BIOMATHEMATICS TRAINING PROGRA~
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STATISTICAL INFERENCE FOR CENSORED BIVARIATE NORMAL
DISTRIBUTIONS BASED ON INDUCED ORDER STATISTICS*
by
Frank E. Harrell, Jr. and Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1178
JUNE 1978
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Statistical Inference for Censored Bivariate Normal Distributions
Based on Induced Order Statistics*
By FRANK E. HARRELL, Jr. and PRANAB KUMAR SEN
Department of Biostatistias~ University of North Carolina
ChapeZ HiZZ~ N.C. 2?5Z4~ U.S.A.
SUMMARY
In life testing studies, where failure times are usually recorded
only for the first
k
(out of
n) units that
fail, one may wish to
study the relationship between the failure time and other concomitant
variates.
The present investigation concerns this study for the case
when these concomitant variates are observable only for the units
•
pertaining to the actual failures, i.e., for the induced order statistics.
For a bivariate normal distribution, estimation of parameters
based on these induced order statistics is considered and a related
test for independence is also studied.
Empirical power studies are
made for this test.
Key Words and Phrases:
Bivariate normal distribution, censoring,
independence, induced order statistics, order statistics, modified
maximum likelihood estimator and likelihood ratio test.
*Work Supported by the National Heart, Lung, and Blood Institute,
Contract NIH-NHLBI-71-2243 .
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1.
Let
INTRODUCTION
(Xl' Y1)' ... , (Xn , Yn )
be independent and identically dis-
Yn,l
tributed random vectors (i.i.d.r.v) and let
the order statistics corresponding to
Y , ... , Y .
l
Bx 1. (y)
F
f
for
. (y),
BX1
B
and
i = 1, ... , n;
X.1
here, the
Y.
given
1
may be
X.
is of
1
p (~l)
- vectors
is also an unknown vector.
as the failure
Y.
1
times, if an experiment is run until the first
of
Yn,n be
with a probability density function (p.d.f.)
In a life testing problem, treating the
•
~
Also, assume that
n
the conditional distribution function (d. f.) of
the form
'"
~
k
failures (out
occur, the likelihood function of the censored sample (and
n)
given the
X.)
_
L
n,k
1
is
= { nk
i=l
f
BXn[i]
(Y.) }{
n,l
nn
i=k+l
[1 - F
BXn[i]
(Y)] }
(1.1)
n,k
where
if
Bhattacharya
Y
.
n,l
= Y.J
(1974) has termed the
for
i, j = 1, ... , n .
Xn[i]
(1. 2)
as the induced order
statistics while David and Galambos (1974) have named these as the
concomitants of order statistics.
Various authors have used (1.1) to draw statistical inference
[on
••
B
and other parameters associated with
p.d.f.
fBx(y)];
may refer to David and Moeschberger (1978) for some of these
we
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procedures.
The conditional likelihood approach of Cox (1972) is
similar to this, but uses fewer parametric assumptions.
These
procedures, however, assume that though
are only
observable, the entire set
Xn[l]' ... , Xn[n]]
(Xl' ... , X )
n
In many other life testing problems, especially,
arising in clinical trials,
i
[or equivalently,
is given prior to experimentation, so that (1.1)
is properly defined.
is so (for
y n, l'···'Yn, k
is observable only when
so that the second factor on the right hand side
~n),
(rhs) of (1.1) is no longer deterministic.
•
if recording of
Yn,1.
X.
This is particularly true
necessitates the failure of the ith unit, so
1
that for the surviving
[n - k]
units, the
X.1
are not observable.
In toxicological experimentation this is a common feature.
The necessity of observing concomitant data for all individuals
no longer exists if we make the assumption that
(X. , Y.) has a
1
bivariate normal d. f. and denote its density by
~
= (llX'
lly' o~, o~, p),
variances and
p
llX' lly
Xn[l]' •.. , Xn[k]'
order statistics
Greenberg (1962).
2
oX'
for the correlation coefficient of
while the estimation of
y
< •.• < y
n,l
n,k
where
cl>f2(x,y)
stand for the means,
Watterson (1959) found linear estimators of
1
llX and
and
0
2
Y
for the
X and
poX using
using the
can be made as in Sarhan and
However, the estimation of
O~
(and p) poses
certain difficulties and constitutes the main objective of the
••
current study .
Y.
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In Section 3, an estimator of O~
is developed.
Likelihood
equations are incorporated in Section 4 for deriving Cramer-Rao
lower bounds for variances of unbiased estimators based on censored
data.
Estimation of
and
~X'
p using both order and induced
order statistics is formulated in Section 5.
In Section 6, the
desirable properties of the classical Pearsonian correlation coefficient as a test statistic for testing H : p = 0 are considered
O
and some power calculations are presented for various combinations
of
•
(k, n).
Foundations of many of these results are laid down in
Section 2.
2.
CONDITIONAL DISTRIBUTIONS AND MOMENTS OF INDUCED ORDER STATISTICS
Since
~~(x,y),
(Xl' Yl)' ... , (Xn ' Yn ) are LLd.r.v. having the p.d.f.
by an appeal to Lemma 1 of Bhattacharya (1974), we conclude
that given
Yn,l' ... , Yn,k'
Xn[l]' ... , Xn[k]
are (conditionally)
independently distributed and the conditional p.d.f. of
the
Yn,l.,
1~ i
~
k
and
-1
Let then
~=cry (Y.-!~y).
given
is univariate normal with mean
~x+pcrx(Yn,j-~Y)/cry and variance
every n~k~l.
Xn[j]
~=
cr~(1_p2),
for
(Xn[l]' ... , Xn[k])"
j=l, ... ,k
and
Y= (Yn,l' ... , Yn,k) ,
The joint (conditional) p.d.f. of
~
given
Y
is then
-.
2
[crX
(1 - p2) 21T]
-kl2 exp{ -
1
2cr 2 (1_p2)
X
}
[~ - ~X! - pcrX~] , [X - ~X! - PcrXk] . (2.1)
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It follows from (2.1) that
Thus, if we let
EZ=u= (un, l' ... , un, k)
~
and
~
E(Z-EZ)(Z-EZ), =V={(vn,1J
.. )),
~
~
~
~
(2.3)
~
then, from (2.2) and (2.3), we obtain that
Note that for
•
k ~ n ~ 20,
u
and
V are tabulated in Sarhan and
Greenberg (1962, pp. 193-205) and approximations for the same for
higher values of n
are considered in David (1970, pp. 65-7).
are quite useful in subsequent sections.
3.
o~
ESTIMATION OF
FROM THE INDUCED ORDER STATISTICS
Watterson (1959) used (2.4) to find estimators of
which are linear functions of
~X
Xn [1]' ••• , Xn [k] .
~
and
poX
His estimator of
-
~
a.~, X -- I 1=
. 10.·1 Xn [.]
1
k
~
-
u =k
K
-l~k
l. . 1\1
estimator of
1=
~X
n,1. •
when
(3.1)
~
a.1 =k-1-u.K(un,1.-u.)/r;,k
leu
-U )2,
K
m=
n,m k
and
~X
is
~X -
•
These
~X
l~i~k,
(3.2)
is the minimum variance unbiased (MVU)
p = o.
~X
losses its MVU property when
and it is not in general possible to find a MVU estimator since
p~0
p
is
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not known.
~X
However, he observed that for a certain range of
p
in (3.1) has a smaller variance than that of the MVU estimator,
p = 1.
assuming
The a.1
where the estimation of
in (3.2) are the same as in Gupta (1952)
~V
based on
Vn, l " " ' V n, k
is treated.
It follows from (2.4) that there is no estimator of
in
X and valid for all
estimator of the form
p.
X'AX.
would depend on the unknown
It is therefore natural to try a quadratic
Again, the variance of such an estimator
p.
might minimize the variance of
minimizing
•
tr~2
linear
Following the Watterson approach, we
-X'AX
--
when
subject to
p
=0, which amounts to
The authors have found
(by extensive simulations) the use of this estimator, which requires
the solution of
k simultaneous equations, has little to gain over a
much simpler estimator which we present below.
= l~1= lC'(X
1 n [.]
1 -il X)2
0X2
where
Let
is defined by (3.1) and the
C.
1
(3.3)
are real constants.
Using
(2.4), (3.1), (3.2), and (3.3), it follows by some some standard steps
that
(3.4)
where
~
= (aI' ... ,
~)
"
e= (b l , ... , bk) "
u.
£ = (c l , ... , ck )'
a. = (k-l)/k+u. {I+2(u . - )}/Lk_l(U
1
K
n,l
K
mn,m
•
-uk )2
k
k
k
b. = - a. - 2 L Ci. v .. + L L CirCis V n , rs + v .. + U2 .
j=l J n,J1
r=l s=l
n,ll
n,l
1
1
with
(3.5)
(3.6)
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for
i
a' c
=1,
Thus, for an unbiased estimator, we require that
... , k.
= 1 and
~'£
= 0 and , motivated by Watterson, we minimi ze
V(cr~)
which minimizes
when
p
=0,
considering llx
given.
£' £
The
result is
= {b.a'b -
c.
1
An
1- -
estimator of p
estimator of
paX
-2
aX
square root of
does not depend on
•
a.b'b}/{(a'b)2 - (a'a)(b'b)},
1- -
-
-
-
-
--
(based on
4.
(3.7)
X alone) and dividing the same by the
in (3.3) and (3.7).
y,
However, such an estimator
and hence, when the latter is given, may not
For this reason, in the next Section, we
proceed to study the e f ficiency of estimators of
(~,
~k.
can also be obtained by considering the Watterson
be a very efficient one.
based on
1~i
2
2
§ = (llX' lly' aX' cry, p)
r).
INFORMATION LIMITS TO THE DISPERSION OF UNBIASED ESTIMATORS OF
r
Note that the joint p.d.f. of
-8
is
(4.1)
where
n [k]
d.f. and
=n
~l
... (n - k + 1),
is its p.d.f.
that the joint p.d.f. of
~
<PI
is the standard univariate normal
Multiplying (2.1) and (4.1), we obtain
and
X
is equal to
(4.2)
•
Yn,l syn,2 ... s y n,k
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where
is the bivariate normal p.d.f. with the parameter
<l>e(x,y)
Wi th the notations in (2.3), we let
w.1
=\In,11
..
+
u2 .,
n,1
1 sis k.
of (4.2) and define
r,
x =-X -
have (on letting
•
~,
-0
(d/d11y)Ln,k
~
= (wI'
""~)
Also, we denote by
and
L
n,
k
~.
where
the logarithm
Z as in before (2.1).
Then, we
11 X-1)
-1
= (n-k)oy
(4.3)
<1>1 (Zn,k)/ [1 - ~l (Zn,k)]
- (1_p2)-IO-1 [pO-lX' 1 - Z' 1],
y
X -0-
-
(4.4)
where
for
•
a, b,
C,
d
k = 1, ... , n.
with respect to
are non-negative numbers and the p.d.f. of
Z k
n,
is
Differentiating each side of each equation in (4.3)
11X' 11y' oX' 0y'
and
p and taking expectations, we
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obtain by some standard steps that
(4.6)
(4.7)
2
115 = E{-(Cl 2/allX ClP)L ,k} = (~'!) [aXCl-p )]-I,
n
1
(4.8)
12
= E{-(Cl 2/Clll Clll y )L ,k} = -kp[a a y (1- p 2)]-I,
X
X
n
(4.9)
14
= E{-(Cl 2/Clll aay )L k} = -p(~'»[aXay(1-p2)rl,
n,
X
(4.10)
1
(4.11)
•
123=E{-(a2/aaX~y)Ln,k} = _p2(~'!)[aXay(1-p2)]-I,
1
1
35
= E{- (a 2/aa ap)L ,k} = p(~' !-2k) [a (1- p 2) ]-1,
X
X
n
(4.13)
34
= E{ - (a 2/aaXaa y ) Ln,k} = _p2 (\I)' D [a a y (1- p 2)] -1,
X
(4.14)
P (l_p2) -2,
(4.15)
ISS = E{ - (a 2/a p 2) Ln,k} = {2kp 2 + (1_p2)~,
1
52
= E{ - (Cl 2/ClpClll y ) Ln,k} = -p(~' 1) [a y (1_ p 2)
r
1,
(4.16)
54
= E{ - (a 2/apaa y ) Ln,k} = -p(~'!) [a y (1-p2)
r
1,
(4.17)
1
1
22
= E{-(a2/all~)Ln,k} = {(n-k) [1JJ(n,k; 2,0,2,0) - 1JJ(n,k; 1,0,1,1)]
+ k (1_ p 2)
1
1
24
44
-1
}/a~,
(4.18)
= E{-(a 2/all y aa y )Ln,k} = {(n-k) [1JJ(n,k; 1,0,1,0) - 1JJ(n,k; 1,0,1,2)
+ 1JJ(n,k; 2,0,2,1] + ~'!(2_p2)/(1_p2)}/a~,
(4.19)
= E{-(a2/a~)Ln,k} = {(n-k) [21JJ(n,k; 1,0,1,1) - 1JJ(n,kj 1,0,1,3)
+ 1JJ(n,kj 2,0,2,2)] -k -
•
(4.12)
and where
I
ji
= I ij
for every
(1_p2)-1(2p2_3)~'p/a~,
j, i= 1, ... , 5 .
(4.20)
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[In passing, we may remark that Des Raj (1953) in the context of
~,
maximum likelihood estimators (MLE) of
when
Y
is truncated instead
of censored, obtained equations similar to those in (4.3).
However
our (4.6) - (4.20) are different from his.]
!
To evaluate
need to evaluate
that given
= ((I. 0))' 0=1
J.{..
u'l
J,.{..
~'!.
and
k- 1
•
the information matrix, we
For this, as in Saw (1958), we note
independent
d.f., truncated from above by
Sl'.'"
5'
the joint distribution of
Zn, k'
reducib Ie to that of
by
, ••• ,
Sk_1'
so that for
Zn, k;
(Z n, 1'···' Zn, k-l)
is
rv' s from a univariate normal
we denote the latter variables
1 ~ i ~k-l,
E(S~IZ
1. n, k)=l-Z n, k<!>(Z n, k)/iIJl(Z n, k)'
E(s·lz
1. n, k)=-<!>l(Z n, k)/iIJ 1 (Z n, k),
(4.21)
and, in general, for
r
~
0,
E(S.r+21 Z k) = (r+1)E(S.rl Z k) - Zr+lk<!>1(Z k)/iIJ (Z k)
1
n,
1. n,
n,
n,
1 n,
(4.22)
Hence,
!:!'! =
k
L E(Z n,1..)
. 1
= E(Z
1=
= E(Z n, k) +
=
w'l =
=
•
~(n,k;
r~=llE{E[S.
11. Iz n, k]}
0,0,0,1) -
L~_1E(Z2
.)
1.n,1.
~(n,k;
k-l
k) + L·_lE[S.]
n,
1.1
(k-l)~(n,k;
= E(Z2 k) +
n,
0,0,0,2) + (k-l)
[Note that, in Saw's notation,
~(n,k;
(4.23)
1,1,0,0);
r~=llE{E(S~IZ
k)}
1.1. n,
[1-~(n,k;
1,1,0,1)].
(4.24) .
a,a,O,b) = ~(k/(n+l), n: a,b)
and these can be easily calculated for given
(k, n)
by numerical integration.]
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For the calculations which follow,
as unity.
aX and ay
are both taken
For three selected sample sizes and five values of
the elements of
! were calculated.
p
The Cramer-Rao lower bounds
for variances of unbiased estimators, which are the diagonal elements
of
! -1 '
are presented in the following table.
These bounds also
represent asymptotic variances of the maximum likelihood estimators
and
us ing (4. 3) ,
p.
TABLE I
•
Cramer-Rao lower bounds for variances
k
p
20
10
o
.2433
.0761
.0500
.0601
.2433
.1
.2417
.0761
.0514
.0601
.2372
.3
.2283
.0761
.0618
.0601
.1916
.5
.2015
.0761
.0775
.0601
.1181
.9
.1078
.0761
.0787
.0601
.0049
o
.4598
.0567
.0250
.0345
.2134
.1
.4557
.0567
.0266
.0345
.2077
.3
.4235
.0567
.0385
.0345
.1653
.5
.3590
.0567
.0562
.0345
.0984
.9
.1333
.0567
.0564
.0345
.0032
o
.6170
.0508
.0100
.0155
.1410
.1
.6114
.0508
.0112
.0155
.1370
.3
.5661
.0508
.0200
.0155
.1081
.5
.4755
.0508
.0330
.0155
.0630
.9
.1584
.0508
.0322
.0155
.0017
100
1000
•
'"
V[p]
n
20
50
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It is interesting to note that for a large censoring proportion (small
kin)
yet
cannot be estimated well at all from induced order statistics
ax can be estimated with more accuracy than ay for small
Also, lower bounds for variance of estimators of
independent of
p as one would expect;
~y
and
p.
~y
and
ay
can be estimated
ay
are
efficiently with a small number of order statistics regardless whether
or not induced order statistics are observable.
The difficulty in estimating
~x
sterns from
being unknown,
p
therefore bringing an unknown location bias to the sample of induced
order statistics due to the selection process on
p
•
were known to be zero
with variance
5.
~x
Y.
For example, if
would be_estimated by the sample mean
a~/k .
MODIFIED MAXIMUM LIKELIHOOD ESTIMATORS OF
An iterative method may be used to estimate
§
e
by using (4.3).
Des Raj (1953) set up the implicit solutions to the likelihood
equations for the case of double truncation on
Y from a bivariate
normal sample which is similar to our case relating to censoring.
Assuming the theorem outlined by Halperin (1952) to be valid for our
bivariate caset , the ML-estimator will be consistent, asymptotically
normally distributed and asymptotically efficient too.
tSince
•
However, the
Y l' ... ,y k are not independent, study of the asymptotic
n,
n,
properties of MLE may demand additional regUlarity conditions [viz. Sen
(1976)]. Nevertheless, for bivariate normal d.f. IS, these hold
under quite general setups.'
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complexity of the solutions in (4.3) makes the study of the properties
of the estimator difficult for any finite sample size.
~y
results obtained by Saw (1961) for estimating
Also, if the
and
cry
by the
MLE from censored data can be extrapolated to our bivariate case, we
can expect the MLE to have undersirable properties for small
k or n.
Particularly, the MLE may have considerable bias (relative to its
standard error).
The linearization of the log-likelihood function
described in Chan (1967), which results in simple and efficient
estimators of
does not work here as, in this case, we
will face 4 equations in 5 unknown parameters.
•
Ln,k
order Taylor-series expansion of
is needed and this will be as
complicated as using the original equations in (4.3) and does not lead
to unbiased estimators.
There are other good estimators of
considered by Chan (1967).
unbiased estimator of
k/ (n+ 1))
~
94%
for
and
cry
besides the one
For instance, Saw (1959) developed an
which is a linear combination of
~y
k/ (n+ 1)
a linear combination of
efficiency for all
as low as 0.25.
\k-1 (Y
~i=l
as an unbiased estimator of
cr~
Y
n,k
p
tion of
n,i
- Y )2
n,k
cry)'
and
{\~=11
(Y . - Y k)}2
~In,1
n,
which is of 100% asymptotic
VCD )
x
and
(and hence', equal to the bounds when
C~Y'
He has also considered
k/Cn+l).
Since the lower bounds for
of
~y
Ck-1)-lL~-ly
. and has asymptotic efficiency Ca function of
1=1 n,1
and
•
Therefore, a second
V(ox)
p = 0) ,
are independent
for the estima-
the induced order statistics may not contribute
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.e
any information.
In view of this, it seems natural to estimate
by some (efficient) procedure [based only on
(lly'Oy)
n, 1"'" yn, k)]
[denote the estimators by
y = (Y
-
(lly*, Oy*)]
and then
to consider only the first three equations in (4.3) wherein
are replaced by
ll~,
0;
and equating these to
equations in three unknown parameters
(llX' oX' p)
as the modified MLE.
solutions
0,
lly,Oy
we get three
and denote these
For this purpose, we
denote by
-
X =k
•
-1
\k
L.i=l Xn [i] ,
(5.1)
and
SXY
-1\k
k L.' 1 (X [.] - X) (Y
=
1=
n 1
-
n,l. - Y) •
Then, for the modified MLE, we have
[or
(5.4)
and the third equation in (4.3) along with (5.2) leads us to
(5.5)
so that by (5.4) and (5.5), we have
(5.7)
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From (5.4) and (5.7), we obtain that
(5.8)
and from (5.2), (5.7) and (5.8), we have
(5.9)
Thus, having estimated
and
~y' a y
~y' ay,
by
we may estimate
p by (5.9), (5.8), and (5.7), respectively.
of these estimators is that as
kin
1
+
~X' aX'
One nice feature
(i.e., the amount of
censoring decreases), the estimators approach the ordinary MLE,
•
which are known to be efficient .
r,
Noting that given
~~
is linear in
~,
we obtain by (2.2),
(5.1), and (5.9) that
(5.10)
and hence,
~~
is unbiased for
~X
if
~;
is unbiased for
~Y'
It
can also be shown that
V(~~)
E(a*2)
X
=a~(l-p2)
[1 + E{(~~ - y)2/S~}]/k + p2a~ay2V(~;),
= a 2 + p2 a 2 [a - 2E(a*2) _
X
Xy
y
1]
+
2 (1_p2) k- 1 [E{a*2 IS 2} - 2]
aX
(5 12)
y
y
••
In order to estimate the expected values of
•
*
*
aX'
a~
.
as t h e var1ances
0
combinations of
(n, k) and five different values of
sets of
f
~X'
and
p*,
(5.11)
and
p*
as well
for each of three different
p, 500 random
(X, Y) were generated and the statistics were evaluated.
--
e
.e
-16-
~~,
a;
were taken as the estimators due to Gupta (1952).
V(~X)
for comparison
(where
~X
is defined by (3.1)-(3.2)) was
u . in (3.2) were approximated
n,l
evaluated and for the latter, the
For simplicity, we have taken
by the method due to Harter (1961).
here
aX
=ay = 1.
Also,
We have also estimated the trace efficiency of the
modified MLE, which is the ratio of the sum of the five Cramer-Rao
lower bounds from Table I to the sum of the estimated variances of the
five modified MLE.
TABLE II
•
Estimated moments of modified maximum likelihood estimators
Trace
Efficiency
k
p
20
10
0
.281
.297
.074
1.021
.077
.078
-.020
.236
.902
.1
.287
.293
.092
.998
.074
.088
.060
.213
.884
.3
.281
.293
.082
1.043
.079
.079
.275
.196
.862
.5
.265
.276
.087
.99R
.102
.086
.449
.172
.749
.9
.123
.126
.093
1. 014
.106
.083
.889
.024
.764
0
.495
.508
.076
1.038
.042
.045
-.044
.169
.955
.1
.548
.564
.087
1.059
.047
.051
.080
.176
.859
.3
.487
.507
.090
1.054
.050
.049
.256
.152
.868
.5
.400
.420
.081
1.020
.057
.046
.442
.115
.865
.9
.202
.209
.087
1.014
.080
.048
.885
.009
.667
0
.663
.685
.078
1.045
.019
.022
-.016
.114
.931
.1
.601
.616
.091
1.044
.018
.025
.092
.105
.983
.3
.624
.648
.094
1.048
.024
.024
.274
.103
.875
.5
.477
.032
.025
.462
.066
.923
.187
.091
.082
1.030
.9
.508
.188
.987
.038
.023
.887
.003
.777
100
1000
•
v[ai] v[a y] E[p*] V[p*]
n
20
50
V[~x] V[ll x ] V[~~]
E [ax]
.
e
.e
-17-
l.l~
From Table II we see that
-
~X
is consistently slightly better than
although a small part of this may have to do with the approximation
used for the
u
..
and 102% according to the bounds in Table I.
is estimated as well as
Oy
variance in some cases.
Also,
is 53-103% efficient.
the cases studied except when
•
~~
has
Ox
On the other hand,
in almost every case and has lower
O~
appears to be not badly biased and
has efficiency between 69%
p*
p
is between 66%
As expected,
large variance for large censoring proportion.
except for moderate
~x*
The simulated efficiency of
n,1
p = .9.
The bias in
(p = .3 or .5
and 130% for
p*
is very low
resulted in maximum estimated
bias of .06) .
6.
Since
TESTS FOR INDEPENDENCE OF
are efficient estimators of
(X,Y)
for all
~Y'
p,
the modified MLE in Section 5 may be incorporated to prescribe a
H : p =0 .
O
modified likelihood ratio test for
* OX'
* P*
~X'
t-'y'
p = 0,
0*·
y'
Over the
the parallel estimators are
0*·
y'
ii*X = X and
are defined by (5.1).
Since the second factor of (4.2) involves only
have the same estimator
•
before,
are the estimators considered in Section 5.
parameter space restricted by
11*
As
(~Y'
Oy)
(6.1)
and we
in both the null and non-null cases,
substituting the two sets of estimates in (4.2), taking the ratio and
e
-18-
.e
we
proceeding by some standard steps)
obtain that
-2 times the
logarithm of the (modified) likelihood ratio statistic is equal to
(6.2)
and the
SX' SY' SXy
are defined by (5.1).
product moment correlation of the set
r*
Thus,
., X [. l)
n,1
n 1
(Y
Thus, as a test statistic (for testing
,
is the ordinary
i = 1, ..., k.
H : p = 0) , we may use
O
r*.
We define
•
Let
2 /{ (k-2) -1 [SXSy
2 2 - SXyl
2 }
T * = (k-2)r *2 / (l-r *2 ) = SXY
(6.3)
*_
2 2 _ - 2\'k
- 2
u -kSy/Oy-Oy l.i=l(Yn,i -Y) .•
(6.4)
Hb.(t; a,b)
(DF) (a,b)
for
and
0::; u < 00.
be the non-central F d.f. with degrees of freedom
non-centrality parameter
b.
and
G k(u) = p{U* ::; u}
n,
Finally, let
H;(t; a,b) =
r
o
p 2U /(1-p2) (t; a,b)dGn,k(u) ,
0::; t <00.
(6.5)
Then, we have the following
Theorem 6.1
For every
t ~0
and given
y*
(6.6)
P{T* ::;tIY*} = Hp2(1_P2)-1u*(t; 1, k-2) .
Hence,
P{T*::;tl=H;(t; 1,k-2),Vt
*
Hp(t;
1, k-2) = HO(t; 1, k-2)
•
£
Under
[0,00).
is the central
F
Ho:P=O,
d.!. with DF (1, k-2) .
e
.e
-19-
Proof:
As
in Section 2, given
X, Xn[l]' ... , Xn[k]
independently normally distributed with means
1 ~ i ~k
a~(l-p2).
and a common variance
are (conditionally)
-1
]..IX + fXJ Xa y (Yn,i - ]..Iy)
Hence, given
Y,
kS
XY
is
conditionally normally distributed with mean
and variance
a~(l-p2)O:-YD'!k(X-Y!)
=
a~a~(l-p2)u*.
k 2S 2 /{a 2a 2 (1_p2)U*}
XY
XY
where
•
X~(6)
Therefore, given
~ X2(p2U*(1~p2)-I)
1
X,
(6.7)
'
stands for the non-central chi-square d.f. with
DF and non-centrality parameter
~.
p
Further,
(6.8)
where
~ = a~u* C!k U = {a 2 a 2
X Y
-Y, U-
and given
k-
1
11 '~ - (X - y !)(X - Y 1) '}
(1_p2) u*}-
(6.9)
12 X
-
is conditionally normally distributed with mean vector
(6.10)
and dispersion matrix
~k-1S;2
k-ls~2Ik.
Further, it can be shown that
is an idempotent matrix and rank of ~ =k-2.
Thus, given
Also,
y'~X =o.
X,
(6.11)
•
Finally, noting that
(Y - Y1)' A = 0,
~
- - -
we obtain by (5.1), (6.8), and
-20-
e
e
(6.9) that given
r,
{S~S~ - S~y}
SXY and
are conditionally independent,
and hence, (6.6) follows from (6.3), (6.7), (6.11) and the fact that
this conditional df depends on
Y ~hrough
U*
alone.
Finally, by
(6.5) and (6.6),
=E{P(T* ~ t
P(T* ~ t)
to
HO(t; 1, k-2)
=H*(t;
Iu*)}
-
P
for all
U*,
1, k-2).
Under
and hence,
H :p
O
H; =H
If follows from Theorem 6.1 that under
O
H :p
O
= 0,
'
Q.E.D.
= 0,
T*
classical variance-ratio distribution with OF (1, k-2)
can be made without any difficulty.
•
point of
H ( ; 1, k-2)
o
of the test based on
significance
F*
a
= 1- a.
Then, the power
is given by
1 - H * (F*' 1 k - 2)
P a'
k < n,
10On%
(or r*) and corresponding to the level of
a(O < a < 1)
In general, for
has the
and the test
be the upper
Ho(F~; 1, k-2)
i.e.,
T*
Let
(6.6) is equal
G k'
n,
(6.12)
,
the d.f. of
U*, is quite complicated,
and hence, analytical solutions for (6.12) are difficult to obtain.
However, one can obtain one or two moment approximations for it, using
t
the following formulae
due to Saw (1958):
t This paper contains additional errors not reported in its
Corrigenda. Let ljIab =ljI(n,k; a,a,o,b), defined by (4.4). Then, noting
Z = (Z 1"'" Z k)' with Z .=(Y . - ].ly)/cry ,
n,
n,
n,l
n,l
1 sis; k relate to the standard normal df, we find after rederi ving Saw's
formulae that
that
..
U*
= (Z-Zl)'
(Z-Zl)
- - -
EU*
where
= k- 1 (k-l) [(k-l)
and the coefficient of
ES~, should be - (3k
2
-
- (k-3)ljIll + ljI02 - (k-2)ljI20]
ljI13
= ES~/cr~
in equation (3.6) of Saw (1958), relating to
14k + 15) instead of (5k 2
-
10k +1).
confirmed these findings in a personal communication.
ourselves to k > 4 (or> 6) for these approximations.
Dr. Saw has
Also, we limit
e
-21-
e
(6.13)
(6.14)
For the one moment approximation, we replace the distribution of
where
A = l;;EU*
defined after (6.4).
distribution of
T*
and
F
has the d.f.
a, b (~)
T*
HLi(t; a,b),
For the two-moment approximation, we replace the
by
aF ,k_2(0)
b
and equating its moments to (6.13) -
(6.14), we obtain that
(6.15)
Note that under
•
p
= 0, a = 1, b =1 and A =0, so that both the
approximations give the correct null d.f.
mations, (a, b) values for different
For the two-moment approxi-
n, k, p
combinations are as
follows.
V(U*)
a
b
3.87797
1
1
0.1
1.035
1.001
0.3
1.339
1.057
0.5
2.144
1. 312
0.9
15.633
3.731
1
1
0.1
1.043
1.002
0.3
1.417
1.085
0.5
2.407
1.443
0.9
18.992
5.174
n
k
P
20
]0
0
100
1000
20
50
0
EU*
3.43246
4.22029
3.60135
1
1
0.1
1.069
1.004
0.3
1.672
1.182
0.5
3.264
1.845
0.9
29.954
9.063
0
6.79174
4.41258
e
e
-22-
For each
n, k, and
P, the power of the r* test under censoring
at the 5% level was estimated by generating 2000
k x 2
random vectors.
The exact upper 5% critical values were used and the power was calculated
by counting the number of r*'s having absolute value greater than the
true critical value.
values are also given.
For purposes of comparison the simulated critical
In order to compare the power of the test using
censored data to that using a complete bivariate sample, the complete
sample size
c
required to achieve the same simulated power using the
Fisher-Yates Z-test is given.
TABLE III
Estimated power of 5% r-test for censored data
=========-===
Power from
One Moment
Approximation
Power from
Two Moment
Approximation
Simulated
Power
.0500
.0500
.0485
.1
.0531
.0531
.0615
13
.3
.0810
.0802
.0810
6
.5
.1570
.1522
.1645
7
.9
.9160
.8331
.8080
7
.0500
.0500
.0445
.1
.0544
.0544
.0555
.8
.3
.0939
.0921
.0915
7
.5
.2025
.1920
7
.9
.9796
.1929
.9468
.9305
9
.0500
.0500
.0535
.1
.0576
.0575
.0595
12
.3
.1266
.1219
.1130
9
.5
.3140
. 322~
11
.9
.9995
.2967
.9985
.9980
14
'
n
k
20
10
100
1000
..
20
50
P
o
o
o
Estimated
Critical Value
of Ir*1
.624
.433
.280
True
Critical Value
of Ir* I
.632
.444
.276
c
-23-
e
. e
Note that a complete sample of about one-fourth the size of the
censored s ampl e would give the s arne power for
n
=1000,
k = 50.
The
two power approximations are very similar but the two-moment approximation appears to be slightly better.
However, because of the
simplicity of the one-moment approximation, it might be preferred.
-24~
e
e
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