Contractor:
Bernard G. Greenberg, University of North Carolina
Project Numbers:
Department of Army Project No. 5B99-01-004
Ordnance R
~
D Project No.
OOR Project No.
TB2-0001
1597
Technical Report No. 1
Title:
"Estimation of Parameters of Distributions by Order
•
Contract No.
Statistics
DA-36-034-0RD-2184
November 21, 1955 through March 31, 1956
by
Ahmed E. Sarhan and Bernard G. Greenberg
Institute of Statistics
Mimeograph Series No. 148
Summary
Tables are provided to calculate the estimate of the mean CJ" from singly and
doubly censored samples of sizes :5. 10 from the one-parameter single exponential
distribution.
Similarly, tables are provided to estimate the minimum value (A..~),
the standard deviation (
cr ) and the mean value
(AI\ +CP
from the same samples
drawn from the two-parameter single exponential distribution,
Interpretations are made for the variation of the coefficients and variances
of the estimates as the number of censored observations var,yo
The following are two general conclusions which can be given concerning the
effect on the estimates of the side from which the sample is censored.
(1)
For the one-parameter single exponential distribution, the estimate of
\I obtain-
ed from singly censored samples from the left has a smaller variance than that based
upon singly censored samples from the right provided that the number of missing
e
observations is equal on both sides.
(2)
For the two-parameter single exponential distribution we have
(a)
f.A. can be estimated more efficiently in samples censored from the right
than in samples censored :'rom the left provided that the number of missir.i.g
observations is equal o
(b)
The estimate of (f in singly and doubly censored samples from this
distribution does not depend upon the side from which censoring takes
place and depends only on the total number of missing observations.
f'
Estimation of the Parameters of the One-and Two-Parameter Single Exponential
Distributions From Singly and Doubly Censored Samples
by
A. E. Sarhan and B. G. Greenberg
Department of Biostatistics, Universi ty of North Carolina
I.
Introduction
Censored samples are encountered because values of some of the observations
in a sample are unknown as a result of their occurrence below a lower bound (or
above an upper limit) imposed by either the observer or the measuring process.
The
values beyond the 1imits are believed to form a continuation of the scale of
measurement.
The censored samples considered here are those in which the total number of
sample elements is known but measurements on some of them at both extremes might
be lacking.
e
The censoring procedure might be performed in one of two ways, viz.
censoring the observation because it falls outside fixed bounds (~e I) or
censoring a fixed percentage of observations at either end of a sample (Type II).
In experimental biology, a known number of individuals might be exposed to an
agent and the responses of some fall outside the 1imi ts •
Thus, if n animals are
injected with the same dose of antigen and blood samples from each animal are
tested for antibody response after a period of time, there may be only n - r l of
the animals with measurable amounts.
This means that r
1
of the animals developed
the antigen to a level which cannot be measured by the prevailing technique.
Of the n items, the smallest rl observations are censored because of fixed
bounds and it is required, for example, to estimate the population mean and standard
deviation by using the largest n - r
observations. This sample is called a singly
l
censored sample from the left and this case was considered by Ipsen (8), Cohen (1),
Ra1d
(6), Gupta (5) and others (2,7,11).
Similarly, one may have n items drawn at random from a population and to save
time and expense, the experiment is discontinued before all items have actually
- 2 developed the phenomenon being observed.
Such a decision to cut off the experiment
is made as soon as the first n - r2 experimental units have responded and the
censoring is based upon a fixed proportion of the observations.
For example, a bio10gis t may perform an experiment on animals to determine the
effect of exposure to a drug by noting reaction times.
an extremely long time to react.
Some animals may require
The experiment might be stopped when a fixed
percentage have reacted, i,.e., the data based on the smallest n - r2 items.
sample is termed a singly censored sample from the right.
This
This occurs in life
testing, incubation periods, and fatigue testing and was considered by Halperin (7),
Ra1d
(6), Gupta (5), Cohen (1), Epstein (3), and
~pstein et al
(4).
Furthermore, the above two situations may occur jointly such that there are r 1
smallest observations in a sample of n items that are missing plus r2 largest obser-
e
vations that are censored.
This is termed a doubly censored sample.
For example,
in certain studies of blood clotting, the speed of the reaction is such that r
l
animals may respond almost spontaneously before individual measurements can be taken
on them whereas some animals barely respond and may require an infinite waiting
period.
In such a situation, censoring on the left is by Type I whereas that on
the right is by Type II.
The case where observations are missing from both extremes is the most general
and the first two illustrations are special instances of it.
The common situations
of censored samples encountered in practice are those which occur with samples drawn
from either an exponential or a normal dis tribution (2, 11).
The main aim of this work is to provide tables for calculating estimates of the
mean and standard deviation from doubly censored samples drawn from the one-and twoparameter exponential distributions given respectively by
fey)
=-(1j
e
r
- 3 and
,
fey) :: -(J"1 e
A second objective is to interpret the foregoing tables by pointing out patterns
and arrangements "ri thin the tables.
These pa tterns can help to throw light upon the
relative worthwhileness of individual observations and to provide guidance in designing an experiment most efficiently.
II.
Estimation of
paramete!:~
The best estimates of the parameters are given as a linear function of the known
ordered observations i.e., the known observations are arranged in ascending order
and the best linear combination of them is obtained.
termed linear systematic statistics (9).
These linear estimates are
These estimates are simple, easy to
calculate and of high efficiency.
The general formulae for the estimates, their variances, relative efficiencies
and examples are presented herein for Type II censoring.
The derivation of these
formulae can be found in (10).
Consider a sample of size n with r 1 smallest missing observations and r 2 largest
missing observations, and denote by Yi the i th observation in ascending order of
magnitude.
For the one-parameter exponential distribution,
(2.4)
()2
=k
- 4-
where
k =
median* = cr* loge 2
V(median* )
(2.6)
<f2
=--k
(log 2) 2
e
where cr* is the best linear estimate of () and median* is the estimate of the
population median.
For the two-parameter exponential distribution"
*
}J-
=c
2
rl+l
(2.8)
V( fl *)
I
r-
=C
i=l
and
([
*
1
(n - i + 1)
- 5(2.10)
where
V(
c
=
cr*) = CO-2
1
n-r1 -r 2 -1
Furthe:are,
(2.11)
*
*
~{ 1
mean = /-l + e:r • c~
C+
n-r2
JI
-1
i=r +1
1
V(mean* )
=
y.
1.
c
and
(2.14) V(median*)
III.
Tables
The tables are provided for all possible combinations where the samples are
of sizes
~
10 since these values are the ones most conunonly encountered.
Extension
of these tables to larger values of n is straightforward.
I
Table I gives the coefficients (wi ) in the best linear estimate of () in the
one-parameter single exponential distribution from singly and doubly censored samples
-
of sizes < 10 such that
(5
*
- 6 The coefficients in each row of Table I must be divided by the common divisor
given in the last column.
This table shows that:
(1)
For a fixed value of rl' the numerator in the coefficient of the largest known
observation increases as r2 increases.
In fact it will be increased by the sum of
the numerators of the coefficients attached to the censored observations.
(2)
For a fixed value of rl' the numerator of the coefficient of the smallest known
observations will be equal regardless of the value of r20
(3)
'!he numerators of the coefficients of the middle elements are always equal.
Table II gives the exact variance (in terms of
0'"
2) of the es timate of 0- in
the same distribution from singly and doubly censored samples of sizes
<
10 for
different values of r l and r •
2
This table shows that:
(4)
The estimate based upon singly censored samples from the left has a smaller
variance than that based upon singly censored samples from the right when the number
of missing observations is equal on both sides.
(5)
The denominator of the variance in the common divisor from Table I.
The numer-
ators in any row (n and rl fixed) are all equal and the same value as the coefficient
of the largest known observation indicated in Table I when r 2
=0
for that specific
n and r •
l
Table III gives the percentage efficiencies of the estimate of 0- from singly
and doubly censored samples of sizes
~
based on the complete uncensored sample.
10 relative to the best linear estimate
By eXamining the entries in this table in
a diagonal fashion (i,e., n, (rl + r2) fixed), it can be seen that the efficiency
declines more rapidly when the censoring is from the right.
e
This means that the
experimenter is sacrificing more precision when censoring is from the right.
On
the other hand, the expected wai ting time for the largest observations may make
this sacrifice a desirable one.
- 7T'i~k
L
Z:,
pro,t.:-:!'~i?nal
tiiO
3:.:..
reduction in expected waiting time for a singly
censored sample from the right of size
D,l ccmbinic.g the z-esnl":,s
fro~
10 with differing values of r •
2
Table III and IV, a measure termed "efficiency
~
per uni t of waiting time" can be calculated to guide the experimen ter in determining
the advisability of censoring a given sample.
From the ap:;;:,aaran::e of the graph in
figure 1, the "efficiency per unit of waiting time" is seen to decline as more observations from the right go uncensored.
This decline is somewhat constant until the
point r 2 = 4. As this point, the rate of decline is accelerated and the additional
waiting time may be ur~emunerative. Consequently, for samples of size 6 to 10, the
observations might be censored profitably when only 3 or 4 of the largest observations are
remain~ng.
Similarly, a more meaningful measure of experimental efficiency can be constructed if the waiting time can be converted to a cost function.
.e
In this event, one
could calculate "efficiency per unit of cost" where the cost would consist not only
of the increment for the extra waiting time but also the original expenses of
setting up the experiment.
Table V provides the exact coefficients (wli ) for the best linear estimate of
the population value of
in the two-parameter single exponential distribution from
)J--
singly and doubly censcred
The values of w are calculated up to samples
li
of size 10 with different values of r 1 and r such that
samp1es~
2
n-r 2
= L
\'
i:=::,+l
w " y.
1.~
~
..!..
All elements in 6&ch row should be divided by the proper divisor given in the last
column.
The divisor'is given by
rl+l
e
(3.3)
1
r
n - r1 - r 2 - 1
n i=l
1
i + 1
•
- 8 -
e
In some instances, the value of the denominator given by (J.,3) may not be identical wi th the one in Table V because a common fac tor was cancelled from both the
denominator and the numerator.
The coefficients (both numerator and denominator)
in this table show the following systematic changes as rl' r 2 , and n vary:
(6) In complete sample estimation (i.e., r = r = 0), the numerator of the smallest
1
2
sample element is given b,y (n + l)(n - 1) and all the other elements have numerator
= -1 while
the divisor is n (n - 1).
For a fixed rl and as r 2 increases, the numerators of the coefficients if the
largest known elements decrease (increase in absolute value). In fact, the actual
(7)
value is equal to the sum of the numerators which were attached to the censored
elements plus that of the largest known observation.
(8)
The numerators of the middle elements are always equal.
(9)
For any fixed value of rl' the smallest known element always has a numerator
which decreases as r2 increases b,y a fixed number that equals the denominator for
the fixed value of r l and the largest r possible. In most cases this decrement is
2
also equal to the denominator of the previous value of r l and smallest r •
2
Table VI is constructed to give the exact coefficients (w2i ) in the best linear
estimate ·of the population standard deviation
a-
for the two-parameter single
exponential distribution from singly and doubly censored samples such that
n-r 2
0-*
==
I
W
2i Yi
i=rl+l
All coefficients in each row must be divided by the proper divisor for that row
given in the last column.
This divisor is calculated from
1
- 9-
e
From this table, the following additional observations can be made:
(10)
The coefficient (w ) of the smallest known element in samples censored only
2i
from the left (i.e., r2 = 0) is always =-1, and all the other coefficients are equal
and have the value of
1
n-rl-l
•
(11)
For a fixed value of rl' the coefficients of the smallest known elements are
equa~
regard?-ess of the value of r 2 •
(12)
For fixed value of nand rl' as r
increases the numerator of the largest
2
known observation will increase and in fact will be equal to the sum of the numerators
of the coefficients attached to the censored elements plus that of the corresponding
element for the largest known observationo
(13)
e
As r 1 increases, the numerators of the smallest elements increase by a differ-
ence equal to the increase in r •
1
(14) The coefficients of the middle elements are always equal.
Since (mean)*
=
J.A.. * + ()*
the exact coefficients (w ) are given in
3i
Table VII for the best linear estimate of the mean in singly and doubly censored
,
samples from the two-parameter single exponential dis tribution such that
n-r
U.6)
mean
*
=
2
L
w3i Yi
•
i=r +1
1
All coefficients in each row must be divided by the proper divisor given in tile last
column.
(15)
From this table tile following additional observations are noted:
For a fixed value of n and rl' the numerator of the coefficient of the largest
known element increases as r2 increases, and in fact the numerator is equal to the
sum of these coefficients of the censored elements plus that of the corresponding
elements for the largest known observation (opposite to that of observation #7 and
same as #12).
- 10 (16)
For a fixed r , as r increases the smallest known observation has an numerator
2
l
which decreases by a number equal to the last divisor for the fixed r and largest
l
r •
2
(17)
(Same as observation #9)
The middle elements always have equal coefficients (Same as observation #8)
Table VIII gives the variances of the best linear estimate of p. * in singly
and doubly censored samples of sizes :: 10 from the two-parameter single exponential distribution for different values of r
and r in terlilS of 0'2. The entries
1
2
in Table VIII were based upon the exact fractional values for the variances but were
converted to decimals to faci1i tate reading of the table.
From this table, the
following points can be noted:
(18)
The variance of the estimate increases as r 2 increases for a fixed value of
r
e
and vice versa. Vfhen the number of censored observations is equal on both sides
l
the variance of the estimate of fA.. is less for singly censored samples from the
right than that obtained from singly censored samples from the left.
(19)
For a fixed nand r , the variance of the estimate of )J. does not undergo
l
much increase as r increases except for the last possible value of r 2 • In other
2
words, the variance of the estimate of f- is roughly independent of r 2 prOVided
that r l is small.
Table IX gives the percentage efficiencies of the best linear estimate of
in singly and doubly censored samples relative to uncensored
parameter exponential distribution for n <
sa~les
from the two-
10 and all possible values of rl and
r 2•
+ r ) is fixed for a
l
2
given n, it can be seen that the efficiency increases as the sample is censored
(20)
Reading this table in diagonal fashion such that (r
from the right rather than the left.
encountered in estimating
(J'"
This is direc tly oppoaite to the si tuation
in the one-parameter exponential distribution.
-11Tables X and XI are constructed to show the exact variance and relative efficiencies of the estimate of the population standard deviation,
() , for the same
sample size and different values of r 1 and r from the same distribution.
2
(21) These tables show that the variances and relative efficiencies of the estimate
of the population standard deviation are independent of the side from which censoring takes place.
The values in both tables show that the effect of censoring depends
only upon the total number of missing observations for any given n.
Table XII gives the variances of the estimate of the population mean for censored samples of sizes
~
10 and with varying values of rl and r • This table is
2
expressed in seven decimal places although the values were calculated exactly as
fractions wiich are available from the authors.
(22)
Table XII shows that the variance of the estimate of the mean increases more
rapidly with censoring from the right than it does with censoring from the left.
This is true in general but for n > 7, the reverse is true for the largest possible
value of r l or r •
2
Table XIII presents the relative efficiencies of the estimate of the population
mean for the same distribution with different degrees of censoring.
IV.
Example
The data for this example are part of an experiment* in which ten rabbits were
inoculated with 0.2 ml of graded inoculum containing varying numbers of treponema
pa11idum.
Each rabbit received six injections from solutions containing 101 , 102 ,
103, 104, 105 and 106 spirochetes per ml, and was then observed for a period of 90
days to observe whether a syphilitic lesion developed at the site of injection.
*The authors should like to thank Dr. Harold J • .ilagnuson, Venereal Disease Experirr.ental Laboratory, United States Public Health Service, for permission to use these
data from Experiment #30.
- 12 ~
The incubation time required for a lesion to appear is an index of the amount
and potency of the inoculation as well as the susceptibility of the individual
rabbi t.
The distribution of incubation periods follows the two-parameter exponen-
tial distribution.
Knowl edge of the reac tion mechanism in rabbi ts has indicated tha t censoring
the observations at 90 days after inoculation is desirable because only an infinitesimal proportion of rabbits will have an incubation period beyond that.
In fact,
the present data will be considered censored at one-half that period, viz.
45
days.
Experience has also showed that with data of this type the reciprocal transformation
not only tends to stabilize the variance but also to make the relationships additive
(12) •
In other words, the harmonic mean is calculated as the measure of central
tendency.
Censoring from the right presents no problem under this transformation
since an inoculation site which does not develop into a lesion is considered to
~
represent an
infini te incubation period.
During the experiment, the rabbits are examined about twice a week for lesions.
Those lesions which develop in the interim between examinations are undetected until
the next period.
Lesions which are one or two days old at time of first observation
can be distinguished by their greater size.
The first examination is performed
approximately one week following inocula tion.
This resul ts in the fac t that certain
observations can be considered censored from the left if the size of the lesion is
large at the first examination period.
In Table XIV, the data from one portion of this experiment are presented.
is a slight deviation from the original data in one respect.
There
For illustrative pur-
poses of censoring from the left, it has been assumed that some of the rabbi ts had
lesions large enough at the time of first examination to presume that the true
incubation period ended a few days earlier.
For example, note tha t when 106 inoculum
was used, rabbits 7 and 9 were considered to have lesions large enough to presume
that the incubation was less than 7 days.
In Table XIV, the values of (mean)* for
- 13 each inoculum have been calculated using the coefficients w3i from Table VII(l).
For comparative purposes, two harmonic means have been indicated below the values
for (mean)*.
Harmonic mean A has been calculated under the assumption that there
was no censoring from the left.
For example, rabbits 7 and 9 were assigned .an
incuba tion period of exac:t,ly 7 days for the 106 inoculum.
The harmonic mean B has
been calculated under the assumption that there was censoring from the left so that
the incubation period for those rabbits at that dose was one day less.
One can see from the table that the harmonic means did quite respectably although
not as well as the exact estimate.
As expected, the harmonic means are slightly
lower than the arithmetic mean until that dose is reached for whiCh a fair proportion of animals do not develop any lesion.
By assigning those animals an infinite
incubation period, the result is to pull even the harmonic mean above the other
mean.
The reciprocal transformation is of little value) however, if the earliest
incubation day which is represented by }L~~ is to be estimated.
The values of
AA *
for each dose are given in the last row of the table using the coefficients wli
from Table V.
The relationship of
jJ-
*
to the mean value is evident and this throws
further light upon the speed of the reaction.
V.
Summary
Tables are provided to calculate the estimate of the mean () from singly and
doubly censored samples of sizes
distribution.
~
10 from the one-parameter single exponential
Si.7nilarly, tables are provided to estinate the minimum value (
fA- ),
the standard deviation ( cr ) and the mean value (}Jc -+ C) from the same samples drawn
from the two-parameter single exponential distribution.
'(l)The ~e of censoring practiced in this example was of Type I whereas the coefficients used to estimate the Farameters were based upon the assumption of Type II censoring. Other authors (Sampford in Biometrics} 1954) as well as sampling investi~a­
tions conducted by tile present authors, indicate that the possible bias caused by
this factor is negligible and of no practical import~
- 14 Interpretations are made for the variation of the coefficients and variances of
the estimates as the number of censored observations vary.
The following are two general conclusions which can be given concerning the
effect on the estimates of the side from which the sample is censored.
(1)
For the one-parameter single exponential distribution, the estimate of 0- obtain-
ed from singly censored samples from the left has a smaller variance than that based
upon singly censored samples from the right prOVided that the number of missing
observations is equal on both
(2)
sides~
For the two-parameter single exponential distribution we have
(a)
}J..
can be estimated more efficiently in samples censored from the right
than in samples censored from the left provided tha t the number of missing
observations is equal.
(b)
The estimate of () in singly and doubly censored samples from this
distribution does not depend upon the side from which censoring takes
place and depends only on the total number of missing observations.
e
e
e
,
TABLE I
Coefficients (wi) in the best linear estimate of (~) for the one parameter single exponential
distribution (2.9.1) from single and doubly censored samples.
(The coefficients in each row
must be divided by the common divisor given in the last column of each row.)
n
r
3
0
0
1
4
r
1
Y1
0
1
1
0
1
0
0
0
1
2
1
1
1
0
1 0
1 1
2 0
5
2
0
0
0
0
0
1
2
3
1 0
1 1
1 2
2 0
2 1
1
1
1
1
1
Y2
1
2
17
1
1
3
34
34
1
1
1
4
57
.57
57
Y
4
Y3
0
0
0
0
0
1
2
3
4
1
1
1
1
2
1
2
3
0
0
0
1
1
1
1
1
1
1
1
1
.5
86
86
86
86
Y
7
V
8
Y
9
Y10
Div.
1
3
13
38
2
1
2
1
2.5
50
95
25
4
3
2
99
14
230
61
1
1
3
1
2
1
41
4l
41
41
123
1282
1282
3 0
6
Y6
Y5
.5
4
3
2
204
163
122
3147
2978
7598
82
769
1538
2951
769
1669
1
1
1
4
1
1
3
61
61
61
244
813
61
61
183
61
122
61
469
469
469
1
2
1
6
5
4
3
2
365
304
243
182
2776
e
e
e
TABLE I (continued)
n r 1 r 2 Yl
2
2
3
3
4
1
2
0
1
0
7 0
0
0
0
1
2
3
4
5
0
0
0
8
1
1
1
1
1
2
2
2
2
3
3
3
4
4
5
1
2
3
4
0
1
2
3
0
1
2
0
1
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
0
Y'2
Y3
813
813
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
6
121
121
121
121
121
1
1
1
1
1
1
7
162
1
1
1
1
5
85
85
85
85
425
6914
6914
6914
6914
Y4
Y6
Y5
469
1407
1682
1682
3451
938
Y9
Y10
Div.
2307
1838
4987
4118
9338
869
1738
1769
869
1
1
3
1
2
1
85
85
85
340
85
85
255
85
170
85
3889
3889
3889
15556
54237
54237
54237
3889
3889
11667
3889
7778
3889
26581
26$81
79743
100LI8
100418
26581
53162
26581
46181
92362
190699
46181
1
1
1,
4
Y8
Y'7
7
6
5
4
3
2
594
509
424
339
254
27005
23116
19227
15338
181504
154923
128342
303043
256862
537842
90281
1
1
1
1
1
6
1
1
1
1
5
1
1
1
4
1
1
3
1
2
113
113
113
113
113
1
113
8
1
6
5
4
3
2
903
e
e
e
TABLE I (continued)
n
r 1 r2 Y'1
8 1
1
1
1
1
2
2
2
2
2
3
3
3
3
4
4
4
5
5
6
9 0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
2
3
4
5
0
1
2
3
4
0
1
2
3
0
1
2
0
1
0
0
1
2
3
4
5
6
1
0
1
2
3
4
5
6
Y'2
162
162
162
162
162
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
8
209
209
209
209
209
209
209
.
Y'~
Y'4
113
113
113
113
618
3259
3259
3259
3259
3259
1
1
1
1
1
1
1
145
145
145
145
145
145
1015
Y'6
Y'5
Y'8
Y'1
Y'9
113
113
113
565
113
113
452
113
339
226
1801
1801
1801
1801
9005
154124
154124
154124
154124
1801
1801
1801
1204
1801
1801
5403
1801
3602
1801
13249
13249
13249
292996
272073
272073
212073
13249
13249
219141
13249
146498
13249
117349
117349
352047
467822
467822
111349
234698
117349
195749
839971
839971
195749
1
1
3
1
2
1
1
1
1
1
6
1
1
1
1
5
1
1
1
4
145
145
145
145
145
870
145
145
145
145
125
145
145
145
580
Y'10
Div.
190
611
564
451
338
14334
12533
10732
8931
1130
511085
503836
430581
351338
904096
186767
669398
1438027
1242278
2454398
372149
1
9
8
1
6
145
145
435
145
290
145
5
4
3
2
1304
1159
1014
869
724
579
434
•
e
e
ThBLE I (continued)
n r 1 r 2 Y'r
2
2
2
2
2
3
3
3
3
3
4
4
4
4
5
5
5
6
6
7
0
1
2
3
4
5
0
1
2
3
4
0
1
2
3
0
1
2
0
1
0
10 0
0
0
0
0
0
0
0
0
1
1
1
1
1
0
1
2
3
4
5
6
7
8
0
1
2
3
4
2
1
1
1
1
1
1
1
1
1
Y2
1
1
1
1
1
1
1
1
9
262
262
262
262
262
Y3
Y4
Y5
Y6
Y7
1'8
1'9
22530
22530
22530
22530
22530
22530
12289
12289
12289
12289
12289
73734
41875
41875
41875
41875
41875
12289
12289
12289
12289
12289
12289
12289
49156
12289
12289
36867
12289
24578
12289
19345
19345
19345
19345
19345
19345
19345
77380
96725
1784516 737641
1784$16 7376~
1784516 737641
1784516 2950564
2919057
2919057
2919057
19345
19345
58035
19345
38690
19345
737641 737641
737641 1475292
2212923
737641
1
1
1
1
1
1
1
8
1
1
1
1
1
1
7
181
181
181
181
181
181
181
181
181
181
61445
1
1
1
1
1
6
181
181
181
181
181
1
1
1
1
5
1134541
1134541
3403823
4759198
4759198
1
1
1
4
Y10
110215
97926
85637
73348
61059
48770
172350
153005
133660
114315
94970
6481205
5743564
5005923
4268282
9698704
8564163
7429622
14896083
13055942
24670622
1134541 1134541
2269082
1840141 18401L!
3680282
8186939 3427741
1
1
3
1
2
Div.
1
10
9
8
7
6
5
181
181
181
181
905
181
181
181
724
181
181
543
181
362
181
4
3
2
1809
1628
1447
1266
1085
e
e
e
TABLE I (continued)
n
r 1 r2 1'1
10 1
1
1
2
2
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
5
5
5
6
6
6
7
7
8
5
6
7
0
1
2
3
4
5
6
0
1
2
4
5
0
1
2
3
4
0
1
2
3
0
1
2
0
1
0
1'2
1'3
262
262
262
181
181
1448
9113
9113
9113
9113
9113
9113
9113
1'4
1'5
181
1267
1086
1'6
1'7
4921
4921
4921
4921
4921
4921
4921
4921
4921
4921
4921
4921
4921
4921
19684
4921
4921
24605
492l
4921
29526
34447
811266 310729 370729 310729
811266 370729 370729 370129
811266 370729 370729 370729
817266 370729 1853645
817266 2224374
1364395 547129 547129
1364395 547129 547129
1364395 54n29 547129
1364395 547129 2188516
1364395 2735616
2165540 801145
2165540 801145
2165540 801145
2165540 3204580
3363585
3363585
3363585
1'8
4921
4921
14163
370729
370129
1112187
1'9
4921
9842
1'10
4921
370729 370729
741458
547129 547129 547129
547129 1094258
1641387
801145 801145 801145
801145 1602290
2403435
1198045
1198045
3594135
5267230
5267230
1198045 1198045
2396090
1903645 1903645
3807290
8758475 3491245
Div.
904
123
542
49088
44167
39246
34325
29404
24483
19562
3681223
3310494
2569036
2198307
1827578
5382774
4835645
4288516
3741387
3194258
7745741
6944596
6143451
5342306
11217256
10019211
8821166
16774491
14870846
27120566
TABLE II
Exact variances of the estimate of ( IT) for the one parameter single exponential
distribution from singly and doubly censored samples (in terms of cr 2 )
en
3
r2
°
1
4
5
25
99
'jIj."
2
61
230
°
1
2
3
4
7
8
e
5
1
1
2"
25
1
1
Ij."
3"
41
41
2QIj."
Ib3
41
122
"5
7ff)
7ff)
3747
2978
1
2"
16ff)
7598
1
1
b
"5
61
61
3Qli:
305
4ff)
4ff)
2776
2307
8ff)
~
1
Ij."
1
1
3"
2"
61
61
243
182
4ff)
•
~
8ff)
m:s
17ff)
9338
P
1
1
"7
"5
1
1
Ij."
3"
2"
1
85
5§Ij."
85
509
424
85
85
339
2
3889
27005
3889
23116
3889
19227
1§~
3
26581
181504
26581
154923
26581
128342
4
46181
303043
46181
2508tQ
5
90281
537842
°
°
6
13
1
3
4
3S
1
°
3
1
3"
°
2
2"
1
2
e6
1
3"
1
Ij."
1
6
°
r1
1
1
S
1
1
"7
1
1
b
"5
85
254
1
Ij."
1
1
3"
2"
7
8
TABLE II (continued)
Exact variances of the estimate of «(J") for the one parameter single exponential
distribution from singly and doubly censored samples (in terms of (7"2)
en
8
r
2
113
903
113
790
7fFf
5b4
45I
~
1801
1801
12533
1801
10732
1801
8931
1801
7130
3
73249
577085
73249
503836
73249
430587
73249
357338
4
117349
904096
~
5
195749
1438027
195749
1242278
6
372149
2454398
°
9
"S
'7
1
"6
1
145
1304
145
1159
145
1014
N9
2
12289
110215
12289
97926
"S5®
3
19345
172350
19345
153005
4
737641
6481205
5
1
2
9
9
10
•
10
2
1
rl
°
117349
113
4
113
5
6
7
1
1
2"
113
117349
bb9398
1
1
3
724
145
145
579
434
lli48
12289
12289
61059
12289
48770
ITIb60
19345
19345
114315
19345
94970
737641
5743564
737641
5005923
737641
4268282
1134541
96)8704
1134541
8564163
1134541
7429622
6
1840141
14896683
1840141
13055942
7
3427741
24670622
12289
145
"5
I2bb
181
1085
4921
39246
4921
34325
4921
29404
21+4'S3
370729
3310494
370729
2939765
370729
256)036
370729
2198307
370729
1827578
547129
5382774
547129
4835645
547129
4200516
547129
3741387
547129
3194258
5
801145
7745741
801145
6944596
801145
6143451
801145
5342306
6
1198045
11217256
1198045
10019211
1198045
8821166
7
1903645
16774491
1903645
14870846
8
3491245
27120566
9
"S
1
181
1809
181
TIi28
I447
2
49OO"B
4921
44Ib7
3
370729
3681223
4
4921
181
1
"5
1
"6
1
1
1
10
°
8
TIS
1
4
1
1
113
3
'7
181
1
145
1
4
3
1
954
181
181
723
542
4921
4921
19562
181
1
2"
e
e
•
j
Table III (continued)
Percentage efficiencies of the estimate of 6' in the one-parameter single exponential distribution
from singly and doubly censored samples.
The efficiencies are calculated relative to the best
linear estimate based on the complete sample.
r2
0
1
2
3
4
5
6
0
1
2
3
4
5
6
100.00
98.89
99.49
98.48
96.30
91.83
82.44
87,50
87·39
86.99
85·98
83·80
79·33
75·00
74.89
74.49
73.48
71.30
62·50
62·39
61·99
60·98
50.00
49.89
49.49
37·50
37 ·39
25,00
9
0
1
2
3
4
5
6
7
100.00
99·92
99.65
98·99
97·63
94.98
89·95
80.01
88.89
88.81
88·54
87·88
86·51
83·89
78.88
77.78
77·70
77043
76.77
75·41
72·77
66.67
66.59
66·32
65·66
64·31
55.56
55·48
55·21
54.55
44.44
44·37
44.10
33·33
33·25
22.22
10
0
1
2
3
4
5
6
7
8
100.00
99·94
99075
99-30
98·38
96.68
93·63
88.12
79·98
90.00
89·94
89·75
89·30
88·38
86.68
83.63
78.12
80.00
79.94
79075
79·30
78038
76.68
73·63
70.00
69.94
69·75
69·30
68·38
66.68
60.00
59.94
59·75
59·30
58·38
50.00
49094
49·75
49.30
40.00
39.94
39·75
30.00
29.94
n
r
8
l
7
8
20.00
•
e
e
Table III
Percentage efficiencies of the estimate of ~in the one-parameter single exponential distribution
from singly and doubly censored samples.
The efficiencies are calculated relative to the best
linear estimate based on the complete sample.
r2
--4
0
1
0
1
100.00
97·44
66.67
4
0
1
100.00
99·00
94.26
75·00
74.00
50.00
5
0
1
2
3
100.00
99·51
97·45
91.05
80.00
79·51
77.45
60.00
59·51
40.00
6
0
1
2
3
4
100.00
99':'7398.65
95.65
87·98
83·37
83·06
81·98
78·98
66.67
66.39
65·32
50.00
49·73
33·33
7
0
1
2
3
100.00
99·83
99·19
97·55
93·75
85.11
85·71
85·55
84·91
83·26
79·46
71.43
71.26
70.63
68·98
57.14
56·97
56·34
42.86
42.69
n
r
3
l
4
5
2
3
5
28.57
6
7
8
Table IV
Proportional reduction in expected waiting time to observe the
first in - r
2
failures in singly censored samples drawn from
the one-parameter single exponential distribution.
r
2
2
4
6
n
0
1
2
1
1
"3
3
1
5
11
2
11
4
1
13
25
7
25
3
25
5
1
77
137
47
137
27
137
12
137
6
1
87
147
57
147
37
147
22
147
10
147
7
1
669
1089
459
1089
319
1089
214
1089
130
1089
60
1089
8
1
1443
2283
1023
2283
743
2283
533
2283
365
2283
225
2283
105
2283
9
1
4609
7129
3349
7129
2509
7129
1879
7129
1375
7129
955
7129
595
7129
280
7129
10
1
4861
7381
3601
7381
2761
7381
2131
7381
1627
7381
1207
7381
847
7381
532
7381
3
5
7
8
9
252
7381
TABLE V
-
The exact coefficients (w1i ) in the best linear estimate of", for the two parameter single
exponential distribution from singly and doubly censored samples. (The coefficients in
each row must be divided by the common divisor given in the last column of each row.)
n
r
3
0
0
1
0
1
0
8
5
0
0
0
1
1
2
0
1
2
0
1
0
15
11
7
0
0
0
0
1
1
1
2
2
3
0
1
2
3
0
1
2
0
1
0
24
19
14
9
6
0
0
0
0
0
0
1
2
3
4
35
29
23
17
11
6
1
1
1
1
2
2
2
3
3
4
0
1
2
3
0
1
2
0
1
0
0
0
0
0
0
0
1
0
1
2
3
4
5
0
1
2
3
4
0
1
2
3
0
1
4
5
e
7
r
1
1
1
2
2
2
2
3
3
1
r
2
Yl
48
41
34
27
20
13
Y2
Y3
-1
-2
11
-1
-1
-1
-3
38
26
-1
-2
-1
-1
-1
-4
87
67
47
Y4
Y5
Y6
Y7
-7
Y9
YlO
Div.
6
3
6
-5
-14
25
Y8
-1
12
-7
4
24
12
12
8
-13
-1
-1
-3
-1
-2
-1
-9
-9
-27
214
154
-9
-18
-9
-47
-94
137
-47
20
15
10
5
60
40
20
120
60
60
-77
-1
-1
-1
-1
-5
-1
-1
-1
-4
-1
-1
-3
-1
-2
-1
30
24
18
12
6
164
134
104
74
-11
-11
-11
-33
-11
-11
-37
-37
-111
234
174
-37
-74
-37
-57
-114
147
-57
120
90
60
30
180
120
60
120
60
60
-1
-1
-1
-1
-1
-6
275
233
191
149
107
-11
-11
-44
291
231
171
-22
-87
-1
-1
-1
-1
-5
-1
-1
-1
-4
-1
-1
-3
-1
-2
-1
-13
-13
-13
-13
-65
1268
1058
848
638
-13
-13
-13
-52
-13
-13
-39
-13
-26
-13
-107
-107
-107
-428
2217
1797
-107
-107
-321
-107
-214
-107
-319
-319
-319
-638
-319
42
35
28
21
14
7
210
168
126
84
42
840
630
420
210
1260
840
e
•
e
TABLE V (continued)
n
r
7
3
4
4
5
2
0
1
0
0
0
0
1
2
3
4
5
6
0
1
8
1
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
3
3
3
3
4
4
4
5
5
6
9
0
0
0
r
Yl
2
Y2
Y3
Y4
Y5
1377
-957
1158
1338
or-
63
55
47
39
31
23
15
2
3
4
5
0
1
-1
-1
-1
-1
-1
-1
-7
426
370
314
258
202
146
2
3
4
0
1
2
3
0
1
2
0
1
-1
-1
-1
-1
-1
-6
-15
-15
-15
-15
-15
-90
1205
1037
e69
701
533
-1
-1
-1
-1
-5
-1
-1
-1
-4
2
80
71
62
-1
-1
-1
-1
-1
-1
' .Y 7
-459
-918
1089
-459
-1
-1
-3
-1
-2
Y8
Y9
Y10
Div.
420
840
420
420
-669
-1
56
48
40
32
24
16
-15
-15
-15
-15
-75
-15
-15
-15
-60
-15
-15
-45
-15
-30
-15
-73
-73
-73
-73
-365
5492
4652
3812
2972
-73
-73
-73
-292
-73
-73
-219
-73
-146
-73
-533
-533
-533
-2132
4749
3909
3069
-533
-533
-1599
-533
-1066
-533
-743
-743
-1023
-2046
2283
-1023
0
0
1
Y6
-1
-1
-1
-1
-1
-1
-743
-143
-2229
3126
2886
-1
-1
-1
-1486
-l443
-1
-1
-1
-2
-3
8
336
280
224
168
112
56
840
672
504
336
168
3360
2520
1680
840
2520
1680
E40
1680
840
840
-1
72
63
54
e
•
e
TABLE V (continued)
n
r
9
0
0
0
0
0
1
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
4
4
4
4
5
5
5
6
6
7
1
r
2
3
4
5
6
7
0
1
2
3
4
5
6
0
1
2
3
4
5
0
1
2
3
4
0
1
2
3
0
1
2
0
1
0
Y1
53
44
35
26
17
Y2
Y6
Y7
Y8
Y3
Y4
-1
-1
-1
-1
-1
-1
-1
-7
-8
623 -17
551 -17
479 -17
407 -17
335 -17
263 -17
191 -119
4170
3666
3162
2658
2154
1650
-1
-1
-6
-1
-5
-4
-17
-17
-17
-17
-17
-102
-17
-17
-17
-17
-85
-17
-17
-17
-68
-17
-17
-51
-17
-34
-17
-191
-191
-191
-191
-191
-1146
3895
3391
2887
2383
1879
-191
-191
-191
-191
-955
-191
-191
-191
-.164
-191
-191
-573
-191
-382
-191
-275
-275
-275
...275
-1375
17596
15076
12556
10036
-275
-275
-275
-1100
-275
-275
-825
-1879
-1879
-1879
-7516
15087
-1879
-1879
-5637
-1879 -1879
-3758
-2509
-2509 -2509
-5018
Y5
12567 -2509
10047 -7527
11738
9218
Y9
Y10
Div.
45
36
27
18
9
504
432
360
288
216
144
72
3024
2520
2~16
-275
-550
-275
-3349 -3349
-6698
7129 -4609
1512
1008
504
2520
2016
1512
1008
504
10080
7560
5040
2520
7560
5040
2520
5040
2520
2520
e
e
e
TABLE V (continued)
n
r
10
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
r
2
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
0
1
2
3
4
5
0
1
2
3
4
Y1
99
89
79
69
59
49
39
29
19
Y6
Y7
Y8
Y9
Y10
-1
-1
-1
-1
-1
-5
-1
-1
-1
-4
-1
-1
-3
-1
-2
-1
-1
-1
-1
-1
-6
-19
-19
-19
-19
-19
-19
-133
-19
-19
-19
-19
-19
-114
-19
-19
-19
-19
-95
-19
-19
-19
-76
-19
-19
-51
-19
-38
-19
-121
-121
-121
-121
-121
-121
-847
22362
19842
17322
14802
12282
9762
-121
-121
-121
-121
-121
-126
-121
-121
-121
-121
-605
-121
-121
.:.121
-484
-121
-121
-363
-121
-242
-121
-1207
-1207
-1207
-1207
-1207
-7242
20735
18215
15695
13175
10655
-1207
-1207
-1207
-1207
-6035
-1207
-1207
-1207
-4828
-1207
-1207
-3621
-1207
-2414
-1207
-1627
-1627
-1627
-1627
-8135
-1627
-1627
-1627
-6508
-1627
-1627
-4881
-1627
-3254
-1627
Y3
Y4
Y5
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-8
-9
872 -19
782 -19
692 -19
602
-19
512 -19
422 -19
332 -19
242 -152
3366
3007
2647
2287
1921
1567
1201
-1
-1
-1
-1
Y2
-1
-1
-1
Div.
90
80
70
60
50
40
30
20
10
720
630
540
450
360
270
180
90
2520
2160
1800
1440
1080
720
360
15120
12600
10080
7560
5040
2520
12600
10080
7560
5040
2520
e
e
TABIE
n
r
10
5
5
5
5
6
6
6
7
7
8
1
r
2
0
1
2
3
0
1
2
0
1
0
Y1
Y2
Y3
Y4
Y5
v
e
(continued)
Y6
Y7
Y8
18604
16084
13564
11044
-2131
-2131
-2131
-8524
15843
13323
10803
-2131
-2131
-6393
-2761
-2761
-[)283
12242
9722
Y10
Div.
-2131.
-4262
-2131
-2761
-5522
-2761
-3601
-7202
7381
-3601
10080
7$60
5040
2520
7560
5040
2520
5040
2520
2520
Y9
-4861
TABLE VI
The exact coefficients (w2i ) in the best linear estimate of the
standard deviation (0-) for the two parameter single exponential
distribution from singly and doubly censored samples.
(The coefficients in each row must be divided by the common
divisor given in the last column of each row.)
n
rl
r
3
0
0
1
0
1
0
-2
-2
0
0
0
1
1
0
1
2
0
1
0
-3
-3
-3
0
1
-4
-4
-4
-4
4
2
5
0
0
0
0
1
1
1
2
2
3
6
e
7
0
0
0
0
0
1
1
1
1
2
2
3
2
Yl
Y2
2
3
4
0
1
2
-5
-5
-5
-5
-5
3
2
0
1
2
2
3
3
4
0
1
0
0
0
0
0
1
2
Y7
Y8
Y9
YlO
Div.
1
1
1
1
1
1
1
3
2
2
1
3
1
1
1
4
-3
-3
-3
1
1
1
1
5
-4
-4
-4
-4
1
2
-1
1
2
1
1
1
1
1
2
1
1
1
1
4
3
2
1
3
1
1
1
2
3
-2
-2
2
1
1
1
4
1
1
1
4
-3
-3
-3
1
1
1
1
1
1
3
1
2
1
2
1
1
-1
1
1
1
1
2
5
1
4
3
3
2
1
1
1
1
2
4
3
1
2
1
3
1
1
3
-2
-2
-6
-6
-6
Y6
2
0
1
0
0
1
Y5
1
-2
2
Y4
1
2
-1
-2
0
1
Y3
1
1
1
1
2
1
3
1
2
-1
1
1
2
1
2
1
1
1
1
1
2
3
1
6
5
4
TABLE VI (continued)
The exact coefficients (w2i ) in the best linear estimate of the
standard deviation (rr) for the two parameter single exponential
distribution from singly and doubly censored samples.
(The coefficients in each row must be divided by the common
divisor given in the last column of each row.)
n
r
7
0
0
0
3
4
5
1
1
1
1
1
0
l
2
2
2
2
3
3
3
4
4
5
8
0
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
3
3
3
3
r
2
Yl
-6
-6
-6
1
2
3
4
Y2
Y3
Y4
1
1
1
4
6
-5
-5
-5
-5
-5
0
1
2
3
Y5
5
-4
-4
-4
-4
2
1
1
1
4
4
-3
-3
-3
1
2
3
4
5
6
0
1
2
3
4
5
0
1
2
3
4
0
1
2
3
7
-6
-6
-6
-6
-6
-6
YlO
Div.
3
2
1
1
1
1
1
1
1
6
1
1
5
4
3
2
2
3
1
1
1
1
1
1
1
4
3
2
1
2
3
1
1
1
3
-2
-2
0
1
1
1
1
1
1
Y9
1
1
1
1
0
-7
-7
-7
-7
-7
-7
-7
Y8
1
1
1
1
1
1
0
Y7
5
0
1
Y6
1
1
1
4
1
3
2
1
2
-1
1
1
1
1
1
2
1
2
1
1
2
1
1
3
5
7
6
5
4
3
2
1
1
1
1
1
1
6
-5
-5
-5
-5
-5
1
1
1
1
1
1
1
4
1
1
1
1
2
3
5
6
5
4
3
2
1
1
1
1
1
..:,
1
1
1
4
1
1
1
1
2
3
5
4
3
2
1
-4
-4
1
1
1
-4
4
1
1
3
1
2
1
4
3
2
1
TABLE VI (continued)
The exact coefficients (w2i) in the best linear estimate of the
standard deviation (~) for the two parameter single exponential
distribution from singly and doubly censored samples.
(The coefficients in each row must be divided by the common
divisor given in the last column of each row.)
n
r
8
4
0
4
4
1
2
5
5
6
0
9
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
4
4
4
4
5
5
5
1
r
2
Yl
Y2
Y3
Y4
Y5
-3
-3
-3
Y
Y8
1
1
1
2
1
3
1
2
-1
1
1
2
1
2
1
1
1
1
1
2
0
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
0
1
2
3
4
5
0
1
2
3
4
-8
-8
-8
-8
-8
-8
-8
-8
1
1
1
1
1
1
1
8
1
1
1
1
1
1
-7
-7
-7
-7
-7
-7
-7
1
1
1
1
1
1
1
1
1
1
1
6
1
1
1
1
5
7
-6
-6
-6
-6
-6
-6
1
1
1
1
1
1
1
1
1
5
1
1
1
4
1
1
1
1
5
1
1
1
4
1
1
1
1
1
4
1
1
-3
-3
-3
1
1
6
-5
-5
-5
-5
-5
-4
7
0
7
6
5
1
1
1
4
1
1
1
2
1
7
6
5
4
3
3
2
1
3
1
8
4
3
1
1
1
1
5
-4
-4
0
1
3
1
1
1
1
1
6
-4
6
6
1
1
1
4
YlO
2
1
1
2
1
2
Y9
7
0
0
7
3
-2
-2
1
Div.
Y6
1
1
6
1
2
5
4
3
3
1
2
1
3
3
1
2
1
3
2
1
4
3
2
1
3
-2
-2
2
1
5
4
1
2
1
3
1
2
-1
1
2
1
2
1
1
1
TABLE VI (continued)
The exact coefficients (w2i) in the best linear estimate of the
standard deviation (or) for the two parameter single exponential
distribution from singly and doubly censored samples.
(The coefficients in each row must be divided by the common
divisor given in the last column of each row.)
n
r
10
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
0
1
2
3
4
5
6
0
1
2
3
4
5
0
1
2
4
3
4
5
5
5
5
6
4
0
1
2
1
6
6
7
7
8
r
2
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
3
0
1
2
0
1
0
Yl
-9
-9
-9
-9
-9
-9
-9
-9
-9
Y2
Y3
Y4
Y5
Y6
Y
Y8
Y
9
YlO
1
1
1
1
1
1
1
1
9
1
1
1
1
1
1
1
8
1
1
1
1
1
1
1
1
1
1
1
6
1
1
1
1
5
1
1
1
4
1
1
3
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
6
1
1
1
1
5
1
1
1
4
1
1
3
1
2
1
1
1
1
1
6
1
1
1
1
5
1
1
1
4
1
1
3
1
2
1
1
1
1
1
6
-5
-5
-5
-5
-5
1
1
1
1
5
1
1
1
4
1
1
3
1
2
1
1
1
1
1
5
1
1
1
4
1
1
3
1
2
1
1
1
1
1
1
3
1
2
1
1
1
3
-2
-2
1
2
1
1
2
-1
1
-8
-8
-8
-8
-8
-8
-8
-8
8
-7
-7
-7
-7
-7
-7
-7
7
6
5
4
3
2
1
1
7
-4
-4
-4
-4
8
7
6
5
4
3
2
1
7
-6
-6
-6
-6
-6
-6
9
8
7
7
1
1
1
1
1
1
Div.
1
7
6
5
4
3
2
1
6
5
4
3
2
1
5
4
3
2
1
4
-3
-3
-3
1
4
3
2
1
3
2
1
2
1
1
TABLE VII
The exact coefficients (w 3i ) in the best linear estimate of the mean for the two parameter single
exponential distribution from singly and doubly censored samples. (The coefficients in each row
e
must be divided by the common divisor given in the last column of each row.)
n
r
3
0
0
1
0
1
0
2
-1
0
0
0
1
1
2
0
1
2
0
1
0
3
-1
-5
0
0
0
0
1
1
1
2
2
3
0
1
3
0
1
2
0
1
0
4
-1
-6
-11
0
0
0
0
0
1
1
1
1
2
2
2
3
3
4
0
1
2
3
4
0
1
2
3
0
1
2
0
1
0
5
-1
-7
-13
-19
0
0
0
0
0
0
1
1
1
1
1
2
2
2
2
3
3
3
0
1
2
3
4
5
0
1
2
3
4
0
1
2
3
0
1
2
6
-1
-8
-15
-22
-29
4
5
6
e
6
7
l
r
2
2
Yl
Y2
Y3
2
4
5
2
3
3
9
14
2
4
4
4
16
27
7
-13
5
5
5
5
25
44
14
-16
-46
Y4
Y6
Y7
Y8
Y9
YlO
Div.
6
3
6
1
3
6
3
5
10
13
5
12
8
4
24
12
12
-1
4
4
12
4
8
4
11
11
33
94
34
11
22
11
13
26
13
77
-17
5
5
5
20
5
5
15
5
10
5
19
19
19
76
111
51
-9
19
19
57
19
38
19
23
23
23
46
23
3
6
87
3
20
15
10
5
60
40
20
120
60
60
30
24
18
12
6
120
90
60
30
180
120
60
120
60
60
69
114
54
6
6
6
6
6
36
65
23
-19
-61
-103
Y5
-27
6
6
6
6
30
6
6
6
24
6
6
18
6
12
6
29
29
29
29
145
428
218
8
-202
29
29
29
116
29
29
87
29
58
29
103
103
103
412
957
537
117
103
103
309
103
206
103
101
101
303
101
202
101
42
35
28
21
14
7
210
168
126
84
42
840
630
420
210
1260
840
420
TABLE VII (continued)
e
The exact coefficients (w3i) in the best linear estimate of the mean for the two parameter single
exponential distribution from singly and doubly censored samples. (The coefficients in each row
must be divided by the common divisor given in the last column of each row.)
n
r
7
4
4
5
0
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
3
3
3
3
4
4
4
0
1
2
3
4
5
6
0
1
2
3
4
5
0
1
2
3
4
0
1
2
3
0
1
2
5
5
6
0
1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
0
1
2
3
4
5
6
7
0
8
e
8
9
1
r
1
2
3
4
5
6
0
1
2
3
4
5
0
1
2
3
4
2
Yl
Y2
Y3
Y4
Y5
918
498
7
-1
-9
-17
-25
-33
-41
8
-1
-10
-19
-28
-37
-46
-55
7
7
7
7
7
7
49
90
34
-22
-78
-134
-190
8
8
8
8
8
8
8
64
119
47
-25
-97
-169
-241
-313
Y6
-39
-78
669
Y
7
Y8
Y
9
Y10
Div.
840
420
420
-39
-249
7
7
7
7
7
42
7
7
7
7
35
7
7
7
28
7
7
21
7
14
7
41
41
41
41
41
246
365
197
29
-139
-307
41
41
41
41
205
41
41
41
164
41
41
123
41
41
95
95
95
95
475
2132
1292
452
-388
95
95
95
380
95
95
285
95
190
95
307
307
307
1228
2229
1389
549
307
307
921
307
614
307
97
97
291
97
194
97
2046
1206
-183
-366
1443
-183
56
48
40
32
24
16
8
336
280
224
168
112
56
840
672
504
336
168
3360
2520
1680
840
2520
1680
840
82
1680
840
840
-603
8
8
8
8
8
8
56
8
8
8
8
8
48
8
8
8
8
40
8
8
8
32
8
8
24
8
16
8
55
55
55
55
55
55
385
1146
642
138
-366
-870
-1374
55
55
55
55
55
330
55
55
55
55
275
55
55
55
220
55
55
165
55
110
55
313
313
313
313
313
1878
1375
871
367
-137
-641
313
313
313
313
1565
313
313
313
1252
313
313
939
313
626
313
229
229
229
229
1145
229
229
229
916
229
229
687
229
458
229
72
63
54
45
36
27
18
9
504
432
360
288
216
144
72
3024
2520
2016
1512
1008
504
2520
2016
1512
1008
504
TABLE VII (continued)
e
The exact coefficients (w3i ) in the best linear estimate of the mean for the two parameter single
exponential distribution from singly and doubly censored samples. (The coefficients in each row
must be divided by the common divisor given in the last column of each row. )
n
9
10
e
10
rl
r2
4
4
4
4
5
5
5
6
6
7
0
1
2
3
0
1
2
0
1
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
2
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
5
5
5
5
6
6
6
7
7
8
0
1
2
3
4
5
6
0
1
2
3
4
5
0
1
2
3
4
0
1
2
3
0
1
2
0
1
0
Yl
Y2
Y3
Y4
Y5
7516
4996
2476
-44
9
-1
-11
-21
-31
-41
-51
-61
-71
9
9
9
9
9
9
9
9
81
152
62
-28
-118
-208
-298
-388
-478
Y6
641
641
641
2564
7527
5007
2487
Y7
Y8
Y9
641
641
1923
641
1282
641
11
11
33
6698
4178
11
22
11
-829
-1658
4609
-829
9
9
9
9
9
9
9
72
9
9
9
9
9
9
63
9
9
9
9
9
54
9
9
9
9
45
71
71
71
71
71
71
71
568
71
71
71
71
71
71
497
71
71
71
71
71
426
71
71
71
71
355
71
284
847
487
127
-233
-593
-953
-1313
239
239
239
239
239
239
1673
7242
4722
2202
-318
-2838
-5358
239
239
239
239
239
1434
239
239
239
239
1195
1313
1313
1313
1313
1313
7878
8135
5615
3095
575
-1945
YlO
10080
7560
5040
2520
7560
5040
2520
5040
2520
2520
-2089
9
9
9
36
9
9
27
9
18
9
71
71
71
213
71
142
71
239
239
239
956
239
239
717
239
478
239
1313
1313
1313
1313
6565
1313
1313
1313
5252
1313
1313
3939
1313
2626
1313
893
893
893
893
4465
8524
6004
3484
964
893
893
893
3572
893
893
2679
893
1786
893
389
389
389
1556
8283
5763
3243
389
389
1167
389
778
389
-241
-241
-723
7202
4682
-241
-482
-241
7~
Div.
-1081 -1081
-2162
4861 -2341
90
80
70
60
50
40
30
20
10
720
630
540
450
360
270
180
90
2520
2160
1800
1440
1080
720
360
15120
12600
10080
7560
5040
2520
12600
10080
7560
5040
2520
10080
7560
5040
2520
7560
5040
2520
5040
2520
2520
e
e
e
Table VIII
Variances of the best linear estimate of){ for the two-parameter single exponential
distribution from singly and doubly censored samples (in terms of
a-
2
)
r2
1
4
n
rl
3
0
1
.1666667 .2222222
1.0555556
4
0
1
2
.0833333
.3437500
1·5972222
.0937500
.5138889
.1250000
5
0
1
2
3
.0500000
.1700000
.5204167
2.1105556
.0533333
.2037500
.8411111
.0600000
·3050000
.0800000
6
0
1
2
3
4
.0333333 .0347222
.1013889 .1125926
.2570370 .3204167
.6926389 1.1161111
2·5938889
.0370370
.1350000
.5105556
.0416667
.2022222
.0555556
7
0
1
2
3
4
5
.0238095 .0244898
.0673469 .0721372
.1530896 .1747241
.2425208 .4391241
.7280669 1.4561338
3.0489909
.0255102
.0801209
.2179932
.7275624
.0272109
.0408163
·3 78005
.0306122
.1439909
0
1
2
3
4
5
6
.0178571 .0182292
.0479911 .0503827
.1015731 .1110137
.2044661 .2380178
.4271051 ·5575021
.9565101 1.7605981
3·4159552
.0187500
.0539700
,1267479
·3051212
.7391762
.0195313
.0599490
.1582164
.5064314
.0208333
.0719069
.2526219
.0234375
.1077806
8
0
2
3
.0~60884
5
6
.0312500
7
8
e
e
e
Table VIII (continued)
Variances of the best linear estimate of A for the two parameter single exponential
distribution from singly and doubly censored samples (in terms of 6'
r
0
1
2
3
2)
2
4
5
6
7
8
n
r
9
0
1
2
3
4
5
6
7
.0138889
.0359347
.0723150
.1357001
.2551495
.5090863
1.1728460
3.8848926
.0141093
.0144033 .0148148 .0154321
.0372621 .0391204 .0419078 .0465535
.0771022 .0842831 .0962512 .1201873
.1505860 .1753958 .2250154 .3738741
·3014804 .3941423 .6721281
.6743011 1.1699455
2.0559244
10
0
1
2
3
4
5
6
7
8
.0111111
.0279167
.0541093
.0966139
.1695255
.3049312
·5887952
1.3207429
4.2706863
.0112500 .0114286 .0116667 .0120000 .0125000 .0133333 .0150000 .0200000
.0287125
.0297737 .0312593 .0334877 .0372016 .0446296 .0669136
.0567991 .0605648 .0662133 .0756276 .0944560 .1509414
.1042610 .1157315 .1348490 .1730841 .2877894
.1903677 .2251048 .2945788 ·5030011
.3645228 .4837058 .8412550
·7888646 1·3890725
2.3417180
1
.0164609 .0185185 .0246914
.0558449 .0837191
.1919958
e
e
e
Table IX
Percentage efficiencies of}U* relative to the best linear estimate for the two parameter single
exponential distribution from singly and doubly censored samples
I:
2
0
1
0
1
100.00
15·79
75·00
4
0
1
2
100,00
24.24
5·22
88.89
16.22
66.67
5
0
1
2
3
100.00
29.41
9·61
2·37
93·75
24.54
5·94
83.33
16·39
62.50
6
0
1
2
3
4
100.00
32.88
12·97
4.81
1.29
96.00
29.61
9.61
2·99
90.00
24.69
6.53
80.00
16.48
60.00
7
0
1
2
3
4
5
100.00
35.35
15,55
6.84
3,27
,78
97·22
33·01
13.63
5,42
1.64
93·33
29·72
10·92
3·27
87.50
24.78
6.85
77.78
16·54
58·33
8
0
1
2
3
100,00
37,21
17·58
8,73
97,96
35.44
16.09
5,50
95.24
33·09
14.09
5·35
91.43
29,79
11.29
3·53
85·71
24.83
7·07
76.19
16.57
n
r
3
1
2
3
4
5
6
57.14
7
8
•
e
e
Table IX (continued)
Percentage efficiencies
of~*
relative to the best linear estimate for the two parameter single
exponential distribution from singly and doubly censored samples
0
1
2
4
5
6
4.18
1.87
·52
3·20
1.01
1.88
0
1
2
3
4
5
6
7
100.00
38.65
19·20
20.60
5.44
2·73
1.18
.36
98.44
37·27
18.01
9.22
4.61
2.06
96.43
35·50
16.89
7·92
3·54
1.19
0
1
2
3
4
5
6
7
8
100.00
39.80
20·53
11.50
6.55
3.64
1.89
.84
.26
98.77
38.70
19.56
10.66
5.84
3·05
1.41
.47
97·22
37·32
18·35.
9·60
4.94
2·30
.80
n
r
8
9
10
1
.68
r2
--
4
5
6
7
93·75
33·31
14.43
6.17
2.07
90.00
29.83
11.56
3·71
84.38
24.87
7·23
75·00
16.59
56.25
95·24
35·55
16·78
8.24
3·77
1.32
92·59
33·18
14.69
6.42
2.21
88.89
29.87
11·77
3·10
83·33
24.90
7·36
74.07
16.61
3
8
55·56
Table X
Exact variances of the estimate of standard deviation «(I) in the
two parameter,single exponential distribution from singly
and doubly censored samples (in terms of cr 2)
r2
n
r1
0
1
.3
0
1
0
1
2
0
1
2
.3
0
1
2
.3
4
0
1
2
.3
4
5
0
1
2
3
4
5
6
1/2
1
1/.3
1/2
1
1/4
1/.3
1/2
1
1/5
1/4
1/.3
1/2
1
1/6
1/5
1/4
1/.3
1/2
1
1/7
1/6
1/5
1/4
1/.3
1/2
1
1/8
1/1
1/6
1/5
1/4
1/3
1/2
1
1/9
1/8
1/7
1/6
1/5
1/4
1/3
1/2
1
1
4
5
6
1
8
1\
9
10
0
1
2
.3
4
5
6
7
0
1
2
3
4
5
6
1
8
2
.3
4
5
6
7
1/2
1
1
1/.3
1/2
1
1/2
1
1
1/4
1/.3
1/2
1
1/.3
1/2
1
1/2
1
1
1/5
1/4
1/.3
1/2
1
1/4
1/.3
1/2
1
1/.3
1/2
1
1/2
1
1
1/6
1/5
1/4
1/3
1/2
1
1/5
1/4
1/3
1/2
1
1/4
1/.3
1/2
1
1/.3
1/2
1
1/2
1/1
1/6
1/5
1/4
1/.3
1/2
1
1/6
1/5
1/4
1/3
1/2
1
1/5
1/4
1/3
1/2
1
1/4
1/3
1/2
1
1/.3
1/2
1
1/2
1
1
1/8
1/1
1/6
1/5
1/4
1/3
1/2
1
1/1
1/6
1/5
1/6
1/5
1/4
1/3
1/2
1
1/5
1/4
1/3
1/4
1/3
1/2
1
1/3
1/2
1
1/2
1
1/1~
1/3
1/2
1
1/2
1
'I
8
1
1
e
e
e
Table XI
Percentage efficiencies of the estimate of (J in the two-parameter single exponential
The efficiencies are calculated
distribution from singly and doubly censored samples.
relative to the best linear estimate based on the complete sample.
r
2
n
r1
0
1
3
0
1
100.00
50.00
50.00
4
0
1
2
100.00
66.67
33·33
66.67
33·33
33·33
5
0
1
2
3
100000
75·00
50.00
25·00
75·00
50.00
25·00
50.00
25·00
25·00
6
0
1
2
3
4
100.00
80.00
60.00
40.00
20.00
80.00
60.00
40.00
20.00
60.00
40.00
20.00
40.00
20.00
20.00
7
0
1
83·33
66.67
50.00
33·33
16.67 ...
66.67
50.00
33·33
16.67
50.00
33·33
16.67
33·33
16.67
16.67
3
4
5
100.00
83·33
66.67
50.00
33·33
16.67
0
1
2
100.00
85.71
71.43
85·71
71.43
57.14
71.43
57·14
42.86
57·14
42.86
28·57
42.86
28.57
14.29
28.57
14.29
2
8
2
3
4
5
6
14.29
7
8
e
e
e
Table XI (continued)
Percentage efficiencies of the estimate of 0- in the two-parameter single exponential
distribution from singly and doubly censored samples.
The efficiencies are calculated
relative to the best linear estimate based on the complete sample
r2
n
rl
0
1
2
3
8
3
4
5
6
57.14
42.86
28·57
14.29
42.86
28·57
14.29
28·57
14.29
14.29
9
0
1
2
3
4
5
6
7
100.00
87·50
75·00
62.50
50.00
37 ·50
25·00
12·50
87·50
75.00
62.50
50.00
37·50
25·00
12·50
75·00
62.50
50.00
37·50
25·00
12·50
10
0
1
2
3
4
5
6
7
8
100.00
88.89
77·78
66.67
55.56
44.44
33·33
22.22
11.11
88.89
77·78
66.67
55.56
44.44
33·33
22.22
11.11
77·78
66.67
55.56
44.44
33·33
22.22
11.11
4
5
6
7
62.50
50.00
37·50
25·00
12·50
50.00
37·50
25·00
12·50
37 ·50
25.00
12·50
25·00
12·50
12·50
66.67
55.56
44.44
33·33
22.22
11.11
55·56
44.44
33·33
22.22
11.11
44.44
33·33
22.22
11.11
33·33
22.22
11.11
22.22
11.11
8
11.11
e
e
e
Table XII
Variances of the estimate of the mean for the two-parameter single exponential
distribution from singly and doubly censored samples (in terms ofo-
2
)
r2
4
0
1
0
1
.3333333
.3888889
.5555556
4
0
1
2
.2500000 ·3437500
.2604167 .3472222
.4305556
.6250000
5
0
1
2
3
.2000000 .2533333
.2033333 .2537500
.2370833 .2605556
.5438889
.3600000 .6800000
.4050000
6
0
1
2
3
4
.1666667
.1680556
.1792593
.2426389
.6938889
.2013889
.2014815
.2037500
.2438889
.2592593
.2683333
.2772222
7
0
1
2
3
4
5
01428571
.1435374
.1483277
.1699622
.2661083
.8632766
.1673469
.1673753
.1683749
.1796003
.2704195
.2040816 .2653061 ·3877551
.2071051 .2865646 ·5249433
.2084694 ·3287528
.2085147
·7551020
8
0
1
2
3
.1250000
.1253720
.1277636
.1372042
01432292
.1432398
.1437518
.1483352
.1687500 .2070313 .2708333
,1700415 .2147109 .3040497
.1703987 .2236926 ·3835743
.1705974 02373838
·3984375
·5720663
n
r
3
1
2
3
5
6
·3750000 ·7222222
.4688889
·7812500
7
8
e
e
e
Table XII (continued)
Variances of the estimate of the mean for the two-parameter single exponential
distribution from singly and doubly censored samples (in terms of 0- 2 )
r2
n
r
-
1
0
1
2
3
4
5
6
7
8
4
5
6
.1707559
·3011529
1.0427409
.1729783
.3248838
.1796457
9
0
1
2
3
4
.1252205
.1252251
.1255149
.1277685
.1377238
.1786661·3979878
.1440329
.1446759
.1447989
.1449725
.1485074
.1786757
.1703704
.1738522
.1769390
.1793804
.1808582
.2098765
.2224794
.2405089
.2826042
.2757202
·3197338
.4340593
.4074074 .8024691
.6114969
7
.1111111
.1113316
.1126589
.1174461
.1323320
.1786630
.3438778
1.2269561
0
1
2
3
4
5
6
7
8
.1000000 .1112500
.1001389 .1112522
.1009347 .1114288
.1036245 .1126737
.1112715 .1175503
.1321138 .1340995
01939634 .1956169
·3954846 .4892016
1.4221381
.1257143
.1260700
.1261204
.1262474
.1280148
.1380709
.2005773
.1450000
.1468148
.1481578
.1488702
.1489439
.1499852
.1720000
.1779321
.1848868
.1941159
.2019504
.2125000
.2297942
.2583449
·3298529
.2800000 .4150000
.3335185 .6446914
.4787191
~
10
8
.8200000
•
e
e
Table XIII
Percentage efficiencies of the estimate of the mean for the two-parameter single exponential
distribution from singly and doubly censored samples. The efficiencies are calculated
relative to the best linear estimate based on the complete sample.
-
r2
n
r1
0
1
3
0
1
100.00
85.71
60.00
4
0
1
2
100.00
96.00
58.07
72.72
40.00
5
0
1
2
3
100.00
96.36
84.36
36.77
78.95
78.81
76.76
55.55
49.38
29.41
6
0
1
2
3
4
100.00
99.17
92.98
68.69
24.02
82.76
82.72
81.80
68.34
64.29
62.11
60.12
44.44
23.08
35.55
7
0
1
2
3
4
5
100.00
99.53
96.31
84.05
53.68
16.55
85.37
85.35
84.84
79.54
52.83
70.00
68.98
68.53
68.51
53.85
49.85
43.45
36.84
27.21
18.92
8
0
1
2
3
100.00
99.10
91.84
91.11
87.27
87.21
86.96
84.27
74.01
13.51
73.36
73.27
60.38
58.22
55.88
52.66
46.15
41.11
32.59
31.37
21.85
72.00
2
3
4
5
6
16.00
7
8
e
e
e
Table XIII (continued)
r2
n
r1
-_.
0
8
4
5
6
73.20
41.51
11.98
0/
0
1
2
3
4
5
6
7
100.00
99.80
98.63
94.61
83.«)6
62.20
32.31
9.06
0
100.00
99.86
99.07
96.50
:0
1
2
3
4
5
6
7
8
8~.87
'!S.69
.Sl.56
25.29
7.03
. ...
1
2
72.26
38.47
69.58
88.73
77.14
76.80
3
4
5
6
7
65.22
63e91
62.79
61.94
61.44
52.94
49.94
46.06
39.32
40.30
34.75
25.60
27.27
16.17
13.85
68.97
68.11
67.49
67 ..17
66.. 59
66.70
58.14
56.20
54.09
51.52
49.52
47.06
43e52
38.71
30.32
35.71
29.98
20.8)
24.00
15.51
8
_~
88.72
88.52
86,,96
80.68
62.19
27.92
76.64
74082
62.18
89.89
89.89
89.74
88,75
85.07
74.57
51.,12
20.44
79.55
79.32
79.29
79.21
78.12
72.43
49.86
76~73
12.20
'l'able XIV
Days of incubation among ten rabbits following inoculation with
graded amounts of treponema pa11idum
Rabbi t number
.106
lOS
Inoculum
104
103
102
lor
7
<7
<11
18
<18
<25
>45
8
11
11
18
18
40
>45
9
<7
(11
14
>45
>45
)45
10
7
11
18
<18
<25
<25
11
11
14
18
25
25
25
12
14
14
18
21
29
25
13
7
11
18
18
29
32
14
>45
35
40
25
>45
>45
15
7
14
18
25
29
40
16
11
14
18
21
29
>45
19.80
21.16
33.10
50.51
(mean)*
9.88
Harmonic mean A
9.45
13.01
18.49
22.86
35.34
56.74
Harmonic mean B
9.04
12.71
18.49
22.52
34.93
56.21
5.54
9.27
13.36
13.80
20.90
18.17
e
e
e
FI 9 u re
EFFICIENCY PER UNIT OF
WITH DIFFERENT VALUES
SAMPLES
I.
WAITING TIME FOR
OF r2 IN SINGLY
n = 10,9 AND 8
CENSORED
FROM
THE
RIGHT OF
THE
ONE-PARAMETER
EXPONENTIAL
DISTRIBUTION.
13
I
I
........ 7c,: '.
12..5
>"".
I
12
Efficie ncy
per
uni t
wQltln~
I
'i~.
KEV
.!!.
10
9
8
9
---~
..........
1
8
7
5
6
r
2
4
3
2
11.5
of
time
References
(1)
Cohen, A. C.
"Estimating the Mean and Variance of Normal Populations
from Singly Truncated and Doubly Truncated Samples"
All..nals-2.! Mathematical Stati~, XXI (1950) pp. 557-569.
(2)
Davis, D. J.
"An Analysis of Some Failure Data", Journal of the American
Statistical Association, XLVII (1952) pp. 113-150.
(3)
Epstein, Benjamin "Statistical Problems in Life Testing Proceedings
of the Seventh Annual Convention of the American Society for
Quality Control, (1953) pp. 385-398~
(4)
Epstein, Benjamin and Sobel, lvI.
_._-
"Life Testing", Journa.l of the American
Stati~~U~£!:!.tion, LXVIII
(1953T;'·pp. 486.·502.
(5)
Gupta, A. K.
"Estimation of the Mean and Standard Deviation of a Normal
Population from a Censored Sample" Biometrika, XXXIX
(1952) pp. 260-273.
(6)
Hald, A.
"Maximum Likelihood Estimation of the Parameters of a Normal
Distribution Wl:.i'Jh is Truncated at a Known Point" Skand.
ill:;-t~r. Tid (1949)
(7)
Halperin, Max. ':Ivhximum Likelihood Estimation in Truncated Samples"
Anr.~s of ~mthematical Statistics, XXIII (1952) pp. 226-238~
(8)
Ipsen, J., Jr.
"A Practical Method of Estimating the Mean anj Standard
De~dQtion
Vol~
(9)
of Trw"'lcated Normal Distributions.
Hi.lJl1an Biology
21 (1949) pp. 1-16.
-
Mosteller, Frederick
"On Some Useful Inefficient Statistics" Annals of
~~at~lematical Statistics, XVII (1946) ppo 377-407-;--
(10) Sarhan, A. E. "Es timation of the IJean and Standard Devia tion by Order
S tc. tis ti cs, Part HIe
g I' 7 "
,
. . Annals
of Ma thelllD.t:.~al S'~,atil3?.i£!!,(Dec. 1955J J,{j. J.(,,)/,. '1/ fro S''}{p-.r"T
(11) Sarhan, A. E. and Greenberg, B. G. "Estimation of Location and Scale
Parameters by Order Statistics from Singly and Dou'bly Censored
Samples", Part I., The Normal Distribution Up to Samples of
Size 10. To be published in the ~Enals of Mathemati~al
Statistics.
(12) Turner, T. B., Kluth, F. C., i\llcLeod, C. and l?insnr, Co P. "Protective
Antibodies in the Serum of Syphilitic Patients"
Journal of Hygiene, 48: 173, 1948"
American