AN ASYMPTOTICALLY EFFICIENT TEST FOR THE
BUNDLE STRENGTH OF FILAMENTS
By
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina, Chapel Hill, N. C.
Institute of Statistics Mimeo Series No. 805
February 1972
AN ASYMPTOTICALLY EFFICIENT TEST FOR THE BUNDLE STRENGTH OF FILAMENTS*
BY PRANAB KUMAR SEN
University of North Carolina, Chapel Hill
Based on a Wiener process approximation, a sequential test for the bundle
strength of filaments is proposed and studied here.
Asymptotic expressions for
the OC and ASN functions are derived, and it is shown that asymptotically the
test is more efficient than the usual fixed sample size procedure based on the
asymptotic normality of the standardized form of the bundle strength of filaments,
studied earlier by Daniels (1945), and Sen, Bhattacharyya and Suh (1972).
*Research sponsored by the Aerospace Research Laboratories, Air Force systems
Command, U. S. Air Force, Contract F336l5-71-C-1927. Reproduction in whole
or in part permitted for any purpose of the United States Government.
1.
INTRODUCTION
Consider a sequence {X.1 ,i>l}
of independent and identically distributed
(iid) non-negative random variables (rv) with an absolutely continuous distribution function (df) F(x), defined on (0,00).
For every
n~l,
the ordered
variables corresponding to X1 "",Xn are denoted by Xn,l<"'<X
-n,n
D = max 1<.< [(n-i+l)X .], Z = n
n
_l_n
n,l
n
-1
Let then
D •
n
(1.1)
When the X. represent the breaking stresses of filaments, D is the maximum
1
n
stress which a bundle of n parallel filaments of equal length can stand, and is
termed the bundle strength [cf. Daniels (1945)].
We assume that F has a finite
second moment, so that
o<
A2 =
f
00
o
x 2 dF(x) <00,
(1.2)
and x[l-F(x)] has a unique maximum at xo(O<xo<oo) i.e.,
(1. 3)
Note that
(1. 4)
Further, the first derivative of x[l-F(x)] vanishes at x=x O' so that
f(x ) = F' (x ) >0.
O
O
Our second assumption is that for 0(>0) sufficiently small,
for all x£[xo-o,xo+o],
x[l-F(x)] ~
e-
k
clx-xol , k~l, C<oo.
(1. 5)
In fact, if x[l-F(x)] is twice differentiable in some neighborhood of
X
o with a
continuous and non-null second derivative, then (1.5) holds for k=2.
If follows
2
from Sen, Bhattacharyya and Suh (1972), Bhattacharyya, Suh and Grandage (1970),
and Sen (1972) that Z almost surely (a.s.) converges to 6 as
n
n~.
We term 6
as the mean (per unit) bundle strength of filaments.
Our parameter of interest is 6, and the unknown df F is treated as a
nuisance parameter (in the family of all dfs satisfying the aforesaid conditions).
We want to test
(1. 6)
where 6
0
and 6 are known.
It follows from DaniAls (1945), and Sen, Bhattacharyya
1
and Suh (1972) that n~(Z -6) has asymptotically (as n~) a normal distribution
n
with mean 0 and variance
(1. 7)
and writing (1.1) equivalently as
<n),
on = (n-rn +1) Xn,r (where 1-<rnn
that the integer valued random variable r
n
is unique with probability 1.
(1. 8)
Then,
from Sen (1972), it follows that
(1. 9)
Thus, as n-?<lO
(1.10)
The weak convergence in (1.10) enables one to construct a large sample test for
(1.6) which achieves asymptotically a specified level of significance.
However,
like the usual fixed sample size procedures, this fails to have any predetermined
power.
3
Now, by the results of Sen (1972) and Strassen (1967), (1.10) can be
strengtened to a Wiener process approximation of a continuous sample path version
of {Dn -ne ;n>l}.
-
Thus, in the same way as a sequential probability ratio test
(SPRT) based on a Wiener process approximation is an improvement over the fixed
sample likelihood ratio test based merely on the asymptotic normality, it should
be possible to obtain, at least asymptotically, a better test for (1.6) based
on the above Wiener process approximation.
With this objective, a sequential
test for (1.6) is proposed in section 2 and its termination probability is
studied.
The allied OC and ASN functions are then studied in section 3.
test is compared with the fixed sample size test in section 4.
sections 3 and 4 are of asymptotic nature, where we let
are valid whenever
~
~+o.
The
Results of
In practice, these
is small.
2.
THE PROPOSED SEQUENTIAL TEST
To motivate the proposed sequential test, we first consider (in Theorem
2.1) a Wiener process approximation of a continuous sample path version of
{D -ne,n>l}.
n
Subsequently, we make use of the results of Dvoretzky, Kiefer
-
and Wolfowitz (1953) on sequential probability ratio tests for Wiener processes.
Let us define DO=O, and for t£[n,n+l], let
D = Dn + (t-n)[D n+ l-D],
n=O,l, ... ,
n
t
so that {D ,t>O} has continuous sample paths.
t
-
(2.1)
Then, along the lines of Strassen
(1967), we frame the following almost sure invariance principle for {Dt-te;t~O}.
Theorem 2.1.
Under (1.2) - (1.5),
1
D -te = vW(t) + o(t~) a.s.,
t
(2.2)
4
where W= {W(t),t~O} is a standard Brownian motion on [0,00).
Proof.
Let c(u) be equal to 0 or 1 according as u is < or
U. = xO[F(xO)-c(xO-X )],
1
i
T =
n
Li~l
U. ,
0, and let
i~l,
(2.3)
and T =0'
n~l,
1
~
°'
T = T + (t-n)U 1
n+
t
n
for
(2.4)
tlS[n,n+l], n>O.
(2.5)
Then, it follows from Theorem 3.2 of Sen (1972) that under (1.2) - (1.5), as
n~,
n- ~ ID -n8-T
n
I~
n
0
a.s.
(2.6)
Also, by (2.1), (2.5) and (2.6),
~
0 a.s., as
n~.
(2.7)
Consequently, it suffices to show that
T = vW(t) + o(t
t
k2
)
a.s., as t~.
(2.8)
Now, the U.,
i~l, are bounded valued iidrv with EU=O, EU 2 =v 2 , and hence, by
1
Theorems 1.5 and 4.4 of Strassen (1967), we have
1
1
T = vW(t) + OCt log log t)~(log t)~) a.s., as t~,
t
which implies (2.8), and completes the proof.
(2.9)
Q.E.D.
Theorem 2.1 provides us with at least a heuristic justification for
replacing the process
{Dt-t8,t~0}
an asymptotic setup (allowing
~~O)
by
{VWt,t~O};
a rigorous justification under
will be considered in section 3.
Thus, if v
5
were known, transmitting the hypotheses H and HI in (1.6) in terms of drifts
O
(per unit of time) of the process {VW(t)+t8,t~0}, and then using the results
in section 3 of Dvoretzky, Kiefer and Wolfowitz (1953), one could have constructed
the following sequential test.
Corresponding to the desired strength (a,S) of the test, we define two
positive numbers (B,A) (where
O<S/(l-a)~B<l<A~(l-S)/a<oo),
and define a stopping
variable N(6) as the smallest positive integer for which the inequality
(2.10)
(where b = log B and a = log A) is vitiated; if for N(6)=n, 6[Dn - ~(80+8l)]
~ v 2 b (or ~ v 2 a), then H
Now v 2
(or HI) is accepted.
O
is unknown, but as has been noted earlier that
~2
v
= Z 2 P (1 -p ) -1 + V 2 a.s., as n+oo.
n
n n
(2.11)
n
Thus, if we start with an initial sample of size n (;n O(6), moderately large
O
for small 6), and replace for
m~nO' V
by vm' we may, by analogy to (2.10), con-
sider the following proposed sequential test for (1.6):
Starting with an initial sample of size n (=n o(6)), continue drawing obser-
o
vations, one by one, so long as
vmb < 6[Dm - ~(8
+8 )]
201
2
if N*(6)=n is the smallest positive integer
n
<
2
vm'
a
(2.12)
m_>n (6);
O
(~nO(6))
A
for which (2.12) is vitiated,
A2
accept H or HI according as 6[D n - 2(8 0+8 1)] is ~ V~b or ~ vna.
o
We first show that for every pair (8 ,6) of positive constants, the pro0
posed test like the Wald (1947) SPRT terminates with probability 1.
6
Theorem 2.2.
Under (1.2) and (1.5), for every 8 >0 and 6>0,
0
(2.13)
that is, the process in (2.12) terminates with probability one.
Proof.
By (2.12), for every
n~no(6),
(2.14)
Consider now the two possible cases (a) 8
by (2.11), for every y>O, as
=
(8 +8 )/2 and (b) 8+(8 +8 )/2.
0 1
0 1
n~,
In-Yv b/61 + 0 a.s., In-Yv a/61 + 0 a.s.,
n
while, by (1.10), n
o mean
-~
A
[D -n8]/v
and unit variance.
(2.15)
n
n
Now,
n
is asymptotically normally distributed with
Hence, in the first case,
+ 0 as
n~
(2.16)
(for every fixed 6>0).
In the second case, we rewrite (2.14) as
(2.17)
where by the results of Sen (1972), n
-1
D
n
= Zn+8
a.s., as
n~,
8 + ~(80+8l)' by (2.15) and (2.17), P8 {N*(6»n} + 0 as n~.
3.
and hence, for
Q.E.D.
OC AND ASN OF THE PROPOSED TEST
For theoretical justifications, here we consider the asymptotic situation
where we let 6+0; in practice, the results provide good approximation when 6 is
small.
In this set up, the situation is similar to the dual problem of sequential
7
bounded length confidence intervals, treated in Chow and Robbins (1965), and
others, where the results are justified in the limiting case when the width of
the confidence intervals is made to converge to O.
As in the SPRT, for small
the excess over the boundaries are negligible,
~,
so that we can take
e
a
=A =
(l-S)/a
and
b
e
= B = S/(l-a).
(3.1)
Secondly, proceeding as in section 2, it can be shown that for every 8(180),
the OC of the proposed test converges (as
or < 8 .
0
~+O)
to 0 or 1 according as 8 is >
Hence, to avoid the limiting degeneracy, we let
(3.2)
K(>~)
where
ASN of
N*(~)
is a finite number.
is proportional to
Finally, it will be shown later on that the
~
-2
as
~+O,
and, up to a first order of approx-
imation, both the OC and ASN are not affected by an initial large sample size
nO(~)' provided ~2nO(~) is small.
On the otherhand, a reasonably large initial
sample size is needed to insure the accuracy of the estimation
n~nO(~).
Vn
of v for all
Hence, we assume that
(3.3)
Let us then denote by
LF(~'~)
the OC function of the proposed test when
8 = 80+~~ and the underlying df is F.
that
LF(~'~)
Then, in the following theorem, we show
is ADF (asymptotically distribution-free) for all
isfying (1.2) - (1.5).
~£I
and F sat-
8
Theorem 3.1.
Under (1.2), (1.3), (1.5), (3.1) and (3.3), for every ¢e:l,
lim
_ {(Al-2¢_1)/(Al-2¢_Bl-2¢),
ll+O Lp (¢ ,ll) -
(3.4)
(log A)/[log A - log B],
Hence, asymptotically (as ll+O), the test has the prescribed strength (a,S) for
all P satisfying (1.2) and (1.4), that is, the test is asymptotically consistent.
Proof.
We only prove (3.4), as the later part of the theorem follows directly
by putting ¢=O and 1, and using (3.1).
By (2.2), for every e:>O.and n>O, there exists a to(e:,n) such that
(3.5)
and by (2.11), there exists an nO(e:,n), such that
P{maxnO ( e:,n )<_n <00 1V"2n -v 21 > ~e: } <~.
(3.6)
Thus, by (3.3), we can select a llo = llO(e:,n) such that
(3.7)
Now, for every 0>0, to>o and ¢e:l, let
W(t)~O-lb+tO(~-¢) for a smaller t
P(¢,o,a,b,t ) = P
O
(~tO)
than any other
+
t(~tO)
for which Wet) > o-la
(3.8)
to(~-¢)
Also, we define a = log A, b = log B,
i
= [1+(-1) e:]b, i=1,2.
(3.9)
9
Then, from (2.12) and (3.5) through (3.9), it follows that for every
e
=
e0+¢~,
~E[O,~O]'
¢El,
(2)
E
P(¢,~/v,a
(1)
,b E
,nO(~))
- n -<
LF(¢'~)
< P(~ ~/v a(l) ~(2) n (~)) +
-
't',
E
,
'
E
'
0
n.
(3.10)
Now, I¢-~I is bounded for all ¢EI, and t~2 ~ E for all tE[O,nO(~)]' ~E[O,~O].
Also,
{W(t),t~O}
is a process of independent increment.
inequality, the probability that
(V/~)b(j)
E
+
[W(t),O~t~nO(~)]
(t~/v)(~-¢) or (v/~)a(i)
E
+
<
-
(2)
E
(1)
,b E
crosses either of the two lines
(t~/v)(~-¢) (for every i,j=1,2) can be
made smaller than n'(>O), where n'+O as E+O.
P(¢,~/v,a
Hence, by the Levy
Thus, by (3.10),
,0) - n - n
P(~'t', ~/v , a(l)
b(2) ' 0)
E
' E
+
,
n
<
LF(¢'~)
+
n'.
-
(3.11)
Now, by the results of section 3 of Dvoretzky, Kiefer and Wolfowitz (1953), for
every 0>0, d<O<c,
_ {(e C (1-2¢)_1)/(e C (1-2¢)_e d (1-2¢)),
P(¢,o,c,d,O) c/(c-d)
,
~-k
't'2·
(3.12)
The proof of the theorem is then completed by noting that P(¢,o,c,d,O) is
continuous in c and d, so that by choosing E and n sufficiently ,small, both
the left and right hand sides of (3.11) can be made arbitrarily close to (3.4).
Q.E.D.
Theorem 3.2.
Under (1.2) - (1.5) and (3.1) - (3.3), for every ¢EI,
(3.13)
10
where E¢ stands for the expectation under 8=8 +¢6,
0
(3.14 )
v 2 is defined in (1.7), P(¢) = P(¢,6/v,a,b,0) in (3.12), and P'(~) = (d/d¢)P(¢)I¢=~'
Proof.
We first consider the case of ¢+~, and let for every 6>0,
(3.15)
where K will be chosen later on.
Then, noting that for
k~O,
(3.16)
we have by (3.3) that as
6~0,
n
16 2 E N*(6) - 6 2 [L ¢,6
P {N*(6»n} + L >
P~{N*(6»n}]
¢
n=n O(6) ¢
n n¢,6 ~
I
= 6 2 n (6) P¢{N*(6»n (6)} ~O.
O
O
Now, for every €>O and
n~n¢,6'
(3.17)
by (1.1) and (2.12),
P¢{N*(6»n} ~ P¢{V~b < 6[D n - I(8 0+8 I )] < v~a}
< p{V~>V2+€} + P¢{V 2 b(1+€) < 6[D
= p{Vn2 >v 2 +d
n
- I(8 +8 )] < v 2 a(1+€)}
o
I
+
p~{n~6(~-¢) + bV (C+€) < n~(Z -8)
2
~
6n 2
n
n~6(~-¢) + av (i+€)}.
2
<
6n~
(3.18)
With the aid of Theorems 3.1 and 3.4 of Sen (1972), it follows that for every
€>O and s (which we select> 1), there exist a finite positive c(€,s) and an
nO(€'s), such that for all
n~nO(€'s),
11
(3.19)
Now specifying in Theorem 2.3 of Sen (1972), a power rate O(n
-s
), as in Lemma 1
of Bahadur (1966), s>l, and then proceeding as in Theorems 2.1 and 3.2 of Sen
(1972), it follows that for all
n~nO(£'s),
~
!s...
-l/4k
-s
pfln (Z -8)-n--r 1 > c*n
(log n)}.::. c(£,s)n ,s>l,
n
(3.20)
n
where k is defined in (1.5) and T in (2.4).
On the other hand, for n~n~,~'
~
~-l
n ~ ~ (n/n~,~) Kvl~-~I ,so that by proper choice of K, v 2 (a-b) (l+£)/~n~ can
n
k:
be made smaller than ~1~_~I~n2. Thus, the second term on the right hand side of
(3.18) is bounded above by
k:
P~ {n 21 Zn -8 1 ~
1
-l-n~~ I~-~ I}
~
k:
~ p~fn2Izn-81 ~ ,v(n/n~,~) }
1
k:
~
k:
~
~ p~{n2IZn-8-n- Tnl ~ ~KV(n/n~,~) 2} + p~{n- ITnl ~ ~KV(n/n~,~)},
Now, Tn(=I~Ui) involves summation over iidrv's, where Iui/xol ~ 1, Vi.
(3.21)
Hence,
by Theorem 1 of Hoeffding (1963),
k:
1
p~fn-~ITnl > ~KV(n/n~,~) 2}
~ 2 exp{-~(n/n~,~)TIo(1-TIo)K2}.
2~2 I >
n
n~,~
2
expf-~(n/n~ A)TI (1-TI )K }
~,D
n
=
(3.22)
2~2 [p(~,~)] ~,~
+1
O
O
[l_P(~,~)]-l
= 2[exp{-~0(1-TIo)K2}][2Xo(~_~)-2fl+0(~2)}]
< ~£,
where £(>0) is arbitrarily small,
(3.23)
12
by proper choice of K in (3.14).
nO(~O) ~
Let us now select a
nO(E,s), defined in (3.19) and (3.20).
~O=~O(E)
(>0) such that
Then, by (3.17) through (3.23),
it follows that for every E>O, there exists a KE«oo), such that for
by (3.14) with K=K , and
E
n¢,~'
defined
O<~<~ ,
- 0
1~2E¢N*(~)
n
-
~2 Ln~~~(~) P¢{N*(~»n}1
o
~E.
<
(3.24)
Let us now define two stopping variables NJl)(~) and NJ2)(~) as the least
positive integer (~nO(~))' for which Tn+n~(¢-~) is not contained in [v2b(1-E)/~,
v2a(1-E)/~],
and [v2b(1+E)/~, v2a(1+E)/~], respectively, and the terminal
decisions are to accept H or HI according as for NJi)(~) = n, Tn+n~(¢-~) is
O
~ v2b(1+(-1)iE)/~ or ~ v2a(1+(-1)iE)/~, i=1,2. Then, by the same technique as
in (3.22) and (3.23), it follows that, parallel to (3.24),
(3.25)
By a two-sided version of (3.19) and (3.20), for every E>O, there exists a
~O(E»O,
such that
nO(~)
n~nO(~)'
by (3.19) and (3.20),
> n (E/2,s) for all
O
O<~~~O(E).
Then for every
P¢{N*(~»n} ~ P¢{V2b(1+E)/~ < Tm+m~(¢-~) < v2a(1+E)/~, n o(~)<m<n}
--
+
Vm2>V 2+E/2, m-~~Izm-8-m- l TmI > E/2,}
P {for at least one m: nO(~)~m~
(2)}
~ P¢{N¢
(~»n
+ 2C(E/2,s)[nO(~)]
-s+l
.
(3.26)
Similarly,
(3.27)
13
Since 6 2 n¢,6 is bounded, while s>l, so that by (3.3), [n (6)]-S+1 + 0 as
o
6+0, we obtain from (3.26) and (3.27) that as 6+0,
(3.28)
where n can be made arbitrarily small by choosing 6 small.
Consequently, by
(3.24), (3.25) and (3.28), it follows that we are only to show that for every
n>O, there exist an £>0 and a 6 (=6 (n»O) , such that for every ¢(+~) £1 and
0
0
0<6~60'
(3.29)
Since NJj) (6) is based on
{Tm+m6(¢-~), m~nO(6)}, where Tm involves a sum-
mation over independent, bounded valued, random variables, the excess over the
boundaries is negligible for small 6, and hence, by the Wald (1947, p. 171)
fundamental identity and Theorem 3.1, we have
(3.30)
Thus, (3.29) follows from (3.30) by letting £(>0) to be arbitrarily small.
Hence
the theorem follows for ¢+~.
For
¢=~,
we can not adopt the above proof.
However, we may note that P(¢)
is continuous and differentiable in some neighborhood of
¢=~,
so that P'
Hence, by considering a sequence of values of ¢, say, ¢r =
exists.
(~)
~± E ,
r>l,
r-
where £ +0 as r+oo, and then using the above proof, we obtain by the L' Hospital's
r
rule that
- v 2 p' (~) (a-b).
Q.E.D.
14
4.
A
Since v
n
COMPARISON OF THE TWO TESTS
stochastically converges to v, for large n, the test for (1.6)
based on the convergence in (1.10) has the same properties as the test based
on n~ (Z -8) assuming v to be known.
n
Had v been known, the sample size needed
to have a test for (1.6) with strength (a,S) is given by
and T
E
n(~),
where
is the upper 100E% point of the standard normal distribution.
Thus,
Now, from Theorem 3.2 and (4.2), we conclude that the asymptotic (as
~+O)
relative efficiency of the fixed sample size procedure [based on (1.10)] with
respect to the proposed sequential procedure is equal to
e<p = ~~{E<p[N*(~)]/n(~)}
= ~(<p,V)/v2(Ta+TS)2
=
{[bP(~) + a{1-p(~))1/[(~-~)(Ta+TS)2],
- p'(~)(a-b)/(Ta+TS)2
<p+~
<p-~
- 2.
(4.3)
Numerical computation of (4.3) for various <P(EI) and (a,S) shows that e<p lies
close to
~
to
·to
indicating a substantial saving in the sample size for the
sequential procedure.
In particular, for <p=0 or 1, the table on page 57 of
Wald (1947), provides the numerical values of (4.3) for various (a,S).
15
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[1]
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[2]
BHATTACHARYYA, B. B., SUH, M. W., and GRANDAGE, A. H. E. (1970). On the
distribution and moments of the strength of a bundle of filaments.
Jour. Appl. Prob. ~, 712-720.
[3]
CHOW, Y. S., and ROBBINS, H. (1965). On the asymptotic theory of fixed
width sequential confidence intervals for the mean. Ann. Math.
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[4]
DANIELS, H. A. (1945). The statistical theory of the strength of bundles
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[5]
DVORETZKY, A., KIEFER, J., and WOLFOWITZ, J. (1953). Sequential decision
problems for processes with continuous time parameter. Testing
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[6]
HOEFFDING, W. (1963). Probability inequalities for sums of bounded random
variables. Jour. Amer. Statist. Assoc. 58, 13-30.
[7]
SEN, P. K. (1972). On the fixed size confidence bands for the bundle
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