1-tl )
R(2t 2 -1)+6b2 -2"'16(1-t 2 )
,
jtl{l-tl )
jt (1-t )
2
2
L
r
-2(R+"'~)(R+t-L~)
- e
'>I.
F [ R(1.-2t1. )+LIb1.
-2\.L~t1.
jtl(l-tl )
, R( 2t2 -1.
)+LIb2-2A~( 1.-t2 ) ; _Pj-
jt2 (1-t 2 )
(continued on next page)
37
-2(R+)..36)(R~36)
- e
x
+ e
F
R(l_2t l )+6bl -2\.136tl
[
,
R(1-2t2)+~b2-2\.13~t2
jtlCl-tl )
j
P
]
jt2 Cl-t 2 )
-26£[R+)..1~)(~1-~2)-(R~~)()..1-)..2)~
)( F [ -R+Abl - 2ki l (1-t1 )-2I-<i\t1 , R+6b 2+2LI( k 1 -k2 )( 1-t2 ) ; -p
i t l (l-tl )
+ e
F
-R+~bl-2)..i~(1-tl )-2\.13~tl
[
,
-R+~b2-2)..1~(1-t2)-~3~t2.
J
p]
/t 2 (1-t 2 )
jtl(l-tl )
-~rR+)..~)(~2-~3)-(R~3~)()..2-)..3)~
xF
l- R+Abl-2(~2-~)!ltl
i t l (l-€l)
- e
jt2 Cl-t 2 )
-~rR+)..1~)(~1-~3)-(R+~3~)()..1-)..3)~
l(
+ e
J
, -R+6b2-2ki\(1-t2 )-2I-<f\t 2 ;
jt2 (1-t 2 )
-2~(R+)..1~)(~1-2\.12+~3)+(R+)"~)(R~~)+~R~~)(~1-2A2+)..3)-7
where
These formulas rapidly become intolerably complex, and we shall
38
not attempt to write down any more of them.
perform calculations with larger
Were it necessary to
k we might well prefer to con-
sider approximations, as indeed we should be forced to do if aCt)
and bet) were not piecewise-linear.
39
CHAPTER III
ASYMP'lIOTIC Zn( t) PROCESS PROBABILITIES
We return now to consideration of the asymptotic power functions
of the Kolmogorov tests, which were defined as
and
In
L-IOJ
Doob suggested that "in calculating asymptotic x (t) pron
cess distributions when n ---i CD we may simply replace the ~ (t)
processes by the x(t) process. II
Z of this paper.)
(Doob uses xn and x for the Zn and
In the case at hand, we might interpret his
suggestion as follows:
assume that the limit
set)
= n-;.CD
lim Sn (toH , Gn )
~
exists, then it is not unreasonable to suppose that
P
(H,fG~~,a) = 1
-
~LS(t)-Q,S(t)+Q,T~
and
P!.(H,{ G -J
n
,ex) = 1 - ~r=
a:>,S(t)+R,T 7
.J
The major aim of this chapter is to justify the "heuristic" procedure which we have just exemplified.
However, we are already able
to present two simple cases in which it yields correct answers:
according to (1.36) ,
40
hence, using our procedure, we might
lim
Now n~co In(~)
= J(Q)
e~ect
that
is given by (1.15), whereas ~EQ,Q,TJ can
be evaluated by sUbstituting 6
=0
into
(2.39); and the reader may
verify that the two formulas are indeed equivalent.
Similarly,
according to (1.41),
J~(Rn) = ~n[= OO,Rn,TJ ,
so that we
e~ect
lim +()
n -+ co Jn Rn
and
= ~L.r:- co, R, T-'7 ;
I
(3.2) may also be verified, using (1.17) and (2.40).
Doob's suggestion was justified under fairly general conditions
by Donsker, whose theorem in
Theorem 3.1.
["9J
may be stated as follows:
( Donsker ) Let A be the space of all real, single-
valued functions f(t) which are continuous for
most a finite number of finite jumps.
t€T except for at
Let F be a functional defined
on A which is continuous in the uniform topology at almost all
points of
:=:,
process.
Then
the space of continuous sample functions of the Z( t)
n~~
at all the
~oints
pr [F[Zn(t)J
~ Q} = pr rF[Z(t)J ~ Q)
of continuity of the distribution function on the
right.
Application of this theorem to the case at hand, following the
41
reasoning which established Corollary 2.2, yields
Corollary 3.2.
If aCt) and bet) are any two functions defined for
t€T, then
at all the points of continuity of (3*( Q).
We see that the preceding corollary is not sUfficiently general
for our purposes; however, it is easy to extend it to
Theorem 3.3.
and {bn (t ) J
If the t"l'O sequences of functions {an(t )}
converge uniformly to functions aCt) and bet) respectively then
(3nLan(t)-~,bn(t)~,T~= (3ta(t)-Q,b(t)+Q,T~
= (3*(Q)
at all the points of continuity of (3*(Q).
Proof.
By uniform convergence, there is an n (t) such that for
l
and let n2 (&) be so chosen that for n
I~
where
[>
0 is arbitrary.
- QI
~
n
2
C
~ 2 ;
Then for n ~ nO
= max
(~,n2) we have,
for all te:T,
a( t) - Q -
6 ~
an(t) - ~ ~ a( t) - Q + e
bet) + Q - E ~ bn(t) + ~ ~ bet) + Q + e
and hence
n~ n
,
l
42
Now if ~*(Q)
= ~ta-Q,b+Q,T_7
is continuous at Q+E
and Q-t
then by
Corollary 2.2
n~~ ':JnLa-QH,b+Q-."· ,TJ
= ~[8.-Q+c, b+Q- c ,TJ
nl;~ ~nLa-Q-s,b+Q+',i7 = ~La-Q-c,b+Q+s,T~
so
lim ffi -Q ,b +Q ,T'J
<
- n-+co ~n a n""n
n--n
~ ~La-Q-E,b+Q+c,T~ •
But since ~*(Q) is a distribution function, it can have at most a
countable number of points of discontinuity, and we can let
£
tend
to zero through values such that ~*(Q) is continuous at Q+6' and
Q-c ; this yields the theorem.
Corollary 3.4.
[Sn(t;H,Gn )}
If (H, {Gnl )e
7 and the
sequence of functions
converges uniformly to some function S(t), then
P (H,{Gn},O:) = 1 - ~LS(t) - Q, Set) + Q, TJ
for all a such that ~LS(t)-Q,S(t)+Q,TJ is continuous at Q = Q(O:); and
1
P+(H, fGn ,0:) = 1 - rpL-:: co,S{t) + R, TJ
for all 0: such that rpL-::co,S{t) +
R,TJ
is continuous at R
= R(o:).
Even Theorem 3.3 is not sufficiently general for some of the
applications we have in mind; and we now undertake to extend it
under slightly less restrictive conditions.
For this we shall re-
quire two lemmas, of which the first is an extension of Lemma 2.4.
Lemma 3.5.
There is a finite postive constant
$
such that if
o :::
t
1
< t II
~
1 and & > 0 then for every positive integer n
ljl(t"-t' )
sup
inf
()}
;
<
2
pr {t' < t < til Zn(t) - t' < t < til Zn t > s
€
- -
and also
pr [ t
Proof.
1
sup
in!
< t < til Z(t) - t t < t < til Z(t) >
e}
<
ljl(t"-t' )
2
E
Let
s = pr fl t'
sup
( )
< t < t" Zn t
in!
()
- t' < t < t" Zn t >
[sup
( )
< t < til Zn t - t
= pr L t'
- -
+ pr { t'
~s~p~
inf
I
()
< t < til Zn t
- -
E}
c
> ,
IZn(t' )- Zn(t " ) I ~ ~ }
til Zn(t ) - t. <i~< til Zn(t) > f; ,
IZn(tt )-Zn(t")1
< ~}
c}
>
-4
+ pr ft'
+ pr[tl
=
say.
el
'5.s~p'5. tOO
'S.in;~ til
Zn(t) > z,,(t') +
Zn(t) < Zn(t')
~,
IZn(t')-Zn(t")1 <
~-J
-~, 1~(t')-Zn(tll)1 < ~}
+ ~2 + 53'
Now by Chebyshev's inequality
Then, since probabilities about Zn(t)
= /D.(FnLp""l(t)J-t)
do not
depend on F, so long as FE: t*, we may as well simplifY matters by
taking
44
(3.4)
F{x)
so that F-l{t)
£2
= pr
= t,
=
a ij
x <: 0
x
0 < x <1
1
x>
teT, and ~(t)
1
= /D. {Fn(t)
f
sup
(t! ::: t ~ t" Zn(t) > Zn(t ,) +
/(j-1)_n{t"_t')1
where
o
rZn(t') = jn (~ - t
= pr
f n(t') = !n ,
n!
= i! ( j - i ) ! (n- j H
~, IZn(t
I ) -
Then
Zn(til) I<
~}
<¥
= pr
F
- tJ.
F
I), Zn(t ")
J
n (t ") = Q.
n1
(t,)i (t"-t I )j-i (l_t" )n- j
and
bi j
= pr
rt <s~p< til
I
Zn(t) > Zn( t i) +
j zn (tf) = In (!n -
t'),
~
zn (til) = Jii
(Q.
n - til)}
i
= pr { t' <SUP
t < til LFn ()
t -t..:..7 > n
- t I + -C-
-
IFn{t l
If j
=i
2~
)
= ~,
Fn{t") = ~ }.
then certainly b.. = 0; so assume j > i.
iJ
But, for F (t ' )
n
and Fn(t") given, that portion of the empirical distribution
function F (t) where t' < t < til is itself, except for a linear
n
-
-
transformation, an empirical distribution function; namely, if we let
45
G. . (u)
J-J..
= J....
.n!
F (t) - ji... .
n
1
u€T ,
,
where
t-t'
U
= til -ili'
,
then Gj_i(u) is the empirical distribution function of a random
sample of size (j-i) from
F(x).
i
And since
=0
F (t' ) = n
n
if and only if
G. i(O)
F (til) =.J.
if and only if
G. . (1) = 1,
J-J..
J-
,
and
n
n
we have
bJ..' '
J
= pr
fsup rj-iG. i(u)+! -tt-U(tll-t')J >! - t'+...L}
(UE:T - n J n
n
2 jn
= 1-pr
~
f1
L
OJ i(U)
-
r
()
1-pr 'l G. . u
J-J..
= 1-pr
[sUPT
u~
.s 2n
< ce
... 8(j-i)
) + 2 ( .rn)
< nu(t"-t'
..
. ,uET 1$
J-J..
J-J..
-
<u +
-
~n
..,
J-J..
u~T
1
)
/~)
{G..
(u)-uJ < 4(
}
J-J.
J-J.
,
where c is the finite positive constant of Lemma 2 in LIl~.
then certainly
b ..
J..J
< 8c(J'-i)
and hence
e 2n
~
g <-
But
2 'O~i<j~n.
I (j_i)..n(tr'_t )f<~0
8e( j-i)
2
c
n
f
<
8e
n~l ~
n~
e2n i=O j=i+l it (j~I~:r)!(n-Jjt
8c(t U .. t l
62 .
)
(t'
)~(tll_t' )j-i(l_t ll )n- j
46
Since a similar argument will give the same upper bound for
£3 we
have
s= s
1
+ l
ll
+ ~ < 16(c+l)(t _t
2
3
£2
1
)
The preceding applies equally well to the process Z(t) also as far
as
(3.3); then, using Theorem 2.1,
J 7~
-00
t
Z -
E
1j:
(~}(i;)
z'+ ~
JJ
00
,
-CD
<
and since
4(t"_tt)
62
Z -
2
E
1j:
;
£3 again may be shown to have the same upper bound, we
have for Z(t) that
£=
s1
+
s2
+
s3 <
24( tll-t' )
. t2
Then from (3.5) and (3.6) we see that the lemma is verified if we
take
4> =
max
f"
1
~ 16(c+l),24 J •
If A ,is any finite subset of T (which we always take to include
47
the points 0 and 1) let M= M(A) be determined by
1
min
I1'_1' 'I
M
= 1'2 T,T'eA
•
Then if f(t) is any functiolf defined for tET, and m > M(A), we define
1
min
T
TeA It- I ::: iii
f(t)
(3. 8 )
Am(tjf,A)
=
sup
ret)
min
TEA It-TI
l·t_T'I<~
= It-T'I <!m
and
min
f(t)
e
Bm(tjf,A)
(3.9)
=
int
/t-T' 1<
TeA
f(t)
min
T€A It-TI
~
>!
-m
It-TI
= It-T'I
<!
m
With these definitions we have
Lemma 3.6.
If aCt) and bet) are defined for teT, and m > M(A), and
Q > Q*La(t),b(t)~ defined by (2.29), then
I~J.i'(
t)-Gl, b( t )+Q,TJ
~ ~nffim(t; a,A )-Gl,Bm( t; b ,A)+Q,TJ I
1
1
< A(m- 3 + n- '2 )
and
j
1
- ~LAm(t;a,A)-Q,Bm(t;b,A)+Q,T~1 < Am- 3 ;
.
I ~Li(t)-Q,b(t)+Q,T~
Then .i\n(t;a,A)
o~
= aCt)
i ~ k+l, where lim =
~(ti;a,A) = t:~~
o~
and Bm(t;b,A)
aCt}
{ t:
= aim
= bet)
teT,lt-til
*
La(t)-Q,b(t)+Q,TJ-f3.n
n
< in11J •
and Bm(t 1 ;b,A) =
i ~ k+l; and let Q-Q [B.(t)3b(t)J = c
(3
unless t€I1m for some i,
>
O.
Let also
t;~~
bet) = b im ,
Then
k+l
}
iAm(t;a,A)-Q,Bm(t;b,A)+Q,TJ=prU
E-rm
{. i=O ~
where the event
f
E
im
=
but
~nfl
e
a(t)-Q
im
and
< Zn (t) <
bet) + Q, tel. ;
~m
-
-
Zn (t) < a.1m - Q or
sup
Z (t) > b
+Q } ,
tel.~m n
1m
hence
(3n~
~(t)-Q,b(t)+Q,T7_(3 ~ (t;a,A)-Q,B (t;b,A)+Q,T7
~
~m
m
<
k+l
k+l
1: A. + 1: B.
- 1=0 J.m
1=0 1m
where
A.
J..m
and
= pr 1
\ Zn(t) >_
a(t)-Q,t€l
im
Z (t) < a. - Q1
J
tel.~m n
J.m
; but inf
= pr
r
inf'
()
supI
Zn t
t e.
- b.~m + Q , t e'.
I ~m Zn t <
~m
r
sup
tel
(
)
> b.~m +
Q ,
- ~}
Z (t) _ inf
7, (t) > em
n
tel im -n
im
SUP
Z (t) > b. + Q ,
~m
t elim n
+ pr
-!
Z (t) _ inf
Zn(t) ~ em 3
tel
n
tel im
im
sup
~ pr (j" t€l
sup
+ pr
I
~1
Zn(t) - inf
Zn(t) > emteI
im
im
1
Z (t)
SUPl
t e.
~m
n
<
-
b.
~
+ Q + em- '3
inf
Z (t) > b. + Q _ emtel im n
~m
~ pr
} sup
I tE:Iim
~ J1
Z (t) - tinfI Z (t) > cmn
e im n
~}
1
+ pr { bim +
l
Q -
em-
1 )
'3 < Zn~-~m
(t.) < b.
+
Q
By Lemma 3.5,
,"
J sup
pr ) tel..
l
~m
J1
Zn(t)
inf
tel
Consider the case where i
or bOill + Q > c, and so
Z (t)
im
= 0;
n
> em
then t
i
~}
= to = O.
Now
+ em-
'3 Jl
50
=
and the same obviously holds for i
= k+l.
°;
But for 1 < i < k we
where uP /-F-l(t.)J is BiD~mial (n,t.); and then by the Berryn~
~
Esseen Uniform Central Limit Theorem
~l
J
we have this probability
not greater than
1
(3.10 )
I
L
1
b +Q+cm- 3'
bim+Q-cm- 3'
)_ cp(
cp( i m
)
/ti(l-t i )
jti(l-ti )
Let 'Y ( A )
where c i s an absolute constant.
o
c
+ --.£
;n
;;: 1
2
2
t 1 +(l-t i )
jti(l-ti )
.
max
1
< . < k ---- ,
_ ~ _
j..-ti....(.....
l-....t-i"'l"')
then, since dd cp(x) < _1_ , the probability above is not greater
-/21f.
x
than
1
('Y c !g)m-
j
and hence
;:r
3' +
1
('Y c )n- '2
0
51
k+l
The same upper bound clearly holds for Z A,
. 0
J,=
and hence the lemma
J
J.ID
is verified for Zn(t) if
l.(A,C) = 2 """
[20~~+2) + kr c
fi '
kr Co }
The proof for the process Z(t) obviously proceeds in exactly the
same manner, except that the second term in (3.10) does not appear.
Theorem 307.
If the two sequences of functions fan(t») and tbn(t)}
are such that for some finite subset A of T and for every sufficiently large m the corresponding sequences of functions
and
fBm(t;bnJA)~
fAm(t;an ,A)1
converge uniformly to l\n(t;a,A) and Bm(t;b,A) res-
pectively, where aCt) and bet) are any two piecewise-continuous
functions, then
n:;~ ~ntan(t)-~,bn(t)+Qn,T~= ~ta(t)-Q,b(t)+Q,T~ = ~*(Q)
for Q ~ Q*ta(t),b(t)~ •
Proof.
(2.29):
Consider first the case where Q < Q*Li(t),b(t)~ defined by
say Q = Q* - c, where c > O.
. supposJ.ng
.
that Q*
generalit y J.n
~
>
There will be no loss in
= lim
8~0
0
<sup
t < 8 a (t) • G'J.ven any
- -
0, choose m = m(~) to be the least integer greater than
r
max M(A), 9$ 2} , where
<P
is the constant of Lemma 3.5.
Then be-
~c
cause the point 0 is in A (as we always assume)
Am ( O;a,A )
= 0
()
sup
<t <! a t
-
m
2:
lim
8~0
0
sup
( ")
<t <8 a t
-
-
and by the uniform convergence of IfAm(t;an ,A)Jl to Am(t;a,A) there
is an no = nJ.m(~)J such that for n 2: no we have both
52
A (O;a ,A) > A (O;a,A) - ~
m
n
- m
::>
and
Then
sup
o < t <! an(t) - ~
m
= Am( 0; an,A
)
- ~
2:
* 3"c -
Q -
Q-
c
c
'3 = 3"
so
< pr
-
r(nZ (t) >- an (t )-0--n ,
"
0 < t <! ~
m )
~ prfo
<s~p< !m Zn(t)~ 0 ~
< ! an(t)-~}
,m
~ pr[0 'S.s~p< ~
but since Zn(O)
=0
Zn(t) >
J} ;
0ni'
with probability 1, and hence 0 <~t <
!
Zn(t)
m
'S.
0
with probability 1,
pr
rla'S.
sup
t <~
c1
j' sup
2n()
t > 3' J 'S. pr) 0 'S. t 'S. ~
()
ini'
() c
Zn t - 0 ~ t 'S. ~ Zn t > 3"
using Lemma 3.5; and thus by the definition of
m(~)
Then it follows that for Q < Q*
lim ~ ~ (t) _ 0 , b (t) + Q , TJ
n..., OD nL "'n
--n n
--n
=0
;
but, by statement (ii) of Theorem 2.6,
~La(t) - Q, bet) + Q, T~
=0
;
and thus the conclusion of this theorem is verified for Q < Q*.
1)
53
Now consider the case where Q > Q*:
c > 0.
say Q = Q* + c, where
Then, for any m > M(A),
I f3nffin(t)-~,bn(t)~,TJ - f3ffi(t)-Q,b(t)+Q,TJI
~ I f3nLan(t)-~, bn(t)~,TJ
- ~nm
LA (t;an,A)-Qnm
,B (t;bn,A)+Q'""n ,T71
J
+ I f3nLAn(t;an,A)-%,Bm(t;bn,A)+~,TJ
• f3LAm(t;a,A)-Q,Bm(t;b,A)+Q,TJI
+ 1f3!A
. L.n.m(t;a,A)-Q,Bm(t;b,A)+Q,T7
J
- f3ffi(t)-Q,b(t)+Q,TJI •
It is not difficult to verify that for any pair of functions fl(t)
and f 2 (t) defined for t€T
Furthermore,
lim
m-ta:>
and hence, for any T) > 0, we can choose an
(i)
(iii)
meT)~
so large that
m > M(A)
Q*ffi(t),b(t)J ~ Q*LAm(t;a,A),Bm(t;b,A):J -
Next, since {"Am(t;an ,A)} converges uniformly to Am(t;a,A) and
{ Bm(t;bn ,A)} converges uniformly to Bm(t;b,A) it follows that
lim
(t;an ,A),Bm(t;bn ,A)J= Q*LAm(t;a,A),Bm(t;b,A)J7
co Q*/I
L:. m
n~
*
54
and hence we can choose an n (m) so large that for n > n
0 - 0
:I.
(1) n- 2A(A,%) <
IQ -
(11)
~I <
*
%
where (iv) follows from Theorem 3.3 and (i1) and (iii) imply that
~>
Q-
%= Q*Li(t),b(t)J
+
.f
~ Q*LAm(t;a,A),Bm(t;b,A)~ + ~
> Q*~
(t;an ,A),Bm(t;bn ,A)J + ~'+
L"""m
But then from (3.11) and (3.12) and Lemma 3.6 we have
where D > 0 is arbitrary, and thus the proof of the theorem is
complete.
Corollary 3. 8 •
f
J and the
If (H, Gn J)E .
sequence of functions
fSn(t;H,Gn)} is such that for some finite subset A of T and every
sUfficiently large m the sequence of functions {B (t;Sn,A)1 converges
m
uniformly to Bm(t;b,A), where bet) is piecewise-continuous in T, then
55
for R
f Q*~OD,b(t)~;
and if also the sequence {Am(t;Sn,A)} con-
verges uniformly to Am(tja,A), where aCt) is piecewise-continuous
in T, then
P(H,[Gn} ,a) = 1 - ~ta(t)-Q,b(t)+Q,T~
fer Q f Q*La(t),b(t)~ •
Since Gn €
J*,
G -let) is monotonically increasing in t and
n
hence cannot have more than countably many points 'of' discontinuity;
thus since FE 1* also,
Sn(t)
= jn
(F[Gn-l(t)J - t)
is necessarily piecewise-continuous in T.
ft s n (t) ~J
But if the sequence
converges uniformly to some function set), then certainly
co
res) c U r(Sn) ,
n=l
which latter is a countable set, and thus set) is piecewisecontinuous in T also.
This shows that Corollary 3.8 actually
includes Corollary 3.4 as a special case.
56
CHAPTER IV
BOUNDS ON THE POWER
Let
j*
be any aubclas s of the clas s
')
defined by (1. 24) •
Then we define the asymptotic least upper bound on the power in
of the
o~e-sided
1*
Kolmogorov test to be
( 4.1)
other asymptotic bounds are defined similarly:
( 4.2)
(4.4)
We shall deal in particular with the following subclasses of
a)
J(~)
= {(H,f Gn} )€)
J
such that
j'D. D(H,Gn ) = 6., n=1,2, ••• }
b)
J+(6.;I') =
c)
j+
f(H,fGn1)€J(~)
such that for some x
",
H(x) = T = G (x) - ~ , n=1,2, ••• J
U
n
jn
(6.) = 0 < T < 1
(~,T)
J+
d) C+(~,T) = f(H,{Gn1)€ j+(~,T)
jD
e)
D+(H,Gn ) = 0, n=1,2, •••
c: +(~) = 0 < T < 1
.~
such that
C+(~, T)
u·~
•
1
57
Pairs (H,G ) abstracted from members of C+(6) have been called
n
"stochastically comparable!1 by Birnbaum and Scheuer L4Jj it is for
distinguishing such pairs that a one-sided test of fit seems most
reasonable, although the one-sided Kolmogorov test is suitable for
the larger class j+(6).
j+ of
Finally, define also, for any subclass
J,
·.J~LJJ= [S(t):
(4.6)
S,t)=Sn(tjH,Gn ) for some (H,[Gn1)e
J *}.
The procedure by which we shall find asymptotic bounds on the
power of the one-sided test is justified by the following simple
Lemma 4.1.
j* be any subclass of
Let
J.
Suppose that for every
SUfficiently large n
':LoJ*J S(t) = Sln(t)
Se
€
J
nL
JJ j
then
+1:.")* ,aCI = 1
U
-
j-- CD, Sl ()
~ •
-lim t3nn-)(X)
n t +Rn , T
Similarly, if for every sufficiently large n
Se sup
.jn
L
J*J
S( t ) = S2n(t ) e·;60nL- J*J
then
Proof.
Obvious.
Then the asymptotic least upper bound on the power of the onesided test in any of the classes defined by (4.5) is given by
Theorem 4.2.
Let
J* be
any of the classes defined by (4.5).
u+C1*,aJ =
J"t
2
le- (R-6)2
6 <R
6
>R
Then
58
Proof.
If (H,: Gnl)€
J*
then jn D(H,Gn )
all the classes defined by
= 6,
n
= 1,2,...
since
(4.5) are subclasses of ;(6). Hence
Sn(tjH,G ) > -6, and if we let
n
-
-t .fi
~<
jD.-
-6
°nf
then certainly Sl (t) < ~ ~
. ~ S(t)c
n
- Se "8fiL1 J
U(x;a,b)
then let H(x)
= U(xjO,l)
=
i
( x° a
b -a
-
a<x<b
1
x
and G (x)
n
["3J,
(The pair
But for every a < b define
x<a
~
= U(Xj
Then it is easily verified that (H,~G
l
n 1)e
= Sln(t)e;5 [" J*J.
<1
t
(H,f Gn1)
b
j
- ~ , 1 - ~ ), ~1,2, ••..
jD..;n
1* and that Sn (tjH,Gn )
is given by Birnbaum in
where he exhibits an exact closed e~ression for the power
of the one-sided test of H against G based on a sample of size n.)
n
4.1
Hence from Lemma
we have
+L- .J,a
-1* :J = 1
U
- n-lim
00
But the sequence of functions [Sln(t)}
Corollary 3.8 with A
=[0,1\,
bet)
satisfies the conditions of
= -6,teT:
hence
and the numerical evaluation follows from (2.40).
We may compare this asymptotic least upper bound with Chapman's
statement in
L-6J that for large nand 6 ~ Rn
(in our notation)
59
-2(R -6) 2
:. e
n
The following asymptotic greatest lower bound holds little
interest in itself and is given mainly for the sake of completeness:
Corollary
4.3.
L+L- ) (6),aJ
Proof.
2
= e- 2 (R+6)
•
Replace Sln(t) in the proof of Theorem 4.2 by
.
6
-
In
O<t<l--
-
1 - ~
jn (l-t)
<t <1 ,
jn -
-
= U(x;O,l) and
which corresponds to the pair (H,fGn} ) where H(x)
G (x) = U(x; ~ , 1 + ~ ) ; the proof then proceeds in a similar
n
In;n
manner.
Theorem
(4.8)
·4.4. For
°<
T
< 1,
L+[- j+(6,T),aJ
=1
.. pr
1
= -~
[~~~
z(t)
(R - 6 ')
/T(l-T)
_2(R+6)2 ['
+ e
~ R+6,Z(T)
~
(
-
-R~ 36
::: R-6 }
\
~\~~ J
(-R(1-2 T)-6-26.(1_T»)
jT(l-T)
+ ~ (-R(2 T-l)-6-26.T )]
jT(l-T)
60
,
and
(4.10)
<1-~
-
Proof.
R - 6
( j,.{l-T)
0+a.
Let
S
(t'T)
3n '
=
(/ minL-6,
l
O<t<,.+~
jn (T-t)J
-
')
m1nL6, jn (l-t)J
T
rn
+ ~
But if we ta.ke
( 4.11)
H(x)
=
0
x<o
x
O<X<T
,.
T<x<l+T
x-l
1
and
-
<
jn -
-
- 1+T<x<2
- x>2
-
t
<1
;
61
o
6
x- -
jn
T+~<x<l+T+2.b.
+ ~
T
.;n
jn -
in
x-l- - 6
jn
x>
1
so that H(l) = T = Gn(l)
-
2 +
~
In
."'A.' n=1,2, ••• , then (H,fGn1)eJ+(6,T)
and Sll(t;H,Gn ) = S3n(t;T)e
JnL- J+(6,T)J.
Hence by Lemma 4.1
) J = 1 • n-+oo
rrm '""n
A LL+L- .)/; +( 6,T,Ct
-oo,S3n ( t;T) +Rn,TJ •
But the sequence of functions {S3n(tjT)1
of Corollary 3.8 with A = {O,T,l~
b(t)
=
1::
J
Hence L+L- +(6, T) ,CtJ = 1 part of (4.8).
satisfies the conditions
and
:.:,Tt fT.
I3L=- CD, b{ t )+R, TJ,
which is the first
The explicit formula is obtained by substituting
the values Al = ~2
= ~l
= ~2
= .b
= 1 into (2.43).
The argument for (4.9) is the same, except that we take
In
(T.t) J
minto, jn
(l.t)J
minf!5,
O<t<T+~
jn
T+~<t<l
In - -
,
62
H(x) as before,
I
(4.12)
Gn(x)
=
I
]
0
x<O
x
O<X<T +~
-
- -
jn
T+~<x<l+T +~
jn
T +~
;n- -
;n
1+T+~<x<2
j'n-
x-l
-
1
x>2
-
,
and
t
=T
bet) =
then we must substitute ~l
= ~2 = ~l = ~2 = 0,
b
= -1,
into (2.43).
Finally, (4.10) follows easily from (4.8) and (4.9).
Birnbaum and Scheuer
the lower bound in
matid
L-4~
C+(L~, T)
give an exact closed expression for
for any sample size n; this was appron-
for large n by Chapman
C6J .
Theorem 4.5.
~2
(i)
(11)
(iii)
L+/: j+(b.),aJ ::; e-S{RM)
for b. < R
L+C J+(b.),aJ ~ 1 - ~ (2R-2.6.)
for b. > R
L+L- J+(b.),aJ ~ 1 - ~ (2R-2.6.) - w (.. 2R-6b.)
+ 2e- 2 (R+b.)
2
t1l(-~)
~ 1 - ~ (2R-2.6.) + a
(iv)
L+C C+(b.),aJ
= e-
2R2
=a
for b. ::: R
for b. < R
63
(v)
(vi)
L+L-c+(I:~),aJ ~ 1 - \tl (2R-26)
for I:::. > R
L+L-e, +(I:::.),aJ ~ 1 - \tl (2R-26) - \tl(-2R-26)+2e~ 1 - \tl (2R-26) + a
Proof,
Since J+(I:::.,T)
Co
J+(1:::.)
2R
2
\tl(-26)
for I:::. ~ R •
j (I:::.) for every T, 0 < T < 1,
c
we have
L+C J +(I:::.,T),aJ ~ L+C J+(I:::.),aJ
~ L+["J (b),aJ
but, from (4.8),
;~o L+L-J+(I:::.,T),aJ
for I:::.
- CD
=
e- 2 (R+I:::.)2
< R: hence (i). Next, if (H, f Gn})€ (:+(1:::.) then Gn(X) ~ H(x),
< X < co, so that Pn+(H,Gn ,a) -> Pn+(H,H,a) = a
by Theorem 1.1, and thus L+L-C+(I:::.),aJ ~ a
C+(I:::.,T)
C
C +(1:::.)
~ L+CC +(I:::.),a
J;
for 0 < T < 1
for n=l,2, •••
from (4.3); but since
certainly L+CC+(I:::.,T),aJ
and, tlt'om (4,9) ,
~::o L+CC+(I:::.,T),aJ = e- 2R2 = a
for I:::. < R:
hence (iv).We can obtain (iii) and (vi) by putting
T = ~ in Theorem 4.4.
follow from (ii).
Also, since
C+(I:::.)
C
j
+(1:::.), (v) will
Finally, we prove (ii) as follows:
1
we have
If (H, fG )€ J+(I:::., T) then for some x
n
0
64
H(XO )
= T = Gn(Xo )
• ~ , n=1,2, ••• ,
jn
and from (1.21)
P~(H,Gn,a) = pr {D~ > Rn }
(F
= pr { ~o::> <si< 00
= Gn )
jn {Fn(X) - H (x)J > Rn }
fIii IJn(xo ) - H(xo )_7 > Rn }
pr lx > "Tn + Rn fD.l
::: pr
=
where x
= nF
n
~ < P < J..
In
-
(x ) is Binomial (n,T
0
+ ~ ) • Write p
In
=T +~
;n
;
Then
P~(H,GN'O:)
::: 1 • pr {X
=1
- pr
:s np
- (i6.-Rn ) jD. }
x-np
< -(i6.-Rn)
,jnp(l-p) - jp(l-p)
f
1.
J
Since i6. > R, we may choose no so large that, for n > n , A > R •
-
0
Then by Chebyshev's inequality
P+(H,G ,0:) > 1 - p(l-p) ;
n
n
(i6.-R )2
n
and by the Berry-Esseen Uniform Central Limit Theorem
where co is an absolute constant; hence
L-1J
n
L+L-J+(A)
u,rY.
J
> .lim
- n
-+00
inf
~
<P <1
-
-
jD.
1 _ W
(I
max
1 - p(l-p)
'2'
(~-R )
n
Co
6. - Rn \
-
f'
j~\l-P)
) -
(Ji 2+(1_P)2,}
\/ll \
jp(l-p) )
and it is not difficult to see that this yields (ii).
The procedure we use for finding asymptotic least upper bounds
on the power of the two-sided Kolmogorov test is justified by
Lemma 4.6.
Let
J* be
any subclass of
1.
Suppose that there
exist two continuous functions Sl(t) and S2(t) defined for t€T such
that for every sufficiently large n
and that there exist two disjoint dense countable subsets
I al ,a2, ...}
and B
i blJ b
=
2
A
=
of the open interval (0,1) such
, ... }
that for every positive integer k there is an n (k) so large that
o
for n > n (k) there exists a function V €
J*J with
-
Vk(O)
k
0
= S2(bi ),
1 ~ i ~ k.
uL- j* ,aJ = 1
Proof.
-
Then
f3LS2 (t) -
Let (H,fGn1) be a member of
Pn(H,Gn,a)
=1
-
~
1 ~ i ~ k, and
= Vk(l) = 0, vk(ai ) = Sl(ai ),
Vk(bi )
i /-
Q,
J*.
Sl (t) + Q, TJ
•
Now by (1.37)
f3nL'Sn(t;H,Gn ) - ~, Sn(t;H,Gn ) + ~,
but since for all (H,{G 1)€ j
n
* we
TJ
i
ha.ve Sl(t) < S (tiH.G ) < S2(t)
- n
' n -
for n SUfficiently large, certainly
66
sup
,
.)
(H'1Gn~ E
1
J
*
and then, by Theorem 3.3, from the definition (4.2) we have
( 4.13)
But consider
~!.Yk-Q,Vk+Q,TJ = pr
{ E!'yk-Q,Vk+Q,TJfi
ELS2-Q,Sl+Q,TJ)~
+ pr? ElVk-Q,Vk+Q,TJ () E@2- Q,Sl+Q,TJJ
~ ~LS2-Q,Sl+Q,T~
+ pr
r
+ pr
i EL-=
00 ;'llk+~,
= ~LS2-Q,Sl+Q,TJ
say.
,TJ nE@2- Q,ro ,TJ j
ELVk-Q,ro
+
m:Jn EL-: 00 ,Sl+Q, TJ {
Ok
+
~k '
Here
Ok = pr { Z(t) ~
~
pr {Z(t)
~
Vk(t)-Q,teT;Z(t) < S2(t)-Q, some teT
Vk(bi)-Q,l
~
i
~
1
k;Z(t) < S2(t)-Q, some teT
= pr : z(t) ~ S2(b i )-Q,1 ~ i ~ kjZ(t) < S2(t)-Q, some teT ~
because Vk(b.)
~
= S2(b.),
~
1 < 1 < k.
--
But then, by Theorem 2.6(i),
since S2(t) is continuous and B is dense in T, for any ~ > 0 there
is a kl ( ~) so large that for k ~ ~ we have
for k ~ k2(~)' say, ~k
Ok < ~.
Similarly,
< ~ ; and then, choosing k(~) ~ maX(kl ,k2 ),
we have
~LVk-Q,Vk+Q,TJ < ~LS2-Q,Sl+Q,TJ + ~ •
...
Next, if n
Vk(t)
~ nJY.(~)J = no(T))
= Sn(t;H,Gkn ),
then vke,xnCfJ ; say
where (H,{Gm})e j
*.
111en cel'ta.1X1ly
67
(su~)
( H,i G )
n
€
J*
Pn(H,Gn,a) ~ Pn (H,G1m ,a)
and thus
Sup
U'r L
r
) *' a:J = nlim
~ ex:> (H, t G ) ) €
n
> lim P (H G
- n-?ex:>
n
= n:~~
{l -
=1
'1m'
4
)
*
Pn(H,Gn-.a)
,
a)
~nLVk-Q'Vk+Q,TJ }
- ~lJk-Q,Vk+Q,TJ
> 1 - ~~2-Q,Sl+Q,TJ - ~ ;
but since
~
is arbitrary we can conclude that
Then (4.13) and (4.14) yield the lemma.
Theorem 4.7.
Let
J * be
any of the classes defined by (4. 5abc ); then
1 if
6~
Q; and otherwise
(4.15 )
co
2)2
1-2 E (_1)k-l e -2k (Q-6
•
k=l
And let
C*
be any of the classes defined by (4.5de); then
(--
1 if 6
\
( 4.16)
uL-C*,aJ = l-~E Q,-6+Q,TJ =1 E00
( k=l
~
Q; and otherwi se
fIT)
J e-2v2k-l
l
72
Q- k6.J
'-
+e- 2L[2k-l)Q-(k-l)6J
_2e- 2k2 (2Q-6)2"} •
2
68
Proof'.
It is easily verified that the conditions of' Lemma
~.6
are
satisf'ied if, f'or example, Sl(t) = -6; S2(t) = 6; A and B are arbitrary disjoint countable dense subsets of (0,1); Vk(t) is the
function whose graph is the broken line passing through the required
2
points; and no(k)
= ~2
where 13 k
= -1 ~ ~~~ ~
k lai - bjl and
a = b = 0, a_ = b_ = 1. Hence we have the first equation of
o
o
l
l
(4.15); and the explicit evaluation follows from (2.39) with
).. = 1.1. =
° and Q replaced by Q-6.
equation of
Similarly we obtain the first
(4.16) with all expressions identif'ied in the same way
except that here S2(t)
= 0;
and the explicit evaluation follows
f'rom (2.39) upon substituting -1 for).. and
..
1.1.,
~ f'or 6, and (Q - ~)
for Q•
We have not succeeded in finding exactly the asymptotic greatest lower bounds on the power of the two-sided Kolmogorov test in any
of'the classes defined by
(4.5); but f'rom the following three lemmas
approximations to the bounds can be obtained which will in no case
be in error by more than a •
The first method of approximation simply makes use of the lower
bounds already found f'or the power of' the one-sided test.
particular, we have
Lemma 4.8.
Proof.
Let
J .
j* be any subclass of
J*
r 1)
) be any member of
Let (H'tG
n
'
Then
•
Then
In
69
~ 1 - f3 nEro"Sn(t;H"Gn ) + ~"TJ
= P~(H,Gn"Rn-l~~(a)~).
Hence
and
LrJ
* a 7 = lim
'L
"n-too
inf
P (H G )
(H,fGn~)e)* n 'n"a
,
where the last step follows without difficulty since
lim
n~
Rn
<X)
-1 C ()J
L~ a
I
= R-lm()
LQ a -I
•
The second method of approximation" which we may call the "onepoint" method, is due to Massey" who used it in ["16J to derive
(4.17) below. The proofs of (4.17), and also (ii) in Theorem 4.5,
involve an extension of it.
Lemma
4.9.
For
0<
T
<1
(4.17)
-Q-6
(
And if 6 > Q then
\
/iTl-T)'j .
70
LLC+(I:~) p.J ~ LL-J (~),aJ
( 4.18)
?: 1
Proof.
- ~(2Q-~) + iP( -2Q-~).
Let (H, fG }) be any member of
n
Pn(H,Gn,a)
= pr
= pr
=1
(
Pn(H,Gn,a)
~1
=1
t -~ ~
pr {
T
= Gn )
f
- H(x)1 >
~}
!In(x)-H(xi~ ~,-co < x < oo}
~ ;n ~
+ ~ ).
jn
- pr { np -
- pr
jn
Then
-~ ~ .;n IJn(Xo)-H(Xo)J ~ ~}
t Tn -
- pr
where x is Binomial (n"
(F
J+(~,T).
<s~p< 00 .;n IFn(X)
-00
- pr
2: 1-
=1
>~}
[Dn
= LL- 1+(~) ,aJ
x
Write p
~
Tn +
=T
~ ;n }
+ ~.
;n
Then
~ jn - ~jn ~ x ~ np + '\tID. -~IID. 1
-~-~
<
x-np
Ip(l-p) - Jnp(l-p)
<
-
'Qn- 6 }
Jp(l-pJ
so
<
x - np
v'np(l-p)
<
~
-6/
jp(l-p) J
and by the central limit theorem
which is the right-hand inequality in (4.17).
The right-hand
71
inequality in (4018) follows from the Chebyshev inequality and the
Uniform Central Limit Theorem just as in the proof of (ii) in
Theorem 405.
That
LL J (I).),aJ "* LL-
J+(6),aJ
is clear from
considerations of symmetry; and the remaining inequalities follow
from the inclusion relations among the classes defined by (4.5).
It can be shown by numerical calculation that neither Lemma 408
nor Lemma 409 yields approximations which are uniformly better (i,e.
larger) than those yielded by the other.
Finally, we approximate the asymptotic greatest lower bounds
from above by considering
~
"almost-worst"
sequence of alternative
distributions.
For 0 < T < 1,
Lemma 4.10.
Lf.- C
(4.19 )
+(6,T),aJ
~ I-pr fI Z(t)1 ~ Q,t€T;Z(T) ~
<l-~(
Q-6\
/T(l_T) )
Q-6}
+a.
Also, for 6 < Q,
(4.20 )
and, for 6
~
1[" C +(6) ,aJ ~
(4.21)
Proof.
Q"
1 - ~(2Q-26) + a .
Let H(x) be defined by (4.11) and G (x), n
n
(4.12). T1ian (H,~Gn1)e C+(6,T).
{Sn(t;H,Gn )}
= 1,2,0'0'
by
Also, the sequence of functions
satisfies the conditions of Corollary 3.8 with
aCt) ~ 0 for teT, and bet)
=0
for teT, tIT, where beT)
= -6
;
72
this yields the first inequality of (4.19).
An explicit evaluation
of the probability could be obtained by substituting Al
= ~2 = a = 0,
b
= -1,
into (2.42).
= A2 = ~l
The second inequality follows
since
I
1 - pr [ Z( t)
I~Q,
teT; Z( T)
~ Q-6 } ~ L!-pr[Z( T) ~
Q-6}
J
+L.I-pr! I Z( t)1 ~ Q, teT}
and
1 - pr
by
T
rI
Z( t) I
(1.18) and (3.1).
~
Q, teT
1:: 1 - J(
Q) ::
ex
We can obtain (4.20) from (4.19) by letting
tend to 0, and (4.21) by setting T
= '21
•
The reader can now verify that, as was stated in the paragraph
preceding Lemma 4.8, the approximations now obtained are in no case
in error by more than
0:.
In particular, the well-known lower bound
(4.17) due to lfJ.assey is within ex of the true value.
J
73
CHAPTER V
ASYMPTOTIC EFFICIENCY;
Let H€
EXAMPLES
J* be a hypothesis distribution, and let
= 1,
Ti = {Tn(i)(a), 0< a < 1, n = 1,2, •.•}, i
2, be any two tests
(actually systems of tests) of it, where T (i)(a) is based on n obn
servations and has Type I error a.
Also, let [G(X;8), 8€f1) be a
class of alternative distributions; then write P (i)(8,a) for the
n
,
power of T (i)(a) against G(x,8), and let n.(8,a,P) be the least
n
~
integer n such that
(5.1)
if no finite n satisfying
(5.1) exists we write
=+
n. (8 ,a,p)
~
CD
•
The relative efficiency of T with respect to T is then defined to be
2
l
For example, if test
Tl
requires twice as many observations as
test T in order to achieve power P against G(x;8) when the Type I
2
error is a, then its efficiency with respect to T at (8,a,p) is ~ •
2
In what follows
will be an interval of the real line which
n
includes 0, and
lim
( x;8 )
8 ~O G
= H(X)
•
74
Then, in agreement with the usual definition, we may say that the
asymptotic relative efficiency of T with respect to T2 is
l
if this limit exists.
Now suppose that expressions of the form
p(i)(6
have been obtained.
,0:
)
= nlim
~oo
Then we conclude that for small 5
Pn(i)(5,0:)
rv
p(i)(5 .;n,o:)
and hence, if 6(i)(ct,p) is determined from the relation
then
thus finally the asymptotic relative efficiency of Tl with respect
to T is
2
In the remainder of this chapter we shall restrict our'attention to the asymptotic power and asymptotic relative efficiency of
T+, the one-sided Kolmogorov test.
Let T. be some competitive test
J
whose asymptotic power function has the form
Then
75
and the asymptotic relative efficiency of T+ with respect to T.
J
(simplifying the notation) is
(5.4)
(p)
E
j
0:,
[~-l(p) - ~-l(a)1
=
J
c(j)n+(o:,p)
2
where A+(O:,P) is determined by the equation
P+( I:!+ ,0: )
If p(j) (6,0:)
=1
=P
•
then we write c(j)
p(j)(n,o:) < 0: we write c(j)
-
=0
= + co
and E.(o:,P)
J
and E. (o:,p)
J
=+
co.
= OJ
and if
Note that if
p+(n,o:) were also of the form
then the asymptotic relative efficiency of T+ with respect to T
j
would be
which is independent of
0:
and Pj this situation is the most familiar
one in statistical literature.
The sign
~
based on the p-fractile consists in rejecting the
hYllothesis if the sign of LH-l(P)-xiJ is positive "too often"j
that is, if FnLH-l(P)J is "too large."
Theorem 5.1.
Assume GL-H-l(t)j5~
=t
+ 5c(t) + 0(5), 0 < t < 1.
Then if T(P) is the sign test based on the p-fractile, and c(p) > 0,
the asymptotic relative efficiency of T+ with respect to T(P) is
E(P)(o: p) >
sup
p(l-p)
,
- 0 < t < 1 t(l-t)
~_l(a)].
[C(t). <Q-l(p)_
C"G?J ~-l(p) + RCa) _
/tCl-t )
2
•
76
Proof.
The random variable nFJ.-H-l(P)J is Binomial (n,t) where
t = FL-H- l (P)_7 and F is the true distribution.
then t
= p;
If F(X) = H(x),
and if
S
n
.;n (FJH-l(P)J - p)
= ----::::::;;:::::::;---- ,
v'p(l-p)
then Sn rJ NLO,lJ; hence the test T(P) 1s asymptotically equivalent
to rejecting if S > ~-l(l_a).
But if F(x)
n
hypothesis t
= P + 6c(p) + 0(.2:.
In.;n
= G(x;
~
.;n
) then by
) and thus S _ 6C(p)
n
/p(i-p}
rv
NLO,l 7
-
so that'the asymptotic power of T(P) is
11
and hence from
(5.3)
(5.5)
E(P)(a,p)
and
(5.4)
=
we have
[v'P~l-P)
c p)
where
•
~-~(P)
.. Cl)-l(a)]
A+(a,p)
2
77
where Zn rv N[O,lJ ; so
p+(6,a) >
and
6+(a p) <
,
-
~(-R(a) + 6
C
(t»)
/t(l-t)"
jt~l-t)
[~-l(p) +
c t)
which yields the lemma. by substitution into
R(a) J/£(1-£)
(5.5), since twas
arbitrary.
Note that it is easily shown by differentiation that the lower
5.1 increases with increasing P.
bound on E(P)(a,p) given by Lemma
Theorem 5.2.
Suppose that we can write
(5.6)
H(x) =
f
x
h(u)dU
1
-co
I
where h(u) is continuous for a-leo) :::: u :::: H-l(l) and identically
zero elsewhere; and that
G
n
(x )
Then, defining set)
en
= H( x + .;n
- ),
lim
n-lCID
en = e >
0 •
= -ehCH-\t)J for tE:T, we have
=1
l3E <:o,S(t)
+
R,T7 .
Consider first the case where en
= e for
P+(H,f GnLa)
Proof.
where
-
- ;n
-e
so that from (5.6) and (1.31)
all n.
Then from
78
H-\t)
Sn(tjH,Gn )
r
= -;n
H-l(t) _
h(x)dx.
-!
;n
Now if hex) is continuous in the closed interval LH-l(O),H-l(l)J
it must be uniformly continuous therej that is, for any
f.
>0
there is a 0o(e) > 0 such that if 0 < 0 < 0o(E) and if the closed
interval Lx,x+oJ does not include H-l(O) or H-l(l) then
Ih(x) - h(x+5) I <
integer m > M(A)
l3
= (1
c. Let A = [O,l} j and, choosing any positive
= 2,
1
- iIi ' lJ.
;
<
define II
= L.O,~ ),
l2
= L-~ ,
1 - ~
J ,
and
Assume n sufficiently large that
min! 80 «), H-l(~) _ H(O), H-1(1) • H-1(1 - ~ ) 1
Then for tEll
Bm(tjSn ,A) - Bm(tjS,A)
= inf S (t) _ in! set)
tEll n
tEll
h(X)dx ,
e
;n
which is obviously non-negative.
Since h["H-l(t)J is continuous
79
sup eh[-H- l ( t) J tell
If t
l
ehLH-\ t 1 )-7
•
> 0, choose n so large that ~ < H-l(t) - H-l(O); then for
jn -
H-l(t)
> jn
f
h(x)dx
H-l(t ) _ ~
1
= ehLH-l(tl )
fD.
- <l>e
jn
J
by the mean value theorem for integrals, and hence, using the
uniform continuity of h(x),we have
(5. 8 )
On the other hand, if t l
= ° then
by the mean value theorem again and
argument obtains if teI •
3
(5.8) still holds. A similar
But for teI 2
80
H-l(t)
= - .;n
f
~x)dx
+
ehL-H-l(t)J
..!
;n
efhLH-l(t)J - hL-H-l(t)
H-l(t) _
=
- <l>e
;n
J}
using the mean value theorem once more" and thus we have established
that for n sufficiently large
Then the lemma is verified by applying Corollary 3. 8 •
But in the general case
elsewhere" it must be bounded by some positive constant K.
Then
and since this tends to zero independently of t the previous argument
can obviously be modified to give the same result.
We remark that Theorem 5.2 can be extended easily enough to the
two-sided test" or to cases where
e<
0" or where the possible points
of discontinuity of h(x) form an arbitrary finite set n; the main
81
difficulty consists in properly defining set) when H-l(t)ei"l.
We now consider five numerical examples of pairs
which satisfy the hypotheses of Theorem 5.2.
(li,fan })
These are presented
in Table 5.1, and the corresponding situations are illustrated in
the accompanying diagram, Figure 5.2.
In words, the asymptotic
power p;(.6,a) of the one-sided Kolmogorov test of Hi(x) against the
sequence of alternative distributions
= Hi(X
ein
in
+-
) and
lim
n..., 00
e.J.n = e.~ >
{a.
~n
(x)} , where
a.
~n
(x)
0, is the probability that a
sample function of the stochastic process Z(t) will croes the
boundary Si (t j 6) + R(0:) j from Theorem 5. 2 it follows that the
relationship between .6 and
6
e is
gi yen
by
( )
= e. sup
x h. x •
~
~
One may note the following partial ordering of the Pi+' s from such a
diagram immediately:
for all 6, 0:
fP~(I:',Ci)
(5.10)
>
1P4(A,Ci)
82
t
Name
i
O
1
{~
x<O
O<x<l
x>l
-
- -
I
Rectangular I
{).
I
- {).(l-t)
2
EJq>onential
I
I
-~t
[
3
-26(l-t)
4
0
~t ~~
Laplace
~~t
:s 1
Normal
1 1
-l()
'2
+ 1C tan
x
Cauchy
Figure 5.2
R \.'" ~ 8 + R
.. 4';"..... /
<~"( Cauchy»)
\
'"''
"
~
"
\
\
\
\
, ...
82+ R?
'.
.
,...,"
1
,,'
J
,
,~
,.
I
I
I
,
,
.,
...,
p
I
I
"
,..,
8 +R(Rectangular
...
J
......"
.",.. ..-
.... ....
,
R-~
~
(Exponential )
,-.?
,
~
I
I
J
...
'. ,
"
,,/~
,/8 4+ R
.... " ' ( Normal )
-=-:=' =...:...
(1,0)
(0,0)
,
8ince 8l(tj~), 82(t;~), and 83(t;~) satisfy (2.30) with
k
= OJ
0, and 1 respectively, the formulas for the corresponding
asymptotic power functions can be obtained immediatelY,by sUbstitution into (2.40) and (2.43); we then have
~<R
~
and
>R
,
84
Some computations based on these formulas are exhibited in Tables
5.3 and 5.4.
Table 5.3
0:
P
.10
.90
.95
.99
.90
.95
.99
.90
.95
.99
.05
.01
,
!
I
(Rectangular)
~
~
(Exponential)
.8435
.9128
1.0021
.9944
1.0637
1.1530
1.2879
1.3573
1.4465
1.0239
1.0491
1.0683
1.1808
1.2029
1.2198
1.!J827
1.5005
1.5141
R
1.0730
1.2239
1.5174
It will be necessary to approximate the power functions which
correspond to S4(t;6) and S5(t;6) because these curves are not
piecewise-linear.
Now if set) ~ A(t),
°
~
t ~ 1, then certainly
1 - ~L:-oo,S(t)+R,T_7 ~ 1 - ~E- oo,A(t)+R,TJ •
T
<
t
<
1 - T
°
~ t
<
T, 1 - T
<
t
<1
for i = 4, 5; then Si(t;6) < A.(t;T) and
-
J.
P~(6,0:)
> 1 - ~r:oo,A.(t;T)
+ R, T-f7 •
J.
LJ.
The actual formula required, which can be calculated using
Corollary 2.8, is .
(5.14)
The lower bounds exhibited in Table 5.4 have been computed
from (5.14) with
T
= .375;
this value of
T
was chosen mainly for
computational convenience, but some numerical work indicates that it
may be near the optimal value for use in such a formula.
We admit
that these bounds are undoubtedly quite crude, particularly in the
case of the Cauchy distribution; some refinement could be obtained
·e
by the use of formula (2. 45 ), but to go further would apparently
require considerable numerical work.
Of course, from (5.10) we have
also that p~(L~,a) ::: p;(a,a), but we see that this bound is even
poorer for all the entries in the table.
Table 5.4
a
.J.3534
I
!
R
1.00
.03405
1.30
.00598
1.60
Lower Bound
+on
Lower Bound
+on
(~lace)
(!lQ;rmal.)
(cauchy)
.74i.i9
.76326
.96028
.99743
.75318
.97902
.99966
.74288
.98968
.99997
.69362
.92445
.99114
.65938
.94735
.99773
.62380
.96437
.99953
I
I
~
1.00
1.50
2.00
1.30
1.95
2.60
1.60
2.40
3.20
p~(~,a)
.94210
.99404
.69966
.96040
.99867
.66905
.97597
.99979
p~JA,a)
P5(~,a)
86
Finally, we shall consider the asymptotic relative efficiency
of the one-sided Kolmogorov test with respect to several competitive tests for the same five examples.
The t-test consists in rejecting if
x
n
is "too small."
In particular, if when x is distributed according
e(x) =
to H we have
t
J
then t n N N[O,l
(tn + ~)
where
t
rv
e is
(x)
IJ. and var(x)
n
=
fil(x
cr
n
-IJ.)
= cr 2 ,
and if
,
and hence the t-test is asymptotically equivalent
to rejecting if t
true then
n
E x
n i=l i
1
=-
n
< 0- 1 (0:).
en
fD.
= IJ.
N{rJ, 1J
- -
;
If for each n the alte~na'':iive G is
n
, var(x)
= cr
2
, and the random variable
hence the asymptotic power of the t-test is
determined from
(5.9). Thus for the t-test the constant
required in equation (5.4) is
,
1<i<4.
The preceding argument does not apply to the Cauchy distribution,
for which
C(x)
does not exist, but it is not difficult to show
that the asymptotic power of the t-test 1)1 here equal to a; hence
we may take
87
(5.16)
c
(t)
= 0 •
5
The U-test, which is actually a one-sample analogue of
"
Wilcoxon1s familiar two-sample test, consists in rejecting if
CD
Ii
Un =
J
[Fn(X) - H(x)J dH(x)
-00
is "too large. 1I
It is easily shown that
U
n
where zi
H€
J* we
e
= 21 -
=-
n
E
1
In
i=l
z.
J.
H(Xi ), and that when x is distributed according to
have
00
E(z)
I
J L-~ -
=
H(x)J dH(x)
=0
- 00
and
CD
JL~ -
=
var(z)
2
H(X)J dH(X)
1
= 12
-co
so that Un N N/35'l~J and hence the u-test is asymptotically
equivalent to rejecting if
U
n
1
> --
/12
~
-l( 1-0 ) •
Now if H(x) has a derivative hex) then under suitable restrictions
(which are satisfied by the five examples under consideration) we
have that if for each n the alternative Gn is true then
88
co
.....
t (z) =
•
8
.;n
(
J
2
h (x)dx
+
1
0
-co
(-)
rn
and
1
= 121 + 0 (-)
var(z)
;
.;n
hence the asymptotic power of the u-test can be shown to be
co
. p(U)(.6.,a)
= <pL-{p-l(a) +
8 112
(h2 (X)dXJ,
J
-co
where 8(6) > 0 is determined from
(5.9). Thus for the U-test the
constant required in equation (5.4) is
r
00
2
h (x)dX
/12
\.1
c. (U)
(5.17)
=
J.
-co
sup h(x)
x
The median test consists in rejecting the hypothesis if m , the
-----
sample median, is "too small."
n
In particular, suppose that H(x) has
a derivative h(x) which is continuous in some neighborhood of 11,
where H(11)
= 12 ' and let
then when the hypothesis is true Mn
N
N{O,lJ
and hence the median
test is asymptotically equivalent to rejecting if M
n
< (p-l(a). But
if for each n the alternative G is true then LMn+28h(
n
and hence the asymptotic power of the median test is
where 8(6) > 0 is determined from
T)J
rv
N[O,lJ
(5.9); that is, for the median test
the constant required in equation (5.4) is
2h (TI)
i
S~p:. 0. (x)
1
,
If the median and mode of H are the same, then c (m) = 2.
The median
test is aapptotically equivalent to a sign test based on the median
.
(the p-fractile with p
1
= 2)
and hence Theorem 5.1 would apply here
also.
The most powerful tests for translation among rectangular and
exponential populations are based on the extreme values; against the
sequences of alternatives considered here their asymptotic power is
1 for all
ex, 6. >
0; thus, the constants needed for equation (5. 4) are
In the Laplace case the most powerful test is equivalent to a sign
test based on the median (cf. Hoeffding and Rosenblatt L13J) and
hence to the median test; in the Normal case the t-test is most
powerful:
these have already been discussed.
One
can easily derive,
using the well-known Neyman-Pearson Lemma, that the test which
rejects for small values of
n
I:
V =
Zi'
n
i=l
where zi =
Xi
--~2
' is asymptotically most, powerful for detecting
1 + Xi
"
infinitesimally small translations among Cauchy distributions.
o
H (X) is true, C (z)
5
=0
and var(z)
1
= 10
so that
w
In
rJ
When
N[6,lJ and
90
hence this test is asymptotically equivalent to rejecting if
Vn < ~
~-l(a).
If for each n the alternative G (x) is true then
5n
etvNLo,l~
and therefore the asymptotic power is
p;(L~.,a) = ~Lw-l(a) + e~
where e(6) > 0 is determined from (5.9), so that we obtain finally
1
= 1C •
The values of the constants needed for equation (5.4) have been
calculated from (5.15) through (5.20) and are given in Table 5.5.
Table 5.6 then gives the asymptotic relative efficiency of the onesided Kolmogorov test (or a lower bound on it), with respect to the
various competitors considered here, for the same five examples,
using the same values of P and a as appear in Tables 5.3 and 5.4.
Table 5.5
Constants Required for Equation (5. 4)
Test
Distribution
t
U
Rectangular
/12
./12
Exponential
Laplace
1
/3
/2
13
/21C
/0
/3
Normal
Cauchy
0
median
most
powerful·
2
1
2
co
2
/~
2
1C
co
2
91
Table 5.6
I
I
I
Distribution
0:
.10
.05
Rectangular
.01
.10
.05
"
~ntial
.01
.1353
Laplace
"(
.0340
.0060
I
Asymptotic Efficiency of T+
with respect to
P
median
most
powerful
t
U
.90
.95
.99
.90
.95
.99
.90
.95
.99
.770
.857
1.080
.722
.797
.989
.6;4
.713
.862
.770
.857
1.080
.722
.797
.989
.654
.713
.862
3.241
2.165
2.391
2.966
1.962
2.140
2.586
0
0
0
0
0
0
0
0
0
.90
.95
.99
.90
.95
.99
.90
.95
.99
6.267
7.782
11.406
6.142
7.480
10.599
5.921
7.004
9.442
2.089
2.594
3.802
2.047
2.493
3.533
1.974
2.335
3.147
6.267
7.782
11.406
6.142
7.480
10.599
5.921
7.004
9.442
0
0
0
0
0
0
0
0
0
1.532
1.589
1.634
1.631
1.685
1.725
1.701
1.750
1.785
1.021
1.059
1.089
1.087
1.123
1.150
1.134
1.167
1.190
.766
.795
.817
.815
.843
.863
.850
.875
.892
.7418
.9421
.9940
.6997
.9604
.9987
.6691
.9760
.9998
2.309
2~570
.766
.795
.817
.815
.843
.863
.85 0
.875
.892
I
i
92
Table 5.6, continued
Distribution
ex
Asymptotic Efficiency of T+
with respect to
p
t
.1353
Normal
(~OWU"
bounds
on
efficiency)
.0340
.0060
.1353
,
Cauchy
(lower
bmmds
on
efficiency)
.0340
.0060
.7633
.9603
.9974
.7532
.9790
.9997
.71l29
.9897
.99997
.526
.577
.605
.593
.623
.641
.623
.644
.659
.6936
.9245
.9911
.6594
.9474
.9977
.6238
.9644
.9995
co
co
co
00
00
co
00
00
co
median
most
powerful
.551
.604
.634
.621
.653
.672
.653
.674
.690
.827
.906
.95 0
.931
.979
1.007
.979
1.012
1.035
.526
.577
.605
.593
.623
.641
.623
.644
.659
.•861
.954
1.005
.985
1.040
1.072
1.042
1.079
1.102
.646
.715
.754
.739
.780
.804
.782
.809
.827
.262
.290
.305
.300
.316
.326
.317
.328
.335
U
In this chapter we have restricted our attention to the onesided Kolmogorov test, but no theoretical difficulties remain which
would prevent extension of the results to the two-sided test, and
also to many other kinds of examples; the only differences would lie
in the numerical work required.
i
93
BIBLIOGRAPHY
L-l~
Berry, A. C., YThe accuracy of the Gaussian approximation to
the sum of independent variates," Transactions of the
American Mathematical Society, Vol. 49 (1941) ,pp. 122-136.
L-2J Birnbaum" Z. W., "Numerical tabulation of the distribution
of Kolmogorov's statistic for finite sample size, Ii
Journal of the American Statistical Association, Vol. 47
(1952), pp. 425-441.
L3J
Birnbaum, Z. W." "On the power of a one-sided test of fit for
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