·
~
UN.rVERSITY OF NORTH CAROl,INA
Department of Statistics
Chapel Hill l N. C.
SOME LIMIT DISTRIBUTIONS CONNECTED WITH FIXED DlTERVAL ANALYSIS
by
J. Sethuraman
Indian Statistical Institute and University of North Carolina
January 1963
Grant No.
AF-AFOSR-62-l69
The proofs of some limit theorems on the limiting
distributions of statistics that enter fixed interval analysis are presented.
This research work was done at the Indian Statistical Institute. The writing of
this paper was supported by the Mathematics Division of the Air Force Office of
Scientific Research.
Institute of Statistics
Mimeo Series No. 349
SOME LIMIT DISTRIBUTIONS CONNECTED WITH FIXED INTERVAL ANALYSIS
by
J. Sethuraman
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Summary:
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1
UNIVERSITY OF NORTH CAROLINA and
INDIAN STATISTICAL INSTITUTE
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The proofs of some theorems (stated in Sethuraman (196;))on the limiting
distributions of some statistics that enter in the method of Fixed Interval
Analysis are presented.
1.
Let
Ek
Introduction
(Y,. X) be a random variable taking values in
,.."
the Euclidean space of k
dimensions and.1
(Yx. ~) where
I\j
is a measurable space.
is
Let
E , E , ••• , E be g disjoint measurable sets in ~ whose union is the whole
2
l
g
space
*- .
x2 ), ... , (y
,x)
are n
<Xl' Xl)' (Y2'
('
",n
n
The number of xis that fall in E is n , j
j
j
independent observations on
= 1, ••• ,
g.
(1, X).
is defined by
the relation
u.
""'J
j
where E
=
E
j
y./n.
""~
j
J
is the summation over all "i"
= 1,
••• g
such that
Xi
is in Ej •
Throughout this paper it is assumed that
V(Y)
'""
<
(1)
a:l
and
where for any random variable rv
Z, v(Z)
,...
of /'.I
Z.
j
= 1,
••• g
(2)
denotes the variance covariance matrix
~his research work was done at the Indian Statistical Institute. The writing of
this paper was supported by the Mathematics Division of the Air Force Office of
Scientific Research.
2
The following theorem is established in section 3.
Theorem 1.
The asymptotic distribution of
(~l'
••• , ~g)
is the distri-
bution of g independent normal distributions.
This theorem plays a fundamental role in the method of Fixed Interval Analysis, for instance see Sethuraman (1963).
Interpreted in Sample Survey language
this theorem, among other things, states that the post-stratified stratum
me~ns
are independently distributed in the limit.
2.
Notations, definitions and preliminaries.
Let Y(E.), called the conditional random variable of Y given that
~
J
N
in Ej , denote a random variable on
X is
~ with the distribution defined by
Probe (Y(E j ) € A) = Probe (Y € A, X € E.)/prob. (X € E ). For any random variable
j
~
~
J
Z, E(Z) denotes the vector of expectations of Z. Define
""
'"
.E(Y(E.) )
=
~j
(3 )
V(Y(E.)
,.., J )
=
E.
(tJ.);
=
nj/n
(5)
'"
J
Pj
J
Fn
(u.
- ,...J
ll.)
"'J
=
:.G (n)
( 6)
In
(p. - ~.)
=
.~(n)
(7)
-J
.....J
j
= 1,
••• g •
be a sequence of families of probability
distributions on the Borelsubsets of E (or more generally, of any topological
m
space) and
e
Definition.
{'S n(
vary in a compact topological space K.
• ,
e)}
is said to converge weakly, uniformly and continuously
(in other
words, in the UC * sense) to S0 ( •
.
,e)
with respect to
e
in K
3
if for every bounded continuous function
S
f
and
g(y)
~n(dY,
e) --->
g(y)
soC dy,
e)
f
g(y)
hey)
on
~o(dy,
Em
e) uniformly in e
is a continuous function of
e.
The following theorem found in Sethuraman (1961) will be used in section ,.
Theorem 2.
where
(Y , X)
Let
n
n
be a sequence of random variables on
S is a complete separable metric space.
measure of Y
given that
n
distribution of X
be
n
X
n
n
m
x S)
Let the conditional probability
be denoted by g ( • ,x)
n
and the marginal
Let Sn(' ,x) converge in the UC*
~.
SoC • , x) with respect to x
weakly to
=x
(E
in any compact subset of S and
~
n
sense to
converge
Then the joint distribution of
Il '
o
the distribution determined by
distribution of
(Y, X)
o
Prob.
0
f Yo e
~
o
(Y , X ) converges weakly to
n n
( • , x) and ~
or, more precisely, to the
0
where
AI
Xo e B
r = f ~o(AIX) ~(dx)
B
The following lemma, which is immediate, is useful in
•
es~blishing
the
UC *
convergence of a special sequence of fami11es of di8tributions
Let
~ll' •••• ,
f lk (Q)
1
.. ,
.
Z l'
. , Znk (0)
..... n
rV
n
........................................
be a triangular scheme of random variables in E where the variables in any row
m
are identically and independently distributed. Assume that E (Z 1) = v
""n
and V(Z
1)
.-n
·-n
= Vn are finite and that Vn -> V as n -> co. Again let
4
inf k
e
(8) ->
n
MN(~,L)
Let
as
co
n",....>
ro
•
stand for the multivariate normal distribution with mean vector
a and variance covariance matrix L.
Lemma 1.
f (~nl+
... +
The sequence of families of distributions of
~nkn(e)
knee) I n )/ jkn(e)
-
1converges in the
UC*
sense to the
distribution MN(O, V).
3. Main theorems.
We first prove the following lemma •
Lemma. 2.
8
E zi
(~(n),
The distributions of
= 0 converges in the
UC
*
••• ~a(n»
"'Q
given that
,en)
,...,
=z
~
,
sense to the distribution MN(O, /\ ) with
1
respect to z
in any closed bounded subset of Eg , where
~
I
E
"1
1
I
=
1t
E
2
...
...
\
I
--];
0
1\
0
t
I
Proof:
ni
I
=,..,z
The event ,..,en)
= (nrc i + In zi]
Probe
f
i
0
,-
2 ~
I
I
I
I
I
._.
0
0
I
I
ri
0
(8)
I-1t
g
,
E
g
is equivalent with probability one to the event
= 1, ••• , g, since
nrci +.;n 'i(n) = _[n1t i +
rn
The conditional distribution of !l' ••• , In
'i(n)] , i
= 1,
given that
ni
.... , g
J = 1-
= (nrci + ;n zJ '
~= ;;E ';:' .~.,i:(:h~,a::::::::::~~f ~li~el~~:e~~.:'fir~es~O;n;iZ:ren~:e···' ng
l
~ g
jn: -In ----g
/V
~
'
.
5
normalized means of these
subset of E
g independent samples.
we note that
g
inf [n11'. +
i
~
,~
Iii
~
11'.,
~
uniformly in
(Xl
as
n ->
Further.,[n11'i +
z in any closed bounded subset of E.
g
(~l(n),
,.,
••• , r-g
~ (n»
given that
sense to the distribution MN(O, /\)
(Xl.
rn ZiJ"'/n
~
ditional distributions of
in the UC*
in a closed bounded
""
z.] ->
all the conditions of lemma 1 are satisfied.
to
For z
Thus
tends
Hence the con-
=Z
,en)
.y_
with respect to
converges
Z in any
,-./
closed bounded subset of Eg •
Theorem 3.
The joint distribution of
weakly to the distribution MN(O, B)
11'1(1-11'2)
where
C
=
""
,r n »
...."g;:Y
converges
where
...
-11'111'2
-11'211'g
-11'l11'g
-11'211'g
11'2(1-11'2)
-11'111'2
-11'111'g
Proof:
(TI1(n), ••• , ~ (n),
...
(10)
11' (1-11' )
g
g
This theorem is an iIlDllediate consequence of Theorem 2, lemma 2 and the
observation that the distribution of
MN(O, C) •
Proof of theorem 1:
-
C(n)
converges weakly to the distribution
Theorem 1 is contained in theorem 3.
4. References
1.
2.
Sethuraman, J. (1961). Some limit theorems for joint distributions, Sankhya,
. Series A, 23, 379-386.
Sethuraman, J. (1963). Fixed Interval Analysis and Fractile Analysis, submitted to the Mahalanobis Birthday Volume.