•
ESTIMATION OF THE MEAN LIFE OF THE EXPONENTIAL
DISTRIBUTION FROM GROUPED DATA WHEN THE SAMPLE
IS CENSORED - WITH APPLICATION TO LIFE-·TESTING
by
.
PETER JAMES KENDELL
and
R L, ANDERSON
t
Institute of Statistics
Mimeograph Series NOt 343
February) 1963
•
ERRATA SHEET
Page
6, Last line:
e" = rz
r
i=l
r
Z t + (n-r)t
i
i=l
r
r
-
&=
Should be:
Page 7, Line 7
t
r
A
e
z
=
i
+ (n-r)t
r
1=1
I
•
•
•
,
..
..
.
..
r
Z t
&=...1....=1
Should be:
Page 14, Line
+ (n-r)t
i
_
r
7:
Should be:
(X~fi
-
X~-lf)(Fi
(X~fi
-
X~-lfi-l){Fi
..
- Fi - 1 ) •• •
- Fi - 1 )
Page 14, 2nd line from bottom:
.
Should be:
• •
< n
..
~ <n
..
n
• •
·
i
·•
··
Page 14" Last line:
Should be:
"
Page
17"
Equation
·• •
··•
n
i
~
= n ..
=n
··
• ··
(3.,.8):
e" = :21 • • •
Should be:
eA 1
1
=-
2
.
• • •
Page 22"
·..
·..
...
Page 2
•
ERRATA SHEET
pt
Page 24, 2nd line of Equation (4.2.9):
::: nQ2e
Should be: =
•
pt
nQ2e
A
R(e;
Page 24, Equation (4.2.11):
8,k) •••
00 1
Q~
Page 26" Last line of Equation (4.3.7)::::
+
=
00"
Q1
nk~
- -t
1
8e8 _ ~t_k~
8
ntk
]2
Q_
e
;:;
+ -e E( -) - 1
•
{ ~-1
~~
~
r
A
e,.,1
Page 35, First line of Equation (4.5.1):
"
A
e1 =e 0
Should be:
"
eo
--r.If
t
g
Page 35, Last line of Equation (4.5.1):
=
" -T
eo { g
eo
+
=
"
eo
c
(1 - 'ij:)' say,
C
A
= eO (1 - 'ij:)' say,
Should be:
C= g \1 +
Should be:
Page 35, Line 7:
*
I: nih2
i
Page 35,
...
Should be:
l'<
Page 37, Line 9:
•
Should be:
.•
•
1
= E(-)
r
.•
{z
1
= E(-)
r
2
Pigihi
• • •
~+1
t*
2
g h
P1 1 i
-
Qk+l
" e
. . relatively small eO"
2
• . • re1ative1y small n eo"
• • •
A
Page 40, Line 4:
•
A
Should be:
1
1
l; .. ·
1
8"k) •
•••
k
1}2
- -8 + nt E(-)
- 1
~+1
2
k r
8e8
Page 27" Last line of Equation (4.3.10): + { e8 _1
Should be:
Should be:
•••
n~~
R(eoj
Should be:
A
e
2
+ '14:
• • •
,
Page 3
ERRATA SHEET
rex)
Page 48, First line:
Page 49, Line 10;
Page 53, Line 14;
lt
s->
Page 55, Line 12;
g(x)
=
Should be:
0
...
~-2,1
Should be:
Should be:
= ••
h
k-2-k-2
rex)
lt
s->
~-2,k-l
~-2,k-2 ~-2,k-l
~-2,1
( _1)k-2 d
11
•
.•
2 + ~,k+l
Should be;
Page 65, Second line:
Should be:
e"-1 =
e'"1 =
...
...
Page 68, Line 3:
Page 69, Reference 5 (EPstein):
Should be:
Add; Detroit, Michigan.
Page 70, References 19 and 20 (Lehman and Mendenhall):
Add:
( Un1versity Microfilms, Ann Arbor)
=
...
g(s)
0
=
• • •
•
iv
TABLE OF CONTENTS
Page
CHAPrER 1 - INTRODUCTION
•• • • • • • • • • • • • • • • • • • • •
•
CHAPrER 2 - REVIEW OF LITERATURE
CHAPrER
3.1
3.2
3.3
3.4
3.5
3.6
• • • • • • •
4
• • • • • • • •
10
• • • • • • • • • •
3 - MAXIMUM LIKELIHOOD ESTIMATION OF a • •
1
Test Procedure • • • '. • • • • • • • • • • • • • •
Derivation of' the Maximum. Likelihood Estimator • •
Existence and uniqueness of the Maximum Likelihood
Iterative Estimation Procedure • • • • • • • •• •
Simplif'ication of' the Maximum. Likelihood Equation
Modif'ied Maximum Likelihood 'Estimator •• • • • •
• • • ••
• • • ••
Estimator
• • • ••
• • • • •
• • • ••
10
10
12
15
15
17
• • • • • • • • • • ••
19
A
CHAPrER 4 -PROPERTIES OF THE ESTIMATOR ~O
A
aO • • • • • • • • • • • • • • • • • • 19
A
'
ao • • • • • • • • • • • • • • • • • • • • • • • • • 22
A
4.1 Mean and Variance of'
4.2
4.3
Bias in
4.4 On
4.5
•••••••••••••••••
24
the Non-Monotonicity Property • • • • • • • • • • • • • •
28
An EXpression f'or Var(e.o)
A
al
• • • • • • • • • • • • • •
35
and the M.L.E. (Equal Spacing) • • • •
46
Eff'ect of' Neglecting Terms in
A ,
4.6 Comparison between
CHAPrER
e.0
5 - OPrIMAL DECOMPOSITION OF THE SAMPLE SPACE • • • • • • • 47
47
47
Introduction. • ~. • • • • • • • • • • • • • • • • • • • •
Determination of' the Decomposition • • • • • • • • • • • • •
A Numerical Exa:rrq>le • • • • • • •
• • • • • • • • • • •
57
CHAPrER 6 - Stnv1MARY, CONCLUSIONS AND RECOO1ENDATIONS FOR FURTHER
RESEARCH • • • • • • • • • • • • • • • • • • • • • • • • • • • •
63
5.1
5.2
5.3
LIST OF REFERENCES
·.
• • • • • • • • • • • • • • •
~
• • • • • • •
• 69
v
•
LIST OF TABLES
Page
4.2.1(a).
E(~) = r=l
~ ~ (~) ~+l ~~~ . / l-~+l
4.2.1(b).
V(~)
=
•• • • • • • • • • 29
E(~) - (E(~»2, where
r
1
• • . . . . . . .
E( '"'2)
30
r
Bias of
"
e.O
N
..
,
"
~qually
,
spaced case • • • • • • •• • • • • 31
Variance of eON equally spaced case • • • • • • • • • • 33
Approximations to the upper bound of the correction
factor
equally spaced 'case (as percentages of e) • • 38
N
,
A
Approximate bias of
e.2
N
equally spaced case • • • • ••
.A
variance of e
41
4.5.3.
App~oximate
equally spaced case. • ••
42
4.5.4.
A
Comparison of the mean square.. errors of eo'
e" 2 and the
maximum likelihood estimator (ungrouped) - -equally
spaced c~se • • • • • • • • • • • • •• • • • • • ••
43
2
N
•
CHAPI'ER 1
INTRODUCTION
The 'longevity of animate and inanimate objects under certain
environmental conditionsis -of the utmost importance in :rna.ny fields.
Actuaries have long been interested in the life-span of human beings,
and, +n more recent times, industrialists and engineers have been concerned with the reliability 'of a product, a component, or a system of
components, under certain decremental stimuli.
In each 'instance the obj ect under study is characterized by a
measurable life-span which varies with the amounttm.d ty,pe of stimulus
applied.
The process is analogous to the variation in crop yield in an
agricultural experiment.
The length of life is determined by some pre-
stated definition of "failure ft , whether it is the inability of the
object to measure up to some prescribed standard or even outright destruction.
To investigate the ft mortality' characteristics of an obj ect under
certain stress conditions we require knowledge of its underlying mortality curve or failure distribution.
Although the form of this dis-
tributionvaries according to the' item studied, most generally it
follows an exponential or modified exponential type, e.g. Weibull distribution [29]:
t
~
t'" a,
g(t) =
M
>0
(1.0.1)
o
t
< t' •
•
If a sample of items is place'!: on test and subjected to a 'stress,
for economic reasons it may be inappropriate to continue testing until
all have failed.
This arises firstly, because the failure time of the
last item may be indeterminably long, and secondly, because in practice
it may be too costly to destroy all of the units on test.
In other
words, exact infol"Illation about some of the units and partial information on the others is often a prer'equisite of a life-test.
procedure is called a censored sampling plan.
guish between censoring and truncation.
Such a
Here, we mustdistin-
A censored sam;ple may be
defined as one in which all variate values beyond a certain range are
unknown, but their number is known; whereas, in a truncated procedure
we have no way of determining either the values or the number of variates beyond a certain range.
It is this partial statistical informa-
tion which, except in the simplest cases, tends to complicate the
statistical techniques associated with censoring.
Most generally life-tests are of the single censoring variety, the
type depending on the particular stopping rule adopted.:
I.
e.g.,
Stop after a prescribed number of units (1') have failed,
(O<r<n),
II.
Stop after a prescribed duration of time (t) has elapsed,
(t > 0),
III.
Stop after a prescribed time t
if
l'
or more units, have
failed, otherwise continue the test until
l'
have
f~led,
and in particular, most of the research yet performed has considered
the effect of a single stress on the life-span of a subject.
A more
•
3 _
practical result" however" would be a study of the effect of several
factors acting together.
Some notable research along this line was
published by Zelen [31; 32]" who considered the estimation and inferential problems arising from such a factorial arrangement under the
assumption of an underlying exponential failure distribution.
However" an objection to almost all of the research yet published
in the field of life-testing is that it presupposes that individual
failure times are recorded.
In certain experiments on electronic
equipment and so forth" this may not present a problem" but in lifetests 'on component parts of machinery" for example, this requirement
may
necessitate an expensive timing arrangement, whether it is human or
mechanical; and,. in certain biological studies where the effect of a
•
stress on a primitive organism is measured in terms of dilution, it may
be absolutely impossible to obtain exact failure times.
The above" then, is the genesis of the problem considered in this
research.
How do we estimate the parameters of a failure distribution
when the data are collected not individ~ly but at certain sampling
points" and what effect does this grouping of info:rma.tion have on the
properties of the estimator?
estimation of the mean life,
In particular, wenll consider the
e,
when the mortality distribution is of
the exponential type:
e1 exp- et '
g(t)
e>0
=
(1.0.2) .
o
e.
t > 0"
otherwise •
4
CHAPrER2
REVIEW OF LITERATURE
The problem of' analyzing f'or essentially continuous variates
grouped data is f'air1y old in the realm of' statistical methodology.
As
long ago as 1898 Sheppard [26] asserted that if' one used equally spaced
intervals 1 corrections could be made to the moments of' the discrete
distribution to bring them closer to those of' the continuous distribution.
In 1934 Wold [30] 1 again f'or the case of'equal spacings 1 gave the
general expression f'or any moment of'the continuous distribution in
terms of' the "raw" moments of' the discrete distribution1 except for an
error term.
Kendall 117] investigated this error term and gave condi-
tions for the validity of Sheppard's corrections.
Gj eddeba~k in a series of papers ([ 7 ]" [8 L [9], [10] 1 [11])
considered the problem of estimating the mean and variance of a normal
distribution from coarsely grouped data" by use of maximum likelihood.
The likelihood equations requir.ed an iterative solution.
The asymptotic
eff'iciencies of the estimators were considered along with
their asymp.
.
totic distributions.
Kul1dorf [18] considered the necessary and suf'fi-
cient conditions for the existence and the uniqueness of estimators f'or
the case of a normal distribution and determined an optimum allocation.
Walker [28] investigated the more general problem of the estimation
of a parameter from a continuous distribution by the use of maximum
likelihood1 when the data are arbitrarily grouped1 and gave a proced'l.:Q:'e
for obtaining a
deco~osition
of' the sample-space which is optimal in
•
5
the sense that it maximizes the information (in the Fisher sense)
inherent in the sample.
In more recent times a great deal of research, as evidenced by the
bibJiogralib.Y of Mendenhall [21], has been carried out on the statistical
theory of censored sa.m;pliDg, and in its application to life-testing
prob).ems.
Although a censored sample differs from one that is trun-
cated, research on both has often appeared jointly in the literature.
One of the earliest papers in this field was that of Hald U5] who
considered maximum likelihood estimation of the mean and variance of a
normal distribution for both truncated and censored samples, and gave
tables for iteration of the estimates and for evaluating the asymptotiC
covariance matrix.
Cohen £2] darived maximum likelihood estimators for siDgly and
doubly truncated normal distributions under fixed time censoriDg, the
solutions being
in
a form suitable for· the use of normal tables.
In a
later paper [3] he obtained moment estimators for the parameters in
truncated Pearson type distributions and stated that these 'Would provide suitable first approximations for iteration of maximum likelihood
estimates.
A more general paper was that of Halperin [16] who considered the
maximum likelihood estimation of a parameter
e from
censored· and. trun-
cated samples, when the underlyiDg probability density function f(x; e),
is subject to certain mild regularity conditions.
He found that the
estimator is consistent, asymptotically norinal and of minimum variance
for large samples.
•
6
Gupta [14] studies the estimation problem for a normal distribution
when censoring occurs after a fixed number of units has failed.
Best
linear unbif¥led estimators of the mean and standard deViation were obtained by· using the method of least squares on the ordered vat"iates
~
< ~ < ••• < xr " r
~
n"
trom a
s~le
of sizen.
The estimators
were of the form
,.
C1
-
the coefficients b , c
i
i
and r = 2(1)n-l.
=
r
1:
i=l
c.x. ,
~ ~
being tabulated for samples of size n = 3(1)10
The variances and covariances are given.
Best linear unbiased estimation was also- used by Sarhan and
Greenberg [25] again for the parameters of a normal distribution when
the sam;pling procedure was such that the smallest k
k·
2
l
and the largest
items were not measured (i.e., doubly censored).
Epstein and Sobel [6] gave procedures for estimation and tests of
hypotheses in the case of. sampling trom
which only the first
measured.
r
an:
exponential distribution in
ordered units from a sample of size n
are
In a later .pa;per Epstein [5]· gave more general results· for
censored sampling with and without replacement of failures.
were given for the case of a fixed number of failures
test termination time t.
r" and a -fixed
In the first instance the :maximum likelihood
estimates of the average life is
e
.
=
~ t i + (n-r)tr
i:i;l
. r
Procedures
'
without :replacement case
•
7
,
.
When r
'With replacement case.
A
is specified,
e has
.
the property of being unbiased, su:f'ficient
2·
A
V(~) =!...
The distribution of 2r ~ is
r ·
and of minimum variance,
1;1
that of chi-square 'With 2r degrees of freedom.
at a fixed time, t, the number of failures
r (r
When censoring occurs
> 0),
being a random
variable, then
A
e
r
= 1:
t
i
r
i=l
A
e
+ (n-r)t
nt
'Without replacement case"
'With replacement case.
= r
The estimator is not unbiased although it has all of the usual asymptotic properties of consistency and minimum variance.
Mendenhall (20] considered the estimation problem for santPling
from a mixed failure population in particular when the probability
density function is of the form:
-t/a2
e -tl/Ol
p + (l-p) !L
~
f(t) = .Ol
o
,
t>o
otherwise
•
He considered' the maximum likelihood estimators of p, Ol and
along 'With their . large sample properties.
~
The estimators are badly
biased and· have large variances for small sample sizes and test termination time
t.
Mendenhall and Hader
a mixture of k
Ha] extended the above results to
sub-populations •
Lehman (19] considered the estimation of the scale parameters of a
Weibull distribution for sampling under a mixture of two stopping rules--
•
8
stop at time t
if r
the test until r
or more units have failed, otherwise continue
have failed.
The bias and variance of this esti-
mator are both non-monotonic as functions of the sample size.
This same non-monotonic property is exhibited by the estimator of
t]:J.e· exponential parameter when censoring occurs at a fixed time (see
e.g. Bartholomew [1]).
When censoring occurs at a fixed time the maximum likelihood
estimators are functions of inverse moments of a truncated binomial
distribution.
Several writers (e.g. Bartholomew [1], Grab a.I+d Savage
[121, Mendenhall and Lehman [23], and Stephan [271) consider large
sample approximations to these moments, and Grab and Savage give the
exact results tabulated for .
p = .01, .05 ( .05) .95, .99 and n
= 2(1)20
~
and for
p
= .01,
.05 ( .05) .50, .99
and n = 21(1)30.
One of the few papers to consider the joint problem of censored
sampling and grouping of the data was that of Grundy [13], who investigated the estimation of the parameters.ofa nornial distribution when
censoring is to the right and the data groUJ;>ed.
Using as class bound-
aries
where
X
and observed frequencies
and defining ~'adjusted" moments of the form
o=
0:;
~
= 00
•
9
he obtained likelihood equations as fmctions of
expanding
M:t
and
~
and
~•
After
as power series, tTl.mcating the series and
M:L and
substituting for
~
~,
the likelihood equations were reduced to
a form which enabled a simpler iterative procedure using tables from
Ha.ld t s [15] paper.
A recent pSsPer by Ehrenfeld [4] considers the estimation of the
mean life of a component when the failure distribution is of the exponential form and sampling stops at some time
being divided into k
t , the interval (0, t )
k
k
For the case of equal spacing the
equal groups.
likelihood permits an explicit solution
-~.....;;A:::...----r--
§ =
In.[ 1+
~
j=l
xj
/
~ sJ'
j=l
~
where
Xj
= observed frequencies
Sj
= n
- (Xl +~ + •••. + x.)
J
The asymptotic variance of
ungrouped case.
....
e
j
= 1, ••• ,k •
•
is derived and compared With that for the
•
CHAPrER 3
MAXIMUM LIKELIHOOD ESTIMATION OF
e
3.1 Test Procedure
A number of items (n} are subjected to a certain stress condition
for a predetermined period of time (t
= t k ).
At certain time intervals
(t _ , t ) during the life.,.test the number of items that do not come up
i
i l
to specification or have failed since the previous inspection are
counted, and removed from the life-test.
Asa direct result of this sampling plan we have a multinomial
situation in which the probability of an item failing in a specified
interval of· time is a function of the unknown parameter (e).
The class boundaries are:
'With observed frequencies:
~, ~,
... ,
~, ~+l;
where the number of items having failed is
k
r
= i~l
3.2
ni
and the number of survivors iS~+l
= n:-r
•
Derivation of the Maximum Likelihood Estimator
The probability of an i tern failing in the interval (t _ , t ) is
i
i l
given by
Since the distribution of failure times is assumed to. follow the
exponential law
11
•
t
g(t)=
o,
-> 0 ,
,
otherwise
then,
fl - exp- t
.~,
G(t) ::
10
t
>0
otherwise •
Putting Xi:: tile
then
hob (ti_l~t<ti) = hob (xi_l~x<xi)
= F(Xi
) - F(x _ )
i l
Xi ~ 0
otherwise •
From the multinomial property the joint density of the sample can
be
wr~tten 11
•hence
Differentiating with respect to the unknown parameter (e) yields
but
dF i
~
=
dFi
dixi
dX.J.
e,
W
::;,
":xif. J..
e-
,
l/For purposes of'notation we will henceforth use
k+l
1:;;: 1:,
1=1
k
1:*
=:;.
~, Fi = F( Xi ),
1=1
f1
= exp- Xi •
•
12
hence
C~.2.,5)
a result analogous to that obtained by Gjeddebaek [7J in estima.ting
from grouped data" the standard deviation of a normal distribution.
A
The maxim:um likelihood estimator (9) is thus a solution" if one
exists" to the equation
3.'
Existence· and tJniqueness of the Maximum Likelihood Estimator
To determine the existence of a root to equation (3.2.6) we must
Gonsider the sign' of the first derivative" .
space" (o~' 9
O~riL"
over the parameter
<'0).
Putting
since Xi
> xi _l " we can write
From the definition of Xi we can see that
~t
xi =
8->0
hence"
~t
00"
i = 2" ••• "k+l ,
Yi = ~t
xi -> 0). xi _l -> 00
X _ [8 - ex,p- x _ (1-8
i l i
i
i l
1 - exp- x i _l (1-8 i '
i
)J
= -00 ,
= 2" ••• , k
•
13
•
= 0
.et Yk;+l
.. .et
*k-> 00
e":"'>
0 . '.
.et xi
.et
•
~->··oo
e,
Sim;tla.:rly, for large
hence,
"k ;: -co
O~nL = 00 if at lea.stone observation exceeds t 1 ,
Thus.et
e->
-
=
0
i
= 1",.,k,
Q)
Yi
xi _1-> 0
.et Y1
~->o
=
X _ (5 -exp- X _ (1-8:i)J
i l
l' - exp- X:i-l( 1-51 )
i 1 i
.et
xi~i-> 0
=.et
x1-> 0
and,
Xl exp- ~
1 - exp- Xl
- ~
.et
=
=
= 1,
1,
0-,
Xk-> 0
from which we can infer that .et·
oW'
= 0-
if at least one obser-
9-> 0 0 "
vation is less than t ,
k
if ~
rn
and ~+1
Hence a root exists to the equation (3.2,6)
r n.
The uniqueness of a root can be determined by consideration of the
2 .
.
.'
olnL
sign of the second, derivative, . . 2 • Now,
. oe
'
d~r =_ ~2 ni [<xi -2;~f~;~~~i~~_1>fH' + (l'~:i:X~~~ri"1>J.
1:
(3.3,1)
-- --
•
14
For values of
e
'021nL
satisfying the likelihood equation (3.3.1), .;;...;;~;...
. ?j32
becomes
Let us consider the expression in the square brackets:
say,
then,
Since Fi - Fi-l = -(fi - f i _l ) when F(x) = 1 - exp- x" x ~ 0 ,
then
~2
l~ . < 0 for all 8, (0 < 8 < GO), which im,plies that the'
de
likelihood eqUation (3.2.6) possesses at most one root which is a ma.xi-
Tl+erefore"
0
mum. " The above may be
summari~ed
as "follows:
Theorem 3.".1
The likelihoodecauation for
1:
ni (
t i exp-
8
is given by
tile -
ti_l,exp- t i _1 /8
exp-t /e _ exp- t /8
)
i-l
" i
=
0"
and possesses a root which is a unique"maximum if n < n and
but has no a.cceptable root if either ."
n, = n
J
or
~+l
= n.
~+l
<n,
15
•
3.4 Iterative Estimation Procedure
Using (3.3.2) and the property of the multinomial distribution
E(ni )
= n(Fi - 'i-l) ,
we obtain
2
-E(~ lnL)
?lJ2
Denoting the left hand side of (3.2.6) by g(e) and putting
2
G(e)
=
(xifi - xi_lfi _l )
1: " F _ F
i
i-l
'
then by the method of scoring (see, e.g. Rac> [24], pp. 166-167) starting
with an initial value of
A
e
we can USe the iterative formula
'"
~j+l=
'"
'"
In practiceG(~j)
will tend to stabilize quickly as ~j->
eA and
hence it will be unnecessaI-y to recompute it after each iteration.
If
we have a large number of classes, however, the' i terat1ve procedure can
become tedious atLd we might ask ourselves the question:"Is there a
,
simp~e:t: wq of obtaining a solution to the likelihood equation? "-
The following is an attempt at, answering this question.
3.5 Simplification of the Ma.x:LmUm Likelihood Equation
Let us define ,gi = (t
e'"
u
and
i
i
= gi/8
+ t _l )/2 ,
i
,
hi =t i - ti~l'
i
= 1,2, .... ,k,
16
then
Xi
= Ui
+ hi/2a
: -X.i~l= ui - hi/28,
from which we can write,
Similarly,
thus
Expanding
coth hi/2a . as a power series we have
xi:f'i- xi_l1'i_l
1'i - 1'i_l '
= u.
J.
_ h 12a{ 28 [1 + !(h 128)2 + O(h 128)4])
i
l hi _ 3 i
i·
= ~ - (1 + h~/12a2) + O(hi/2a)4 •
(3.5 •.2)
The likelihood equation (3.2.6) can be written as
*
Xi1'i - xi_l:f'i_l
E. ni (
1'i -1'1_1
) + ~+l xk
=
0,
and substituting (3.5.2) into (3.5.3) we obtain, except for O( h /28 )4,
i
=
But
r.
ui = gila, hence (3.5.4) becomes
~* nigi + (n-r) t k
* nill2i
2
8 .- ( .
r .'
) 8 +~
12r
=
0
•
17
and putting
*
A
~O
=
1:_ niSi. (n-r) t k
r
it reduces further to
The solution to this equation.,
is an approximation to the maximum likelihood estimator of
A
observe from (;.5.6) that ~o
exists if r > 0
A
from (;.5.8) that ~l exists if
If ~
2
~o
A
*
e.·
We
(Le., ~ < n), and
2
ni 11i
..... r:: "> 1:, 3r or max [hi] < V; eo •
= n, from (;.5.6) we can see that
"-
eO
= Sl and from (; •.5.8)
that 81 <V3~, hence ~l does not exist for ~. ~ n.
;.6 Modified Maximum. Likelihood Estimator
From (;.5.8) we observe that
and if in this expression max [hi]
is small relative to
"eo"
such that
we may neglect the correction term., the approximate maximum. likelihood
estimator reduceS to
18
•
We know (Section 3.3)thS.t a non-zero m. 1. estimator is nonexistent for the case when z:t= n, however, in general an experimenter
would desire an estimate when this occurred, even though it would not
be m. 1.
:From thedefinitioIi of 8"0 (3.5.6), we see that we can obtain
an estimate when n=z:t, since "80
exists and is given by
the mid-point of the first time interval.
As a result, it is suggested that
"" would be a suitable esti8.0
mate of 8, which would include the case when n
='.ll..
19
CHAPrE'R 4
PROPERTIFS OF
-Tim
FSTIMATOR
~0
~
4.1 Mean and Variance of ~o
Since ~'"0 is defined eJtcept for
r
= 0 we Will derive the mean and
variance conditional upon the event r > O.
The mean value of ~
,0
is
in consequence given by
But,
E(8A 0 I!)
- .
=r1
E(E* nigi I!) + n-r tIt '
..
r
(4.1.1)
. where
Hence,
A
Pigi
1
E(8 ) = E* - + nt E(-Ir> 0) - t
,0
_ \:+1
k
r ~
21
It
.-
(4.1.2)
/
In a similar fashion:
var(~o) = Var("E*_ nir
g
g
i ) + var(ntk ) + 2nt,. cov(!, E* ni i ) •
A
r
. r
' r.
(4.1.;)
Considering each term of (4.1.;) separately we obtain:
!/Where two expectation signs appear together the inner one will
denote eXPectation over the ni(i=l, ••• ,k) such that I:*'ni=r, and
. the outer one over r given that r >0 •
g/For simplicity we will denote . E(~I! > 0) by E(~).
20
but,
Hence,
n·gi
Va:r(r.*....L.)
,r
.
* PigJ.·2
=, [r. -
Qk+l
-
P g
2
(r.*.J:..l) 1 E(!) •
- ~+l
r
Simila:rly,
and,
(4.1.6)
'e
21
Thus, stibstituting(4.1~4), (4.1.5) and (4.1.6) into (4.1.3) we obtain
var(~o) = (Jrtk )2
p
var(f) + £1:*
2
~~.
P
(F.*
2
~:~) 1 E(f).
(4.1.7)
A.
....
From (4.1.2) and (4.1.7) we observe that bothE(~O) and var(~o)
are functions of certain truncated inverse moments of r, which can be
obtained directly from:
~+1 ~~~
n·
1
n
1 n)
E(~) = Z ~ (r
r
r=l r
For large values of n
( 4.1.8)
1 - Pk+l
and Qk+l' such that
n~+l
> 10, Grab and
Savage [12] suggest the approximation
and Bartholomew [1],
(4.1.10)
Mendenha;J..l and Lehman [25] suggest approximating (4.1.8) by equatins
.'
the momeniis of a Beta distribution.
Their solutions for the mean and
variance are
1
E(;)
n-2
= n(a-l)
(4.1.11)
where
a
= (n-l) ~+1 •
22
•
A
4.2 Bias in ~o
We have seen from (4~1.2) that
It is of interest to obtain an explicit,solution where possible for the
A
In general, of course, the bias
A
B(e o ; n,tk,.!!)
A
= E(~o)
-
B(~O; n, tk'~)
is given by
e,
* Pigi
= (1:_ ~+l
- t k + ntk E(
1
r»
-e •
(4.2.1)
However, in the special case of equally spaced time intervals, the
expression for the bias
~
be reduced to a simpler form.
i -_ e -(i-l)h/e
.
- e -ih/e -_ e.-ih/e ( eh/e -1) ,
Since
'-12·
1-,
, •.•, k
.p
and·
gi
= (2i-l)
h/2 ,
then
(4.2.2)
But,
., .
* -ih/e
1: e
-t /e
_ -h/e· (l-e k )
- e . . 1 _ e- h/e
'
where t k
= kh ,
and
ehIe - 1
•
•
23
Combining these results and substituting in equation (4.2.:H we obtain:
/,
*
-tie
/,
heh/8(1_e k )
E, Pigi =
eh(e
_ 1 -
- tke
Hence
heh/e
h/6 -1 '
e
t e
k
-t
k
/,
h '
-tk/ 8
- ;:;
(l-e
).
c;
/e
h
, -t J6
l-e ' k
-
-tk/ 8
(4.2.4)
- '2 "
as
Thus the bias may be written as
B(§ . n,t ,h)
k
.0 '
Putting 8 = h/e
"
R(~O; n,8,k)
=
h
~ {he
/
' hie
e
e
-1
'~,'~'
- Pk+ltk - t + nt E(!)3 0-+1
2
k
k r
~
the relative bias in E(e"
" n,tk,b)
B(~O;
e'
o)
m.ay be written
8 ,tk 8 ntk,l, 1
I8e
= T""' - '\: e - 2' + T E(r)f e -1
e •
+1
'
1 •
( 4.2.6)
Expanding the first t,erm of (4.2.6) as a power series and neglecting terms of 0(8 4) we obtain:
,2
t
8
k
12 - ':""\:""';+1;;;";'9
As
8--> 0,
With t
R(e; n,t) =
k
~
nt
+, k E(l)
T
r
•
= t
fixed, the relative bias becomes
-: +
ntE(~) }/e
,
(4.2.8)
which a.part from changes in notation agrees With the result given by
•
e
Bartholomew [1] for the case of estimating
using a fixed censoring
time and the indiVidual failure times are known.
In particular using approximation (4.1~10), for large n (4.2.8)
reduces to
R(~j
n,t) == {-t + nt·nQ,+P }
I Q.
(nQ.{J·
Ie
.. Pt
_.--r
·
nQe
(4.2.10)
and hence asymptotically it is approximated by -
(4.2.11)
which is negligible for small h
On
compared with
the other hand, as· 5--> co,
that R(~oj n,5,k) ->R('oj n,oo,k)
4.3 An
e•
with k fixed we see from (4.2.6)
= (l)
•
...
EXpression for var(~o)
r
--
When the intervals are' equally spaced we
~proceed
in a manner
analogous to that adopted in Section 4.2 and obtain 'an~ression for
A
Var(~o) in terms of
.
8 =
hie •
25
Thus>
Now>
since t
k
= kh
>
and
CL
'"'k+1
=1
- P
k+1
=1
- e-tk/e
.
•
Hence (4.3.1) reduces to
Squaring expression (4.2.4) and subtracting from (4.'.2) we obtain: .
* Pig~
1: -
-
* Pi gi
- (1:.Q
.. _..
~+l '""1(+1
2
)
=
¥Jeh / e
..~e~/e _1)2
Substituting (4.'.3) into (4.l.T) yields
-
t~Pk+1
~+1·
26
(4.,.4)
Hence
the result for the case when the exact failure times are known and t
fixed, (see e.g. Mendenhall and Lehmann [23J).
.
As
. .'
n-.-> oo~
•
nPk +l ~+l
; .. 1.
va.r{;);:
1 '.
n~+l
.
(n~+l)
!.§.2 cosech ~}
2
whereas (4.3.6 ) becomes:
::
/
2
_..
1
nQ
Ii"' and hence (4.3.5) becomes:
(4.;.8)
The. result in (4.3..7) d:1f~ers from that given by Ebr~nfeld [ 4],
which is for large n
<.
where 9"" t
is the m.l. estimator of 9 when the data is grouped at
equal intervals' and censoring is the Same as for §o.
The two results have the same limiting form when 8
when 6
--;> 00,
since then Var( ~0)-> 0
and
--;> 0,
var(~') ->
but not
The
00.
reason for this apparent con:tradi.ction is presumably because ~0
.
·A
reality a linear approximation to a non-linear estimator
9'.
is in
•
By ad.d-
"-
ing extra terms to ~o (Section 4.5) we tend to reduce the bias but
increase the variance.
increasing 8
Even though§o has a decreasing variance for
such a situation is hardly reaiistic and a more
pertinent investigation would be to consider the mean square error
A
(m.s.e.), of S.O for small 8.
Denoting the m.s.e. of ~o by D(§O; n,8,k) then
A
D(~o; n,8,k)
=
9'2
(4.3.10)
28
From the results "given in Tables. ~"4.2.2) and (4.2.;) we see that
A
for small samples both the bias and variance of"
~d
hence so is the m. s .e.
eo
are non-monotonic
As the·' sample size increases the bias and
variance and thus the m.s.e. become monotonic, as would be expected.
This non-monotonicity property is also exhibited by the m.l.e.
for the ungrouped data, and has
occurr~d
A
e,
in other research (see e.g.
Lehman .U9], Bartholomew [1J).
4.4 On the
Non-Monoto.~city Property
~
A
The reason for the non-monotonicity of E(~O) and var(t1 0 ) is
that they are functions of certain inverse moments of a truncated
binomial" distribution.
The restriction that for small samples r > 0,
means that even for large values of P +l = p( t > t k ) we 'Will consider
k
only those cases where there has been at least one failure. Thus for
n
A
= 1 we Will obtain an estimate whose expected value [E( e01)'
S8\Y]
'Will be less than t • .When n = 2, we now permit one of the variates
k
to range over the entire real line, or at least that part of it which
occurs in the experiment.
Hence
Depending on the various
t i (and hence the Pi) this ~ard trend
of E(3 ) presumably continues with increasing n" until the effect of
0
the law of large numbers becomes dominant and
The initial bias may be negative or positive depending on t , and k.
k
The variance of
A
~O
behaves in an analogous manner.
Ii '
-
e
1
Table 4.2.1(a). E(iJ
n
i'
"~+1
==
.329680
<e
n 1r'r)
(n Qk+1
rlf-r
k+1'/ 1-1f
k+1
r==l
,1:
.393469
.550671
.698806
.798103~:
.864665
1
1
1
1
,1
1
1
2
3
4
5
6
7
8
9
10
11
.90131238
' .80942789
.72507005 '
.64861487
.58011211
.51933488
.46584491
.41906207
.37832805
.34295963
.31228779
.24246364
.17327547
.10957792
.08030070
' .06342971
.04469338
.03098312
.87754071
' .76641101
.66765587
.58146167
'.50735368
.44436690
.39126013
.34669306
.30935466
.27804279
' ~2517P301
.19430063
.13987827
.08983081
.81002551
.65279861
.52804560
.43184100
.35873204
.30331261
., .21600917
.22828267
.20253180
.,18189569
.16505753
.12926419 '
.09511688
.06234503
.04639278
.03694586
.02625679
.01831187
.73147505
.53929008
,.4093Q372
.32256906
.26365407
.22222668
.19192280
.16891432
.15087522
.13635158
.12440153
.69855789
.07325811
.04843228
.03617996
.02887664
.02057236
.01437295
.66798193
.46377298
.34301829
.26916248
.22099519
.18747295
.16284980
.14398990
.12907109
.11696815
.10694883
.08510359
.06350723
.04213678
.03153020
.02519017
.01796568
.01256187
.61920264
.41580268
.30615197
.241112104
.,19933807
.16983438
•1479a451
.13113857
..li774762
.10684446
.09779312
.077984:72
.05831073
.03876098
.02902991
.02320470
.01655930
.01158349
12
15
20
30
40
50
70
100
~06628375
.05254350
.03715780
.02582377
~
•
,e
e
Table 4.2.1 ()
b.
n
~1
r1)'
V(
(1 ). «1)2 .. :, . (:'1) .' n. 1,' (n\:"Q~ " -n-r/ Jf' .
=E,~,'- E r'. ".Where E:7: ::: rfi r2"'7:'i:~+1' ·J:."k+l. 1- k+l
.329680
.393469
.03960453
.06466577
.07802955
.08257731
.04623341
.01094299
.01990829
.01853496
.07126243
.•550671
.698806
, .798103
.864665
.06215686
.05708293
.03756936
.0218814a
.01229315
.00698811
.00413267
.00251465
.00:).69320
.00116991
.00084316
.OOP37896
.00014330
.00003858
.00001551
.00000177
.00000275
.00000092
.05517306
-.03152801
.01875935
·°°889049
, .00441134
.00238622
.00141315
.00090552
.00061653
.00043986
.000325112
, .00015358
.00006018
.00001665
.00000680
.00000341
.00000121
.00000041.
.04539210
.02281990
.00923883
,,00393120
.00190692
.00105401
.00064485
.00042465
.00029513
.00021368
.00015980
.00007692
.00003062
.00000858
1
2
!
3
4
5
6
.08095~4
10
11
12
15
20
30
.07542173
.06773183
.05918153
.05064396
.011265472
, .03549806 .
.01961121
.00714687
.00131142
40
.•00045848
50
70
100
.00020963
.000068112
.00002119
7
8
9
~06134091
.05089506
,,04114367
.03266204 '
.02561132
.01994749
•.00935535
.00291504
.00060041
...00021121
.00010281
.00003458
.00001121
.05889695
.07345688
.06'9741J2
.05~7056
.03684411
.02531248
,.01113100
.01152503
.00781669
•.00538923
.003797112
.00154185
.00052036
.00012962
.00005082
.00002491
.00000870
•00000289
, ~OO000352
.00000117
.00000063
.•00000021
VI
o
..-/"
T
e
,e
e
Table
4.2.2. Bias of. e'o
- Eq~Spaced Case
8=1000
tk
h'
400 200
100
Ungrouped
500 250
Ungrouped
800 .400
200
Ungrouped
1200 600
400
200
Ungro~ed
1600 800
400
200
Ungro~ed
2000 1000
500
400
Ungrouped
"Sample size en)
5
.6
-52.4
-53.1
-~9.9
87.3
84.8
83.9
179.8
179.0
1~.3
244.~
2eo.7 298.7
241.7" 278.2 " 296.2
240.9 z/7·4 295.4
-115.9
-121.1
69.8
64.6
188.1
J,.82.•9
256.5
251.3
289.7
284.5
299·5
294.3
294.6 281.2
289.4. 276.1
-639.5 -143.4
-649.4" -153.4
-652·8 -159.7
·12~2
117.3
113.9
250.3
240.3
237.0
287.9
277.9
274.6
282.4
272.5
269.1
2591.1
249.1
"245.8
231.0
221.0
227.7
204.2
194.2
190.9
180.8
170.8
166.5
-487.4
-503.9
-513·9
-517.2
68.1
51.6
41.7
38.3
254.1
237.5
227.6
.224.3
277.3
260.7
250.8
210~9
2~7.5
248.0
231·5
221.5
218.2
179·3
162.8
152.8
149.5
155.1 136~9
138.5 120.4
128.6' 1l0.4
125.3 107.1
123..•1
106.6
96.6
93.3
--352.0
-391.5
-401.4
-404.8
185.6
146.1
136.1
132.8
274.1
234.7
224.7
221.4
243.3
203.9
193.9
190.6
161.9
151.9
148.5
~O;I..'
169.6 147.7
130.1 "108.2
120.1
98.3
116.8
94.9
132.5 ·121.5
82.0
93.0
83.1
72·0
68.7
79·7
113.2
73'.7
63.7
60.4
-231.2
";292.4
-299·7
-313.0
245.6
"184.4
177.1
163.8
263.6
202.4
195.1
181.8
218.0
156.8
149.5
136.2
183.0
121.8
114.6
101.2
160.9
99.6
92·3
79.0
1
2
3
4
-810.0
-812.5
-813.3
-488.9
-491.4
-492.2
-238.7
-241.2
':'242.0
-765.5
-388.0
-393.2
~770.7
194.4
184.4
180.1
7
8
146.5 136.6
85.3 " 75.3
68.0
77.9
64.6
54.7
9
129.3
68.1
60.8
47.5
10
30'.~
300.9
300.0
123.8
62.5
55·2
41.9
~
f-I
e
,-
e
Tah1e 4.2.2 (continued)
8=1000
tk
h
11
12
200
100
Ungrouped
299.1
296.6
295.8
250
Ungrouped
Sa.m;p1e size (n)
30
40
15
20
286.5
285.7
244.8'
242.2
241.5
176.2
173.7
172.9
105·9
102.5
101.6
7U..8
72.4
71·5
263.7
258.5
244.7
239.5
191.7
186.5
133.2
128.0
81.9
76.7
5#-~O.
42:;;6·
400
200
UJ:lgrouped
161.2
151.2
147.9
145.1
135.1
131.8
111.7
101.7
98.4
82.4
72.4
69.1
56.8
46.8
43.5
45.1
35.1
31.8
1200 600
400
200
Ungrouped
112.4
95.9
86.0
82.6
104.0
87.5
77.5
74.2
86.6
70.1
60.2
.56.8
70.8
54.3
44.3
41.0
56.2·
39.6
29.7
26.4
1600 800
400
200
Ungrouped
106.7
67.2
'57·2
53.9
101.4
62.0
52.0
48.7
90·5
51.0
41.1
37.7
80.2
40.8
30.8
27.5
70.6
31.1
21.1
17.8
1(.00..
500
800
2000 1000
500
400
Ungrouped
119.4
58.2
50.8
37.5
2~9.0
115.9
54.6
47.3
34.0
108.4
47.1
39.8
26.4
101.3
40.0
32·7
19.4
94.5
33.3
25.9
12.6
50
70
~1oo
58.6
56.1
55.3
!+1.1J.
38.9
38.1
29,•.926.9
26.0
.3.3
0.8
48.0
35.0
29.~S
25.7
20.5
5.2
0
38.4
28.4
25.1
30.9
20.9
17.6
25-.5
15.5
12.2
13·3
3.3
0
49.2
32.7
22.7
19.4
45.2
28.7
..18.7
15.4
40.7
24.2
14.2
10·9
37.4
20.8
10.9
7.6
29.8
13.3
3.3
0
66.0
26.5
16.5
13.2
63.2
23.8
13.8
10.5
60.2
20.7
10.7
7.4
57.9
18.4
8.5
5.1
52.8
13.3
3.3
60.1
91.2
30.0
22.7
9.4
89.3
28.1
-20.7
7.4
87.1
25.9
18.6
5.3
00
-
85.5
24.3
17.0
3·7
0
0
81.9
20.6
13.3
0
\}I
I\)
#e
e
e
Table 4.2.3.
Variance of § 0 .. equally spaced case
e=1000
tk
h
2
400 200
100
Ungrou;ped
500 250
Ungrou;ped
,-'800 400
200
Ungrouped
1200 600
400
200
Ungrouped
1600 800
400
200
Ungrouped
2000 1000
500
400
Ungrouped
3
4
Sample size
5
6
7
8
9
10
34270
36518
37269
101132
103150
103825
59732
64287
;rn.~:I2
175389
333368
~~990:?
li99781 649165 15~60 ~0'31 858!i.90 821307
502805 651798 760567 822368 840289 822913
181914
189934
192629
448206
454669
456840
695876
701104
702861
836929 862693 807340 711719 606232 508053
841205 866245 810343 714303· 608492 510058
842641 867438 811352. 715171 609249 510731
418268
430148
437390
439824
784211
792969
798309
800103
899309
905956
910008
911371
814298
819537
822730
823804
658992
663274
665884
666762
511431 396673
515040 399790
517240 401690
517979' 402329
314218
316961
318633
319195
256246
258696
260190
260692
66~562
688241
6948;54
697075
928132
945961
950553
952094
815338
828524
831920
833061
605835
616182
618847
619742
436799
445295
447483
448218
324988
·332195
334051
334675
207477·
213013
214438
214917
175497
180459
181737
182166··
847982
884317
888945
891136
903247
927646
930754
936254
651463
669428
611716
675766
440515 313779 239967 194170 163364
454741 325476 249933202853 171059
456546 326966 251202 203959 172039
459~9 ~~96P3 ~53449 205917 J.73774
206934 336730 472068596447 698185 771075 814049
208742· 338348 473515 597742 699347 772186 814992
209346 338888 473998 598175 699735 772469 815307
253919
260179
261792
262333
141195
148104
148984
159542
~
e
,e
e
Table 4.2.3. (continued)
e=1000
tk
h
Sam,p1esize
20
40
30
70
100
459115 198569 118165 84479
459547 198842" 118365. 84637
459691 i98933 118432 84690
54083
54194
54232
35175
35253
35279
299654 136487 (81902 65099
300380 136953 '88246 65372
42935
43128
28436
28570
356314
'357949
358498
226994 '136869 77058 53820 41371
228274 137811 77675 54280 41737
228704 138127 ',77882 ,'54434 41860
28295
28555
28642
19194
19376
19437
215084
217298
218648
219102
185080
i87101
188332
188746
130898
132499
133475
133803
38853 30348
3944130817
39799 3110?
35?919 31199
21107
21441
21645
21713
1600 800
400
200
Ungrouped
152261
156757
151915
158304
134599
138711
139770
140125
100112
103383
104226
104509
70311 44129
72753 45749
73381 46166
73593 ,46306
32162
33374
33686
33791
25299
26268
26517
26601
17729
18419
18597
18657
2000 1000
500
400
Ungrouped
124J125
130695
131493
132907
111266
117004
117735
119029
( 8~557
89133
,89716
90748
60456
63878
6431465085
28289
29992
30209
30593
22346
23707
23881
24188
15739 10900
16711 11579
16834, 11666
17054 '11819
1l
12
15
400 200
100
Ungrouped
829191
830046
830331
820967
821746
822006
708404
709009
709211
500 250
Ungrouped
779198
780641
721979
723285
529224
530233
800 400
200
Ungrouped
424333
426134
426739
1200 600
400
200
Ungrouped
50
\~
88572
89762
90487
. 90731
53990
54776
55256
55417
38538
40812
'41102
41615
~,
14483
14717'
14859
14907
12225
12708
12832
12874
•
\X
-t="
'.
35.
A
4.5
Effect o~ Neglecting Terms in.~l
From (.3.5.8) we ~bserve that
i[8 + 80(1 . . .0. . . -
§1 =
2i
E*ni~)]
2
*
nihi
g=E - ,
Putting
" .3re"2
.31'9
if max hi <
V3 ~o -.
0
then
,0.
A
A
$o{
~1 =~()- "4"'". g
A
= ~0(1
C
"'4),
where
Putting
s>l
then
Ts<
gs
s i'·nce.
. .
1·.3.5.··
. . (2s-1). < 1 •
s
2
(s+l)~
Hence an upper bound for the factor. C can be obtained' by rep1acing th.eterms of
C by those of a geometric series which is term by
term greater than those of
C.· Thus
equality being attainedorily when' the intervals are equally· spaced.
36
Hence the correction to
§o
is less than
whi ch for large sa.JIWles may. be approximated by
6
147·
F Ihi~
1h~ ~
2
36 - max
•
hence the percentage error is less than
An
i~roved
bound on C can be obtained by noting that
Hence the bound on the· correction factor becomes for the equally spaced
case
and thus the percentage error- is less than
2
2
255 (4-5 )
4(3-52 ), .
..
37
However, in general the important term in the bias reduction will
2
be the first, namely
* nill.i . . .
1::
The amount of this reduction is given
12~o
by
=
1
* nih~/r
--12 E(1:: ...
)
. e
9
COy
(eo'
..., 1::*n.ll.
21)
r
J. J.
j
When the inte:rvals are, equally spa.ced (4.5.5) reduces to
E(
h~ ) = .
2
h ...
12eO" : 12E(eO)
,
....
..1. 1 +
v~:eo)
[E(e
.o
~
,
'
2
1::
h ... ' ,
)]2., 12E(e )
(4.5.6)
O
2
which :t'or,la:rg.e n_' __["bea~te~)b~~r:9:' ;. 'tpe
,d
p'~ce~t~e
a.pproximation being
The tb:ree a.pproximations to th.e bound are comp8J:'ed in th.e following table.
Table 4.5.1.
Approximations to the upper bound of the correction
factor - equally spaced case (as percentage of e)
.05 .10 .20 .25
.40
.50
.60
.80
1.00
1.50 .
.02 .08 .34 .53 1.41 '2.27 3.41 6.78 12.5
75.0
(4.5.4) .02 .08 .33 .52 1.35 2.13 3.10 5.69 9.37 32.81
(4.5.7) .02 .08 .33 .52 1.33 2.08 3.00 5.33 8.33 18.75
APirox. (4.5.3)
II
From the above table it is apparent that for
a<1
almost all of
"-
the correction to ~o is a~coUnted for by using the first term of (4.5.:1)
only.
It is to be expected that for
a>1
the first term is insuffi-
cient since the series expansion of exp- 8 converges only slowly for
8 > 1.
For the. equally spaced case denoting
then
For e:: 1000 and certain values of n, t
k and'h expression (4.5.9)
is tabulated in ,Table 4.5.2, and the results compared with the ungrouped
case.
Even for Widely grouped data the addition of the extra, term to
tJ0 is su,fficient to reduce the bias to almost that' of the ungrouped
da.ta.
39
The effect of the extra ter.m on the variance of the estimator is
derived below.
A
Var(8,0)
Since
[E(e )]4
o
and
....
then
"'"
A
-var(~o)
[E(~0)]2
"'" f
4
If-"'"
Var(8 ) :: var(8) 1 +
+
. h "'" . ] •
2
,0 . . 6[E(8 )]2
l44[E(8 )]4
,0
. _0
Hence the addition of the bias reducing ter.m has an increasing effect
on the variance, the extent of this increase can be seen by comparing
T~bles
4.2.3 and 4.5.3. A feature of the variance of 8"'"2 is that it
is an increasing function of 8
and hence is more in line with the
:maximum likelihood estimator (4.3.9).
A more realist'ic comparison of
"'"
.
the estimators would be through their mean square errors, D(8 0 ; n,8,k)
"'"
and D(~2; n,8,k}, where
"'" n,8,k)
D(~2;
by
A
D(~o; n,8,k)
'
is given by (4.3.10) and
40
or directly
'" n,8,k)
= D(~O;
'"
...
From Table 4.5.4 it is evident that for relatively small" ~O' ~2
'" have approximately the same mean
and the m.l.e. for ungrouped data, e,
square errors.
In other words, no significant in:t'ormation is lost in
grouping the data..
As
8 ->.:1, the mean square error of
A
~u
tends
to exceed the others, .indicating tnat
for fairly coarse grouping the
I
addition of the bias correction term not only reduces the bias but
also reduce.sthe mean square error.
preferable to 'Work With
'"
~2
•
In general, therefore, it would be
e
,e
·e
Table 4.5.2. Approximate bias. of ~2 ... equally spaced case
tk
h
. .' Sample size' (n)
20
30
40
50
10····
100
00
400
200
100
Ungrom>ed
172.5
172.8
172.9
101.5
101.6
'101.6
71..4
71.5
71.5
55.2
55.3
55.3
38.1
38.1
38.1
26.1
26.0
26.0
0
0
0
.-,.500
250
Ungrouped '
127.5
128.0
76.6
76.7
54.8
54.9
42.7
42.8
29.8
29.8
20.5
20.5
0
0
200
Ungrouped
400
68.7
',68.9
69.1
43.3'
43.4
43.5
31.7
31.7
31.8
25.1
25.1
25.1
17.6
17.6
17.6
12.2
12.2
12.2
0.1
0
0
600
400
2PO
'lJ'ngrOuped
40.6
40..,6
.40.8
41.0
26.4
26.1
26.3
26.4
19.6
19.3
19.3
19,.4
15.7
15·4
15.4
15.4
Ii.3
10.9
10.9
10.9
8.1
7.6
7.6
7.6
0.7
0.2
0
0
800
27·9
. 27·1
27.4
27·5
18.9
17.6
17.7
17.8
14.6
13.1
13·1
13.2
11.9
-10.5
10.5
10.5
9·1
7.4
7.4
7.4
6.9
5.2
5.1
5.1
2.;1
0.1
0
0
15.9
12.4
12.4
12.6
13.0
9.2
9.3
9.4
11.4
' 7.4
7.4
7.4
9.4
'5.3
5.3
5.3
8.0
3.7
3.7
3.7
0.2
. 0.1
0
800
1200
1600
400
200
Ungrouped
2000
1000
500
400
Ungrouped
21.9
18.8 '
19·0
19.4
Values of .the bias below those of the ungrouped data are due to rounding
E!rrors and the approximations used.
4~9
~
-
e
Table
,e
,.
4.5.3. Approximate variance of e.2 ... equally spaced case
9=1000
i;c
h
400
200
100
500
20
30
S4(f'leSize (n)
o '"
. 50
,.
70
100
199668
199116
118848
118537
84982
84764
54416
5~78
35397
35309
250
461330
. 460104
302090
137704
~.·8'81J..9
65718
43354
28n8
800"
400
200
140002
138611
78909
78148
)55142
54618
112400
42000
29009
28737.
19684
19501
1200
600
400
200
93267
91928
91041
56933
56136
55604
40999
40433
40053
32028
31598
31303
22293
21990
21786
15302
15096
14956
.1600
800
400
200
76886
',14555
73843
48331
46903
46461
35252
·34224
33904
27757
26940
26689
19451
18894
18719
13418
13036
"12916
2000
1000
500
400
69050
66363
65932
44086
42420
42150
32387
31183
30984
25595
24651
24496
18037
17379
17270
12496
12044
11969·
is
e
e
Table 4.5.4.
·e
:
,
A
A
Comparison of the mean square eITors of 8 , 8 and the maximum
0 2
likelihood estimator (ungrouped) equaJ.1yspa.ced case!'/
8:::1000
tk
h
20
S~:Le~~ize (n)
.4Cf~···----
A
400
200
100
Ungrouped.
250
_.- 70"
100
i
.
3~40
A
490161
491086
209594
209970
123760
123946
87913
88029
55797
55867
36078
80
48971~:
2093~8
87784
489964
209438
123607
123650
55708
55730
35976
35990
81748'55.6B3
67403
44160
67541.
44242
35955
29097
29139
. e.2
A
e.2
8.
"'"0
9
~
9
2
A
u
_
489585
gQ~)9_
317396
143195
318346143572
Ung1;'o.!lp.~_
9,
.. 31~164
400
0
g2
:~800
50
.
eo
A
500
" 3·0·_·- .
...
9
--..lli5_lt:5._
' 9;Ji514
91722
87~
H..
143659
144722
142836
80284
80784
9l?60
55854
56147
67203
42845
43030
44016
29249
29319'
28990
19845
19833
79865
80032
55512
55623
42543
42630
28991
29047
19616
19650
7977555445
42490
. 28952
19585.
A
200
e.2
143052
143358
~Jli>-ed
$
.142902
9
A
0
!:IVa.lues of D(~o;n,~,k) below D($; n,5,k)
a.pproximationS used.' '
are due to rounding errors.a.nd the
&"
e
e
Table 4•.5.4.
9=1000
h
tk
1200
·e
(continued)
20
30
90
...
92
93585
94916
57148
57630
a2
92710
93577
90
...
92
go
...
600
400
Utlgrouped
.16QC
800
400
Vngrouped
100
41384
22764
22420
15882
15368
56345
56817
40510
40805
31641
31812
22027
22109
15149
15154
92450
·92706
56138
56296
40314
.40425
31453
31540
21847
21904
149718
15014
·92412
76743
77664
;>6114
49113
48688
40296
36518
35465
31436'
29294
27898
21832
21353
19534
14965
15577
13465
74417
75289
46716
47213
34076
34396
26834
27050
18848
18948
13046
13063
8;
74330
74593
46611
46775
33958
34075
26708
26800
18712
18773
12904
12942
e
74349
46623
33965
2671.1
18712
12900
-a0
--
....9
."
92
A
90
#2
...
200
·-10__
32391
32284
...
200
S4?le size (n)
o
. 50
",0
9.2
....
4~74
t:
•
e
Table 4.5.4.
·e
(continued)
9=1000
t
k
2000
h
'20
30
,..
1000
500
400
Ungrouped
9 '
,,0
92
,..
S4?le size (n)
o
50
70
100
10718
69529
41468
44339
36606
32556
30293
25725
23325
1~125
18210
12560
65478
- 66716
41921
42574
30892
-31261
24491
24706
175§2
17407'
12170
12057
e,..0
e;2
65383
66293
4177342304
30725
31011
24309
24551
17180
17298
11955
11980
§
65462
41773
30682
24243
17082
-
!o
-8.2,..
"
11833
~
46
A
_.
4.6 Comparison between e.oand the M.L.E. (Equal.Spacing)
Ehrenfeld
[4] shows that for the equally spaced case the maximum
likelihood estimator has the form:
e=
-h
In(l _
(4.6.1)
*
r _
)
I: in + k(n-r)
i
= (2i-l)
g.
For the same case, since
J.
h/2, then
..
I:*in + (n-r)k
h .
i
')h - - •
A
e.0 = ( .
r
2
(4.6.2)
Substituting in (4.6 •.1) we obtain
e
-_h=::--_ _
=
In(l _
A
2h
)
2e.O+ h
·e
2
h
A
:: eo -
(4.fj.; )
A
6(2e.o+
h)
Hence,
A
If
h
is relatively smaJ.lthen
e.2
A
and
e
are almost identical.
47
CHAPrER 5
OPI'IMAL DECOMPOSITION OF TEE SAMPLE SPACE
5.1
Introduction
The experimenter in fixing the number of time intervals (k) and
the censoring time (tk ) for his experiment would" in some cases prefer
k
equal time intervals as a matter of technical convenience.
However"
in general" a k-fold decomposition of the interval (O"t ) which would
k
:maximize the 'iinformation" provided by the experiment would be more
desirable.
The problem" therefore" is to find that decomposition into
k-abutting intervals which is optimal among all admissible decomposi;;"
tions.
Of the ma.ny aVailable criteria upon which to base such a decompo-
sition" a reasonable one would be that which would minimize the variance
of "9 • The intractability ,of this criterion leads us to· consider the
0
use of the asymptotic variance of the maximum likelihood estimator
(:5.3.3) in its place. Empirically it has been shown (Table 4.2.3) that
for relatively small 5
as
n,-;->
00"
var(ao)
-> var(G)" thus in the'
A
neighbourhood of the minimum of var(9) a small change in the decom,position would not greatly alter the variance" i.e. it is suggested that
the optimal decom,position for
e
"
would be
".
almost II optimal for
~ O.
A
5.2 Determination of the Decomposition
It has been shown (3.4.1) that the information (in the Fisher
sense) intrinsic in the sample is given by
48
e
"
•
The problem then, is one of finding that decomposition or partition X = (~, ••• '~-l) which maximizes the information
I*(X), where
equivalently
But
f~
2
I(X) or
= -fi'
hence (5.2.3) can be written
*.I*
(e )
= f [Xi+lfi +l - xifi _ Xifi - Xi_lfi _l ]
n
Xi
i
fi-fi +l
fi_l-f i
.]
" [Xifi .- xi_lfi _l . Xi+lfi+l-xifi .
..
f
-f
+
f -f·
- 2(1-xi ) •
i-l i
i i+1
dI*
.
Setting ~ = 0 (i=l, ••• ,k-l), yields the equations to be solved for
oX
i
the
.
Xi' and hence the
ti •
The following mathematical ; argwnent is
similar to that adopted by KUlldorf [18] and Walker 1.28] for the case of .
the normal distribution.
In (5.2.4) we observe that
f
> 0, (i=l, ... ,k-l), and that the
i
second term is negative and non-vanishing. This last point II1S\Y be
demonstrated as follows:
Putting
(s) =(X+Stf~X+S) - xf(x)
f x - f(x+s)
,
g,
(a) 11#,
g(s) = 1-x
s..:.-> 0
then
(b)
g( s) is a decreasing function of s,
(c) g(s)
~
-(s+.x-1),
0
< s < 00
0
< s < 00
,
•
By application of l' Hepital t s rule we see that
1t
g(x)
s-> 0
=,1t
s-> 0
(x+s )f(x+s) - xf)X)
f(x) - f(x+s
= 1t
-(x+s )f(X+s) + f(x+s)
s-> 0
f(x+s)
If we can show gt(s) < 0 for
0 <s <
00,
= 1-x •
then we have ,proved (b).
Now
gt(s) =
but
,f(x+S),' 2 {Xf(X) - (x+s)f(x+s) - (s+.x-1)[f(x) - f(X+S»)) ,
[f(x)-f(x+s)]
,
:
'f(x+s)
> 0 for all S > 0, hence the sign of g(s)
[f(x) - f(x+s»)2
depends on the sign of
h(s)
=x
Since h(O)
rex) - (x+s)f(x+s) - (s+.x-1) [f(x) - f(x+s») •
=0
and
ht(s) = (x+s)f(x+s) - f(x+s) - (s+.x-l)f(x+s) - f(x) + f(x+s)
=
f(x+s) .. f(x)
then h( s) < 0 for all S
< 0 for all s > 0 ,
> 0 and consequently so is, g t ( S ) .
50
Part (c) has been proved in the process of showing h( s) <
Putting x = 'x _
i l
>
-
and s = xi-x _
i l
we then find g(s) becomes
by (c) above.
x
1 -
o.
i
by (a) and. (b) above.
Thus from (5.2.5) and (5.2.6) we see that
g(s') < l-x < g(s) ,
i
and hence
g(s') - g(s) < 0
Expression (5.2.7) implies that equation (5.2.4) vanishes if, and only
if, the term in the last bracket vanishes, which gives the resulting
system of equations:
Putting
d i +l
+ di
- 2e
which may be Written
i
= 0,
(i=l, ••• ,k-l) ,
51
i-l,
()i-l
Di = di + - 2 1: (_l)J ei - , + -1.
cL = 0 ,
l
J
.1=0
-:L
(i=l, ••• ,k-l) •
(5.2.10)
Di = Di(xl'~' ••• ,xi +l ) = 0 may be solved by
the generalized Newton procedure. Then
The system of equations
*-l?
~(m) = ~(m-l) - D(m_l) ~(m-1)
where starting 'With a trial solution
~(m)'
*-1
'With D
~t
D* =
d~ :
aD2
aD
2
yields
•
~-----I
dX;
•
•
•
•
•
•
•
•
itera~ion
the mth
aD1 :
~
'~
~(O)'
and D each eValuated at ~(m-l)' and
= (~,D2' ••• '1\:-1)
aDl
,
aD2 ;
di3:
I
•
•
0
0
•
•
•
I
I
\.--
• • • • • • • • • • • • • •
•
•
0
•
0
•
•
•
•
•
•
•
0
•
---
aD _
k 2
d~_l
~
~
However, the matrix inversion may be simplified if we note from
(5.2.10) that
52
e.
j ~ 2 •
In a.ddition, if we put
adi
~
.oxj
= dij
,
then
j=i
j=i-1
o,
j < i-1, or
j
>i ,
Thus
and
. It should be noted that
= 1, ••• ,k-1
i
Also
aDi
~
oX
-1
= 2( -1) i-1 ·+ i(-1)
1
and
d-
~1
,
'.
•
i ,= 2, ••• ,k-1
i
=3, ••• ,k-1
j
= 1, ••• "i-2
•
* to tha.t preceding it we
Hence if we add each successive column of D'
53
obtain a matrix H of the form:
o
, 0 ••••••••••• • • •
23 L __ ,
o
'~I
~l
~2
h
~2
Ij3
~41
h43
h44
0
L,.._,
~l
,..---
h
:
41
0
I
I·
•
H =
••
hii
•
,
~
o
h 45 :
•
•
I
I
•
• • •
•
•
I
•
•
•
•
•
•
I
,
'.
L_--a
0 •••••••••
•
•
,
•
I
I
I
I
I
•
,
••••••••••••• •
~-2,1:
0
~-l,l :
0
•
•
I
---..
IJ __
~-2,k-3
-:- _ _•
• • • • • .0.
'0
• • • • • • •
I
I
L_
0
-- --
~-2-k-2 ~-2"k-i
~-1,k-2 ~-l,k-l
The elements of H are given by
OD
i
+
dXj
h
ij
=
dXj +l
= 1" ••• "k-l
\~ =1" ••• ,k-2
•
cD!,
d
obi-_
'
~-l
{~
= 1, ••• "k-l
= k-l
Hence,
i
= 2, ••• "k-2
i
= k-l
54
e.
"
h
OD
OD
i + i
- d
. 1 - "':::':':"""~
i ,1OXi
OX1 _l - i+1 ' i
=f
( -1 ) i-l <L
~l
i=l, .. .,k-2 •
0
.
' -f
i
[e d
]
-f·
i i+l . i- 1+1 '
= ()1-l
-1
i=4,
00
l
-f
l-f
[ e -<L,
]
~
H
i=3, ••• ,k-2
-h.
-It-l,k-2
=2.
o,k-l
that need to be calculated
i=2
h..
00
•
In8\V be reduced by using the following relations':,
--k-l,k-l
i=3,
o,k-l
j=2,.oo,j-2
The number of' elements of the matrix
1
l
55
e
Thus the matrix H is really of the form:
1 +
2
H=
~
du
d
~1
22
0
1
d
11
d
-~1
-0
~1
0
43
0
• • • • • • • • • • • • • •
0
0
• • • • • • • • • • • • • •
0
1
d
0,4
1
0
d
0 • • • • • • • •
44
-.
• • • •
0
•
•
•
65
•
•
•
•
•
•
•
•
(-1)k-1
du
0
0
• • • •
(_:l.)k-2 d
11
0
0
• • • • • • •
..•
0
•
~-1 k-2
0
1
~-1 k';'l
!\,k-1 2~,k+1
0
This matrix manipulation may be expressed by the equation
DA=H,
*
Where A is a matrix of the form
1
0
1
1
0
0
1
1
• • • • • • • • • •
• • • • • • • •
0
• • • • • •
•
•
•
•
•
Since
0
0
•
•
•
•
•
•
•
A =
0I
1
0
• • • • • • • 0
1
1
0
0
• • • • • • • • • 0
1
1
•
D*-l = Al:I- 1 we may obtain D*-l by computing
B- 1
and
adding successively the (i_1)st row of B- 1 to the i th (i~, ... ,k-1).
56
·The number of iterations may be reduced by a judicious choice of !(Or
Perhaps the simplest approac:p.. which would tend to simplify the imtial
matrix algebra would be to start with equal spacings.
Wben this occurs
we have:
xi
x -x _
i
i 1
f
i
= i8
=8
= e- i8
i=1" ••• "k •
Hence
1 ,
i=2" ••• ,k-2
e-(k.-l)8 .. ' [. 8e- k 8 _ ]
1'.
e -(k-1)1) -e-k5 e.,-{k-1'5-k8
-e
8
=
[5
e -Ie -1
+2
J + 2 = 2-~1"
i=k-1
~-r-1
e
hi ,i+1
=
.L
[1 -
:::i] ~. hn-
1
,
i=l" ••• "k-2
.(l.5~~ -~ -/1 [1 ".:~~]
...
h
i"i-1
=
5
..
e
[ ~
I)
~
e - I e -1
(-ll-l
e5 _1
= 1-2hn'
j
-1. = -hr1
•
•
i=2
1::=3" ••• "k-1 •
1=3" ••• "k-1 •
5-1
From the above we see that starting with equal spacings reduces all of
the non-zero el.ements of the matrix H( 0) to functions of the first
<.
element
~1 •
matter.
The calculation. of ~( 0 )
The computation of H(0)
consequen~ly a
is
simple
is similarly simplified since:
i=l" ••• "k
i=l" ... "k-1 •
_
i. -1
Di+1 - d1+1 -j~
2 't'
(
- 1)
j
i 1
ei _j + (-1) - ....d..
~ ..•
i=l" ••• "k-1
8
8e
= ""8
(1+(-1) i+1 ] - 8[i+1+(-1) i+1 ]-2 ~1[1-i81~[l-(i-1)8]
e -1 -
-
--
1
+••• +(-1)~+1[1-8]
i
odd,
i
even.•
5.3 ANumerical..Ex:am:ple
We will il1l.\S_trate the above procedure by considering the optimum
spacings .for a 6-:f'old decomposition of the interval (0,,1200) when
f(t)-=
f 1;00 0l\P- l~O
t
0
t>o
ot4erwise
•
58
Connnencing with eql,18.l spacings the initial solution is
!( 0)
=
[.2, .4, .6, .8, 1.0] •
Consequently,
h ll ·=
1
2 .2
[1 - ,e2 ]
-1·
e' -1
1 + .2e
~2
= ~3 = h 44 = 1
~5
= 2-~1 = 1.46671
= .53329
hi,i+l = ~l-l = - .46671
h
. l 1-2~1 = -.06958
i,i-1
=
.
.
= 1, ••• ,k-2
i
=2
i
= 3, ••• ,k-1
-
= ·~,53329
-~1
i
i
= 3, ••• ,k-1
Bence the matrix B ) is·
eO
·5333
-.4667
•
•
•
-.0666
1.0000
-.4667
.
•
H(O) = -.4667
-,5333
1.0000
-.4667
•
,4667
•
-.5333
1.0000
-.4667
-.4667
•
•
-.5333
1.4667
and its inverse
•
59 m_
2.8893
2.0920
1.3943
.7837
.2494
1.1589
2.3905
1.5933
.8955
.2850
-1
H(O) = 2.0703
2.6805
3.2147 . 1.8068
.5750
.2225
.9199
1.5304
2.0645
.6570
1.0000
1.0000
1.0000
1.0000
1.0000
·-1 then
Since D*-1 =AH
2.8893
2.0920
1.3943
.7837
.2494
4.0482
4.4825
2.9876
1.6792
.5344
*-1
D(O) = 3.2292
5.0710
4.8080
2.7023
.8600
2.2928
3.6004
4.7451
3.8713
1.2320
1.2225
1.9199
2.5304
3.0645
1.6570
The elements of B( 0) are given by
.00666,
Di
= 1,3,5
i
= 2,4
=
hence
'B(o}
Using the
i
=
~elation
.00666
.00000
.00666
.00000
.00666
•
...
_
60
tit
we obtain
'.
2nd Iteration:
.~alculation of
the elements of
"~
and B( 1) •••
f
'. ;t
""0" "". 1
.1;1-.1;1+1
i
1
2
3
4
5
6
.8438
.7050
.5823
.4748
.3814
.3012
.1698
.3496
.5408
.7449
.9640
1.2000
=1
~1
h
22
°
=
.1562
.1388
.1227
.1075
.0934
.0802
5.4020
5. 0792
4.7457
4.4167
4.0835
3.7556
+ 5.4020 [.8302 - .9174]
Il;3
= h
44
=
6.0792
5.7457
5.4167
5.0835
4.7556
:--~io
"""
f
•
J."
.1433 .9174
.2465 .7435
.3149 .5574
.3537 .3609
.3677 .1499
.3614 - .0786
".8302
.6504
.4592
".2551
.0360
= .5290
1
~5 = 2 - 4.7556 ~ .0360 + .0786] = 1.4550 •
"
h21 = -5.74~7 [.6504 - .5574] - 5.4020 [.8302 - .9174] = -.0633
.
.
~3 = - .5290 + .0633 := - .4657 •
Il;2 = -5.4167 [.4592 - .3609] = - .5325
11;4 = -I'· + .5325 = - .4675 •
43 = -5.0835 [.255i - .1499] = -.5348
h
45 = -~.+ .5348 =-.4652 •
h
61
.
"
~4
= 1.4550 - 2 = - .5450
;hil
=
(-1)i1(
- .
.5290) ,= (-1) i (.4710)
•
Also,
:01
= ~ - 2e1 + c;.
:02
= CJ
= d4
3
:0
:04 =
- 2( e2~el)
= .0005
-
c;.
,
2( e -e2+e1 ) + c;.
3
~
.".
= -.0004
=
a., - 2(e4-e3+e2-e1 ) - 'c;.
.0003
= .0003
Hence
H(I) =
and
·f
.5290
-.4729
-.0633
1.0000
-.4657
-.4710
-.5325
1.0000
-.4675
.4710
•
-.5348
1.0000
-.4652
- .4710 .
•
•
- .5450
1.4550.
its inverse
-1
H(I) =
2.9363
2.1550
1.4394
.8149
.2605
1.1700
2.4107
1.6102
~9116
.2915
2.1132
2.7363
3.2619
1.8467
.5904
.2293
.9360
1.5541
2.0908
.6685
1.0364
1.0482
1.0481
1.0470
1.0220
Using D(l) and proceeding in the s~ way as for the previous
iteration we find after iterating 4 times the solution (to the 4th
significant digit)
~(4)
= [.1695, .3492, .5403, .7445, .9636] •
63 _
CHAP.J:IER 6
SUMMARY" CONCLUSIONS AND RECOMMENDATIONS·
FOR FURTHER RESEAiwH
This dissertation considers the estimation of the mean life of
i tams for the case of censoring from the right and grouped data when
the mortality follows the exponential law given by
t
1
=:
:r( t)
[
exp-
t ~ 0" e > 0
e
otherwise
•
Censoring occurs at some presta-ted time (t ) where
k
k
is the number
of inspections made in the interval (0" t ) •
k
Maximum likelihood estimation is considered and the conditions for
the existence and uniqueness of a- solution examined.
Xi
Using the notation
= tile and f i = exp- Xi it is found that the likelihood has the
form
k+l
(1:
E
1:"
i=l
where
~
=n
k
*
and 1: == 1: ) ,
i=l
n = number of items failing in the interval (ti_l"t ).
i
i
or
~+l
= n
If
the solutions are 0 and ex> respectively, both of
which are of' no practical value.
The method of maximum likelihood has
the, drawback therefore that With positive probabilitythere are two
instances where the sample Will provide little information.· If
and
~ =t
~+l =t n" the solution to the likelihood equation is unique but
n
n
cannot. be obtained wi thout .iteration.
Fisher' s method of scoring is
adopted which gives the (j+l)st iteration as
.....
...
_ ...
g(e j )
e j +l - ej(l x
)
.
.
G(e )
.j
where
.....
,
e=e .
.J
and
•
The problem of a non-explicit solution to the likelihood equation
motivates the search for an tlal.mosttlmaximum likelihood estimator
(m.l.e. ) which can be wri.t.ten in an explicit form.
By defining
hi
=
t.J. - t'J.- l
and
i = 1, ••• ,k ,
we may expand the elements of the likelihood equation as power series
..
giving:
xifi - xi_lfi _l
h~. hi 4
f _ - ff
= ui - (1 + 12e2) + 0(2e)'
i l
h
Thus to
0(2~ )
i
= l.t~ •• ,k.
4
the likelihood equation becomes
* .
1:. n u
i i .+ '\:+1~ -
2
* niJii
1: 12e
2 = r ,
*
1: ni = r
•
The solution to this equation is
where
Thus the initial problem of iteration has been reduced to that of
finding the solution to a quadratic equation.
In general the experimenter will have some idea as to the approximate dimensionality of e and using this prior information can attempt
,
hi
to make the ratios 2e' (i = 1, ... ,k), comparatively small. If this
is so,· the estimation problem may be reduced still further by using
as the estimator.
lI
If the intervals are relatively
,..
Sma11~o
,..
~o
will be
almost" the m.l.e. and at the same time will possess the practical
advantage that it is defined for the case of
~
= n,
,..
~0
= gl
~
= n.
For the case
the mid-point of the ,first time interval, which is a
reasonable and intuitive estimator.
,..
The properties of
~+1
f
e.o
are investigated conditional upon the event
n (i. e., r > 0) and it is found that both the bias and variance
are non-monotonic.
The reason for this appears to lie in the condition-
ing restriction which means .that both the mean and variance are functions of certain inverse mO!llents of a truncated binomial distribution.
"
.
The variance of ~o is a decreasing function of the interval width
(equally spaced case), whereas the m.1. e. is an increasing function.
The reason for this contradiction appears to be that
a linear estimator whereas the m.1.e. is non-linear.
for bias by
~ubtracting
,..
~o
is essentially
If we adjust
,..
~o
a correction term, the variance pattern for the
66
A
new estimator ~2 (equally spaced case)
e".2 = "e.0 is similar to that of the m.l.e.
That is to say by
a.ddingona~term of
a Taylor series expansion (so that the estimator is no longer linear)
the variance pattern changes to that of the m.l.e.
"eo' "e
2
However, since
and the m.l.e.'s for both the grouped and ungrouped data are
biased, a comparison of their mean square errors is carried out (Table
4.5.4).~ ~.From Table
4.5.4, it appears· that even for fairly broad group-
ing (of the order of e) there is little loss of information, which means
that the extra cost involved in a continuous sampling case -may be unnecessary and that periodic inspection would suffice.
In other words,
the amount of information per unit cost would be higher for the grouped
data than for a continuous inspection plan where costly timing mechanisms
are required.
The optimal decomposition of the time interval is investigated using as the criterion the minimal asymptotic variance.
The reason for
this choice is that it leads to a fairly simple series of equations which
may be solved by the generalised Newton procedure.
Such a criterion
has the drawback in that the optimal decomposition is a function of the
unknown parameter.
However, for tests of hypotheses a decomposition
could be made under the null hypothesis.·The use of a sma.J.J.. number of inspection periods leads quite
naturally to other possible fields of research:
67
,.
1.
The distributional properties .01'
~othesesusing
~O
the distribution of
and the power of tests of
,.
~O
compared with tests based
on other criteria" e. g. , the multiple contingency table approach
is of some importance.
2.
Often the test equipment is a matrix of n "test compartments"
which is best operated when
·~h:e~·
a:1".'e always full.
Thismeans
at each inspection perlod the failures are replaced by new items.
If the items are fairly inexpensive the true saving of this procedure as compared with the non-replacement case would be of interest.
3. This type of estimation procedure can be used for other stopping
criteria and problems.
For example it may be applied to the case
when we stop the experiment when at least
failed.
l'
of the items haVe
It would also be useful to apply this to the problem of
mixed failure distributions.
4. A practical application of this estimation procedure is in the field
of factorial life tests" where except in the simplest of cases the
estimation problem is different.
Thus in two-factor expe:t"iments
defining
i = 1" ••• "ca
j = 1" ... "b
where (Xi" ~ j correspond to the main effects and· '1ij
to the
interaction" .then eiJ may be estimated ..USing the procedures of this
dissertation and applying the restrictions ~ (Xi ~.~ ~j
n '1ij = 1
j
we obtain
=~
'1ij =
e•
68
~..i
=
[It
.j
e ]lIb / ~
.ij
.
A
~ij
•
(
•
e•
..
LIST OF REFERENCES
1.
Bartholomew, D. J. 1957. A problem in life-testing.
Stat. Assoc. 52:;50-;55.
2.
Cohen, A. C., Jr. 1950. Estimating the mean and variance of.
normal populations from singly and doubly truncated samples.
Ann. Math. Stat. ~:557-569.
.
;.
Cohen, A. C., Jr. 1951. Estimation of parameters in truncated
Pearson frequency distributions. Ann. Math. Stat. 22 :256-265.
4.
Ehrenfeld, S. 1962. Some experimental design problems in
attribute life testing. J. Am. Stat. Assoc. ~:668-679.
5.
Epstein, B. 1954. Life-test estimation procedures. Unpublished
Technical Report No.2, Dept. of Mathematics, W~e State
University.
6.
Epstein, B. and Sobel, M.
Assoc. 48:486-502.
7.
Gjeddebaek, N. F. 1949. Contribution to the study of grouped
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Murthy, V. K. On the general renewal process. June, 1961.
Ray-Chaudhuri, D. K. Application of geometry of quadrics of constructing PBIB designs. June, .1961.
Bose, R. C. Ternary error correcting codes and fractionally replicated designs. May, 1961.
Koop, J. C. Contributions to the general theory of sampling finite populations without replacement and with unequal
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Foradori, G. T. Some non-response sampling theory for two stage designs. Ph.D. Thesis. November, 1961.
Mallios, W. S. Some aspects of linear regression systems. Ph.D. Thesis. November, 1961.
Taeuber, R. C. On sampling with replacement: an axiomatic approach. Ph.D. Thesis. November, 1961.
Gross, A. J. On the construction of burst error correcting codes. August, 1961.
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Hoeffding, Wassily. The strong laws of large numbers for u-statistics. August, 1961.
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Fukushima, Kozo. A comparison of sequential tests for the Poisson parameter. September, 1961.
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Bhattacharya, P. K. Use of concomitant measurements in the design and analysis of experiments. November, 1961.
Anderson, R. L. Designs for estimating variance components. November, 1961.
Guzman, Miguel Angel. Study and application of a non-linear model for the nutritional evaluation of proteins. Ph.D.
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Koop, John C. An upper limit to the difference in bias between two ratio estimates. January, 1962.
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Potthoff, Richard F. Use of the Wilcoxon statistic for a generalized Behrens-Fisher problem. February, 1962.
Potthoff, Richard F. Comparing the medians of the symmetrical populations. February, 1962.
Patel, R. M., C. C. Cockerham and J. O. Rawlings. Selection among factorially classified variables. February, 1962.
Amster, Sigmund. A modified Bayes stopping rule. April, 1962.
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Sathe, Yashwant Sedashiv. Studies of some problems in non-parametric inference. June, 1962.
Hoeffding, Wassily. Probability inequalities for sums of bounded random variables. June, 1962.
Adams, John W. Autoregressive models and testing of hypotheses associated with these models. June, 1962.