Page 396 - Contributed Paper Session (CPS) - Volume 6

CPS1995 Daniel B. et al.
2. Methodology
Assume that a random sample of size n from a fixed distribution F is
observed, where the observations are confined to intervals given by pre-
defined fixed points within the support of F. In this section we consider
estimation of parameters using interval information.
Formally, let ( , … , ) be independently and identically distributed (IID)
1

latent random variables with distribution F and consider fixed points , … ,

1
in the support of F such that −1 < , = 1, … , , where = −∞ and

0
= ∞ . Let represent the count or frequency of the (unobserved) ′
0

falling in the ith interval ( −1 , ], i = 1, . . . , k. In order to account for

information from subjects who prematurely discontinue from the trial, we also
consider right-censored observations, i.e., intervals of the form (, ∞) where a
is in the support of F. Such intervals will overlap the fixed intervals ( −1 , ], i

= 1, . .k. Denote the count for the interval ( , ∞) by , s = 1, . . . , k-1 and

the total count for the non-overlapping intervals by 0. Thus, the data consists
of two sets of frequencies, (01, … , 0) for the non-overlapping intervals and
(1, … , −1) for the overlapping intervals. From these let = ∑ −1 and
0
=1
0
= ∑ −1 so that the total number of observations is = 0 + 1.

1
=1
In the presence of right-censored observations, the log-likelihood is given
by () = ∑ =1 log[( ; ) − ( −1 ;)] + ∑ −1 log[1 − ( ; )].

0

=1
Note that () = () + () is just the respective sum of the log-likelihood
0

for uncensored intervals and for right-censored intervals. Similarly, the score
function () Hessian matrix () and information matrix () admit the
same decomposition. Thus, () = () + (), () = () + () and
1
0
0
1
() = () + () , where , and are the quantities corresponding to
0
0
0
0
1
the non-censored intervals and 1 , 1 and 1 are the quantities corresponding
to the right-censored intervals.
Estimation of the unknown parameter can be done by maximizing the
log-likelihood (). In general, maximization of the log-likelihood will not

admit a closed solution for the estimating equation () = 0. Hence, the
̂
maximum likelihood estimator (MLE) , must be obtained by using an iterative
approach such as a Newton-type method, a modified Fisher scoring algorithm,
or a modified EM-type algorithm. Using the Newton-Raphson method, the
estimate of at the ( + 1) iteration is given by
ℎ
(+1) = − −1 ()()| = (1)
Where () () is the Fisher score function and () = 2 () is the
′

Hessian matrix. Iteration is terminated when | (+1) − | ≤ , for some pre-
specified amount . Under reasonable assumptions on (; ) and sufficiently
accurate initial value, the sequence of estimates () enjoys local quadratic
̂
convergence to a solution . This solution is in fact the MLE of if () is
concave and unimodal.
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