..
)
AN INVARIANT SEQUENTIAL TEST FOR ZERO DRIFT
BASED ON THE FIRST PASSAGE T!MES IN BROWNIAN MOTION*
by Nanak Chand**
Univerasity of Norath CaraoZina at ChapeZ HiZZ
•'.
Institute of Statistics Mimeo Series #987
March, 1975
* This research was partially supported by the Army Research Office, Durham,
under contract DAHC04-74-C-0003.
** Currently at the North Carolina Department of Human Resources.
AN INVARIANT SEQUENTIAL TEST FOR ZERO DRIFT
)
BASED ON THE FIRST PASSAGE TIMES IN BROWNIAN MOTION*
by
Nanak Chand
University of North Carolina
and
North Carolina Department of Human Resources
1.
Let Wet)
INTRODUCTION
be a separable Brownian Motion process with W(O)
a, and diffusion constant
~2.
= 0,
drift
It is well known (see, for example, Cox and
Miller [3, p. 210] that the first passage time T to the state A > 0 has the
density function
pet)
~,.
where ~
= (A/t3)~~(A~(l-~t)/t~),
is the standard normal density function,
t >0
~
= a/A
(1.1)
and
2
A = A /o 2 •
We shall construct an invariant sequential test for the hypothesis of zero
drift against the alternative of positive drift, based on the observations
t l ,t 2 , •..
on the first passage times.
Nadas [5] has given a fixed sample size
test for this problem, Seshadri and Shuster [6] have evaluated the exact critical region based on Nadas' test statistic.
Section 2 contains the derivation of an invariantly sufficient sequence
of statistics for testing the hypothesis of zero drift.
tonicity of and an asymptotic
sections 3 and 4 respectively.
forrr~la
A proof for the mono-
for the likelihood ratio are given in
It is proved in section 5 that the probability
is one that the sequential test comes to a conclusion with the acceptance of
either the null or the alternative hypothesis.
This research was supported in part by the Army Research Office - Durham
under Contract DAH C04 74-C-0030.
2
•
2.
DERIVATION OF THE TEST SEQUENCE
Since A > 0, the drift is zero if and only if T = ~A =
o.
We shall
first develop an invariant sequential test for testing HO: T = 0 against
HI: T = ~, where ~ is a positive number. It shall be shown in thp. sequel
that the test remains valid for testing HO against the alternatives Hi: T ~~.
For a positive number c, the density function of cT is the first passage
density (1.1) with
~
replaced by
and with A replaced by AC.
~/c
Thus
the problem of testing H vs. HI remains invariant under the group G
O
of positive scale transformations and for a sample of size n,.(tl/tn, .•• ,tn/tn )
is a maximal invariant on Rn , the n-dimensional Euclidean space. The induced
group G on the parameter space also consists of all positive scale transformations on the quadrant
e-
(~ ~
O,A > 0)
A maximal invariant under G is T.
on the sequence {x. = t./t ,
lIn
e·
n=1,2, .•.•
be!) = r~=lti
(tI, ••. ,tn)
Making the change of variables from t.1
where
g(~,A)
=
(~/C,AC)
for c >
o.
An invariant test is, therefore, based
i=l,.~.~n};
C = (A/2Tr)n/2, rr(~) = rr~=lti3/2,
Then the joint density of
so that
We shall use
~he
notation
and d(~) = r~=ltil .
is, from (1.1),
to x.l=It./t
, i=l, ••• ,n, we obtain
n
n
n
v = b(x)d(!J = b(!)d(.!) = L t. L t~l
n
i=l Ii=l 1
(2.2)
•
3
It follows from the factorization theorem that
the joint distribution of
v
n
is sufficient for
L
in
(xl, ••• ,xn).
Thus the sequence {vn } is an invariantly sufficient sequence f~the family of distributions (1.1) under the
group G of transformations on Rn . An alternative way is to obtain a sufficient sequence, to employ an invariant reduction on this sufficient sequence
and then to prove that the resulting sequence is, in fact, invariantly sufficient.
An elegant theory has been developed which gives sufficient condi-
tions for the resulting sequence to be an invariantly sufficient sequence.
(See Hall, Wijsman and Ghosh [4]).
Since we derived vn directly, we do not
resort to this theory in the present context.
fo(xl""'~)'
To obtain an analogous expression for
e·
T=
O{~=O,A>O),
we note that for
the density function of T is
t
Proceeding as above, the joint density of
> 0 .
(xl, ••. ,xn)
(2.3)
is obtained as
00
ri/2
f o = fO(xl, ... ,xn ) = [II{x)b
The likelihood ratio
observed values
f~/fO
(tl, ..• ,tn)
n
-n/2f s -2 -1e
(X)](41T)
o
vn
-4s
ds.
.
at the n-th stage of sampling depends on the
only through vn ' and is given by
Ln(vn ) • 6neXP{nh}f=h1 (S)d,;j=h o(S)ds
o
where hl(s) =
h(s,~)
n
= s-
(2.4)
2 -lexp{_s _
(2.5)
0
2
V~~}
and hoes) = exp{s}h(s,l).
It is easily verified that (2.5) is the ratio of the density functions of v
n
at T = ~ and at T = O. The continuation region of the sequential test is
(2.6)
4
~
where b
= ~n(a/(l-a))
and a
= ~n((l-S)/a).
the test terminates with probability one.
excess of
~n[Ln(vn)]
It shall be proved later that
Then, on the assumption that the
over the decision boundaries is negligible on the
termination of the test, the test has the desired strength (a,S).
first show that
Ln (v)
n
3.
Let
~n(vn)
We shall
is a decreasing function of vn •
MONOTONICITY OF THE LIKELIHOOD RATIO
denote the log likelihood ratio
at the n-th stage of sampling.
The derivative of
(~n(vn)
~n(vn)
= ~n[Ln(Vn)])
with respect to
vn is obtained from (2.5) as
0000
f f h1(s)hO(t) (t-
1
2 l
_6 s- )ds dt
o0
ff
=
1 2 1
hl(s)hO(t) (t- _6 s- )ds dt
{S>A 2t}
+
ff
h1 (s)hO(t) (t- l _A 2S- 1)dS dt
{S<A 2t}
If we substitute A2t
= u,
s
=v
in the first integral and
in the second integral, we obtain
ff
{v>u}
e.
(u -1. -v -1 )g(u,v)du dv ,
s
=u
, 6 2t
=v
5
v 60 2
)
n
n
-1-1
~I - -4-- (u +v ) (e-v_e- u )
2
where g(u,v) = (A /uv)
g(u,v)
e
over the whole range of integration.
It follows that
is negative
and hence
n (v)
n
is a strictly decreasing function of vn • Thus the decision rule
~
Ln(vn)
of the test can be written as
(H ):
1
where
!n
and v
v :
v < vn < n
-n
for n=2,3, ..•
(H )
O
are the solutions of
n
=a
~
)
n (v
~
~
and
n (vn )
=b
.
The above monotonicity property of Ln(vn) also implies that the test is
valid for testing HO against all alternatives Hi: T ~ A •
4.
ASYMPTOTIC FORMULA FOR THE LIKELIHOOD RATIO
co
We consider first the integral
II
=/
o
s = 2-lv~6o exp{S} , we obtain
hl(s)ds.
Substituting
co
= (2/V:A)n/2f
II
exp{-v:A cosh S -
n~}dS
-co
where cosh(o) is the hyperbolic cosine function.
Thus
II
may be written
as
(4.1)
co
where Kv(Z) = (%)/ exp{-z cosh S - vS}dS , is the modified Bessel function
-co
of second kind (see Abramowitz and Stegun [1]).
co
1 =
0
fo
s
n
--2 -
Vn
1e - -4s
ds =
SVn
s2 -le- ""4 dS
co
f
Also, we have
n
0
= 2nrCn/2)/vn/ 2
n
(4.2)
6
Substituting (4.1) and (4.2) in (2.5), we obtain
-!!.1
L (v ) = 2 2 ~n/2en6vn/4K
(v~~)[r(n/2)]-1.
n n
n n/2 n
(4.3)
2
' 2
2
We shall write wn = 4vn In , -i'l
w = 4v-i'l In and wn = 4vnIn , and shall denote
the likelihood ratio by Mn(Wn) and its logarithm by mn(wn). It is easily
seen that
(4.4)
Now Stirling's asymptotic formula for the Gamma function is
(4.5)
as v
~
00,
and an asymptotic formula for
Kv(Vz) (see Abramowitz and Stegun [1])
is
K (vz)
v
as v
~ 00,
where n
=. (~/2V)~(1+z2)-~e-vn{1
= (1+z2)~
totic forms of rev)
+ O(v- 1)}
+ log{z/(1+~+z2)}.
(4.6)
Substituting these asymp-
and Kv(Vz) in (4.4) and taking logarithms, we obtain,
after a routine calculation,
(4.7)
G(W)
= log[1+(1+~2)~]
5.
- log 2 - (1+w~2)~ + 1+2~ •
(4.8)
TIlE TERMINATION PROPERTY
To prove that the above sequential test terminates with probability
one, we shall first show that there exists a unique number t
lying in
a bounded open interval (a',b') such that
e·
and
(i)
mn (w)
~
00
for W < t
(ii)
mn (w)
~
_00
for w >
~
•
(5.1)
7
Writing a'= 4(6- 1+1)
G(a')
and b'= 2(a'+4), it follows from (4.8) that
= log(l+~)
and G(b') =
Using the inequality log(l+x)-x < 0 for
log(l+2~)-2~
•
(5.2)
x > 0, it follows from (4.7) and
(5.2) that
lim m (a') =
n-+oo n
and
00
lim mn (b.') = _00
(5.3)
•
Jl'"l'OO
Also, from (4.8) ,
(5.4)
This implies that
lim
n-+oo
d:[~(W)]
for all
< 0
The existence and uniqueness of t
need an explicit form of
~
seen that
is a
solutio~
We shall follow the
a decision is one.
~
w>0 •
follows from (5.3) and (5.5).
for the present purpose.
of
(5.5)
G(~)
a~$UI1lent
=0
We do not
However, it is easily
•
of Cox [2] to show that the probability of
The probability that a terminal decision is not reached
by stage n
is less than or equal to the probability that vn lies in
the open interval (~,vn) which, in turn is equal to the probability that
wn lies in the interval (~,wn). We call this probability Pn .
It is easily seen that the maximum likelihood estimator En of T- l
is (wn /4) - 1. The asymptotic variance of n~En
is V = T- 2 (T- I +2). Thus the asymptotic normality of the maximum likelihood
estimator implies that n~En has asymptotically a normal distribution with
e.
mean n~T -1 and variance V.
Thus, for large values of n,
(5 ;(5')
where
En
= (~/4)-1
and E =
n
(w /4)-1.
n
As
n tends to infinity, wand
n
8
4Il
~
n
both tend to
+ ~
~.
Thus, unless
~ = 4(1+T- l ),
Pn tends to zero as
since both terms in (5.6) approach either one or zero.
The above argument does not apply when
~
= 4(I+T -1 ).
It follows from
(4.7) and (5.4) that the derivative of mn(w) with respect to w is
d~[mn(W)] = (n/2)[g(w) + O(n-l)] ,
where g(w) =- A2/2{l+ (1+~2)~}
(5.7)
Since mn (wn ) = band mn (~) = a, we have,
by the mean value theorem,
-)
(a-b)/n~ (w
-w
'-fl n
for some w
€
~,wn)'
n +~, since g(w)
= (n~/2)g(w)
+ O(n -~)
(5.8)
The right hand side of (5.8) has the limit
is negative for all wand all A > O.
has the limit zero as n
+~.
-~
as
Thus n~(wn-n
-w )
Observing that
1~(c)-~(d)1 < /c-dl,
for -~ < d <
C
<~
(5.9)
we obtain, from (5.6),
(5.10)
Thus the test terminates with probability one also for the case
~
= 4(1+T -1 ).
REFERENCES
[1]
Abramowitz, M. and Stegun, LA. (Ed.), Handbook of MathematiaaZ, Funations,
National Bureau of Standards, Applied Mathematics Series No. 55,
Washington, D.C.: U.S. Government Printing Office, 1964.
[2]
Cox, D.R., "Sequential Tests for Composite Hypotheses," Proaeedings of
the Cambridge PhiZosophiaaZ Soaiety, 48, (1952), 290-99.
[3]
Cox, D.R. and Miller, H.D., The Theory of Stoahastia Proaesses, London:
Methuen and Co. Ltd., (1965).
[4]
Hall, W.J., Wijsman, R.A. and Ghosh, J .K., "The Relationship between
Sufficiency and Invariance with Applications in Sequential Analysis,"
AnnaZ,s of MathematiaaZ Statistias, 36, (1965), 575-614.
•
e.
9
[5]
Nadas, A., "Best Tests for Zero Drift Based on First Passage Times
in Brownian Motion," Technometrias~ 15~ (1973), 125-32.
[6]
Seshadri, V. and Shuster, J. J ., "Exact Tests for Zero Drift Based on
First Passage Times in Brownian Motion," Technomet1'ics, 16, (1974),
133-34.