•
ASYMPTOTIC SEQUENTIAL TESTS FOR REGULAR
FUNCTIONALS OF DISTRIBUTION FUNCTIONS
PRANAB KUMAR SEN
.
~;
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 740
April 1971
•
•
•
ASYMPTOTIC SEQUENTIAL TESTS FOR REGULAR
FUNCTIONALS OF DISTRIBUTION FUNCTIONS*
PRANAB KUMAR SEN
~.
The theory of asymptotic sequential likelihood ratio tests for
composite hypotheses, developed by Bartlett [3] and Cox [6], among others, is
extended here to cover a broad class of regular functionals of distribution
functions.
Various properties of the proposed sequential tests are studied and
compared with those of some alternative procedures.
A few applications are
sketched.
~.
Consider a sequence {X 'X ' ... } of independent and identI 2
ically distributed random vectors (i.i.d.r.v.) having a
p(~l)-variate
bution function (d.f.) F(x), XERP , the p-dimensional Euclidean space.
distriAssuming
that the form of F is specified and it involves only a set of unknown parameters,
Bartlett [3] and Cox [6] considered large sample sequential likelihood ratio
tests (SLRT) for a parameter while treating the others as nuisance parameters.
Basically, these procedures are quasi-sequential; starting with an initial sample
(of at least moderately large size), observations are drawn sequentially until a
terminal decision is reached.
In a broad class of nonparametric problems, the form of F is not known; it
is only assumed that F belongs to some suitable family ~ of d.f. on RP •
In
this setup, (estimable) parameters are defined as functionals of F, defined on
:r;
•
we may refer to Halmos [7], von Mises [9] and Hoeffding [8], for details .
*Work supported by the National Institutes of Health, Grant GM-12868.
2
•
The fact that the functional form of F is unknown makes it difficult to adapt
the probability ratio or the likelihood ratio principle underlying the classical
sequential testing theory.
null hypothesis H :
O
Nevertheless, it is shown here that for testing a
8(F)=8
0
against HI:
8(F)=8 l , where 8(F) is a regular
functional (estimable parameter) of F, by the same motivation as in Bartlett [3]
and Cox [6], sequential tests can be constructed, which for every pair (8 ,8 ):
0 1
8 +8 , of parameters, terminates with probability one, and which for 8 approach0 1
1
ing to 8 , enjoy all the basic properties of the SLRT.
0
Along with the preliminary notions, the proposed tests are described in
section 2.
Sections 3 and 4 deal with the properties of the proposed tests.
In
section 5, an alternative procedure based on the Anscombe [2] and the Chow and
Robbins [5] theory of fixed-width (sequential) confidence intervals is considered
and compared with the earlier ones.
A few applications are briefly sketched in
the last section.
Consider a functional 8(F)
defined on
g:.
8 (F) is said to have the degree m(~l), when m is the smallest
positive integer for which a kernel ¢(Xl, ... ,X ) unbiasedly estimates 6(F) i.e.,
m
J...
J ¢(xl,···,xm)dF(xl)···dF(xm)
~m
8 (F) ,
for all FE~, where we assume, without any loss of generality, that
metric in its arguments.
For every
O~c~m,
•
c
(F)
¢ is sym-
define
O·,
t/J c (xl'···,x c )
r,;
(2.1)
O.
(2.2)
(2.3)
3
•
It follows from the results of Hoeffding [8] that
(2.4)
and, 8(F) is termed stationary of order zero whenever
~l (F)
> 0
and
~
m
(F) <00.
(2.5)
We desire to provide a sequential test for
(2.6)
Our proposed tests are based on the natural estimates of 8(F), considered
earlier by von Mises [7] and Hoeffding [8], among others.
For a sample (XI, ... ,X )
n
of size n, define the empirical d.f.
F (x) = nn
l
\ n c(x-X.), XER P ,
Li= l
1
(2.7)
where for a p-vector u, c(u)=l if all the p coordinates of u are non-negative
and c(u)=O, otherwise.
Then, von Mises' differentiable statistical function
8(F ) is defined as
n
8(F )
n
f···f
Rpm
¢(XI'···,X )dF (xl) •.. dF (x )
m n
n m
(2.8)
•
4
•
so that 8(F ) is the corresponding regular functional of the empirical d.f .
n
F.
n
F
n
Note that 8(F ) is not necessarily an unbiased estimator of 8(F) though
n
unbiasedly estimates F.
Hoeffding [8] considered the unbiased estimator
(U-statistic)
¢ (X . , ••• , X . ); C
1
1
n,m
1
m
U
n
{l<i < ... <i <nL
- l
m-
(2.9)
We motivate our proposed sequential tests by the same principle underlying
the asymptotic sequential likelihood ratio tests of Bartlett [3] and Cox [6].
If F is of specified form involving as parameters 8(under test) and o(nuisance),
these authors working with the ratio of the two maxima of the likelihood function
were able to approximate the same, at the n-th stage, by
(2.10)
"'-
where 8
n
is the (unrestricted) maximum likelihood estimator of 8, and 1
Fisher information for 8.
88
is the
The asymptotic reduction in (2.10) along with the
normality and other properties of the maximum likelihood estimates enables us
to claim that the SLRT has asymptotically (as 6+0) all the properties of the
classical Wald [16] sequential probability ratio test (SPRT) where 0 is assumed
to be known.
In the nonparametric setup, U
n
is the optimal (minimum variance) unbiased
estimator of 8(F), and 8(F ) is asymptotically equivalent, in probability, to
n
k<2
P
U Le., n [U -8(F )]+0 as n~ (cL [8]).
n
n
n
•
k<
k<
Moreover, n 2 [U -8(F)] (or n 2 [8(F )n
n
8(F)]) is asymptotically normally distributed with zero mean and variance
5
•
mZsl(F)(>O), which may be consistently estimated as follows.
. = (
V
n,l
For n>m, let
n-l -1 \'*
1)
L.. cP (X. ,X. , ... , X. ),
m-
1
1
1Z
1
(,c
i! i
m
where the summation I~ extends over all possible l<iZ< ... <i <n with i.fi,
1
Z~j~m.
J
nr-
-
Then, from Sen [10],
s
Z
n
\' n [V ._U]Z
n_ll L.i=l
n,l n
Ysl(F)
as
n~.
(2.12)
Thus, looking at (Z.lO) and the asymptotic SLRT, we are, at least heuristiA
-1
ZZ
cally, in favor of replacing en by Un' lee by m s , and thereby, considering
n
the following procedure.
Corresponding to our desired first and second kinds of error a, S (where
we take
O<a,S<~),
O<B<l<A<oo.
we consider two numbers
A(~(l-S)/a)
and
B(~S/(l-a»,
Then, we start with an initial sample of size nO(=nO(l'l),
large for small 6), and define a stopping variable N(=N(6»
positive integer
(~nO)
so that
rnodL'l H.
:''-
as the smallest
for which
n6[Un-~(eO+el)] ~ m2s~ log B, re accept HO: 8(F)=8 0 ,
while if n6[Un-~(80+8l)] ~ mZs~ log A, we accept HI: 8 (F)=8 =8 +6, 6>0.
l 0
is vitiated.
If for N(6)=N,
A parallel sequential procedure based on 8(F ) may be posed as follows.
n
_ -(m-l) \' n
\' n
f ¢ (X"
x ' ... ,x )
Let us define V* . - n
L.i =1···L.~ =1 ¢(X~,Xi , ... ,X~ ) =
1
m
2
n,l
Z.Lm.L
Z
.Lm
RP(m-l)
dF (xZ) .•• dF (x ), l~i~n, and let
J•••
n
•
n
m
6
•
(n-l)
-1 t n
2
[.. l[V* .-e(F )] .
1.=
n,1.
n
(2.14)
Then, in the second procedure, we replace, in (2.13), U by e(F ) and s by
n
n
n
s*, while the rest remains the same.
n
procedures have asymptotically (as
It will be seen later on that the two
6~0)
the same properties.
The heuristic proposal made above is justified theoretically in the next
two sections.
~~.
~.
We have the following theorem.
Under (2.5), both the procedures terminate with probability
one i.e., for every (fixed) e(F) and 6(>0), P{N(6»nle(F)}
+
0 as n~.
By definition in (2.13), for every n>n '
O
Proof.
2 2
n
Pe { r6 (log B) < [Ur-~(eO+el)] <
m s
P {N(6»n}
e
2 2
r
r6 (log A), n~r~n}
m s
(3.1)
where P stands for the probability computed for e(F)=8.
e
Now, we have three
(a) e(F) > ~(80+el)' (b) 8(F) < ~(eO+el) and (c) 8(F) = ~(eO+el)'
situations:
We know that {U } being a reverse martingale sequence (cf. Berk [4]) converges
n
almost surely (a.s.) to 8(F) as
(2.12) to
2
sn~sl (F)
a.s. as
a.s. converge to zero as
n~.
Also, Sproule [14] has strengthened
n~.
n~;
on the other hand, in cases (a) and (b),
[Un-~(eO+el)]~[e(F)-~(eO+el)](+O)a.s. as n~.
n~.
2 2
In case (c), m s (log
n
Hence, (3.1) tends to 0 as
B)/(6~) and m2 s 2 (log A)/(6~) both a.s. converge
n
1<
to 0 as n~, whereas n2[Un-~(80+8l)] is asymptotically normally distributed
2
•
with zero mean and a finite (positive) variance m sl (F); hence, (3.1) again tends
7
•
to 0 as n~.
Finally, it follows from Sen [12] that 18(F )-U 1+0 a.s. as n~,
n
n
and hence, the proof for the second procedure follows on parallel lines. Q.E.D.
4
For theoretical justifications, we now consider
the asymptotic situation where
~+O;
this is comparable to the asymptotic sit-
uation in (sequential) fixed-width confidence interval problems, treated in
Chow and Robbins [5], and others.
In practice, the results are valid whenever
we have the access to choose small 6.
For 6+0, we concentrate ourselves to a
range of possible values of 8(F) also contracting to 0 as 6+0.
we assume that 8(F)EI
6
Specifically,
where
{~:
I~I<K},
(4.1)
We may remark that if 8(F)~I6' then asympto-
where K(>l) is a finite constant.
tica11y the OC function will be either very close to zero or to one, and hence,
will cease to be of any statistical interest.
Further, looking at (2.13) and
the a.s. convergence of (i) Un to 8(F) and (ii)
s~
to sl(F), we are confident
that as 6+0, a terminal decision will not be reached (in probability) at an
early stage.
Consequently, there is no harm in letting
at a slower rate as compared to the ASN of N(6).
ASN of N(6) increases at a rate of 6
00
-2
but
as 6+0.
nO(=nO(6»~
as 6+00, but
We shall see later on that the
Hence, we assume that
o.
(4.2)
These assumptions are also implicitly needed to justify rigorously the procedure
of Bartlett [3] and Cox [6] .
•
8
•
Consider now a standard Brownian motion W:
O~t<oo,
t
P
P(¢,y)
W first crosses the line y
-1
log B +
t
{ before crossing the line y-l log A +
P (<p)
{<p:
and denote by
yt(~-¢)
(4.3)
yt(~-¢):
/<p/<K}.
(4.4)
Let then the OC function of the sequential procedure in (2.13) be
(4.5)
Theorem 4.1.
Under (2.5) and (4.2),
(4.6)
(4.7)
Remark.
The last equation specifies the asymptotic (as 6+0) consistency
of the proposed sequential test.
Proof.
For every E>O, let us define
[{(log AB
where [s] denotes the largest integer contained in s.
-1
2
)/E6} ],
(4.8)
Then, by (2.13),
P {N(6) < n(1)(6)}
e
- 0
2 2
2 2
m s (log B)
m s (log A)
r
P {
r
< [U -~(e +e ] <
6r
r
0 1
6r
e
Since,
Pe{s;
•
(4.9)
s~+Sl (F) a.s. as n+oo and (4.2) holds, we conclude that for every n>O,
~ (l+n)Sl (F), nO(6)~r~n(1)(6)} + 1 as 6+0.
above by
Consequently, (4.9) is bounded
9
•
where n(lI)-+O as lI-+O.
Now, for all 8(F)E:l , 18(F)-~(80+8l)
ll
I.::.
(K+l)lI, and hl,nci
for every K«oo), we can always select an E>O, such that 18(F)-~(eO+8l) I ~
2
l
l
(m /2l1r)l,1 (F) (l+n) log AB- for all r.::.nb ) (lI).
~
lI(E log AB
l
[Note that for r.::.nb ) (lI), (rll) --1
-1 -1
)
can be made large by proper choice of E>O.]
Hence, (4.10) is
bounded above by
(4.11)
Now, using the reverse martingale property of U-statistics (cf. [4]), and thereby
applying the Chow extension of the Hajek-Renyi inequality (as in Sen [11],
(3.6», the first term of (4.11) can be easily shown to be bounded by
.
l
2 2
2
[4l1 /m (1+n) (log AB-l)C,l (F)] [Cm C,m(F) {nb ) (lI)-nO(lI)}]
l
l
2
.::. nci ) (lI) [l,m(F)/l,l (F)] [4ClI /(1+n) log AB- ]
.::. COE .::. E', where Co <00, independently of lI, E and n.
Thus, for every 8>0, there exists an E>O, such that
(4. L3)
Again, as in (3.1), for every 8(F) EI ,
lI
(4.14)
n=n(2)(lI)
o
•
10
1
•
2
k
k
Now, sn+Sl (F) a.s. as n-xJo, and n 2[U -e(F)]/m s {(F) is asymptotically normally
n
k
Therefore, n2[Un-~(eO+el)]/msn
distributed with zero mean and unit variance.
k
is asymptotically normally distributed with mean [e(F)-~(eO+el)]/m[L;l (F)] 2 and
unit variance.
On the other hand, m(log AB
k
almost surely to mE[Sl (F)] 2, as 6+0.
-1
~
)sn/6n, for n=n
(2)
O
(6), converges
Hence, the right hand side of (4.14) is
asymptotically bounded above by
-k
k
(2n) 2mE [Sl (F)] 2 <E',
which can be made smaller than
~o
(4.15)
by proper choice of E(>O).
This leads us to
for every e(F)EI , and for every 0>0, there exists an E(>O),
6
the following:
such that
limA~O P {n(1)(6) < N(6) < n(2)(6)} > 1-0.
LF
eo
-
-0
(4.16)
Consider now a stopping variable N*(6) defined as the smallest positive
integer
n(~nO(6))
for which
(4.17)
is vitiated; if
n[Un-~(eO+el)] ~ m2sl (F) (log B)/6 accept HO' and accept HI if
n[Un-~(eO+el)] ~
2
m sl (F) (log A)/6.
By the same technique as in above, it
follows that (4.16) also holds for N*(6).
hence, as 6+0, for all
Thus, if we denote by
•
2
Further, sn+Sl (F) a.s. as n-xJo, and
n6l)(6)~n~n62)(6), IS~-Sl (F)/+O,
L~(¢),
with probability one.
the OC function of the procedure in (4.17), then
from (2.13), (4.17) and the above discussion, we conclude that
11
•
o
ncil)(6)~n~n62)(6), n~6
Since for
for all
r
(1)
r t nO
(2)
~r~nO
h
2
between two consecutive
(6), we may, after using the results in Sen [12], conclude that
the process weakly converges to a Brownian movement process
E{~(t;6)}
=
(4.18)
is bounded above by a positive constant
K «00), if we linearly interpolate 6r[U -8(F)]/m[sl(F)]
E
¢EI.
O,~t
6' = 6/m[sl (F)]
h
2,
2
and E{~(s;6)~(t;6), s~t} = s6 ,O<s<t<oo.
~(t;6)t
where
As such, if we let
we have from (4.13), (4.17) and the above that
lim + IL~(¢)-P(¢,6')1 = 0
6 0
for all
(4.19)
¢EI,
and hence, by (4.4), (4.18) and (4.19)
P(¢)
For testing a simple H :
O
8=8
0
for all
against a simple HI:
¢EI.
(4.20)
8=8 =8 +6 (F known)t
1 0
consider the Wald sequential probability ratio test (SPRT) of strength (a,S).
For small 6 t the excess over the boundaries can be neglected t so that
A = (1-S)/a+0(6), B = S/(1-a)+0(6)t and on denoting by L~(¢) the OC function
of the SPRT, we have from Wald [16]t
I-a
and
lim + L~(l)
6 0
s.
(4.21)
On the other hand, the sequence of logarithm of Probability ratio forms a
martingale sequence on which Theorem 4.4 of Strassen [15] yields the weak con-
•
vergence to a Brownian movement process.
Consequently, as in (4.19) and (4.20),
12
•
P(~)
~EI.
for all
(4.22)
Thus, (4.7) follows directly from (4.20), (4.21) and (4.22).
Q.E.D.
!.:
In passing, we may remark that by Sen [12], n 2 [8(F )-U ]+0 a.s. as n+oo
n
2
and Is -(s*)2 1+0 a.s. as n+oo.
n
n
Consequently, by some routine steps it follows
n
that Theorem 4.1 remains valid for the alternative procedure based on 8(F ) and
n
s*.
n
We now proceed to study the ASN of N(6).
It is quite clear from (4.8) and
(4.16) that as 6+0, E {N(6)}+oo for all 8(F)EI . However, we shall see that
6
8
2
under certain regularity conditions, 6 E {N(6)} converges to some limit (depend8
ing on 8(F) and F) as 6+0.
This result is used in the next section to compare
the asymptotic efficacy of the proposed procedures.
In addition to (2.5), we assume that for some 8>0,
for all
V(F)
(4.23)
It may be remarked that in the classical SPRT, Wald [16], while computing the
ASN assumed the existence of the moment generating function of the logarithm
of the density ratio, and (4.23) is no more restrictive than that.
theorem relates to the rate of growth of the ASN when 8(F) +
Theorem 4.2.
l
(F){P(~) log B+[l-P(~)] log A}
(~-~)
•
E~
~(80+8l).
Under (2.5), (4.2) and (4.23), for every ~(+~)EI,
m2s
where
Our next
stands for the expectation under 8(F)
=
80~6, ~EI •
ljJ (~),
(4.24)
13
•
Proof.
We only consider the case of
¢>~;
the case of
¢<~
follows similarly .
For some arbitrarily small E(>O), we define
[E LJ.A- 2 ]
an d
n (A)
LJ.
2
[m
2
~l(F)(log
AB
-1
)1'1
-2 -1
-1
E (¢-~) ].
(4.25)
Then, we have
(4.26)
where obviously,
t
(4.27)
Now, by using the fact that Ln>knP{N=n}
Let now n(>O) be an arbitrarily small positive number.
(I'1n)
-1 2
-1
m (log AB
)(l+n)~l (F)<I'1E(¢-~)
(4.14), we have for
for all
n~n2(1'1),
Then, upon noting that
and proceeding as in
n~n2(1'1),
2
2
2
2
P¢{N(I'1»n} ~ P{m (log B)Sn<nl'1[Un-~(eO+el)]<m (log A)sn}
2
2
~ P{m (log B)(l+n)~l(F)<nl'1[Un-~(eO+el)]<m(log A)(l+n)~l (F)}
+
•
P{s~>(l+n)~l (F)}
< p{[u -e(F)]>(¢-~)6(1-E)} + p{s2>(1+n)~1(F)}.
n
n
(4.29)
14
•
Now, from Sen [13], for every r~2, if E/¢I
r
<00,
E{!· [u -e(F)] Ir } < C n- r / 2 , C <00,
n
-
r
(4.30)
r
and therefore by the Markov inequality, under (4.23),
P{[U -8(F)] >
n
C¢-~)6(1-s)}
2
P{IU -8(F)I>(¢-s)6(1-S)}
n
2
C[(¢_~)6(1_S)]-4-8n-2-8/2, C<oo.
Again, it has been shown by Sproule [14] that s
(4.31)
2
in (2.12) can be expressed
n
as a linear combination of several U-statistics, for each of which, moment of
the order 2+8/2 exists, for some 8>0 (implied by (4.23».
Hence, using (4.30)
and the Markov inequality,
n
where 8'=8/4>0 and C(n)<oo for every n>O.
-1-8'
(4.32)
From (4.25), (4.28), (4.29), (4.31)
and (4.32), we have by routine computations that
lim \'
6-+0 L. n >n (6)nP¢{N(6)=n}<s',for all
z
where s'(>O) can be made arbitrarily small, depending on s(>O).
it suffices to
•
sho~
that for
¢>~(SI),
(4.33)
¢sI,
Therefore,
15
•
For this, we define
and
(4.35)
and let N.(6), i=1,2, be two stopping variables, defined to be the smallest
1
positive integer
n(~nO(6))
for which the following
(4.36)
is vitiated, where n(>O) is arbitrarily small, and for N.(6)=n, if
1
mn
2
i
6[Zn- ~(80+8l)] ~ m (l+(-l~ n)~l (F) log B, we accept HO: 8(F)=8 , while if
0
mn
2
i
6[Zn- ~(80+81)] ~ m (l+(-U n)~l (F) log A, we accept HI: 8(F)=8 1 ; i=1,2.
Now, the Z
n
involve summations of i.i.d.r.v., and for small 6, the excess over
the boundaries can be neglected.
Further, proceeding precisely on the same
line as in the proof of Theorem 4.1, it follows that the DC function for each
N.(6) satisfies (4.6) and (4.7), i=I,2.
1
Wald [16], we have for
Hence, by the fundamental identity of
¢>~(EI),
(l+(-l)in)~(¢), i=1,2.
(4.37)
Also, by arguments similar to in (4.25) through (4.33),
(4.38 )
•
16
"
•
for all
¢>~(€I).
Lemma 4.3.
Let us now consider the following .
For every n (>0), under (2.5),
l
o.
Proof.
(4.39)
It follows from Berk [4] that {U ,n>m} forms a reverse martingale
n
-
sequence, and the same proof applies to {n -1 Z ,n>l } ; both these are defined
n
with respect to a common sequence of a-fields.
also a reverse martingale sequence.
-
Hence, by (4.35), {W ,n>m} is
n
-
Now, for {Wn,nO(6)<n~n2(6)}, we reverse
the ordering of the index set {i} to {n (6)-i+l, 1~i~n2(6)-nO(6)}, and thereby,
2
convert the sequence into a (forward) martingale sequence.
Then, we use the
Chow extension of the Hajek-Renyi inequality (cf. [11, (3.6)]), and obtain that
2
as some routine computations yield that E{W } ~ Cls (F)n
l
n
~
•
Cls (F)[n
l
-2
-(n+l)
-2
] (where Cl<oo).
-2
and E{W2-W 2 }
n n+l
Since the right hand side of (4.40) con-
verges to 0 as 6+0 (for every fixed n>O), the lemma follows .
17
•
Also, by (4.32), for every n >O, there exists a positive C(n ) «00), such
1
1
that
O.
where
Therefore, by (4.35), (4.39), (4.41) and the definition of
2
< P¢ {m (log B) (l+n)Sl (F) <
(4.41)
N(~),
for every n
rm
~[Zr- ~(eO+e1)]
2
< m (log A) (l+n)sl(F) , nO(~)~r<n} + V(~,n)
(4.42)
where
1im~+O v(~,n)=O.
Essentially, by similar steps,
(4.43)
for all n E[n
we get that
•
l
(~),n2(~)]
and
~+O.
Consequently, by (4.34), (4.37) and (4.38),
18
•
:~mO
u~
1:: .
2
E~N(A)
~
U
<
(l+n)l"(~)
+ 1:::.-+0
lim{Au 2V (Au,n )[ n (A)
~ ~
u -n (A)]
u
2
Since, by (4.25), (4.40) and (4.41), liml:::.-+O
I:::.
(4.44)
l
2
v(l:::.,n)n (1:::.)-+0 and n(>O) is
2
arbitrary, the proof follows.
A similar proof follows for the alternative procedure based on {e(F )}
n
and {s*}.
n
The above proof fails when
¢=~.
(d/d¢)P(¢)I~_o
However, if
= P'(¢) exists,
(d/d¢)P(¢)I
(4.45)
~+o
by using (4.24) and the L'Hospital's rule, we obtain that
(4.46)
Derived from the theory of
bounded-length (sequential) confidence intervals of Chow and Robbins [5],
Albert [1] has considered a second type of sequential tests in linear models,
which may be extended as follows.
~
~
Had sl(F) been known, n 2 [U -8(F)]/m s ;(F) would have asymptotically a normal
n
distribution with zero mean and unit variance.
Hence, for a test for (2.6),
based on U , if for small 1:::., we want to have the strength (a,S), we could have
n
used a fixed-sample size procedure with n specified by
(5.1)
•
19
•
where T is the upper lOOa% point of the standard normal distribution.
a
s~,
n~,
defined by (2.12), converges a.s. to Sl (F) as
Since
we may consider the
following sequential procedure:
define a stopping variable
n for which
< T
a
N*(~)
s~m2(Ta+TS)2/~ ~
as the smallest positive integer
n; accept H :
O
8(F)=8
0
if
n~[Vn-80]/msn
and reject H ' otherwise.
O
Along the same line as in Sproule [14], who extended the Chow-Robbins [5]
theory of bounded-length confidence intervals to V-statistics, it follows that
(5.2)
1, VepsI,
(in fact,
EepN*(~)
does not depend on ep).
Thus, on comparing (4.24), (4.46),
(5.1) and (5.3), we obtain that the asymptotic relative efficiency (A.R.E.) of
N*(~)
with respect to
e(ep)
N(~)
is
lim[E N(~)]/[E N*(~)]
ep
ep
~-+O
P(ep)log B+[l-P(ep)]log A
(ep-!i) {T +T S}
a
2
if
For
ep=O
or 1,
peep)
ep=~.
is known, and hence, e(ep) can be easily computed.
(5.3)
For
example, when a=S=.05 and ep=O(or 1), e(ep) equals to 0.48, which is considerably
less than one.
In fact, the table on page 57 in Wald [16] is quite appropriate
here and reveals the supremacy of the earlier procedure over the one considered
in this section.
•
Because of the remark made before (4.45), the two procedures
20
•
of section 2 are asymptotically equally efficient too .
(i)
Sequential test for variance of a distribution.
Let {X ,X , ••. }
1 2
be iidrv with a univariate d.f. F(x), defined on the real line (_00,00), and let
00
J
]J (F)
_00
where we assume that O<cr(F)<oo.
I.
n
1=
2
l(X.-X
) In
1
n
(6.1)
We want to test
2
cr o (nown
k)
Since for the kernel ~(X,Y)
2
2
x dF(x)-]J (F),
_00
= ~(X_y)2, E~
and
2
cr (F)
H1:
vs.
2
crO+L'., L'. known.
2
cr (F), by (2.8) and (2.9), we have
2
[n/(n-1)]cr (F ), where X
n
n
U
n
(6.2)
=!n L.~ i=l
n
X.• (6.3)
1
Also, by (2.11), (2.12) and some routine steps, we get that
s
2
n
(6.4)
(s*) 2
n
1
-4
n
~n
L·
1=
-
l(X.-X)
1
n
4
1
2
2
(6.5)
- '1;[cr (F n )] .
Hence, we have no problem in following the sequential procedures in (2.13) or
the alternative one after (2.14).
Incidentally, here ~1 (F)
= ~{E(X-]J) 4-cr 4 (F)},
and hence, the computations in (4.24) or (4.46) pose no problem.
(ii)
•
Sequential tests for independence.
Let X.
be iidrv with a bivariate d.f. F(x ,x ), -OO<x ,x <00.
1 2
1 2
1
We want to test the
21
•
hypothesis that x(l), x(2) are stochastically independent i.e.,
(6.6)
Various alternative hypotheses may be framed to test for departure from independence.
This may be the usual correlation between x(l), x(2), or the grade
correlation {for definition etc., see [8 ]}, or some sort of association of
the two variates.
We consider here a simple case; the other cases follow on
parallel lines.
Let us define the probability of concordance by
n(C)
(6.7)
(F)
where we assume that F is continuous.
(C)
n(F»(or
iates.
<)~
(6 • 6) , n(C)-k
.
Th en, un d er H0 ln
(F)-2, while
indicates a positive (or negative) association between the two var-
In fact, the tau correlation coefficient, proposed by Kendall and others,
is given by reF) = 2rr(C) (F) - 1.
Thus, we want to test for H in (6.6) against
O
an alternative that reF) = 6, 6>0.
otherwise.
Then,
(6.8)
U
n
unbiasedly estimates reF).
In a sample of n observations, let C . be the
nl
number of X.,
l<j(~i)<n, which are concordant with X., l<i<n, so that
J
T 1
--
•
Un
=
4
\ n
1
n(n-l) Li=l Cni - .
Then, by (2.11) and (2.12), we have here
22
•
2
1, n
-1
2
s n = ---1
n- L·1=1{2(n-l) Cn1.-Un } .
Thus, we can again proceed as in section 2.
for small
~,
(iii)
we may replace
s~
by
t
(6.9)
Since under (6.7), 1';1 (F)
in the definition of
= 9'
1
N(~).
Let {X 'X ' .•. }
l 2
Sequential tests based on generalized U-statistics.
be iidrv with a d.f. F(x), and let {Y ,Y , .•. } be an independent sequence of
l 2
iidrv with a d.f. G(x).
Consider a functional
J... J ~(xl'···'xm
8(F,G)
l
'Yl'···'ym )dF(xl)···dF(xm )dG(yl)···dG(Ym ) (6.10)
l
Z
Z
(6.11)
U
n
be the U-statistic corresponding to 8(F,G), while 8(F ,G ) is defined by
n
replacing F and G in (6.10) by Fn(x) = n
c(y-Y.), respectively.
]
.
.
n,O]
Li=l c(x-Xi ) and Gn(y) = n
~
Lj=l
(X. , X. , ... , X. , Y. , •.. , Y. ) ,
1 12
1
J1
J
ml
mZ
(n)-l( n-ll)-l 'C
'j* ~(X. , ... ,X. ,Y.,Y. , ... ,Y. ),
m
mL
L
1
1m J J Z
1m
n,m
l
Z
1
l
2
l
where the summation L~ (L~) extends over all possible l<iZ< ... <i
1
•
-1 , n
Also, let
vn,10
.
v
n
-1 , n
J
-
<n with
m~
i.fi, j=Z, ... ,m l (1~i2< .•• <im ~n with jifj , i=Z, ... ,m2 ), and finally set
]
2
(6.lZ)
(6.13)
23
•
(6.14)
s
2
n,l
__1__
n-l
Li=l
n [V
_U]2
n,io n '
s2
= ~
n,2
n-l
Lj=l
n [V
_U]2
n,oj n .
(6.15)
Then, we may consider a sequential procedure for testing
(6.16)
2 2
based on the same stopping variable as in (2.13), wherein we replace m s by
n
2
A similar procedure follows for e(F ,G ).
v.
n
n
n
Let then cjJ10(x.) = E{cjJ(x .. X. ,H',X. 'Yo ,H"Y . )}, cjJ01(Y1')
1
1 1
1m J
J
2
11m
2
E{cjJ (X. ,. H ,X. ,y .. Y. ,H" Y . )}, and leE
11
1m
J J2
1m
2
1
(6.17)
Then, by direct generalization of Theorems 4.1 and 4.2, it can be shown that
(4.6), (4.7), (4.24) and (4.46) hold; the only change is to replace m2S1 (F)
2
by V (F,G).
c(~2)
The above results also can be easily extended to functiona1s of
independent distributions.
We conclude this section with an example of e(F,G) having a lot of practical
importance.
Consider the functional
e (F ,G)
-r
p{X<Y}
f
F(x)dG(x) ,
and suppose we want to test the hypotheses in (6.16).
•
ways of constructing sequential tests for the same •
(6.18)
There are two possible
24
•
(a)
i>l.
The Wald SPRT.
Define r. as 1 or 0 according as X.<Y. or X.>Y.,
J
~
~
~-
~
Thus, P{r.=l} = E{r.} = 8(F,G), and as the r. are iidrv with a binomial
~
~
~
distribution, we may directly use the Wald [16] SPRT.
Thus, theorem 4.1 holds
here, while the ASN, as given in Wald [16, p.s7] reduces (as 6+0) for 8(F,G) =
P(¢)log B + {l-P(¢)}log A
(6.19)
(b)
Asymptotic sequential test based on the Wilcoxon statistic.
The
generalized U-statistic corresponding to 8(F,G) is
n- 2 ,.nl{NO of X.<Y., l<i<n}.
~
LJ=
n- l ,.n
LJ = l
J
F (Y.),
n
(6.20)
J
where Fn(x) is the empirical d.f. for Xl' ... ,X .
n
By definition, in (6.12)-
(6.15), we have
V2
n
1
= ---1
n-
{I . n l[l-G
~=
n
(X.)-U] 2 +
~
n
where G is the empirical d.f. for Yl, ... ,Y .
n
n
•
(2.13) with the change suggested after (6.17) .
I .n l[F
J=
n
(y.)-u] 2} ,
J
n
Thus, we may proceed as in
(6.21)
25
•
Note that by (6.17)
(6.22)
so that the ASN as
V
~+O
2
is given by
(F,G)
~~~2 {P(¢)log B + [l-P(¢)]log A}{l+o(l)},
(6.23)
(¢-~)~
where e(F,G) =
eO+¢~'
¢eI.
From (6.19) and (6.23), we obtain the A.R.E. of
the Wald SPRT with respect to the proposed test equal to
(6.24)
eO=~'
In particular, when we want to test F=G against shift alternative,
1 2
v2
(F,G)=s, so that (6.24) equals to 3' which clearly indicates the supremacy
of the second procedure.
In general, (6.24) is less than unity; this is
because of the fact that whereas U compares every X. with every Y., the Wald
n
l
J
SPRT does not do so, and hence, looses some information.
REFERENCES
•
[1]
A. Albert, Fixed size confidence ellipsoids for regression parameters.
Ann. Math. Statist. 37 (1966), 1602-1630.
[2]
F. J. Anscombe, Large sample theory of sequential estimation.
Cambridge Phil. Soc. 48 (1952), 600-607.
[3]
M. S. Bartlett, The large sample theory of sequential tests.
Cambridge Phil. Soc. 42 (1946), 239-244 .
Proc.
Proc.
26
•
•
[4]
R. H. Berk, Limiting behavior of posterior distributions when the model
is incorrect. Ann. Math. Statist. 12 (1966), 51-58.
[5]
Y. S. Chow, H. Robbins, On the asymptotic theory of fixed-width sequential
confidence intervals for the mean. Ann. Math. Statist. 36 (1965), 457-462.
[6]
D. R. ~ox, Large sample sequential tests of composite hypotheses.
Sankhya, Ser. A ~ (1963), 5-12.
[7]
P. R. Halmos, The theory of unbiased estimation.
(1946), 34-43.
[8]
W. Hoeffding, A class of statistics with asymptotically normal distribution. Ann. Math. Statist. ~, (1948), 293-325.
[9]
R. V. Mises, On the asymptotic distribution of differentiable statistical
functions. Ann. Math. Statist. 18 (1947), 309-348.
Ann. Math. Statist. 17
[10]
P. K. Sen, On some convergence properties of U-statistics.
Statist. Assoc. Bull. 10 (1960), 1-18.
[11]
P. K. ~en, The Hajek-Renyi inequality for sampling from a finite population.
Sankhya, Ser. A ~ (1970), 181-188.
[12]
P. K. Sen, Convergence of sequences of regular functiona1s of empirical
distributions to processes of Brownian motion. lnst. Statist., Univ.
N. Carolina Mimeo Report No. 726, January 1971.
[13]
P. K. Sen, On LP-convergence of U-statistics.
(submitted, 1971).
[14]
R. N. Sproule, A sequential fixed-width confidence interval for the mean
of aU-statistic. Ph.D. Thesis, Univ. N. Carolina, Chapel Hill, 1969.
[15]
V. Strassen, Almost sure behavior of sums of independent random variables
and martingales. Proc. 5 th Berkeley Symp. Math. Statist. Prob. 2 (1967),
315-343.
[16]
A. Wald,
Sequential Analysis.
Calcutta
Ann. Math. Statist.
John Wiley, New York, 1947 .