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Work supported by (i) the Army Research Office, Durham, Grant AROD-DA-3l-l24-G746 and (ii) Office of Naval Research, Contract No.
N00014-68-A-0187.
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LIMITING BEHAYIOUROF THE EXTREMUM OF
CERTAIN SAEfLE
~CTIONS
by
P. K. SEN, B. B. BHATTACHARYYA.AND M. W. SUR
University of North Carolina, Chapel Hill, and
North Carolina State University, Raleigh
Institute of Statistics Mimeo Series No. 628
June 1969
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1.
~~EE~~~~E~~~~~~~~~~~:Y.
Consider a bundle of n parallel filaments
z' ...
of equal length and let the non-negative random variables Xl' X
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denote the strength of individual filaments and Y
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the corresponding ordered random variables.
~
Y
Z
~ ... ~
X
n
Y represent
n
Now if we assume that the
force of a free load on the bundle is distributed equally on each individual filament and the strength of an individual filament is independent
of the number of filaments in a bundle, then the minimum load Bbeyond
n
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which all the filaments of the bundle give way is defined to be the
strength of the bundle.
Now if a bundle breaks under load L then all the inequalities
Lin
~
Y , L/n-l
l
~
z' ...
Y
L
~
Y are simultaneously satisfied.
n
Conse-
quently the bundle strength can be represented as
(Ll)
Daniels I1J has investigated the probability distribution of B and has
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shown by very elaborate and complicated analysis that the large sample
distribution of B , properly standardized converges to N(O,l) distribution
n
when {X.} is a sequence of iid (independent and identically distributed)
1.
random variables having continuous distribution function and certain
other regularity conditions are satisfied.
The major objective of the present paper is to obtain similar result
by probabilistic argument for a class of statistics of the form
sup{~(x,S (x»}
n
::x
to which B belongs (under centain simple conditions)
n
where S (x) is the empirical distribution function of a sample of size n
n
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from an m-dependent stochastic process and
~
is a member of a suitably
chosen family of non-negative valued functions.
In sections 2, 3, and
4, the basic regularity conditions and some preliminary results are
derived.
In section 5, the asymptotic normality of the statistic defined
in section 3 is proved.
In section 6, under a concavity assumption,
moment convergence of the statistic is proved.
discussed the special case when
~(x,S
probability.
(x»
max
sample is equivalent to B In if
n
n
= x(l-S n (x» which in large
X,/n
1 < i < n
In section 7, we have
1.
converges to zero in
We have shown that if {Xi} is a sequence of non-negative
interchangeable random variables then B In forms a reverse semi-martingale
n
sequence
ifE(X,) <00.
1.
MOreover if {X.}
is a sequence non-negative iid
1.
.
random variables with its moment generating function M(t) < 00 for all
It I < T then for any 0 < a < 1,
where Yo is maximum of the function X(l-F(x»
x '
o
attained at a unique point
In section 8, some possible generalizations have been discussed.
variables forming an m-dependent stochastic process, not necessarily
stationary.
The marginal cdf of X, is assumed to be continuous and
1.
is denoted by Fi(x), and the joint cdf of (Xi' X +h ) is denoted by
i
Fi,h(x,y), for h
= 1, "', m, i = 1,2, •••• The empirical cdf Sn(x)
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when the sample is (Xl' •.. , Xn ) is defined by
S (x) = n
(2.1)
-1
n
n
L. 1 c(x-X.) where c(u) = {10' u > 00 } .
~=
, u
~
<
Also, the average cdf F(n) (x) is defined by
(2.2)
= ~(x,
Consider a non-negative and real valued function hn(x)
F(n)(X)),
where ~(x, F(n) (x)) assumes a unique (and finite) maximum h~ at x
= x no
We assume that
(2.3)
Further, we let Pn
= -F(n)(x n0 ), and assume that
o<
(2.4)
inf
n p
<
n-
sup
n p
n
< 1.
Our primary concern is to provide a suitable estimator of h
sample (Xi' .•. , Xn ).
Zn~. =
(2.6)
Zn*
~(Xi'
max
= l<i<n
based on the
O
n
•
Let
Sn (X.)),
i = 1, ... , n;
~
Z
n,n
where Zn, 1 -< .•. -< Zn,n
· to d
·
Our centra1 prob 1em ~s
er~ve
t he
Since the random variates Znl' ••. , Z
nn
m
n
Since Sn unbiasedly estimates F(n) at all points,
we are intuitively led to the following estimator of h
(2.5)
O
.
asymptot~c
norma 1·~ty
1
of n 2 [z* - h O ] .
n
are not all independent (even when
= 0), nor necessarily identically distributed, the usual techniques of
deriving the distribution theory for sample maximum fails to provide our
desired result.
n
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1
a_
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proportional to
3.
]
is
00),
Basic regularity conditions and the main theorem.
We assume that
~-----------------------------------------------
for all (x,y):
-
00
<x <
00,
0 < Y < 1,
absolutely continuous in x and y.
(3.1)
(i)
~(x,y
+ 0) -
~(x,y)
if non-negative and
Also,
~(x,y) = O~Ol(x,y
+ 80), 0 < 8 < 1 ,
where for all 10/ ~ 00 (> 0), ~Ol (x, Y + 0) is continuous in 0 at y
I [",~Ol (x,y )]Y = F(n) (x) I -<
(3.2)
() lim sup
g x, n l~i<n _
r
00
g
= F(n) ,
2 ( ) dF ( ) <
x
i x
and g(x) is continuous and uniformly integrable,
are arbitrarily small,
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if I(n)(x~)
I
+
n
and then using the central limit theorem for the later variable.
where
,.
n2[Sn(x~) - F(n)(X~)], in probability (as n
O
1
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Our task is accomplished here by showing that
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2
n rz* n
(3.3)
~lO
and
(3.4)
o
~Ol
< inf
n
Since ~(X,F(n)(X»
(3.5)
=
stand for the partial derivatives, and (iii)
I~Ol(x,
0
n
p)
n
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sup
-< n
I~Ol(x,
0
n
p n)
I
<
00
assumes a maximum at x~, it follows from (2.4) that
(d/dx) F(n) (x) Ix~ > 0 and is continuous at x~; then
00
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and if f(n) (x) is continuous at x~ then hn(x) = ~(x, F(n) (x)) has a
continuous first order derivative at x = xo.
n
Since h (x) attains a
n
maximum at x° ' for all x in the neighborhood of x° we have
n
n
h (x) = h (xo) + (x-xo) h~ (xoe + (l-e)x), 0 < e < 1
n
n n
n
n
n
(3.6)
We also assume that
lim sup
(3.7)
n~
Concerning {F }, we assume that F has a continuous density function
i
i
f.(x) at x = x o for all i = 1, •.. , n.
Let then
n
1
-f (n ).J (xn°)
(3.8)
(3.9)
r n+1+m-j
n+1
n
-1 ",m+1
£,.
J=
°
-
1 n. f ( ). (x ),
J n J n
where (m+1)n.
J
Then we assume that
o
(3.10)
< inf
n f (n) (0)
x n _< sup
n
I (n) (0)
xn <
Now, we require the following notations.
00
•
Let
(3.11)
(3.12)
for h = 0, 1,
Sn,O = n
-1
... , ro,
where of course
n
l:
i=l
(Pn,i - Pn
)2
CI.
n,O
Then, let
= P (l-p ) and
n
n
~
]
n •
' j=1, ••. ,m+1,
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(3.13)
(3.14)
Finally we assume that excepting in the neighborhood of a finite number
of points (Say,
(3.15)
n
< ••• <
-1
n
~i=l
t(k) )
n
'
-
Fi(x) [l-Fi(X)] > 0 for all 0 < F(n) (x) < 1 and all n •
Then the main theorem of the paper is the following.
. .
state d a b ove, 1.'f inf
n y
Un der t h e cond1.t1.ons
1
Th eorem 3 ••
>0, t h en
n,m
1
~-!m p{n2 [z*n - h nO ] Yn,m -<
(3.16)
uniformly in
-
00
< x<
00
1 2
exp(- "2 t ) dt ,
x}
•
The proof of the theorem is postponed to section 5.
(4.1)
Yn1. = 1jJ (X.,
1.
Y 1 < ••• < Y
n,
-
n,n
F( n ) (x.1. )),
i = 1, ""
the order statistics.
n
Now, Y ' ••. , Y form an
nl
nn
m-independent process with the marginal cdf's Gin) (x),
respectively.
... ,
G(n) (x)
n
'
We have by assumption
h~
G~n) (h~)
(4.2)
sup
x ",(
'Y x, F(n) (x)) =
Lemma 4.1.
If hn(x) = 1jJ(x, F(n)(X)) has a continuous first order derivative
in the neighborhood of x
when (3.6) holds.
<
00
-'>
then for every E > 0,
= 1 for i
lim
n~
p{Y
=
1, ... , n .
> h O _ E/n} = 1 ,
n,n - n
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Proof.
a
n
By the hypothesis of the lemma, for any {a } such that
n
-+ 0 as n-+ oo ,
(4.3)
h (xo + a )
n n
n
= h n (x°n ) + an h"n (xon + 8an ) (0
= h n (x°n ) + o(an ) as h"n (xo)
0
n =
< 8 < 1)
.
Thus, for every E > 0,
(4.4)
= P {x.1
< Xo - E (n)} + P {x. > Xo + E (n)} , i
n
1
n
1
2
-
where E (n) and E (n) both -+ 0 as n -+
1
2
nE. (n) -+
(4.5)
00
j = 1,2, as n -+
00
J
where E(n) = E (n) + E (n).
1
2
,
00
1, ... , n,
but
=> n E(n) -+
00
as n -+
00
,
From (4.2) and (4.4), we have
(4.6)
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- P {x. > xo + E (n)} ~ E(n) f.(xo), i = 1,
2
1 n
1
n
..., n
•
Let then
(j) = max IY nj' Ynj+m+1'· •. , Ynj+(n.-1) (m+1) ] , J' = 1 , ... , m + 1 .
Y (n)
J
Then, Y
> y«j)) for all j = 1, ••. , m + 1.
n,n n
(4.7)
Now,
(4.8)
P {y
< hO
n,n - n
-
Thus,
E/n} _< min P {y(j) < h O
(n) - n
-
E/n} .
y~~j is the maximum over n j independent random variables. Hence,
(
O
(n) -
n
P {Y j) < h
-
E/n}
=
n -1
j
IT
k=O
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n
~
(n)
r
k=O
j
s (n)
[1 -
n.
0
Gj +k (m+1)(h n - sin)] J
-
0
n.
f ( ). (x )]
n J n
J
=
j
1, ... , m+ 1.
.
svp ( 0)
( 0)
Slnce,
J f(n)j x
n > fen) Xn > 0, by (3.9) and ns(n)
~
00
as n
~
00
,
we have
(4.9)
m!n
P{Y~~~ ~ h~-
sin}
~
II - sen)
f(n)(X~)]nj ~
° as n ~
00
•
Hence, the lemma follows from (4.7) and (4.9).Q.E.D.
A direct consequence of lemma 4.1 is the following:
o
- Y
]
n
n ,n
= 0 p (1),
(4.10)
n[h
Lemma 4.2.
For every s > 0, there exists a finite c(s) >
uniformly in {F.} •
1
° such that
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n -1
j
< 1_1_
sUP
p{ ~
n'2 ISn(x) - -F(n) (x) I > c(s)} <
S ,
uniformly in {F.} , satisfying (3.15).
1
Proof.
Let us define
(4.11)
for j
=
1, ••• , m + 1.
Then, by definition,
I
(4.12)
s~p
2
I
In {Sn(X) - F(n) (x)}/
~
I
r;:i (n~-)2{S~p
In;{Sn,j(X) - F(n)j(x)}I}.
J
Since (m + l)n.
J
theorem.
~
n, j
= 1,
••• , m + 1, it suffices to prove the following
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Theorem 4.3.
Let {Xi} be a sequence of independent random variables
with continuous cdf's {F }, and let Sn(x) and F(n) (x) be defined as in
i
Then, if (3.15) holds, for all A > 0,
(2.1) and (2.3).
(4.13)
where the equality sign holds when F = ••• = F = F for all n.
l
n
i
1, ... , n.
Thus, n
-1
Define G*(t) =
n
Li =l Gni(t) = t: 0 < t < 1.
t . , i = l , ••• ,n.
1
n
Also let
1
(4.14)
Un (t) = n2" [Gn* (t) - t], 0
~
t
~
1.
Thus, it suffices to show that for every A > 0,
(4.15)
lim p{ sup
n -+ 00
O<t<l
Iu
n
(t)
I
uniformly in {F.} satisfying (3.15).
1
(4.16)
2 2
00
> Al < 2 L (_1)k+l e -2k A
k=l
Direct computations yield
E{U (t)} = 0, E{U (s)U (t)ls < t}
n
n
n
-
n
= n-1 E.1=
ls.(l-t.),
1
1
0 _< s _< t _< 1;
(4.17)
(4.18)
E{U (t) - U (s)}4 < 3(t - s)2 for all 0 <
n
n
-
l)Note that Y . has nothing to do with the Y . of
n1
n~
used later on.
S
(4.1)
< t < 1 .
and will not be
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Since the sample paths of U (t) are not continuous (a.e.), we define
n
*
(4.19)
U (t) =
n
for (k - l)/(n + 1)
~
t < k/(n + 1), k
=
Then U*(t) is
1, •.. , n + 1.
n
a process with continuous sample paths, and using (4.18) we find
(4.20)
E{U*(t) - U*(s)}4 < 27(t - s)2, 0 < s < t < 1
n
n
-
v
~
Hence by a theorem of Ko1rnogorov (cf. Hajek and Sidak 13, pp. 177-179]),
it follows that for every E > 0
lim p{
max
+ 0
It-sl< 8
(4.21)
8
IUn*( t )
* ( )I
} = 1•
- Un s
< E
Using (4.14) and (4.19), we have
1
sup
0< t<l
(4.22)
IUn (t)
- U* (t)
n
I -< 1<k<n+1
max
IU\
k )
n n+ 1
- U* (k- 1) I + 2(n+1) 2"
n n+1
and hence, from (4.21) it follows that for every E > 0,
It
Iun (t)
lim inf p{ sup
n
O<t<l
(4.23)
- U*(t)
n
I
< E} = 1.
therefore follows from (4.15) and (4.23) that we are only to show that
(4.24)
For every
n
n(~
lim p{ sup IU*(t)1 > A} < 2
+ 00
O<t<l n
1), let now 1Z (t); 0 < t < 1] be a Gaussian process
=0
with EZ n (t)
n
--
and E1Z n (s)Zn (t) Is -< tJ
= n-1 L.n1.=1s.C1-t.),
1.
1.
0 -< s -< t -< 1.
1This sequence of Gaussian processes can be conceived of as an average
of n independent Gaussian processes.]
EIZ (t) - Z (s)J
n
n
2
=
n
-1
Since, by definition
n
L.1.=
1Ct. - s.)(l - t. + s.) < (t - s) and
1.
1.
1.
1.-
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E[Zn(t) - Zn (s)]
4
2 2
2
= 3{E[Z n (t) - Zn (s)]}
-< 3(t - s) , according to
v
the same theorem of Ko1mogorov (cf. Hajek and Sidak 13, p. 177]),
such a process exists (in the space of all continuous functions).
Now, for any m(.:. 1) and real
the cdf of
m
~
1
A.
J
*
U (t.),
n
J
~ = (AI'
••• , A ), let H (x; A, t) be
-
m
n
~
~
.
where 0
~
t
l
< ••• < t
< 1.
m-
Also, let
m
~ (x;
n
A, t) be the cdf of
~
~
1
Aj Z (t.).
n
J
Then by (4.23) and the multi-
variate central limit theorem we have
(4.25)
_00<sup<00 IH (x; A, t) - ~ (x; ~, E) I
x
n
- ~
n--
for all A and t.
+ 0
as n
+
00 ,
Hence, on making use of (4.23), (4.25) and a theorem
in Hajek and Sidek [3, p. 180], we may conclude that Ilu*(t) - Z (t)1
n
n
I
converges in the Prohorov-sense to zero i.e.,
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(4.26)
O~~~l
Iu:(t) - Zn(t)1 converges in law to zero as n
+
00 •
Hence, it suffices to show that for every A > 0
(4.27)
Consider now the Brownian bridge 1Z(t), 0 < t
and E1Z(s)Z(t)ls ~ t]
= s(l-t),
0 ~ s ~ t ~ 1.
~
1] with EZ(t)
=0
For finitely many
and D(m) be the covariance
Points 0 < t(l) < ••• < t(m) < 1 , let D(m)
~n
matrices for Z (t) and Z(t) respectively.
n
It is then easy to verify
that
(4.28)
t(j))(t~5I,)
1
(51,)
t
*(m)
)))J' , 51,=1 ,m = D
-n
,
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*(m).1S pOS1. t·1ve sem1. de f··
. p.d.
h
D
were
1n1te ( p.s. d)
. an d D(m) 1S
_n
Also, by the well-known Kolmogorov theorem
(4.29)
So the desired result will follow if we can show that for every
m and 1 < t (1) < ••• < t(m) < 1
- ,
(4.30)
(4.30) really follows from (4.28) and the following lemma.
Lemma 4.4.
Let
~p
be a convex space (in p-dimensions) with the origin
origin as an inner point.
Let X and Y be independent normally distri-
buted random variables (p-vectors) with null means and dispersion
matrices
~l
and
~2
are at least p.s.d.
(4.31)
p{X E:
-
~2
is p. d. and
-e p } -< p{y-
E:
~3
••• , V
*
E2'
By hypothesis 0 -< VI' ""
*
X -+ DX = X , Y -+ DY = Y , let
convex so is
-
e p*•
-
~2
is a null matrix.
DB
D~ = -p
V
__ 2_
-1
~3 = ~l
-ep } .
There exists a non-singular matrix D such that
roots of El
and
Then
where the equality holds only when
Proof.
~l
respectively, where
Let then
V
e p* be the image
VI'
" ' , Vq
> 0, q
P
-- -
I and
-p
are the characteristic
< 1.
pof
DBID~ =
Under the mapping
..e.
P
Since
ep
is
2. p, while the remaining
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p - q of the v's be equal to O.
Then, for the q-dimensional distribu-
tion of X* and Y*, let C * be the intersection of C * with the
-q
-q
pq
P
= Xp* =
q-dimensional hyperplane X*+ =
q l
O.
Since VI' ""
... , Xq* (or Yl* , ... , Y*q )
*
are all positive and Xl'
Vq
are all independent,
well-known results on the multivariate normal distribution yield that
p{X *
-q
(4.32)
E
-e pq* } -< p{y-q*
E
e pq* },
where the equality sign holds only when VI
is a sub-space of
by virtue of Vq+l
--e p* ,P {X-q* -e pq* }-> p{X- *
E
= ... = Vp = 0, Yq*+l =
1, and hence, p{y *
-q
E
~
*
pq
. *
= p{y
}
-
J?* }
E ~
p
= ... = Vq = 1. SinCe~;q
*
p{X
-
E.e} =
p
0 },
E -V
p
while
= Yp* = 0, with probability
= p{y
-
E
0 }
~
p
•
Hence, ( 4 . 31)
follows from (4.32). Q.E.D.
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Now, by definition, Yn,n in (4.1) is equal to ~(X(r)' F(n)(X(r»),
where 1
~
Lemma 4.5.
Proof.
r
~
nand X(l)
(r/n - p )
(4.34)
~
0, as n
X(n) are the order statistics.
+
00
•
o
By the continuity of hn(x) at x ' (4.1)
n
we have IX(r) (4.33)
n
p
+
~
x~1 ~
O.
Hence,
,
(4.10) and (3.7),
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14
where by lemma 4.2, the first term on the right hand side of(4.34) is
1
o
P
"2
(n).
Hence the lemma follows from (4.33) and (4.34). Q.E.D.
Let us now consider a sequence {a } of real numbers, where
n
an
+
° as n
00, and let
+
(4.35)
I
n
= { x: x0n - an-< x -< xo
+}
a
n
n
1
= n"2{[Sn(x)
(4.36)
Gn(x)
Theorem 4.6.
·Th~re e~ists
.
Proof.
It
- F(n)
(x~)]},
x£I n •
two
sequences . {£ n } and
{o n } of real and
. ..
-
positive numbers, such that (i) £
p{supIIG (x)
n
ISn(x~)
- F(n)(x)] -
I : XcI n ]
n
+
0, 0
n
+
° as n
+
00, and (ii)
> £ } < 0
n
n
follows directly from (4.21) and (4.23).
From lemma 4.5 and theorem 4.6, we have the following.
1
"2
0
0
n [Sn(X(r)) - F(n)(X(r))] ~p n [S n (x n ) - F( n )(xn )J
1
2"
Lemma 4.7.
For later convenience, we define
1
( 4.37)
* 2"
0
W = n ISn(x )
Lemma 4.8.
If inf
n
-
n
n
V
n,m
0
- F(n)(Xn )].
r *Iv
> 0, then o-.(W
)
n n,m
+
.N(0,1).
For proof, see lemma 2.3 of Sen IS].
1
Lemma 4.9.
Proof.
max
(X
) _
(2)
Un d er (3 • 2) 'l~i~n
g (i) - 0p n •
Let g.
implies that
1
= g(X.)
1
and the cdf of g. be H., i
1
1
= 1,
••• , n.
Then, (2.2)
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sup
l<i<n
(4.38)
f
00
g2 dH .(g) <
o
00
•
1
00
Now g2[1 - Hi(g)] <
J
Hence, for any
g
{ gn*} for which lim g * =
n-+oo n
00,
we have
(4.39)
Now, let
so that
g~j)
= max[gj' gj+m+l' ••• , gj+(n.-l)(m+l)]' j
max
i g(X(i»
max
(j).
= l2j m+l[gn ' J = 1, ""
2
p{m ax g (X(1.»
i
(4.40)
1, ... ,m+l,
J
m + 1].
Consequently,
_> gn*} < ~m+l p{g(j) > g*}
n
- n
- j=l
= ~m+l II - p{g(j) < g*}j
j=l
n
n
Now, we let g* = s~, where s(>O) is arbitrarily small.
n
Then the
difference between the right hand side of (4.40) and (m + 1)1 1 - exp
{- a(g *)/(m + l)s 2 }] converges to zero.
Also
n
*
a(&n)
(4.41)
*) /(m+l)s 2} ] < -"':;:2~
(m+l)1l - exp. {
- a(g
Thus, p{m~ g(X(i»
n
~ s~} -+ 0 as n -+
s
00,
-+
0, as n
-+
00
for every s > O. Q.E.D.
•
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Lemma 4.10.
Under (3.2)
(4.42)
Proof.
The L.R.S. of (4.42) is bounded above by
(4.43)
and hence, the lemma directly follows from lemma 4.2 and lemma 4.9.
(5.1)
Note that
Z
*
n
max
* and Y
. V.
n1
n,n
1
max V ••
=.
1
n1
Also let an(1) ,an(2) be
so defined that
1
(5.2)
h (x o - a(l»
nn
n
= h nn
(x o + a(2»
n
Then, both a(l) and a(2) + 0 as n+
n
OO
n
,
O
= hn
-
:n2 10g
n, c > O.
and by lemma 4.1,
asn+
(5.3)
OO
•
Consider now the two subsets
(5.4)
= {X(l.): xnO -
O
a(l) < X
< x + a(2)}
n
- (i) - n
n
and
Now, using (2.3), (2.4) and theorem 4.6, we have
1
(5.5)
sup
I 2 (V *. - V .) - 1/J (x0 , p)W *1 : l·c-s(l)J = o (1).
. [n
1
n1
n1
01 n
n n
n
p
Renee, from (5.3) and (5.5), we have
c;.
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17
_ 1
1
max
*
Z
0
*
-Z
Z*
=
n,l
.1£ 8 ( 1) vni = Yn,n + n ~Ol(xn' Pn)Wn + 0p(n )
n
(5.6)
1
1
Now, by lemma 4.1, nZlh o - Y ] = o (~ Z), and hence, from (5.6),
n
n,n
p
we have
(5.7)
Consequently, by lemma 4.8,
1
"2"
(5.8)
cC(n
*1
Iz n,
- hOl/y
n
n,m
) +)(0,1),
and hence, for any c > 0,
*
p{Z
(5.9)
n,
1
1 >
-
Z log n) c/2}
h - (n
n
0
-
+
1, qS n
+
00
•
80 the proof of the theorem will be completed if we can show that
*
0
p{ sup
V . < h - (n
i£8(2) n1 - n
(5.10)
1
2'
log n) c/2}
+
1, as n
+
00
,
n
1
(as then n2'(Z * - Z* )
n
n,l
P
+
0). With this end in view, we divide 8 (2)
n
into two subsets
(5.11)
8 (2) = {X(i)£8 (2):
n,l
n
g(x(~))
~
< go < oo}
8(2) = 8(2) - 8(2)
'n,2
n
n,l
where g(X) is defined by (3.2) and go > s~p 1~01 (x~, Pn)l.
1
(5.12)
nZ/vni - v:il
~
Note that
1
Z
g(x(i))ln {8 n (x(i)) - F(n) (x(i))}1 •
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18
ap (1),
Hence, for all i £ s(2 ), the right hand side of (5.12) is
n, 1
by
1
lemma 4.2.
O
On the other hand, V . < h
n~ n
c~ 2
-
log n for all i £ S(2)
n,l'
Consequently, with probability approaching to unity,
1
*.
Vn~ -<
(5.13)
i
£
hO _
n
1
cn- 2 log n + ap (n- 2) _< n - 2c n- 2 log n, for all
S (2)
n,l .
Thus,
sup
p{
V* . < h 0 - (c/2) -n
i£S(2) n~ - n
n,l
(5.14)
1
hO
1
2' log n}
+
1, as n
+
00
•
o
g(X) are continuous functions in the neighbourhood of (x , p ) and
n
g
o
can be an arbitrarily large but finite real number.
lemma 4.10 the supremum of
Hence, Z *
n,l
= Zn*
conditions:
(a)
(6.1)
where (b)
(6.2)
and (c)
(6.3)
V:i
over
in probability.
~(x,
~(x,
n
Hence the theorem.
+
~
y~(x, 1),
~
y
Vy
1)
i.e.,
£(0, 1),
0)-, 0 = 0, 1 are non-nega tive,
0) is t in x and
square integra1abi1ity of
sup
Therefore, by
is less than z:,l in probability.
y) is concave in yeO
~(x, y) ~ (l-y)~(x, 0)
~(x,
s~~~
n
00
f
a
~2(x,
~(x,
1) is
~(x,
0) dF (n) (x)
0),0
~
in x:
= 0,1,
~ ]J2 (0) <
00
-
00
< x <
i.e.,
for 0 = 0, 1.
00
,
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19
Theorem 6.1.
Proof.
(6.4)
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n
Let X(l)
~
~
•••
n
+
0 as n
-
X(n) be the order statistics.
+
00
•
Then, by (6.1),
Z
n,i
Now, by (6.2)
(6.5)
nn-
I
I
I
Under (6.1)-(6.3), IE(Z*) - hOI
i
ljJ(X(1..)' 0) -<.!
0).
n ~~
J=1.. ljJ(X(J.)' 0) -< 1:.
n ~nj=l '''(X(J..)'
0/
=.! ~n 1 ",(X., 0)
n j= 0/ J
1) < 1:. ~i
",(X
1) <.! ~n
ljJ(
1)
- n j=l 0/ (j)'
- n j=l
X(j) ,
ni ",(X (i)'
(6.6)
0/
=
1:. ~n
n
j=l
ljJ(X ., 1) •
J
Therefore, on writing ~ = n
n
O<Z * <ljJ
(6.7)
Now,
ljJ
n
-
z*n
n -
n
-1
n
l{ljJ(X., 0) + ljJ(X., I)},
1.=
1.
1.
~.
we have
•
converges in probability to h
is uniformly (in n) integrable.
O
n
(by theorem 3.1), while by (6.3)
Hence, the theorem follows by using
the dominated convergence theorem (cf. Loeve 14, p. 125]). Q.E.D.
The next problem is to study the conditions under which in theorem
*
3.1 we can replace h o by E(Z).
n
theorem on this.
n
We have failed to provide a general
However, in many cases this can be done.
In the re-
maining of the paper, we consider the special case of iidrv and a
simple ljJ, where all these results apply directly.
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7.
A
special case.
'V""''''''''''''''''''''''''V'''V __ I'U_'V'''''
statistic B* which describes the average strength of a bundle of
n
n filaments.
,.
I
In the present section we shall show that under certain
sup x(l - -F(n) (x»
restriction, Bn* = O<x<oo
in probability as n
+
00
*
we shall also study the moment convergence of the statistic B.
n
and
We
know that
(7.1)
where Y , Y , ••. Y are the ordered random variables obtained from
2
l
n
n non-negative random variables
XI,
X ' .•• Xn which are the individual
2
strength of the filaments in the bundle.
Lemma 7.1.
Let {X.} be a sequence of independent non-negative random
].
variables with distribution functions {F } and let
i
lim
i
(7.2)
and x
2
Proof.
00
J
0
x
2
dF (x) <
i
00
1
be uniformly integrable then
lim p{ max . {X.} > En2 } =
].
n+oo
l<i<n
It follows directly from lemma 4.9.
Lemma 7.2.
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In the introduction, we have discussed the
Under the conditions of lemma 7.1.
1
lim
n+oo
Proof.
(7.3)
Let leu)
21 B* -
n
= 1 or 0 according as u
1
S (x) = n
sup x(l - S (x»
x
n
n
n
n
~
i=l
I(x - X.),
].
I =p 0
> 0 or not.
Then
o for
every E > O.
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21
B* = 1m<~<x {Y.(l - S (Y.»} = sup {x(l - S (x)}
n
1 n
1
n 1
X
n
(7.4)
since the sup occurs at one of the jump points.
Now at all points
which are not jump points of the function S (x), x(l - S-(x»
n
x(l - S (x»
=
and at jump points
n
X. (1 - S-(X »
1
n i
(7.5)
n
- X. (1 - S (X.»
1
n
n
1
Hence
1
1
n21B: _ s~p x(l _ Sn(x»1 = m~x Xii n2 ~ 01
(7.6)
I.
This leads us to study the special case when
Consider a set of n + m non-negative real numbers
An ineqa1ity:
xl' x 2 ' .•• xn+m with Y1 ~ Y2 ~ ••• ~ Yn+m as the corresponding
ordered set.
Let (xl( i) , x (i), ••• x(i»; i
n·
2
=
1, 2, ..• (n+m) be the
n
all possible combinations of n tuples that can be formed from the
n+m
XIS
and let
(i) <
Y1
-
(i) <
Y2
~ Y~~~
denote the corresponding
ordered set.
(7.7)
and
(7.8)
(i)
(i)
b n (x1 ' x
2 '
Lemma 7.3.
x (i»
n
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22
Proof.
There are (n+m-j+l) (j-l) combinations where
n-j+k
k-l
b (x (i)
n 1
'
(7.9)
xCi)) > (n-j+k) y.
J
n
Hence
(7.10)
>
(n-j+k)
= n(n+m-j+l) y.;
J
(j
n+m-j+jl (j-l)
( n-j+k
k-l
= 1, 2, ••• n+m) •
Therefore
xCi)) > n max{(n+m-j+l) y.}
(7.11)
n
-
J
II .
Theorem 7.4.
If the sequence of random variables {Xi} is a sequence of iid
random variables then the r.v. 's {B *} with probability one form a reversen
semi-martingale if E(X ) <
i
Proof.
00.
MUltiplying both sides of (7.11) by the conditional distribution
of ~, X 2 ' ••• ]{n+m given Bn+m' Bn+m+l'
observing the interchangeability of the
B
(J .12)
n+m
n+m
=
,
...
Bn+m+k
and integrating and
random variables we get
Bn+m+k)
n+m+k
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23
Moreover
o~
(7.13)
B~
<
X => E(B~) <
n
00
•
Therefore
lim
B
n+m
n+m
k~
=
<
lim
k~
EC~ IB~:
with prob. one.
E(
Bn
Ii
I Bn+m'·
Bn+m+k\
n+ m
• • n+m+k)
.)
That the limiting process is valid follows from
Doob 12, p. 332].
Corollary:
Under the conditions of theorem 7.4, E (
B~) is
a monotone
decreasing function of n.
Proof.
Taking m = 1, multiplying both sides of (7.11) by the joint
distribution of the interchangeable random variables (Xl' X , ••• X +l )
2
n
and integrating we obtain the result immediately.
Theorem 7.5.
Let {X.} be a sequence of iidrv with E(X7) <
1.
B* ~.y , where y
n
0
0
1.
00.
Then
by assumption is the unique maximum of the function
x(l - F(x».
Proof.
By theorem 3.1 and lemma 7.2 we know
* -+p
that B
n
Y .
0
Moreover
{B *} is a reverse semi-matingale sequence and B* < -X which is uniformly
n
n - n
integrable.
Therefore, (cf. 14, p. 393]), there exists a random variable
Z such that B* ~ Z.
o
n
0
follows
II.
Hence by the equivalence theorem the result
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Let Z
ni
i =
= X.[l-S(X.)],
1
n 1
1,
... ,
with the common cdf F(x) , such that M(t)
.!i M(t)
Theorem 7.6.
< 00,
V It I
(7.14)
nal [E(Z*) _ Y
]1
Proof.
Let c(u) = 1 if u
~
n
n
(7.15 )
-1
0
As E(Z *) is
+ in
n
have E(Z *) > y
n
-
0
tx
= E(e )
It I
" T.
x.
1
= max x[l - F(x)].
x
0
Then,
0 and 0, otherwise.
J~I
for all
co
< T, then for any a: 0 < a < 1,
-
n
1, ... , n.
c(Xj - Xl')]' i
n and by theorem 6.1, E(Z *)
n
for all n.
<
~ 0 as n ~ 00, where y
[X. + L. 1
1
n, where X 's arc iidrv
i
~ y
0
as n
~
00, we must
Hence, it suffices to show that for any
a: 0 < a < 1,
(7.16)
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Now, for any positive integer k
(7.17)
= n
-k
k n
_ X )]k-s}
k
(k)
1
Ls=O s E{X1 [L j =2 c(X.J
r£
r
k k
Lk~ (n-1) L* E{Xk [c(X -X )] 1 ... [c(X
-\)]
},
(k)
L
n
=
H1
Q,
2 1
1
Q,=1
s=O s
where r. > 1, j = 1,
J -
... ,
*
Q,
L. 1 r. = k - s, and the summation L
Q"
J=
J
extends over all possible choice of r , .•. , rQ,'
1
Xl' the random variables c(X
j
-
Now, conditioned on
Xl)' j = 2, ... , n
are iid with
2, ... , n - 1, and r
:;>
1
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25
and hence,
(7.18)
= ~k
s=O
<
~k
_ ~s=O
I
R-=1
(k) n-s k-s
s
~R-=1
as (n_1)[R-] = (n-1) ••• (n-R-)
(7.19)
.e
f
~k-s n-(k-s) (n:1)~*
00
xk-R- yR- dF(x)
0
N
(where y = x[l - F(x)] -< Y0 )
I_
I
I
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I
I
I
I
(k)
n-s
s
k = k
~
R- (k-s-R-) (1 _
Yo ~k-R- n
~ (n _R-~:)R-.
R-+1)R-(.!.~*
2n
R-I ~
)
1
,
Let us now take
kn+l)kn . -10 n
-1
= I'; 2n log n ] -'> ( 1 - - 'V e
g = n
•
n
2n
Then, from (7.18) and (7.19), we have
(7.20)
k
k
E(Z n) < n- 1y n II +
n~
0
k
k
k
-1 n I n
~ (
)]
}]
{
n
ny ~1 1 + Qn2 '
2n y o 1 + QuI + n Yo
o
where it can be shown that
(7.21)
lim Q = 0 i = 1, 2,
n+oo ni
'
Thus, on using the fact that (Z:)k
(7.22)
*
*k
11k
E(Z ) < [E(Z n)]
n
n
n
k
=
[n E(Z ~)]
n~
11k
n
when M(t) <
~ ~~=1 Z~i'
< [E(~~
~=1
k
00
•
~
0, we have
k
11k
Z ~)]
n
n~
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26
11kn
Consequently, for any a: 0 < a < 1,
3 ll{
1
2
< -.,-==-- [1 + "3
Qnl + "3 Qn2] +
I-a
2n
y
(7.23)
°(n-l+a)
-+ 0
o
Hence the theorem.
Let X.
_1
=
(X .. , ••• , Xi ); (p
1J
P
~
1), i
= 1,
••• , n be independent r.v.
with continuous cdf's F , ••• , F respectively.
n
l
For the jth variate,
we denote the marginal cdf's by F
] , i = 1, ••• , n; j = 1, ••• , p;
ifj
and let
(8.1)
j = 1, •.. , p.
Consider now a function (positive valued)
(8.2)
o
and assume that h (x) assumes a unique maximum at x , where h (xo)
n _
-n
n -n
h
o <00.
n
(8.3)
(8.4)
Let then
=
an
-+ 00 •
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... , y p )
We denote the partial derivatives
o
-
with
0
respect to Yl' ••• , Yp at x = ~n' Yj = Fn(j) (x nj ) , j = 1, ••• , p
by
snl..' "" snp ,respectively.
Finally, let
1
2"
(8.5)
= n
0
-
0
ISn(j) (x nj ) - Fn(j) (xnj )], j = 1, ••• , p •
Then, by the same technique as in theorem 3.1, it can be shown that
(8.6)
Hence,
1
'P 2"
ot.-(n
(Z
(8.7)
where y
*-
n
h 0 » -+ J(O, Y2 ) ,
n
n
2
n
(b) Vector case.
... , h (b) (x)]
Suppose now that h_n (x)
where h(j)(x) assumes a unique maximum at x
n
n
O
.,
~
j = 1, •.• , p.
The
1
2"
0
asymptotic multinormality of n [h (x) - h ] follows along the same
~n
-n
line as in theorem 3.1.
(c)
(8.8)
Extension to linear ordered estimators.
We let
* = rth maximum of. Zn l' ••• , Zun ,
Zn,r
r = 1, ""
n.
Let us consider a set of weights a l' ••• , a
satisfying the fo11ownnn
ing conditions: (a)
(8.9)
a
> 0, V r
nr-
and
Ln
a
= 1 ,
r=l nr
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and (b)
I
nr
Z* is essentially an extreme statistic, so that a
nr
nr
rapidly converges to zero as r increases.
Specifically, we assume
that
1
(8.10)
L
where k
n
r>k
a
n
-2
nr
= o(n
) ,
is so chosen that
1
h (xo ± kin) > h O
n n
n
- n
(8.11)
_
c~ 2"
log n •
1
2"
Thus, by (3.6), k = o(n). Then, proceeding as in the proof of
n
theorem 3.1, we have
*
(a)
all belong to S(l)
Znl' ••• ,
I_
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I
I-
L a
n
k
(8.12)
a
r=l
and (c)
by (8.10),
k
L" n
*
nr IZ nr
a
r=l
n
- y
nr
n
L
]
a
r=l
nr
W
lex no ,
0
+
1, as n
n
1
n
Lr=k +1
a
n
nr
2"
* < Z*
nr - nk +1 (n
Z
n
a
Lr>k
n
nr
)
= 0p (1).
0(1)
Consequently,
(8.13)
Along the same line as in lennna 4.1, it can be shown that
1
(8.14)
for all r < k
n
=
0
2
(n ) .
+
00
,
wn*1 = op (1)
p )
Further
nr
1
2
with probability
k
n
L
(b)
'
o (1).
p
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Hence, from (8.12), (8.13), and (8.14), we have
1
n2 [En
(8.15)
r=l
a
nr
Z* _ h O ] tV
nr
n p
(0
)
*
'Yol x n ' Pn Wn
,',
Consequently,
1
(8.16)
(e)
2
O
J:.(n lEn
a
Z* - h ] /y
)
r=l nr nr
n
n,m
A Jackknife estimator:
where a <
(8.17)
00,
+
$(0,1)
As by theorem 7.5, IE(Zn*) - y 0
1 < a/n,
-
it may be better to consider the alternative estimator
*
*
**
1 n.
Z = - E {n Z - (n-1) Z 1.}
n
n i=l
n
n- ,1
*
where Zn_l,i
is the maximum of XiII - Fn _1 (X )], in a sample of size
i
n - 1 where the ith observation of the sample of size n is omitted.
.
Obviously, the bias of Zn** is less than that of Z*
I t is easily
1 n
-2
seen (on using (5.7» that Iz ** - Z* I = o (n ), and hence, the
n
n
p
1
normality of n
9.
2
(z** - h
n
O
n
9~~:~~~~~~_E:~~E~~.
)
would follow from theorem 3.1.
In order to obtain the asymptotic normality
1
2
of the statistic s~p t/J(x, Sn (x», we have shown that s~p n ISn(x) - F(n) (x) I
is bounded in probability.
but not thier continuity.
We have relaxed the identity of F , ••• , F
n
1
In a subsequent paper various limiting
properties of the above statistics for possibly discontinuous cdf's
will be presented.
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REFERENCES
[1]
DANIELS, H. E. (1945)
of bundles of threads.
405-435.
12J DOOB,
J. L.
(1953)
v
The statistical theory of the strength
Proc. Roy. Soc. London Ser. A. 183,
Stochastic process, New York:
13J HAJEK, J. & SIDAK, z. (1967)
Wiley.
Theory of rank tests, New York:
Academic Press.
[4]
LOf:VE, M. (1960)
Nostrand Co. Inc.
15]
SEN, P. K. (1968) Asymptotic normality of sample quanti1es
for m dependent processes. Ann. Math. Statist. 39, 1724-1730.
Probability theory, New York:
D. van