estimation of the mean life of the exponential
TRANSCRIPT
•
..
ESTIMATION OF THE MEAN LIFE OF THE EXPONENTIALDISTRIBUTION FROM GROUPED DATA WHEN THE SAMPLEIS CENSORED - WITH APPLICATION TO LIFE-·TESTING
by
PETER JAMES KENDELL
and
Rt L, ANDERSON
Institute of StatisticsMimeograph Series NOt 343February) 1963
• ERRATA SHEET
Page 6, Last line:
" re = zi=l r I • • •
Should be:
rZ t i + (n-r)t
&= i=l rr - , .. . .
Page 7, Line 7A re = z
1=1
t i + (n-r)t
r .. . .
Should be:
rZ t
i+ (n-r)t
&=...1....=1 _r . .. ..
. . .
Page 14, Line 7:
Should be:
(X~fi - X~-lf)(Fi - Fi - 1 ) •• •
(X~fi - X~-lfi-l){Fi - Fi - 1 )
Page 14, 2nd line from bottom:
.. • • ni < n .. · •
Should be: • • · ~ <n .. · ·Page 14" Last line:
· • • n = n .. · ·i
Should be: · · • ~ = n • · ·"Page 17" Equation (3.,.8):
" 1e =:2 • • •
Should be:A 1e =-1 2 • • •
Page 22"
· . .
· . .
Page 2
ERRATA SHEET
Page 24, 2nd line of Equation (4.2.9):• pt
::: nQ2e •
Page 24, Equation (4.2.11): R(e; 00 1 8,k) •••
Should be: = ptnQ2e
A
Should be: R(eoj 00" 8"k) •
11 + '14: • • •
"" A eoe =e --r.- g1 0 If
Should be:
Should be:
Page 26" Last line of Equation (4.3.7):::: Q~ ••• Should be: = Q1 •••n~~ nk~
{8e8 t k 8 1}2Page 27" Last line of Equation (4.3.10): + - - - - - + nt E(-) - 1
e8_1 ~+1 2 k r
+{
8e8 _ ~t_k~ 8 ntk 1 ]2Q_ e - ;:; + -e E(-) - 1 •
~-1 ~~ ~ rA
,., " eo {e1 = eo - T g
t1 + l; .. ·" c= eo (1 - 'ij:)' say,
Page 35, First line of Equation (4.5.1):
Page 35, Last line of Equation (4.5.1):
A CShould be: = eO (1 - 'ij:)' say,
. . .
Page 35, Line 7:
Page 35,
Should be:
Should be: C= g \1 +
* 2I: nihi
{z2
l'<
1 PigihiPage 37, Line 9: • . • =E(-)~+1
• • •r
21 t* P1
g1
hiShould be: • . • = E(-) - • • •r Qk+l
" A
Page 40, Line 4: • . . relatively small eO" e2A A
Should be: • . • re1ative1y small n eo" e2
rex) = ••
,Page 48, First line:
ERRATA SHEET
Should be: rex) = . . .
Page 3
Page 49, Line 10; lt g(x) =s-> 0
Should be: lt g(s) =s-> 0
• • •
Page 53, Line 14; ~-2,1 . . . h ~-2,k-lk-2-k-2
Should be: ~-2,1 ~-2,k-2 ~-2,k-l
Page 55, Line 12; ( _1)k-2d • . • 2 + ~,k+l11
Should be;
Page 65, Second line: "-e =1 . . .
Should be: '"e =1
. . .
Page 68, Line 3: Should be:
Page 69, Reference 5 (EPstein): Add; Detroit, Michigan.
Page 70, References 19 and 20 (Lehman and Mendenhall):
Add: (Un1versity Microfilms, Ann Arbor)
•iv
TABLE OF CONTENTSPage
CHAPrER 1 - INTRODUCTION •• • • • • • • • • • • • • • • • • • • • 1•
CHAPrER 2 - REVIEW OF LITERATURE • • • • • • • • • • • • • • • • • 4
CHAPrER 3 - MAXIMUM LIKELIHOOD ESTIMATION OF a • • • • • • • • • • 10
3.1 Test Procedure • • • '. • • • • • • • • • • • • • • • • • •• 103.2 Derivation of' the Maximum. Likelihood Estimator • • • • • •• 103.3 Existence and uniqueness of the Maximum Likelihood Estimator 123.4 Iterative Estimation Procedure • • • • • • • •• • • • • •• 153.5 Simplif'ication of' the Maximum. Likelihood Equation • • • • • 153.6 Modif'ied Maximum Likelihood 'Estimator •• • • • • • • • •• 17
A
CHAPrER 4 -PROPERTIES OF THE ESTIMATOR ~O ••• • • • • • • • •• 19A
4.1 Mean and Variance of' aO ••••••• • • • • • • • • • • • 19A '
4.2 Bias in ao • • • • • • • • • • • • • • • • • • • • • • • • • 22- A
4.3 An EXpression f'or Var(e.o) ••••••••••••••••• 244.4 On the Non-Monotonicity Property • • • • • • • • • • • • • • 28
A
4.5 Eff'ect of' Neglecting Terms in al
• • • • • • • • • • • • • • 35A,
4.6 Comparison between e.0 and the M.L.E. (Equal Spacing) • • • • 46
CHAPrER 5 - OPrIMAL DECOMPOSITION OF THE SAMPLE SPACE •
5.1 Introduction. • ~. • • • • • • •5.2 Determination of' the Decomposition5.3 A Numerical Exa:rrq>le •••••••
• •• •· .
• •• •• •
• • • • • •
• • • • • • • • •• • • • • • • • •• • • • • • • • •
47
474757
• •
CHAPrER 6 - Stnv1MARY, CONCLUSIONS AND RECOO1ENDATIONS FOR FURTHERRESEARCH • • • • • • • • • • • • • • • • • • • • • • • • • •
LIST OF REFERENCES • • • • • • • • • • • • • • • ~ • • • • • •
• • 63
69
•v
LIST OF TABLES
Page
4.2.1(a). E(~) = ~ ~ (~) ~+l ~~~ / l-~+l •• • • • • • • • • 29r=l .
4.2.1(b). V(~) = E(~) - (E(~»2, wherer
1E('"'2)
r• • . . . . . . . 30
4.5.3.
4.5.4.
" ,Bias of e.O N ~qually spaced case • • • • • • •• • • • • 31,.. "
Variance of eON equally spaced case • • • • • • • • • • 33
Approximations to the upper bound of the correctionfactor N equally spaced 'case (as percentages of e) • • 38
, A
Approximate bias of e.2 N equally spaced case • • • • •• 41.A
App~oximate variance of e2 N equally spaced case. • •• 42
A "Comparison of the mean square ... errors of eo' e2 and the
maximum likelihood estimator (ungrouped) - -equallyspaced c~se • • • • • • • • • • • • •• • • • • • •• 43
• 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 con
cerned 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 "failureft, whether it is the inability of the
object to measure up to some prescribed standard or even outright de-
struction.
To investigate the ftmortality' characteristics of an obj ect under
certain stress conditions we require knowledge of its underlying mor
tality 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 dis
tribution [29]:
g(t) =
o
t ~ t'" a, M > 0
t < t' •
(1.0.1)
• 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 informa
tion on the others is often a prer'equisite of a life-test. Such a
procedure is called a censored sampling plan. Here, we mustdistin
guish between censoring and truncation. 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 var
iates 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.: e.g.,
I. 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 inferen-
tial 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 life
tests '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? In particular, wenll consider the
estimation of the mean life, e, when the mortality distribution is of
the exponential type:
e.
1 te exp- e 'g(t) =
o
t > 0" e > 0
otherwise •
(1.0.2) .
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
intervals1 corrections could be made to the moments of' the discrete
distribution to bring them closer to those of' the continuous distribu-
tion.
In 1934 Wold [30] 1 again f'or the case of'equal spacings1 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 pro
vide 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 ob
tained 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
,. rC1 = 1: c.x. ,- i=l ~ ~
the coefficients bi , ci being tabulated for samples of size n = 3(1)10
and r = 2(1)n-l. 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 kl and the largest
k· items were not measured (i.e., doubly censored).2
Epstein and Sobel [6] gave procedures for estimation and tests of
hypotheses in the case of. sampling trom an: exponential distribution in
which only the first r ordered units from a sample of size n are
measured. In a later .pa;per Epstein [5]· gave more general results· for
censored sampling with and without replacement of failures. Procedures
were given for the case of a fixed number of failures r" and a -fixed
test termination time t. In the first instance the :maximum likelihood
estimates of the average life is
e= ~ t i + (n-r)tr. i:i;l . r '
without :replacement case
•7
, 'With replacement case.
. A .
When r is specified, e has the property of being unbiased, su:f'ficient2· A
and of minimum variance, V(~) =!... The distribution of 2r ~ isr· 1;1
that of chi-square 'With 2r degrees of freedom. When censoring occurs
at a fixed time, t, the number of failures r (r > 0), being a random
variable, then
A re = 1:
i=l
A nte = r
t i + (n-r)t
r 'Without replacement case"
'With replacement case.
The estimator is not unbiased although it has all of the usual asymp-
totic 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:
e -tl/Ol -t/a2p - + (l-p) !L
.Ol ~f(t) =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 termina
tion time t. Mendenhall and Hader Ha] extended the above results to
a mixture of k 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 or more units have failed, otherwise continue
the test until r 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
( .05)~
and for p = .01, .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 investi
gated 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 Xo = 0:; ~ = 00
and observed frequencies
and defining ~'adjusted" moments of the form
•9
he obtained likelihood equations as fmctions of ~ and ~ • After
expanding M:t and ~ as power series, tTl.mcating the series and
substituting for M:L and ~, the likelihood equations were reduced to
a form which enabled a simpler iterative procedure using tables from
Ha.ldt 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 expo
nential form and sampling stops at some time t k, the interval (0, t k )
being divided into k equal groups. For the case of equal spacing the
likelihood permits an explicit solution
§ = -~.....;;A:::...----r--
In.[ 1+ ~ xj / ~ s J 'j=l j=l ~
where
Xj = observed frequencies j = 1, ••• ,k •
Sj = n - (Xl +~ + •••. + x.) •J....
The asymptotic variance of e is derived and compared With that for the
ungrouped case.
• 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 = tk
). At certain time intervals
(ti
_l , t i ) during the life.,.test the number of items that do not come up
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
kr = i~l ni and the number of survivors iS~+l = n:-r •
3.2 Derivation of the Maximum Likelihood Estimator
The probability of an i tern failing in the interval (ti
_l , t i ) is
given by
Since the distribution of failure times is assumed to. follow the
exponential law
•11
t > 0 ,- ,
g(t)=o , otherwise
then,
fl G(t) :: 10
texp- .~, t > 0
otherwise •
Putting Xi:: tile then
hob (ti_l~t<ti) = hob (xi_l~x<xi)
= F(Xi
) - F(xi
_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
butdFi dFi dX. ":xif .
J. . J.
~= dix
e,
W::;, e- ,
i
l/For purposes of'notation we will henceforth use
k+l k1:;;: 1:, 1:* =:;. ~, Fi = F(Xi ), f 1 = exp- Xi •
1=1 1=1
•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" . O~riL" over the parameter
space" (o~' 9 <'0).
Putting
since Xi > xi _l " we can write
From the definition of Xi we can see that
~t xi = 00" i = 2" ••• "k+l ,8->0
hence" ~t Yi = ~t
xi -> 0). xi _l -> 00
Xi
_l [8i
- ex,p- xi
_l (1-8i
)J
1 - exp- xi _l (1-8i '= -00 ,
i = 2" ••• , k •
•13 -- --
= 0
.et Yk;+l .. .et - "k ;: -co •
*k-> 00 ~->··oo
Thus.et O~nL = 00 if at lea.stone observation exceeds t 1 ,e":"'> 0 . '.
Sim;tla.:rly, for large e,
.et xi = 0 i =1",.,k,e-> Q)
hence, .et Yi =x
i_1-> 0
.et
xi~i-> 0
Xi_1
(5i-exp- Xi _
l(1-8:i)J
l' - exp- X:i-l(1-51 )= 1,
= 1,Xl exp- ~
1 - exp- Xlx1-> 0
.et Y =.et1~->o
and, .et - ~Xk-> 0
= 0-,
from which we can infer that .et· oW' = 0- if at least one obser-9-> 00"
vation is less than t k, Hence a root exists to the equation (3.2,6)
if ~ rn and ~+1 rn.
The uniqueness of a root can be determined by consideration of the2 .
. .' olnLsign of the second, derivative, . . 2 • Now,
. oe '
d~r =_~2 1: ni[<xi -2;~f~;~~~i~~_1>fH' + (l'~:i:X~~~ri"1>J.
(3.3,1)
•14
'021nLFor values of e 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
Tl+erefore" 0 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-
mum. " The above may be summari~ed as "follows:
Theorem 3.".1
The likelihoodecauation for 8 is given by
t i exp- tile - ti_l,exp- t i _1/81: ni ( exp-t /e _ exp- t /8 ) = 0"
i-l " i
and possesses a root which is a unique"maximum if nJ < n and ~+l <n,
but has no a.cceptable root if either ." n, = n or ~+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
G(e)
Denoting the left hand side of (3.2.6) by g(e) and putting
2(xifi - xi_lfi _l )
= 1: " F _ F 'i i-l
then by the method of scoring (see, e.g. Rac> [24], pp. 166-167) startingA
with an initial value of e we can USe the iterative formula
'"~j+l=
e'"
'" '" AIn practiceG(~j) will tend to stabilize quickly as ~j-> e 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 = (ti + t i _l )/2 ,
ui = gi/8 , i = 1,2, .... ,k,
and hi =ti - ti~l'
then Xi = Ui + hi/2a
:-X.i~l= ui - hi/28,
16
from which we can write,
Similarly,
thus
Expanding coth hi/2a .as a power series we have
xi:f'i- xi_l1'i_l = u. _ h 12a{ 28 [1 + !(h 128)2 + O(h 128)4])1'i - 1'i_l ' J. 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_lE. ni ( 1'i -1'1_1 ) + ~+l xk = 0,
and substituting (3.5.2) into (3.5.3) we obtain, except for O( hi /28 )4,
= r.
But ui = gila, hence (3.5.4) becomes* 2
2 ~ nigi + (n-r) t k * nilli8 . - ( . 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 e.· WeA
observe from (;.5.6) that ~o exists if r > 0 (Le., ~ < n), and
2A A 2 * ni 11i ..... r:: "-
from (;.5.8) that ~l exists if ~o > 1:, 3r or max [hi] < V; eo •"-
If ~ = 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 non-
existent 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 80 (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 8.0 would be a suitable esti-
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.
in consequence given by
But,
A I 1 * I n-rE(80 !) =r E(E nigi !) + r tIt '- -. ..
The mean value of ~ is,0
(4.1.1)
. where
Hence,
A Pigi 1 21E(8 ) = E* -- + nt E(-Ir> 0) - t .-
,0 _ \:+1 k r ~ It
/
In a similar fashion:
var(~o) =Var("E* nig
i ) + var(ntk ) + 2nt,. cov(!, E* nig
i ) •_ r ' r. A r . r
Considering each term of (4.1.;) separately we obtain:
(4.1.2)
(4.1.;)
!/Where two expectation signs appear together the inner one willdenote 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, 2 2n·gi . * PigJ.· P g
Va:r(r.*....L.) =, [r. - - (r.*.J:..l) 1 E(!) •,r Qk+l - ~+l r
Simila:rly,
'e
and,
(4.1.6)
21
Thus, stibstituting(4.1~4), (4.1.5) and (4.1.6) into (4.1.3) we obtain
p 2 P 2
var(~o) =(Jrtk)2 var(f) + £1:* ~~. (F.* ~:~) 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 ~~~E(~) = Z ~ (r n·
r r=l r 1 - Pk+l(4.1.8)
For large values of n 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 n-2E(;) = n(a-l)
where a = (n-l) ~+1 •
(4.1.11)
•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 B(~O; n, tk'~) is given by
A AB(eo; n,tk,.!!) = E(~o) - e ,
* Pigi 1= (1:_ ~+l - t k + ntk E(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. Since
.p _ -(i-l)h/e -ih/e _ .-ih/e ( h/e -1) '-12· ki - e . - e - e e , 1-, , •.• ,
and·
gi = (2i-l) h/2 ,
then
., .But,
-t /e* -ih/e _ -h/e· (l-e k )
1: e - e .. . 1 _ e-h/e ' where t k = kh ,
and
(4.2.2)
hIe •e - 1
•23
Combining these results and substituting in equation (4.2.:H we obtain:
/, -tie /, /,* heh/8(1_e k) -tk/8 h ' -tk/8E, Pigi = - - - tke - ;:; (l-e ).eh(e _ 1 c;
Hence
h/eheh/6 'e -1
-t /et e k
k- , -t J6
l-e ' k
h- '2 " (4.2.4)
as
Thus the bias may be written as
B(§ . n,t ,h) ~ {heh
/e
- Pk+ltk - '~,'~' - t + nt E(!)3 - e •0 ' k ' hie 0-+1 2 k k r. e -1 ~
"Putting 8 = h/e the relative bias in E(eo) m.ay be written
"" B(~O; n,tk,b) I8e8 ,tk 8 ntk,l, 1
R(~O; n,8,k) = e ' = T""' - '\: e - 2' + T E(r)f - 1 •e -1 +1 '
(4.2.6)
Expanding the first t,erm of (4.2.6) as a power series and neglect
ing terms of 0(84) we obtain:
8,2 t nt
k +, k E(l)12 - ':""\:""';+1;;;";'9 T r •
As 8--> 0, With t k = t fixed, the relative bias becomes
R(e; n,t) = ~ -: + ntE(~) }/e , (4.2.8)
which a.part from changes in notation agrees With the result given by
• Bartholomew [1] for the case of estimating e 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 } IeI Q. (nQ.{J·
.. Pt_.--r ·nQe
(4.2.10)
and hence asymptotically it is approximated by -
(4.2.11)
r
which is negligible for small h compared with e •
On the other hand, as· 5--> co, with k fixed we see from (4.2.6)
that R(~oj n,5,k) ->R('oj n,oo,k) = (l) •
...4.3 An EXpression for var(~o)
--When the intervals are' equally spaced we ~proceed in a manner
analogous to that adopted in Section 4.2 and obtain 'an~ression forA .
Var(~o) in terms of 8 = hie •
25
Thus>
Now>
since t = kh > and CL = 1 - P = 1 - e-tk/ek '"'k+1 k+1 . •
Hence (4.3.1) reduces to
Squaring expression (4.2.4) and subtracting from (4.'.2) we obtain: .
* Pig~ * Pi gi2
¥Jeh/e t~Pk+11: - - (1:.Q.. _.. ) = -
- ~+l '""1(+1 ..~e~/e _1)2 ~+1·
Substituting (4.'.3) into (4.l.T) yields
26
Hence
(4.,.4)
the result for the case when the exact failure times are known and t
fixed, (see e.g. Mendenhall and Lehmann [23J).
. . .' ; .. 1. nPk+l ~+lAs n-.-> oo~ va.r{;);: . Ii"' and hence (4.3.5) becomes:
(n~+l)
2• 1 '. !.§. cosech ~}
n~+l 2 2
whereas (4.3.6) becomes:
/1:: _..nQ (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 --;> 0, but not
when 6 --;> 00, since then Var(~0)-> 0 and var(~') -> 00. The
reason for this apparent con:tradi.ction is presumably because ~0 is in. ·A •
reality a linear approximation to a non-linear estimator 9'. By ad.d-
"-ing extra terms to ~o (Section 4.5) we tend to reduce the bias but
increase the variance. Even though§o has a decreasing variance for
increasing 8 such a situation is hardly reaiistic and a more
pertinent investigation would be to consider the mean square errorA
(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 thatA
for small samples both the bias and variance of" eo are non-monotonic
~d hence so is the m. s .e. As the·' sample size increases the bias and
variance and thus the m.s.e. become monotonic, as would be expected.A
This non-monotonicity property is also exhibited by the m.l.e. e,
for the ungrouped data, and has occurr~d in other research (see e.g.
Lehman .U9], Bartholomew [1J).
4.4 On the Non-Monoto.~cityPropertyA ~
The reason for the non-monotonicity of E(~O) and var(t10 ) 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 Pk+l = p( t > t k ) we 'Will consider
only those cases where there has been at least one failure. Thus forA
n = 1 we Will obtain an estimate whose expected value [E(e01)' S8\Y]
'Will be less than t k• .When n = 2, we now permit one of the variates
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(30 ) presumably continues with increasing n" until the effect of
the law of large numbers becomes dominant and
The initial bias may be negative or positive depending on t k, and k.A
The variance of ~O behaves in an analogous manner.
Ii ' - e <e
i'
1 n 1 (n rlf-r / 1fTable 4.2.1(a). E(iJ == ,1: r'r) Qk+1 k+1' 1- k+1r==l
"~+1 .329680 .393469 .550671 .698806n
1 1 1 1 ,1
2 .90131238 .87754071 .81002551 .731475053 ' .80942789 ' .76641101 .65279861 .539290084 .72507005 ' .66765587 .52804560 ,.4093Q3725 .64861487 .58146167 .43184100 .322569066 .58011211 '.50735368 .35873204 .263654077 .51933488 .44436690 .30331261 .222226688 .46584491 .39126013 ., .21600917 .191922809 .41906207 .34669306 .22828267 .16891432
10 .37832805 .30935466 .20253180 .1508752211 .34295963 .27804279 .,18189569 .1363515812 .31228779 '~2517P301 .16505753 .1244015315 .24246364 .19430063 .12926419 ' .6985578920 .17327547 .13987827 .09511688 .0732581130 .10957792 .08983081 .06234503 .0484322840 .08030070 ~06628375 .04639278 .0361799650 ' .06342971 .05254350 .03694586 .0288766470 .04469338 .03715780 .02625679 .02057236
100 .03098312 .02582377 .01831187 .01437295
.798103~:
1
.66798193
.46377298
.34301829
.26916248
.22099519
.18747295
.16284980
.14398990
.12907109
.11696815
.10694883
.08510359
.06350723
.04213678
.03153020
.02519017
.01796568
.01256187
.864665
1
.61920264
.41580268
.30615197
.241112104
.,19933807
.16983438•1479a451.13113857..li774762.10684446.09779312.077984:72.05831073.03876098.02902991.02320470.01655930.01158349
~
• e ,e
() (1)' (1 ). «1)2 .. :, . (:'1) .' n. 1,' (n\:"Q~ " -n-r/1 Jf' .Table 4.2.1 b. V r =E,~,'- E r'. ".Where E:7: ::: rfi r2"'7:'i:~+1' ·J:."k+l. - k+l
!
n ~1
1
23456789
101112152030405070
100
.329680
.03960453
.06466577
.07802955
.08257731
.08095~4
.07542173
.06773183
.05918153
.05064396
.011265472, .03549806 ..01961121.00714687.00131142.•00045848.00020963.000068112.00002119
.393469
.04623341
.01094299
.01990829
.01853496
.07126243~06134091.05089506,,04114367.03266204 '.02561132.01994749•.00935535.00291504.00060041
...00021121.00010281.00003458.00001121
.•550671
.05889695
.07345688
.06'9741J2
.05~7056
.03684411
.02531248,.01113100.01152503.00781669•.00538923.003797112.00154185.00052036.00012962.00005082.00002491.00000870•00000289
.698806
.06215686
.05708293
.03756936
.0218814a
.01229315
.00698811
.00413267
.00251465
.00:).69320
.00116991
.00084316
.OOP37896
.00014330
.00003858
.00001551
.00000177
.00000275
.00000092
, .798103
.05517306-.03152801.01875935·°°889049
, .00441134.00238622.00141315.00090552.00061653.00043986.000325112
, .00015358.00006018.00001665.00000680.00000341.00000121.00000041.
.864665
.04539210
.02281990
.00923883,,00393120.00190692.00105401.00064485.00042465.00029513.00021368.00015980.00007692.00003062.00000858
, ~OO000352.00000117.00000063
.•00000021
VIo
..-/"
e e ,eT
Table 4.2.2. Bias of.e'o - Eq~Spaced Case
8=1000t k
h' "Sample size en)1 2 3 4 5 . 6 7 8 9 10
400 200 -810.0 -488.9 -238.7 -~9.9 87.3 1~.3 244.~ 2eo.7 298.7 30'.~100 -812.5 -491.4 -241.2 -52.4 84.8 179.8 241.7" 278.2 " 296.2 300.9Ungrouped -813.3 -492.2 ':'242.0 -53.1 83.9 179.0 240.9 z/7·4 295.4 300.0
500 250 -765.5 -388.0 -115.9 69.8 188.1 256.5 289.7 299·5 294.6 281.2Ungrouped ~770.7 -393.2 -121.1 64.6 J,.82.•9 251.3 284.5 294.3 289.4. 276.1
800 .400 -639.5 -143.4 ·12~2 250.3 287.9 282.4 2591.1 231.0 204.2 180.8200 -649.4" -153.4 117.3 240.3 277.9 272.5 249.1 221.0 194.2 170.8Ungrouped -652·8 -159.7 113.9 237.0 274.6 269.1 "245.8 227.7 190.9 166.5
1200 600 -487.4 68.1 254.1 277.3 248.0 210~9 179·3 155.1 136~9 123..•1400 -503.9 51.6 237.5 260.7 231·5 194.4 162.8 138.5 120.4 106.6200 -513·9 41.7 227.6 250.8 221.5 184.4 152.8 128.6' 1l0.4 96.6Ungro~ed -517.2 38.3 .224.3 2~7.5 218.2 180.1 149.5 125.3 107.1 93.3
1600 800 --352.0 185.6 274.1 243.3 ~O;I..' 169.6 147.7 132.5 ·121.5 113.2400 -391.5 146.1 234.7 203.9 161.9 130.1 "108.2 93.0 82.0 73'.7200 -401.4 136.1 224.7 193.9 151.9 120.1 98.3 83.1 72·0 63.7Ungro~ed -404.8 132.8 221.4 190.6 148.5 116.8 94.9 79·7 68.7 60.4
2000 1000 -231.2 245.6 263.6 218.0 183.0 160.9 146.5 136.6 129.3 123.8500 ";292.4 "184.4 202.4 156.8 121.8 99.6 85.3 " 75.3 68.1 62.5400 -299·7 177.1 195.1 149.5 114.6 92·3 77.9 68.0 60.8 55·2Ungrouped -313.0 163.8 181.8 136.2 101.2 79.0 64.6 54.7 47.5 41.9
~f-I
e e ,-Tah1e 4.2.2 (continued)
8=1000t k h Sa.m;p1e size (n)
11 12 15 20 30 40 50 70 ~1oo 00
1(.00.. 200 299.1 2~9.0 244.8' 176.2 105·9 7U..8 58.6 !+1.1J. 29,•.9- .3.3100 296.6 286.5 242.2 173.7 102.5 72.4 56.1 38.9 26.9 0.8Ungrouped 295.8 285.7 241.5 172.9 101.6 71·5 55.3 38.1 26.0 0
500 250 263.7 244.7 191.7 133.2 81.9 60.1 48.0 35.0 25.7 5.2Ungrouped 258.5 239.5 186.5 128.0 76.7 5#-~O. 42:;;6· 29.~S 20.5 0
800 400 161.2 145.1 111.7 82.4 56.8 45.1 38.4 30.9 25-.5 13·3200 151.2 135.1 101.7 72.4 46.8 35.1 28.4 20.9 15.5 3.3UJ:lgrouped 147.9 131.8 98.4 69.1 43.5 31.8 25.1 17.6 12.2 0
1200 600 112.4 104.0 86.6 70.8 56.2· 49.2 45.2 40.7 37.4 29.8400 95.9 87.5 70.1 54.3 39.6 32.7 28.7 24.2 20.8 13.3200 86.0 77.5 60.2 44.3 29.7 22.7 ..18.7 14.2 10.9 3.3Ungrouped 82.6 74.2 .56.8 41.0 26.4 19.4 15.4 10·9 7.6 0
1600 800 106.7 101.4 90·5 80.2 70.6 66.0 63.2 60.2 57.9 52.8400 67.2 62.0 51.0 40.8 31.1 26.5 23.8 20.7 18.4 13.3200 '57·2 52.0 41.1 30.8 21.1 16.5 13.8 10.7 8.5 3.3Ungrouped 53.9 48.7 37.7 27.5 17.8 13.2 10.5 7.4 5.1 0
-2000 1000 119.4 115.9 108.4 101.3 94.5 91.2 89.3 87.1 85.5 81.9
500 58.2 54.6 47.1 40.0 33.3 30.0 28.1 25.9 24.3 20.6400 50.8 47.3 39.8 32·7 25.9 22.7 -20.7 18.6 17.0 13.3
Ungrouped 37.5 34.0 26.4 19.4 12.6 9.4 7.4 5.3 3·7 0
\}II\)
e e #e
Table 4.2.3. Variance of §0 .. equally spaced case
e=1000
t k h Sample size2 3 4 5 6 7 8 9 10
400 200 34270 101132 206934 336730 472068596447 698185 771075 814049100 36518 103150 208742· 338348 473515 597742 699347 772186 814992Ungrou;ped 37269 103825 209346 338888 473998 598175 699735 772469 815307
500 250 59732 ;rn.~:I2 ~~990:? li99781 649165 15~60 ~0'31 858!i.90 821307Ungrou;ped 64287 175389 333368 502805 651798 760567 822368 840289 822913
,-'800 400 181914 448206 695876 836929 862693 807340 711719 606232 508053200 189934 454669 701104 841205 866245 810343 714303· 608492 510058Ungrouped 192629 456840 702861 842641 867438 811352. 715171 609249 510731
1200 600 418268 784211 899309 814298 658992 511431 396673 314218 256246400 430148 792969 905956 819537 663274 515040 399790 316961 258696200 437390 798309 910008 822730 665884 517240 401690 318633 260190Ungrouped 439824 800103 911371 823804 666762 517979' 402329 319195 260692
1600 800 66~562 928132 815338 605835 436799 324988 253919 207477· 175497400 688241 945961 828524 616182 445295 ·332195 260179 213013 180459200 6948;54 950553 831920 618847 447483 334051 261792 214438 181737Ungrouped 697075 952094 833061 619742 448218 334675 262333 214917 182166··
2000 1000 847982 903247 651463 440515 313779 239967 194170 163364 141195500 884317 927646 669428 454741 325476 249933202853 171059 148104400 888945 930754 611716 456546 326966 251202 203959 172039 148984Ungrouped 891136 936254 675766 459~9 ~~96P3 ~53449 205917 J.73774 159542
~
e e ,e
Table 4.2.3. (continued)
e=1000
t k h Sam,p1esize1l 12 15 20 30 40 50 70 100
400 200 829191 820967 708404 459115 198569 118165 84479 54083 35175100 830046 821746 709009 459547 198842" 118365. 84637 54194 35253Ungrouped 830331 822006 709211 459691 i98933 118432 84690 54232 35279
500 250 779198 721979 529224 299654 136487 (81902 65099 42935 28436Ungrouped 780641 723285 530233 300380 136953 '88246 65372 43128 28570
800 400 424333 356314 226994 '136869 77058 53820 41371 28295 19194200 426134 '357949 228274 137811 77675 54280 41737 28555 19376Ungrouped 426739 358498 228704 138127 ',77882 ,'54434 41860 28642 19437
\~
1200 600 215084 185080 130898 88572 53990 38853 30348 21107 14483400 217298 i87101 132499 89762 54776 3944130817 21441 ~, 14717'200 218648 188332 133475 90487 55256 39799 3110? 21645 14859Ungrouped 219102 188746 133803 .. 90731 55417 35?919 31199 21713 14907
1600 800 152261 134599 100112 70311 44129 32162 25299 17729 12225400 156757 138711 103383 72753 45749 33374 26268 18419 12708200 151915 139770 104226 73381 46166 33686 26517 18597 12832Ungrouped 158304 140125 104509 73593 ,46306 33791 26601 18657 12874
2000 1000 124J125 111266 (8~557 60456 38538 28289 22346 15739 10900500 130695 117004 89133 63878 40812 29992 23707 16711 11579400 131493 117735 ,89716 64314- '41102 30209 23881 16834, 11666Ungrouped 132907 119029 90748 65085 41615 30593 24188 17054 '11819
•
\X-t="
'.
Putting
A
4.5 Effect o~ Neglecting Terms in.~l
From (.3.5.8) we ~bserve that
2i§1 = i[80 + 80(1 - E*ni~)] if max hi < V3 ~o -.
. ..... - .31'90
2* nihig=E -, then" .3re"2
,0.
A A $o{~1 =~()- "4"'". g
A C= ~0(1 "'4),
where
35.
Putting s>l
thenT < ssg i'· . 1·.3.5.·· (2s-1) < 1s nce .. . •
. . 2s (s+l)~
Hence an upper bound for the factor. C can be obtained' by rep1ac-
ing 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
which for large sa.JIWles may. be approximated by
6 - F Ihi~147· 362 - max 1h~ ~
•
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
2552 (4-52 )4(3-52 ), .
..
37
However, in general the important term in the bias reduction will
2* nillibe the first, namely 1:: .... The amount of this reduction is given12~o
by
1 * nih~/r= --12 E(1:: ... ). e9
(... * 21) jCOy eo' 1:: n.ll. r, J. J.
When the inte:rvals are, equally spa.ced (4.5.5) reduces to
.... '
E( h~ ) = . h2
... ..1. 1 + v~:eo) 1:: h2
... ' , (4.5.6)12eO" : 12E(eO) [E(eo)]2., 12E(eO)
, ~ ., 2 ,d
which :t'or,la:rg.e n_' __["bea~te~)b~~r:9:' ;. 'tpe p'~ce~t~e
a.pproximation being
The tb:ree a.pproximations to th.e bound are comp8J:'ed in th.e follow-
ing table.
Table 4.5.1. Approximations to the upper bound of the correctionfactor - equally spaced case (as percentage of e)
.05 .10 .20 .25 .40 .50 .60 .80 1.00 1.50 .
APirox. (4.5.3) .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
II (4.5.7) .02 .08 .33 .52 1.33 2.08 3.00 5.33 8.33 18.75
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.
Since
A
Var(8,0)
[E(eo)]4
and
then
A
-var(~o).... [E(~0)]2
"'" "'" f If- h4
]Var(8 ) :: var(8) 1 + "'" + . "'" . •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 82 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(80; n,8,k)"'" A '
and D(~2; n,8,k}, where D(~o; n,8,k) is given by (4.3.10) and
"'"D(~2; n,8,k) by
40
or directly
'"= D(~O; n,8,k)
'" ...From Table 4.5.4 it is evident that for relatively small" ~O' ~2
'"and the m.l.e. for ungrouped data, e, have approximately the same mean
square errors. In other words, no significant in:t'ormation is lost inA
grouping the data.. As 8 ->.:1, the mean square error of ~u tends
to exceed the others, .indicating tnat for fairly coarse grouping theI
addition of the bias correction term not only reduces the bias but
also reduce.sthe mean square error. In general, therefore, it would be
'"preferable to 'Work With ~2 •
e ·e ,e
Table 4.5.2. Approximate bias. of ~2 ... equally spaced case
t k h . .' Sample size' (n)20 30 40 50 10···· 100 00
400 200 172.5 101.5 71 ..4 55.2 38.1 26.1 0100 172.8 101.6 71.5 55.3 38.1 26.0 0Ungrom>ed 172.9 '101.6 71.5 55.3 38.1 26.0 0
.-,.500 250 127.5 76.6 54.8 42.7 29.8 20.5 0Ungrouped ' 128.0 76.7 54.9 42.8 29.8 20.5 0
800 400 68.7 43.3' 31.7 25.1 17.6 12.2 0.1200 ',68.9 43.4 31.7 25.1 17.6 12.2 0Ungrouped 69.1 43.5 31.8 25.1 17.6 12.2 0
1200 600 40.6 26.4 19.6 15.7 Ii.3 8.1 0.7400 40 .. ,6 26.1 19.3 15·4 10.9 7.6 0.22PO .40.8 26.3 19.3 15.4 10.9 7.6 0'lJ'ngrOuped 41.0 26.4 19,.4 15.4 10.9 7.6 0
1600 800 27·9 18.9 14.6 11.9 9·1 6.9 2.;1400 . 27·1 17.6 13.1 -10.5 7.4 5.2 0.1200 27.4 17.7 13·1 10.5 7.4 5.1 0Ungrouped 27·5 17.8 13.2 10.5 7.4 5.1 0
2000 1000 21.9 15.9 13.0 11.4 9.4 8.0 4~9
500 18.8 ' 12.4 9.2 ' 7.4 '5.3 3.7 0.2400 19·0 12.4 9.3 7.4 5.3 3.7 . 0.1Ungrouped 19.4 12.6 9.4 7.4 5.3 3.7 0
Values of .the bias below those of the ungrouped data are due to roundingE!rrors and the approximations used.
~
e -Table 4.5.3.
,.Approximate variance of e.2 ... equally spaced case
9=1000
i;c h S4(f'leSize (n) , .
20 30 o '" .50 70 100
400 200 461330 199668 118848 84982 54416 35397100 460104 199116 118537 84764 5~78 35309.
500 250 302090 137704 ~.·8'81J..9 65718 43354 28n8
800" 400 140002 78909 )55142 112400 29009 19684200 138611 78148 54618 42000 28737. 19501
1200 600 93267 56933 40999 32028 22293 15302400 91928 56136 40433 31598 21990 15096200 91041 55604 40053 31303 21786 14956
.1600 800 76886 48331 35252 27757 19451 13418400 ',14555 46903 ·34224 26940 18894 13036200 73843 46461 33904 26689 18719 "12916
2000 1000 69050 44086 32387 25595 18037 12496500 66363 42420 31183 24651 17379 12044400 65932 42150 30984 24496 17270 11969·
,e
is
e e ·e
Table 4.5.4.: , A A
Comparison of the mean square eITors of 80, 82 and the maximum
likelihood estimator (ungrouped) - equaJ.1yspa.ced case!'/
8:::1000
t k h 20S~:Le~~ize (n)" 3·0·_·- . .4Cf~···---- 50 _.- 70" 100
400200
A . . ieo 490161 209594 123760 87913 55797 3~40A. e.2 491086 209970 123946 88029 55867 36078
10080 48971~: 2093~8 123607 87784 55708 35976Ae.2 489964 209438 123650 87~ 55730 35990A
Ungrouped. 8. u _ 489585 gQ~)9_ --..lli5_lt:5._ H.. 81748'55.6B3 35955
500250
"'"90 317396 143195 ' 9;Ji514 67403 44160 29097~
92 318346143572 91722 67541. 44242 29139A
Ung1;'o.!lp.~_ 9, ..31~164 142836 9l?60 67203 44016 28990:~800
400
...90 143659 80284 55854 42845 29249 19845
g2 144722 80784 56147 43030 29319' 19833
19616289914254355512A
90 143052 79865Ae.2 143358 80032 55623 42630 29047 19650
200
~Jli>-ed $ .142902 7977555445 42490 . 28952 19585.
!:IVa.lues of D(~o;n,~,k) below D($; n,5,k) are due to rounding errors.a.nd the
a.pproximationS used.' '
&"
e e
Table 4•.5.4. (continued)9=1000t k h S4?le size (n)
20 30 o . 50 ·-10__ 100...
227641200 90 93585 57148 4~74 32391 15882600 ...
94916 57630 41384 32284 22420 1536892-a 92710 56345 40510 31641 22027 15149400 0a 93577 56817 40805 31812 22109 151542...
92450 5613890 40314 31453 21847 149718200 ...
92 ·92706 56296 .40425 31540 21904 15014--....·92412 ;>6114 40296 31436'Utlgrouped 9 21832 14965
.16QC go 76743 49113 36518 29294 21353 15577800 ."
77664 48688 3546592 27898 19534 13465A90 74417 46716 34076 26834 18848 13046
400#2 75289 47213 34396 27050 18948 13063...
74330 46611 26708 18712 129048; 33958200 ",0
9.2 74593 46775 34075 26800 18773 12942....
74349 46623 33965 2671.1 18712Vngrouped e 12900
·e
t:
• e
Table 4.5.4. (continued)
9=1000t k h S4?le size (n)
'20 30 o 50 70 100,..
2000 9 ' 10718 41468 36606 30293 23325 182101000 ,,0
69529 44339 32556 1~125 1256092 25725-,..65478 41921 30892 24491 175§2!o 12170
5008.2 - 66716 42574 -31261 24706 17407' 12057-,..e 65383 41773- 30725 24309 17180 11955
400 ,..042304e 66293 31011 24551 17298 11980;2
§ "
Ungrouped 65462 41773 30682 24243 17082 11833
·e
~
46A _.
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= -hIn(l _ * r _ )
I: ini + k(n-r)
(4.6.1)
For the same case, since g. = (2i-l) h/2, thenJ. ..
A I:*ini + (n-r)k h .e = ( . ')h - - •.0 r 2
Substituting in (4.6 •.1) we obtain
(4.6.2)
·e
e= -_h=::--__
In(l _ A2h )
2e.O+ h
A
:: e o2h
A
6(2e.o+ h)(4.fj.; )
Hence,
A A
If h is relatively smaJ.lthen e.2 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"tk ) which would
: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 90• The intractability ,of this criterion leads us to· consider the
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,posi-
tion would not greatly alter the variance" i.e. it is suggested that
" ". II Athe optimal decom,position for e would be almost optimal for ~O.
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 parti
tion X = (~, •••'~-l) which maximizes the information I(X) or
equivalently I*(X), where
But f~ = -fi' hence (5.2.3) can be written
(e2) *.I* = f [Xi+lfi +l - xifi _ Xifi - Xi_lfi _l ]
n Xi i fi-fi +l fi_l-fi
" [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
oXi .
the Xi' and hence the t i • 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 i > 0, (i=l, ... ,k-l), and that the
second term is negative and non-vanishing. This last point II1S\Y be
demonstrated as follows:
Putting
(s) =(X+Stf~X+S) - xf(x)g, f x - f(x+s) ,
then (a) 11#, g(s) = 1-xs..:.-> 0
(b) g( s) is a decreasing function of s, 0 < s < 00 ,
(c) g(s) ~ -(s+.x-1), 0 < s < 00 •
By application of l'Hepital t s rule we see that
1t g(x) =,1t (x+s )f(x+s) - xf)X)s-> 0 s-> 0 f(x) - f(x+s
=1t -(x+s )f(X+s) + f(x+s)s-> 0 f(x+s)
= 1-x •
If we can show gt(s) < 0 for 0 < s < 00, then we have ,proved (b).
Now
gt(s) = ,f(x+S),' 2 {Xf(X) - (x+s)f(x+s) - (s+.x-1)[f(x) - f(X+S»)) ,[f(x)-f(x+s)] , :
but '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 rex) - (x+s)f(x+s) - (s+.x-1) [f(x) - f(x+s») •
Since h(O) =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) < 0 for all s > 0 ,
then h( s) < 0 for all S > 0 and consequently so is, g t ( S ).
50
Part (c) has been proved in the process of showing h( s) < o.
Putting x = 'xi _l and s = xi-xi _l we then find g(s) becomes
> 1 - x- i
by (c) above.
by (a) and. (b) above.
Thus from (5.2.5) and (5.2.6) we see that
g(s') < l-xi
< g(s) ,
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
di +l + di - 2ei = 0,
which may be Written
(i=l, ••• ,k-l) ,
i-l, ()i-lDi = di +l - 2 1: (_l)J ei -
J, + -1. cL = 0 ,
.1=0 -:L
51
(i=l, ••• ,k-l) •
(5.2.10)
The system of equations Di = Di(xl'~' ••• ,xi +l ) = 0 may be solved by
the generalized Newton procedure. Then
*-l?~(m) = ~(m-l) - D(m_l) ~(m-1) ,
where starting 'With a trial solution ~(O)' the mth itera~ion yields
*-1~(m)' 'With D and D each eValuated at ~(m-l)' and
~t = (~,D2' ••• '1\:-1) •
aDl aD1 :
~ d~ :~-----I
aD2 aD2 aD2 ;'~ dX; di3:
I
• •• •• •• •
* • 0
D = • 0
•••
• • • • • • • • • • • • • •
• • 0 • 0 • • • • • • • 0 •
II
\.-- - - -aD
k_2
d~_l
~~
However, the matrix inversion may be simplified if we note from
(5.2.10) that
e.52
j ~ 2 •
adiIn a.ddition, if we put ~ = d , then.oxj ij
j=i
j=i-1
o , j < i-1, or j > i ,
Thus
and
. It should be noted that
i =1, ••• ,k-1 •
AlsoaDi i-1 ·i-1~ = 2( -1) + (-1) d- 1, i ,= 2, ••• ,k-1oX1 ~ '.
and i =3, ••• ,k-1
j = 1, ••• "i-2 •
*Hence if we add each successive column of D' to tha.t preceding it we
obtain a matrix H of the form:
53
H =
hii~l
~lh41••
'~I 0 •••••••••••••L,.._,
~2 h23 , 0 •••••••••••L __ ,
~2 Ij3 ~41 0 •••••••••,..--- I ~, ,: 0 I h43 h44 h45 :I L_--a II· I
•I •I •I
• ••
• • •
• • •
o
o
o
•
••• •
• •• I • •
I • •• ,I 0'. •, ---.. L_ - - - -
~-2,1: 0 • • • • • .0. I~-2,k-3 ~-2-k-2 ~-2"k-iJ __ -:- __•
~-l,l :0 • • • • • • • '0 I
~-1,k-2 ~-l,k-lI
The elements of H are given by
ODi obi-_\~
=1" •••"k-ldXj
+ dXj +l =1" ••• ,k-2
hij =cD!,
{~= 1, •••"k-ld '
~-l = k-l
Hence,
•
i = 2, ••• "k-2
i =k-l
e. " .i=l, .. 0 .,k-2 •
54
ODi ODi ' -fih - + - d = [e d ]i . 1 - "':::':':"""~ - i+1 i f -f· . i- 1+1 ',1- OXi OX1_l ' i i+l
( )i-l ()1-l f l [ ]-1 <L = -1 - e -<L, i=3, 00 o,k-l~l l-fl 1 ~
i=4, 00 o,k-l
j=2,.oo,j-2 •
The number of' elements of the matrix H that need to be calculated
In8\V be reduced by using the following relations':,
i=2
i=3, ••• ,k-2
h.. -h. =2.--k-l,k-l -It-l,k-2
55
e Thus the matrix H is really of the form:
1 + du d22
0 0 • • • • • • • • • • • • • • 0
2~
~1 1 0 • • • • • • • • • • • • • • 0
d11 d43 1 d44 0 • • • • • • • • -. • • • • 0
-~1 -0 0,4 1 •
H= ~1 0 0 d65 •
• • • •.. • • •• • •
• 0
(-1)k-1du 0 0 • • • • 0 ~-1 k-2 1 ~-1 k';'l
(_:l.)k-2d 0 0 • • • • • • • 0 !\,k-1 2~,k+111
This matrix manipulation may be expressed by the equation
*DA=H,
Where A is a matrix of the form
1 0 • • • • • • • • • • 0I
1 1 0 • • • • • • • • 0
0 1 1 0 • • • • • • 0
• •A • •= •• •
• •• •• •
10 • • • • • • • 0 1 1 0
0 • • • • • • • • • 0 1 1
Since 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:
Hence
x = i8i
xi-x
i_1
=8
f = e-i8i
i=1" ••• "k •
1 ,
e-(k.-l)8 .. ' [. 8e-k8_1]-(k-1)1) -k5 -{k-1'5-k8 '. + 2e -e e., -e
e
8[5 J=~-r- -1 + 2 =2-~1"e -Ie -1
i=2" ••• ,k-2
i=k-1 •
i=l" ••• "k-2 •
...
hi ,i+1 =.L [1 - :::i] ~. hn-1 ,
.(l.5~~ -~ -/1 [1 ".:~~] =1-2hn' i=2
h =i"i-1
e5
[ I) j~ ...~ -1. = -hr1e -Ie -1
1::=3" ••• "k-1 •
(-ll-l
e5_11=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• The computation of H(0) is consequen~ly a simple
matter. The calculation. of ~(0 ) is similarly simplified since:
i=l" ••• "k
i=l" ..."k-1 •
_ i.-1 j i 1D d 2 't' ( 1) + (-1) - ....d.. •i+1 - 1+1 -j~ - ei _j ~ .. i=l" ••• "k-1
8e8 i+1 i+1 ~ )= ""8 (1+(-1) ] - 8[i+1+(-1) ]-2 1[1-i81~[l-(i-18]e -1 - - --
+•••+(-1)~+1[1-8]1
i odd,
i even.•
t>o
ot4erwise •
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 1
0
;00 0l\P- l~Of(t)-= t
Connnencing with eql,18.l spacings the initial solution is
!(0) = [.2, .4, .6, .8, 1.0] •
Consequently,
1 2 .2hll ·= 1 + .2- [1 - ,e2 ] = .53329
e -1· e' -1
~2 = ~3 = h44 = 1
58
~5 =2-~1 = 1.46671
hi,i+l = ~l-l = - .46671
. l 1-2~1 =-.06958h = .i,i-1 . -
-~1 = ·~,53329
i = 1, ••• ,k-2
i = 2
i =3, ••• ,k-1
i = 3, ••• ,k-1 •
Bence the matrix BeO) is·
·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
2.8893 2.0920 1.3943 .7837 .2494
1.1589 2.3905 1.5933 .8955 .2850-1 2.0703 2.6805 3.2147 . 1.8068 .5750H(O) =
.2225 .9199 1.5304 2.0645 .6570
1.0000 1.0000 1.0000 1.0000 1.0000
Since *-1 ·-1 thenD =AH
2.8893 2.0920 1.3943 .7837 .2494
4.0482 4.4825 2.9876 1.6792 .5344*-1 3.2292 5.0710 4.8080 2.7023 .8600D(O) =
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
59 m_ ... _
D =i
.00666, i = 1,3,5
i = 2,4
hence.00666.00000
'B(o} = .00666 •.00000.00666
Using the ~elation
tit we obtain
'.
2nd Iteration:
.~alculation of the elements of "~ and B( 1) •••
60
if
'. ;t f""0" "". 1 :--~io •.1;1-.1;1+1 """ J."
1 .1698 .8438 .1562 5.4020 6.0792 .1433 .9174 ".8302
2 .3496 .7050 .1388 5.0792 5.7457 .2465 .7435 .6504-
3 .5408 .5823 .1227 4.7457 5.4167 .3149 .5574 .45924 .7449 .4748 .1075 4.4167 5.0835 .3537 .3609 ".2551
5 .9640 .3814 .0934 4.0835 4.7556 .3677 .1499 .03606 1.2000 .3012 .0802 3.7556 .3614 - .0786
~1 =1 + 5.4020 [.8302 - .9174] = .5290
h22 ° = Il;3 = h44 = 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 •
h43 = -5.0835 [.255i - .1499] = -.5348
h45 =-~.+ .5348 =-.4652 •
". ~4 =1.4550 - 2 = -.5450
i(. i;hil = (-1) 1 - .5290) ,= (-1) (.4710) •
Also,
61
:01 =~ - 2e1 + c;.
:02 = CJ - 2( e2~el) - c;.:03 = d4 ~ 2( e3-e2+e1) + c;.
:04 = a., - 2(e4-e3+e2-e1) - 'c;.
= .0005
=-.0004,
= .0003
= .0003
.".
Hence
.5290 -.4729
-.0633 1.0000 -.4657
H(I) = -.4710 -.5325 1.0000 -.4675
.4710 • -.5348 1.0000 -.4652
- .4710 . • • - .5450 1.4550.
and its inverse
2.9363 2.1550 1.4394 .8149 .2605
1.1700 2.4107 1.6102 ~9116 .2915·f -1
H(I) = 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 _ n
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
[
1 t
:r( t) =: exp- et ~ 0" e > 0
otherwise •
Censoring occurs at some presta-ted time (tk ) where 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. Using the notation
Xi = tile and f i = exp- Xi it is found that the likelihood has the
form
k+l(1: E 1:"
i=l
k *and 1: == 1: ) ,i=l
where ni = number of items failing in the interval (ti_l"ti ). If
~ = n or ~+l = n 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 ~ =t n
and ~+l =t n" the solution to the likelihood equation is unique but
cannot. be obtained without .iteration. Fisher' s method of scoring is
adopted which gives the (j+l)st iteration as
.......... _ ..... g(e j )ej +l - ej(l - x ). . G(e )
.j
where
and
.....e=e .
.J
,
•
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. - t' lJ. J.-
and i = 1, ••• ,k ,
..we may expand the elements of the likelihood equation as power series
giving:
xifi - xi_lfi _l h~. hi 4fi
_l
- ff = ui - (1 + 12e2) + 0(2e)' i = l.t~ •• ,k.
h 4Thus to 0(2~ ) the likelihood equation becomes
2* . * niJii *
1:. ni ui .+ '\:+1~ - 1: 12e2 = 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 approxi-
mate 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 ~o
,..as the estimator. If the intervals are relatively Sma11~o will be
lIalmost" the m.l.e. and at the same time will possess the practical
advantage that it is defined for the case of ~ = n. For the case,..
~ =n, ~0 = gl the mid-point of the ,first time interval, which is a
reasonable and intuitive estimator.,..
The properties of e.o are investigated conditional upon the event
~+1 f 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 func-
tions 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 ~o is essentially,..
a linear estimator whereas the m.1.e. is non-linear. If we adjust ~o
for bias by ~ubtracting a correction term, the variance pattern for the
66A
new estimator ~2 (equally spaced case)
" "e = e .2 .0
is similar to that of the m.l.e. That is to say by a.ddingona~termof
a Taylor series expansion (so that the estimator is no longer linear)
the variance pattern changes to that of the m.l.e. However, since
" "eo' e2 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 un-
necessary 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' ~O and the power of tests of,.
~othesesusing the distribution 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 proce-
dure 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 l' of the items haVe
failed. 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" ••• "caj = 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 = ~ '1ij =
n '1ij = 1 we obtainj
e• ~.. i = [It e ]lIb / ~
. j .ij .
68
•
(
A
~ij•
e•
..
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..
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70
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e--
•
INSTITUTE OF STATISTICS
NORTH CAROLINA STATlcOtLEGE
(Mimeo Series available for distribution at cost)
283; Schutzenberger, M. P. On the recurrence of patterns. April, 1961.
284. Bose, R. C. and 1. M. Chakravarti. A coding problem arising in the transmission of numerical data. April, 1961.
285. Patel, M. ~Inve~tig.ations on factorial designs. May, 1961.
286. Bishir, J. W. Two problems in the theory of stochastic branching processes. May, 1961.
287. Konsler, T. R. A quantitative analysis of the growth and regrowth of a forage crop. May, 1961.
288. Zaki, R. M. and R. L. Anderson. Applications of linear programming techniques to some problems of production plan-ning over time. May, 1961.
289. Schutzenberger, M. P. A remark on finite transducers. June, 1961.
290. Schutzenberger, M. P. On the equation aB+" = bl'+mc;S+p in a free group. June, 1961.
291. Schutzenberger, M. P. On a special class of recurrent events. June, 1961.
292. Bhattacharya, P. K. Some properties of the least square estimator in regression analysis when the 'independent' variablesare stochastic. June, 1961.
293. Murthy, V. K. On the general renewal process. June, 1961.
294. Ray-Chaudhuri, D. K. Application of geometry of quadrics of constructing PBIB designs. June, .1961.
295. Bose, R. C. Ternary error correcting codes and fractionally replicated designs. May, 1961.
296. Koop, J. C. Contributions to the general theory of sampling finite populations without replacement and with unequalprobabilities. September, 1961.
297. Foradori, G. T. Some non-response sampling theory for two stage designs. Ph.D. Thesis. November, 1961.
298. Mallios, W. S. Some aspects of linear regression systems. Ph.D. Thesis. November, 1961.
299. Taeuber, R. C. On sampling with replacement: an axiomatic approach. Ph.D. Thesis. November, 1961.
300. Gross, A. J. On the construction of burst error correcting codes. August, 1961.
301. Srivastava, J. N. Contribution to the construction and analysis of designs. August, 1961.
302. Hoeffding, Wassily. The strong laws of large numbers for u-statistics. August, 1961.
303. Roy, S. N. Some recent results in normal multivariate confidence bounds. August, 1961.
304. Roy, S. N. Some remarks on normal multivariate analysis of variance. August, 1961.
305. Smith, W. L. A necessary and sufficient condition for the convergence of the renewal density. August, 1961.
306. Smith, W. L. A note on characteristic functions which vanish identically in an interval. September, 1961.
307. Fukushima, Kozo. A comparison of sequential tests for the Poisson parameter. September, 1961.
308. Hall, W. J. Some sequential analogs of Stein's two-stage test. September, 1961.
309. Bhattacharya, P. K. Use of concomitant measurements in the design and analysis of experiments. November, 1961.
310. Anderson, R. L. Designs for estimating variance components. November, 1961.
311. Guzman, Miguel Angel. Study and application of a non-linear model for the nutritional evaluation of proteins. Ph.D.Thesis-January, 1962.
312. Koop, John C. An upper limit to the difference in bias between two ratio estimates. January, 1962.
313. Prairie, Richard R. and R. L. Anderson. Optimal designs to estimate variance components and to reduce product vari-ability for nested classifications. January, 1962.
314. Eicker, FriedheIm. Lectures on time series. January, 1962.
315. Potthoff, Richard F. Use of the Wilcoxon statistic for a generalized Behrens-Fisher problem. February, 1962.
316. Potthoff, Richard F. Comparing the medians of the symmetrical populations. February, 1962.
317. Patel, R. M., C. C. Cockerham and J. O. Rawlings. Selection among factorially classified variables. February, 1962.
318. Amster, Sigmund. A modified Bayes stopping rule. April, 1962.
319. Novick, Melvin R. A Bayesian indifference postulate. April, 1962.
320. Potthoff, Richard F. Illustration of a technique which tests whether two regression lines are parallel when the variancesare unequal. April, 1962. .
321. Thomas, R. E. Preemptive disciplines for queues and stores. April, 1962. Ph.D. Thesis.
322. Smith, W. L. On the elementary renewal theorem for non-identically distributed variables. April, 1962.
323. Potthoff, R. F. Illustration of a test which compares two parallel regression lines when the variances are unequal. May,1962.
324. Bush, Norman. Estimating variance components in a mUlti-way classification. Ph.D. Thesis. June, 1962.
325. Sathe, Yashwant Sedashiv. Studies of some problems in non-parametric inference. June, 1962.
326. Hoeffding, Wassily. Probability inequalities for sums of bounded random variables. June, 1962.
327. Adams, John W. Autoregressive models and testing of hypotheses associated with these models. June, 1962.