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Hybrid Censoring Know-How
Designs and Implementations
First edition
N. Balakrishnan
McMaster University, Department of Mathematics and Statistics, Hamilton,
ON, Canada
Erhard Cramer
RWTH Aachen, Institute of Statistics, Aachen, Germany
Debasis Kundu
Indian Institute of Technology Kanpur, Department of Mathematics and Statis-
tics, Kanpur, India
2

Table of Contents
Cover image
Title page
Copyright
Dedication
Preface
Chapter 1: Introduction
Abstract
1.1. Historical perspectives
1.2. Type-I and Type-II censoring
1.3. Need for hybrid censoring
1.4. Antecedents
1.5. Burgeoning literature
1.6. Scope of the book
1.7. Notation
Bibliography
Chapter 2: Preliminaries
Abstract
3

2.1. Introduction
2.2. Order statistics
2.3. Progressively Type-II right censored order statistics
2.4. Generalized order statistics
2.5. Sequential order statistics
Bibliography
Chapter 3: Inference for Type-II, Type-I, and progressive censoring
Abstract
3.1. Introduction
3.2. Type-II censoring
3.3. Type-I censoring
3.4. Progressive Type-II censoring
3.5. Progressive Type-I censoring
Bibliography
Chapter 4: Models and distributional properties of hybrid censoring designs
Abstract
4.1. Introduction
4

4.2. Preliminaries
4.3. Type-I hybrid censoring
4.4. Type-II hybrid censoring
4.5. Further hybrid censoring schemes
4.6. Joint (hybrid) censoring
Bibliography
Chapter 5: Inference for exponentially distributed lifetimes
Abstract
5.1. Introduction
5.2. General expression for the likelihood function
5.3. Type-I hybrid censoring
5.4. Type-II hybrid censoring
5.5. Further hybrid censoring schemes
Bibliography
Chapter 6: Inference for other lifetime distributions
Abstract
6.1. Introduction
5

6.2. Weibull distributions
6.3. Further distributions
Bibliography
Chapter 7: Progressive hybrid censored data
Abstract
7.1. Progressive (hybrid) censoring schemes
7.2. Exponential case: MLEs and its distribution
7.3. Progressive hybrid censored data: other cases
Bibliography
Chapter 8: Information measures
Abstract
8.1. Introduction
8.2. Fisher information
8.3. Entropy
8.4. Kullback-Leibler information
8.5. Pitman closeness
Bibliography
6

Chapter 9: Step-stress testing
Abstract
9.1. Introduction
9.2. Step-stress models under censoring
9.3. Step stress models under hybrid censoring
Bibliography
Chapter 10: Applications in reliability
Abstract
10.1. Introduction
10.2. Competing risks analysis
10.3. Stress-strength models
10.4. Optimal designs
10.5. Reliability acceptance sampling plans
Bibliography
Chapter 11: Goodness-of-fit tests
Abstract
11.1. Introduction
7

11.2. Progressive censoring
11.3. Hybrid censoring
Bibliography
Chapter 12: Prediction methods
Abstract
12.1. Introduction
12.2. Point prediction
12.3. Interval prediction
Bibliography
Chapter 13: Adaptive progressive hybrid censoring
Abstract
13.1. Introduction
13.2. Adaptive Type-II progressive hybrid censoring
13.3. Inference for adaptive Type-II progressive hybrid censored data
13.4. Adaptive Type-I progressive hybrid censoring
Bibliography
Appendix
8

A. Geometrical objects
B. Distributions
C. B-splines and divided differences
Bibliography
Bibliography
Bibliography
Index
9

Copyright
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10

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11

Dedication
In the loving memory of my mother
N. B.
Für Katharina, Anna, Jakob & Johannes
E. C.
Dedicated to my mother
D. K.
12

Preface
N. Balakrishnan ; Erhard Cramer ; Debasis Kundu McMaster
University, Hamilton, ON, Canada
RWTH, Aachen, Germany
IIT, Kanpur, India
Though the notion of hybrid censoring was introduced in 1950s, very little
work had been done on it until 2000. During the last two decades, however,
the literature on hybrid censoring has exploded, with many new censoring
plans having been introduced and several inferential methods having been
developed for numerous lifetime distributions. A concise synthesis of all
these developments were provided in the Discussion paper by N. Balakr-
ishnan and D. Kundu in 2013 (Computational Statistics & Data Analysis, Vol.
57, pages 166–209). Since the publication of this Discussion paper in 2013,
more than 300 papers have been published in this topic. The current book
is an expanded version of the Discussion paper covering all different as-
pects of hybrid censoring, especially focusing on recent results and devel-
opments.
As research in this area is still intensive, with many papers being pub-
lished every year (see Chapter 1 for pertinent details), we have tried our best
to make the bibliography as complete and up-to-date as possible. We have
also indicated a number of unresolved issues and problems that remain
open, and these should be of interest to any researcher who wishes to en-
gage in this interesting and active area of research.
Our sincere thanks go to all our families who provided constant support
and great engagement through out the course of this project. Thanks are
also due to Ms. Susan Ikeda (Senior Editorial Project Manager, Elsevier) for
her keen interest in the book project and also in its progress, Ms. Debbie
Iscoe (McMaster University, Canada) for her help with typesetting some
parts of the manuscript, and Dr. Faisal Khamis for his diligent work on liter-
ature survey in the beginning stage of the project. Without all their help and
cooperation, this book would not have been completed!
We enjoyed working on this project and in bringing together all the re-
sults and developments on hybrid censoring methodology. It is our sincere
hope that the readers of this book will find it to be a useful resource and
a b c a
b
c
13

guide while doing their research!
N. Balakrishnan (McMaster University, Hamilton, ON, Canada)
Erhard Cramer (RWTH, Aachen, Germany)
Debasis Kundu (IIT, Kanpur, India)
14

Abstract
In this chapter, some historical comments are made about the devel-
opments on conventional censoring schemes, viz., Type-I and Type-II
censoring schemes. Then, formal definitions and some basic prop-
erties are provided for data obtained from both these censoring
schemes, as well as a comparison of their expected termination times.
A detailed review of the historical literature on hybrid censoring
schemes is then provided, explaining various inferential results devel-
oped under these censoring schemes. Finally, the extensive literature
that has developed on hybrid censoring, especially in the last two
decades, and the need for a study of hybrid censoring schemes are
both described.
Keywords
Type-I censoring; Type-II censoring; Order statistics; Expected termination
time; Hybrid censoring schemes
Chapter Outline
1.1 Historical perspectives
1.2 Type-I and Type-II censoring
1.3 Need for hybrid censoring
1.4 Antecedents
1.5 Burgeoning literature
1.6 Scope of the book
1.7 Notation
16

1.1 Historical perspectives
Reliability of many manufactured items has increased substantially over
time due to the ever-growing demands of the customers, heavy competition
from numerous producers across the world, and strict requirements on
quality assurance, as stipulated by ISO9000, for example. As a result, pro-
viding information on the reliability of items (such as mean lifetime, median
lifetime, percentile of the lifetime distribution, and so on), within a rea-
sonable period of time, becomes very difficult under a traditional life-test.
So, because of cost and time considerations, life-tests in these situations
necessarily become censored life-testing experiments (often, with heavy
censoring). The development of accurate inferential methods based on data
obtained from such censored life-tests becomes a challenging task to say
the least. This issue, therefore, has attracted the attention of numerous re-
searchers over the last several decades!
Early scenarios requiring the consideration of censoring seemed to have
originated in the context of survival data. For example, Boag (1949) dis-
cussed the estimation of proportion of cancer patients in UK who were
cured following a treatment, and while doing so, did consider censoring
corresponding to all those cancer patients who were alive at the end of the
clinical trial; one may also refer to Harris et al. (1950), Berkson and Gage
(1952) and Littel (1952) for further detailed discussions in this regard. Note
that the form of censoring considered in this context is time-censoring,
meaning the trial ended at a certain pre-fixed time; consequently, individuals
who were alive at the end of the trial ended up getting censored at this spe-
cific termination time. However, according to David (1995), the word cen-
soring explicitly appeared for the first time in the works of Hald (1949) and
Gupta (1952). In fact, Hald (1949) made a clear distinction between trun-
cation and censoring, depending on whether the population from which the
sample is drawn is truncated or the sample itself is truncated, respectively.
Yet, in many of the early works, the word truncation got used in place of
censoring; see, for example, Cohen (1949, 1950). It was Gupta (1952) who
pointed out that censoring could arise in two different ways, one when the
observation gets terminated at a pre-fixed time (with all observations up till
that time point being observed and all those after being censored as in the
study of Boag (1949) mentioned above) and another when the observation
gets terminated when a certain number of smaller observations is achieved
17

(with the remaining larger observations being censored). Then, for the pur-
pose of distinguishing between these two different cases, he referred to
them as Type-I censoring and Type-II censoring, respectively.
What is clear from the descriptions and explanations of Hald (1949) and
Gupta (1952) is that truncation is a feature of the population distribution
while censoring is inherently a feature of the sample observed. In spite of
this clear distinction, still today, some authors mistakenly refer to “censored
distributions” and “truncated samples”.
Following the work of Gupta (1952), Epstein and Sobel (1953, 1954) pub-
lished pioneering results on censored data analysis from a life-testing and
reliability viewpoint, basing their results on exponential lifetime distribution.
They utilized some interesting distributional properties of order statistics
(more specifically, on spacings) from the exponential distribution to de-
velop exact inferential results for the case of Type-II censored samples.
Though their distributional results on spacings from the exponential distri-
bution were already known from the earlier works of Sukhatme (1937) and
Rényi (1953), the approach taken by Epstein and Sobel (1954) in developing
exact inferential methods based on censored data from a life-testing view-
point attracted the attention of many researchers subsequently, resulting in
numerous publications in the following years; the works of Deemer and
Votaw (1955), Cohen (1955), and Bartholomew (1957) are some of the early
noteworthy ones in this direction.
During the last six decades or so, since the publication of these early
works, the literature on censored data analysis has expanded phenomenally,
by considering varying forms of censored data, dealing with a wide range of
lifetime distributions, and developing many different methods of inference.
For a detailed overview of various developments on inferential methods for
truncated distributions and based on censored samples, interested readers
may refer to Nelson (1982), Schneider (1986), Cohen (1991), and Balakr-
ishnan and Cohen (1991).
18

1.2 Type-I and Type-II censoring
In the preceding section, while describing different forms of censoring that
had been considered, it becomes evident that are two basic forms of cen-
soring:
(i) Type-I censoring and
(ii) Type-II censoring.
As the historical details in the last section clearly reveal, both these forms of
censoring have been discussed extensively in the literature, and this con-
tinues on to date. Though the form of censoring schemes in these two
cases may look somewhat similar, with slight variation in the resulting like-
lihood functions, there is a significant difference when it comes to devel-
opment of inferential methods based on these two forms of censoring, as
will be seen in the subsequent chapters.
Type-I censoring would naturally arise in life-testing experiments when
there is a constraint on time allocated for the reliability experiment to be
conducted. This may be due to many practical considerations such as limi-
tations on the availability of test facility, cost of conducting the experiment,
need to make reliability assessment in a timely manner, and so on. It is evi-
dent that the duration of the life-test is fixed in this censoring scheme, but
the number of complete failure times to be observed will be random. These
result in advantages as well as disadvantages, the fixed duration being a dis-
tinct advantage in the sense that the experimenter would know a priori how
long the test is going to last, while the random number of failures to be ob-
served being a clear disadvantage. For example, if the duration was fixed to
be too small compared to the average (or median) lifetime of the product,
then a rather small number of complete failures would be realized with a
high probability, and this would in turn have a negative impact on the preci-
sion or accuracy of the inferential methods subsequently developed.
In a Type-II censoring scheme, on the other hand, the number of complete
failures to be observed is fixed a priori; consequently, the experimenter can
have a control on the amount of information (in the form of complete fail-
ures) to be collected from the life-testing experiment, thus having a positive
impact on the precision of subsequent inferential methods. However, it has
a clear disadvantage that the duration of the life-test is random, which
would pose difficulty in the planning/conducting of the reliability
19

experiment, and also has the potential to result in unduly long life-test
(especially when the product under test is highly reliable).
For the purpose of illustration, let us consider the distribution.¹
Let us further suppose we have units available for the life-test, and that
we choose as the pre-fixed termination times under Type-I
censoring scheme, and as the pre-fixed number of complete fail-
ures to be observed under Type-II censoring scheme. In this situation, the
following facts are evident:
(a) The random number of failures that would occur until the pre-
fixed time T, say D, will have a -distribution;
(b) Consequently, the number of failures that would be expected to
occur by time T will simply be , under the Type-I cen-
soring scheme;
(c) Under the Type-II censoring scheme, as stated above, the
termination time will be random, say Y,² and will in fact equal the
m-th order statistic from a sample of size n from the -dis-
tribution, and hence is known to have a -distribution.
To get a clear idea about the difference between the two censoring schemes,
we have presented, in Table 1.1, the exact values of the following quantities:
(i) Termination time T, expected number of failures , and
, corresponding to Type-I censoring;
(ii) Expected termination time , number of failures m, and
, corresponding to Type-II censoring.
From binomial probabilities, it is then easy to verify that the cumulative dis-
tribution function (cdf) of Y is, for ,
(1.1)
where denotes the incomplete beta ratio and (for ) de-
notes the complete beta function. It is also easy to see that .
20

Furthermore, we may also observe in this situation that
(1.2)
a fact that is readily seen in the last column of Table 1.1.
Table 1.1
Scheme
parameters
Type-I censoring Type-II censoring
T m T m
4 0.4 4.0 0.6177 0.3636 4 0.3823 0.4
4 0.5 5.0 0.8281 0.3636 4 0.1719 0.5
4 0.6 6.0 0.9452 0.3636 4 0.0548 0.6
4 0.7 7.0 0.9894 0.3636 4 0.0106 0.7
4 0.8 8.0 0.9991 0.3636 4 0.0009 0.8
5 0.4 4.0 0.3669 0.4545 5 0.6331 0.4
5 0.5 5.0 0.6231 0.4545 5 0.3770 0.5
5 0.6 6.0 0.8338 0.4545 5 0.1662 0.6
5 0.7 7.0 0.9527 0.4545 5 0.0474 0.7
5 0.8 8.0 0.9936 0.4545 5 0.0064 0.8
6 0.4 4.0 0.1662 0.5455 6 0.8338 0.4
6 0.5 5.0 0.3770 0.5455 6 0.6231 0.5
6 0.6 6.0 0.6331 0.5455 6 0.3669 0.6
6 0.7 7.0 0.8497 0.5455 6 0.1503 0.7
6 0.8 8.0 0.9672 0.5455 6 0.0328 0.8
7 0.4 4.0 0.0548 0.6363 7 0.9452 0.4
7 0.5 5.0 0.1719 0.6363 7 0.8281 0.5
7 0.6 6.0 0.3823 0.6363 7 0.6177 0.6
7 0.7 7.0 0.6496 0.6363 7 0.3504 0.7
21

7 0.8 8.0 0.8791 0.6363 7 0.1209 0.8
8 0.4 4.0 0.0123 0.7273 8 0.9877 0.4
8 0.5 5.0 0.0547 0.7273 8 0.9453 0.5
8 0.6 6.0 0.1673 0.7273 8 0.8327 0.6
8 0.7 7.0 0.3828 0.7273 8 0.6172 0.7
8 0.8 8.0 0.6778 0.7273 8 0.3222 0.8
22

1.3 Need for hybrid censoring
From the results presented in Table 1.1, the following essential points need
to be emphasized, which should justify the coverage/exposition of this book
being focused on the importance and need for hybrid censoring method-
ology:
(1) First issue relating to Type-I censoring is concerning the occur-
rence of no failures. This would especially be the case if the termi-
nation time T had been chosen to be unduly small (in other words,
if the length of the test compared to mean life time of the product
under test is too small). For example, in the case of , had T
been fixed as 0.2, then there is almost a 11% chance that the life-test
would not result any complete failure. However, if the termination
time T gets increased to 0.3, then the chance of zero failures being
observed would decrease to about 3%. Moreover, if the number of
units tested becomes larger, say then even with the termination
time being , the chance of zero failures being observed would
be as small as 1%;
(2) Another issue with the case of Type-I censoring is that if the
termination time T is pre-fixed to be large, though one has a certain
required number of complete failures m in mind (at the planning
stage of the experiment), then the corresponding values of
suggest that, with high probability, the actual number of failures ob-
served would be at least m. For example, had the experimenter cho-
sen the termination time of the experiment to be and in fact
had number of complete failures in mind, then there is almost
a 85% chance that the number of failures observed would be at least
6. Similarly, if the termination time had been chosen to be and
had preliminarily in mind, then there is almost a 88% chance
that the number of failures observed would be at least 7. This sug-
gests that if the experimenter has an idea on the number of failures
to be observed and chooses the termination time T to be large, then
a Type-I censoring scheme would end up result in an unnecessarily
long life-test with high probability;
(3) Furthermore, in the case of Type-I censoring, it can also be ob-
served that, with the preliminary value of m one has in mind, if the
termination time T had been chosen to be small, then with high
23

probability the actual number of complete failures that will be ob-
served will end up being less than m. For example, if the experi-
menter had in mind but had chosen the termination time to be
, then there is almost a 62% chance that the number of fail-
ures observed would be at most 7, being less than that was in
the mind of the experimenter prior to conducting the experiment.
Thus, it is more likely that the life-test would be concluded by time T
in this case;
(4) The final point worth mentioning is that in the case of Type-II
censoring, the test duration would be long if one were to choose m
to be large enough in comparison to n, but still may not exceed a
large value of T the experimenter may preliminarily have in mind be-
fore conducting the experiment with a high probability. For example,
suppose units are under test and the number of failures to be
observed has been fixed to be . If the experimenter had an idea
of having the duration of the test to be at least , for example,
the chance that the Type-II censoring scheme would result in a test
exceeding 0.8 would be only about 32%.
Let us consider the first two points above and discuss their ramifications in
statistical as well as pragmatic terms. With regard to Point (1) above, it be-
comes evident that, at least when the number of test units n is small, there
will be a non-negligible probability that one may not observe any complete
failure at all from the life-test in the case of Type-I censoring. In such a situ-
ation, it is clear that a meaningful inferential method can not be developed
(whether it is point/interval estimation or test of hypothesis) uncondi-
tionally and, therefore, all pertinent inferential methods in this case need to
be developed only conditionally, conditioned on the event that at least one
complete failure is observed.³ In fact, this is the basis for the comment that
“there is a significant difference when it comes to development of inferential
methods based on these two forms of censoring” made in the beginning of last
section!
With regard to Point (2) that if the termination time T is pre-fixed to be
large under Type-I censoring scheme, then with a high probability, the actual
number of failures observed would be larger than the preliminary number of
failures (say, m) the experimenter would have had in mind, which is what
formed the basis for the original proposal of hybrid censoring by Epstein
24

(1954). It is for this reason he defined the hybrid termination time of
in order to terminate the life-test as soon as the prelim-
inary number of failures the experimenter had in mind is achieved, and
otherwise terminate at the pre-fixed time T. We refer to this censoring
scheme here as Type-I hybrid censoring scheme, adding the phrase Type-I
to emphasize that this scheme is based on a time-based guarantee (viz., not
to exceed a pre-fixed time T). It is then evident that
where Y has its cumulative distribution function as given in (1.1), which
readily yields the mixture representation for the hybrid termination time as
(1.3)
with the mixture probability as in (1.2). Strictly speaking, in the
above mixture form, T may be viewed as a degenerate random variable at
time T. With time , it is of interest to associate a count random variable
(analogous to D) corresponding to the number of complete failures ob-
served in the life-test. It is then clear that
(1.4)
which is in fact a clumped binomial random variable, with all the binomial
probabilities for m to n being clumped at the value m; see, for example,
Johnson et al. (2005) for details on this clumped binomial distribution.
From (1.3) and (1.4), the values of and can be readily computed,
and these are presented in Table 1.2 for the purpose of comparing Type-I
censoring and Type-I hybrid censoring schemes in terms of termination
time and expected number of failures.
25

Table 1.2
From Table 1.2, we observe the following points:
Scheme
parameters
Type-I censoring Type-I HCS
T m T
4 0.4 4.0 0.6177 0.3228 3.3960 0.3823 0.4
4 0.5 5.0 0.8281 0.3498 3.7618 0.1719 0.5
4 0.6 6.0 0.9452 0.3604 3.9312 0.0548 0.6
4 0.7 7.0 0.9894 0.3632 3.9877 0.0106 0.7
4 0.8 8.0 0.9991 0.3636 3.9991 0.0009 0.8
5 0.4 4.0 0.3669 0.3653 3.7649 0.6331 0.4
5 0.5 5.0 0.6231 0.4158 4.3848 0.3770 0.5
5 0.6 6.0 0.8338 0.4422 4.7649 0.1662 0.6
5 0.7 7.0 0.9527 0.4522 4.9403 0.0474 0.7
5 0.8 8.0 0.9936 0.4544 4.9927 0.0064 0.8
6 0.4 4.0 0.1662 0.3877 3.9312 0.8338 0.4
6 0.5 5.0 0.3770 0.4612 4.7617 0.6231 0.5
6 0.6 6.0 0.6331 0.5107 5.3980 0.3669 0.6
6 0.7 7.0 0.8497 0.5360 5.7901 0.1503 0.7
6 0.8 8.0 0.9672 0.5422 5.9599 0.0328 0.8
7 0.4 4.0 0.0548 0.3967 3.9859 0.9452 0.4
7 0.5 5.0 0.1719 0.4861 4.9337 0.8281 0.5
7 0.6 6.0 0.3823 0.5592 5.7803 0.6177 0.6
7 0.7 7.0 0.6496 0.6077 6.4397 0.3504 0.7
7 0.8 8.0 0.8791 0.6305 6.8390 0.1209 0.8
8 0.4 4.0 0.0123 0.3994 3.9982 0.9877 0.4
8 0.5 5.0 0.0547 0.4964 4.9883 0.9453 0.5
8 0.6 6.0 0.1673 0.5861 5.9476 0.8327 0.6
8 0.7 7.0 0.3828 0.6595 6.8225 0.6172 0.7
8 0.8 8.0 0.6778 0.7068 7.5168 0.3222 0.8
26

(1) The intended purpose of Type-I hybrid censoring scheme, as
introduced by Epstein (1954), is clearly achieved in certain circum-
stances. For example, if the experimenter was planning to conduct
the life-test for a period of and had an interest in observing 7
complete failures out of a total of units under test, then under
the Type-I hybrid censoring scheme, the test on an average would
have lasted for a period of 0.61, and would have resulted in 6.44
complete failures on an average (instead of 7 the experimenter had
in mind). Instead, had the experimenter planned to conduct the life-
test for a period of in the same setting, then the test would
have lasted on an average for a period of 0.63, and would have re-
sulted in 6.84 complete failures on an average;
(2) However, if the time T had been chosen to be too small for the
value of the number of complete failures interested in observing,
then the life-test, with high probability, would end by time T. For
example, if the experimenter had chosen to conduct the life-test for
a period of , but had an interest in observing possibly 7 com-
plete failures from the life-test, then the test on an average would
have lasted for a period of 0.49, and would have resulted in 4.93
complete failures on an average (instead of 7 the experimenter had
in mind). This follows intuitively from the fact that, in this case,
there is about 83% chance that the failure would occur after time
(see the value reported in the last column of Table 1.2);
(3) A final point worth noting is that, like in Type-I censoring
scheme, the case of no failures is a possibility in the case of Type-I
hybrid censoring scheme as well! This then means that inferential
procedures can be developed only conditionally, conditioned on the
event that at least one complete failure is observed, just as in the
case of Type-I censoring!
While Point (5) above highlights the practical utility of Type-I hybrid cen-
soring scheme, Point (6) indicates a potential shortcoming in Type-I hybrid
censoring scheme exactly as Point (3) indicated earlier in the case of Type-I
censoring, viz., that if the test time T is chosen to be too small in compar-
ison to the number of complete failures the experimenter wishes to observe
from the life-test, then with high probability, the test would terminate by
time T, in which case few failures will be observed from the life-test, leading
27

to possibly imprecise inferential results.
It is precisely this point that led Childs et al. (2003) to propose another
form of hybrid censoring based on the hybrid termination time of
, called Type-II hybrid censoring scheme. It is of interest
to mention that the phrase Type-II is incorporated here in order to empha-
size the fact that this censoring scheme provides a guarantee for the num-
ber of failures to be observed (viz., that the observed number of complete
failures would be at least m). It is then clear in this case that
(1.5)
where Y has its cdf as given in (1.1), which readily yields the mixture rep-
resentation for the hybrid termination time as
(1.6)
with the mixture probability as in (1.2). Here again, T may be
viewed as a degenerate random variable at time T. With time , it will be
useful to associate a count random variable (analogous to D) corre-
sponding to the number of complete failures observed in the life test, with
support and probability mass function as
(1.7)
which is in fact a clumped binomial random variable, with all the binomial
probabilities for 0 to m being clumped at the value m. Observe the differ-
ence in the two clumped binomial distributions that arise in the cases of
Type-I hybrid censoring scheme and Type-II hybrid censoring scheme here;
in the former, it is clumped on the right at the value m and in the latter, it is
clumped on the left at the value m. It is instructive to note here that the
28

event would occur in both cases listed in (1.5): in the first case when
, the termination of the life-test would occur at Y resulting in exactly m
complete failures, and in the second case when , the termination would
occur at T, but if no failure occurs in the interval then also exactly m
complete failures would be realized. Now, from (1.6) and (1.7), we can read-
ily compute the values of and . These are presented in Table 1.3 for
the purpose of comparing Type-II censoring and Type-II hybrid censoring
schemes in terms of termination time and expected number of failures.
Table 1.3
Scheme parameters Type-II censoring Type-II HCS
T m
4 0.3636 0.3823 0.4408 4.6020 0.4
4 0.3636 0.1719 0.5138 5.2384 0.5
4 0.3636 0.0548 0.6032 6.0688 0.6
4 0.3636 0.0106 0.7004 7.0123 0.7
4 0.3636 0.0009 0.8000 8.0010 0.8
5 0.4545 0.6331 0.4893 5.2351 0.4
5 0.4545 0.3770 0.5388 5.6154 0.5
5 0.4545 0.1662 0.6123 6.2351 0.6
5 0.4545 0.0474 0.7024 7.0597 0.7
5 0.4545 0.0064 0.8002 8.0073 0.8
6 0.5455 0.8338 0.5577 6.0688 0.4
6 0.5455 0.6231 0.5842 6.2384 0.5
6 0.5455 0.3669 0.6347 6.6020 0.6
6 0.5455 0.1503 0.7095 7.2100 0.7
6 0.5455 0.0328 0.8013 8.0401 0.8
7 0.6363 0.9452 0.6396 7.0141 0.4
7 0.6363 0.8281 0.6502 7.0665 0.5
7 0.6363 0.6177 0.6772 7.2197 0.6
7 0.6363 0.3504 0.7286 7.5603 0.7
7 0.6363 0.1209 0.8058 8.1610 0.8
29

Thing to note with Type-II hybrid censoring scheme is that all pertinent
inferential methods based on it will be unconditional, just like in the case of
Type-II censoring, due to the fact that at least m failures are guaranteed to
be observed. Additional advantages and the intended purpose of Type-II hy-
brid censoring scheme, as stated originally by Childs et al. (2003), become
clear from Table 1.3. If the experimenter had chosen the test time T to be
small compared to the number of complete failures to be observed from the
life-test, then the life-test in all likelihood would proceed until that many
complete failures are observed. For example, for the choice of , if the
experimenter had chosen the time T to be 0.4, then there is about 99%
chance that the failure would occur after and so the expected dura-
tion of the test becomes 0.728 and consequently the expected number of
complete failures observed becomes 8.002, as seen in Table 1.3. On the
other hand, had the experimenter chosen the test time T to be large com-
pared to the number of complete failures to be observed from the life-test,
then the life-test in all likelihood would proceed until time T. For example,
for the choice of , if the experimenter had chosen the time T to be 0.8,
then there is about 3% chance that the failure would occur before ,
in which case we see that the expected duration of the test is 0.801 and the
expected number of failures observed becomes 8.04. Thus, in either case,
the Type-II hybrid censoring scheme provides guarantee for observing
enough complete failures from the test to facilitate the development of pre-
cise inferential results, whether it is point/interval estimation or hypothesis
tests. Of course, this advantage naturally comes at a price of having a longer
life-test than under any of Type-I censoring, Type-I hybrid censoring, and
Type-II censoring.
8 0.7273 0.9877 0.7279 8.0017 0.4
8 0.7273 0.9453 0.7308 8.0118 0.5
8 0.7273 0.8327 0.7412 8.0524 0.6
8 0.7273 0.6172 0.7678 8.1776 0.7
8 0.7273 0.3222 0.8205 8.4832 0.8
30

1.4 Antecedents
As mentioned earlier, Epstein (1954) was the first one to introduce a hybrid
censoring scheme to facilitate early termination of a life-test as soon as a
certain number of failures the experimenter had in mind is achieved, instead
of carrying on with the test until the pre-fixed time T. He then considered
the case of exponential lifetimes and derived expressions for the mean
termination time as well as the expected number of failures under the Type-I
hybrid censoring scheme form that he introduced. In addition, he also con-
sidered a “replacement case” in which failed units are replaced a once by
new units drawn from the same exponential population, and derived explicit
expressions for the same quantities so that a comparison could be made be-
tween the two cases. Later, Epstein (1960b) developed hypothesis tests con-
cerning the exponential mean parameter, while Epstein (1960a,c) discussed
the construction of confidence intervals (one-sided and two-sided) for the
mean lifetime of an exponential distribution based on a Type-I hybrid cen-
sored data, using chi-square distribution for the pivotal quantity and using a
chi-square percentage point approximately even in the case of no failures.
These procedures were subsequently adopted as reliability qualification
tests and reliability acceptance tests based on exponential lifetimes as stan-
dard test plans in MIL-STD-781-C (1977), wherein the performance require-
ment is specified through mean-time-between-failure (MTBF). Harter (1978)
evaluated the performance of these confidence bounds for the MTTF
through Monte Carlo simulations. A formal rule for obtaining a two-sided
confidence interval for the MTTF, in the exponential case, was given by Fair-
banks et al. (1982) who demonstrated that their rule is very close to the ap-
proximation provided earlier by Epstein (1960c) and also provided a vali-
dation for their rule.
In the paper by Bartholomew (1963), the exact conditional distribution of
the maximum likelihood estimator of the mean of an exponential distri-
bution under a time censored life-test (i.e., under Type-I censoring) was de-
rived through conditional moment generating function (Conditional MGF)
approach, conditioned on the event that at least one complete failure is ob-
served. This method was adopted by Chen and Bhattacharyya (1988) to de-
velop the exact distribution theory for the maximum likelihood estimation of
the exponential mean lifetime under Type-I hybrid censoring scheme, and
the conditional moment generating function approach has since become a
31

standard tool for developing exact distribution theory for maximum like-
lihood estimators of parameters under various forms of hybrid censored
data, as will be seen in the ensuing chapters.
The prediction of times of future failures, based on a Type-I hybrid cen-
soring scheme, for the case of exponential distribution, was discussed by
Ebrahimi (1992) for both cases when the failed units are not replaced and
replaced by new units. All the works mentioned so far dealt with a scaled
exponential distribution, involving only the mean lifetime parameter. A two-
parameter exponential distribution, consisting of a threshold parameter
(interpreted as guarantee period in reliability literature) and a scale param-
eter (relating to the residual mean lifetime parameter), was considered by
Ebrahimi (1986) who then developed point and interval estimation methods
as well as hypothesis tests for both cases when the hybrid life-tests involved
without and with replacement of failed units.
All these early developments were on various inferential aspects based on
data observed from Type-I hybrid censoring scheme, as introduced by Ep-
stein (1954). But, as mentioned in the preceding section, the Type-II hybrid
censoring scheme, guaranteeing at least a pre-specified number of complete
failures to be observed in the life-test, was introduced by Childs et al. (2003)
to overcome some of the shortcomings of Type-I hybrid censoring scheme;
but, this advantage comes at a price of having a longer life-test, as men-
tioned earlier. Since then, the literature on hybrid censoring has exploded
with varying forms of hybrid schemes, for many different lifetime distri-
butions, and the development of a wide range of inferential methods. The
following section gives an account of the recent growth in this area of re-
search!
32

1.5 Burgeoning literature
As mentioned above, the literature on hybrid censoring has grown signif-
icantly in recent years. For example, a quick search on zbmath, using hybrid
censoring, hybrid censored, and truncated life test, truncated life tests, truncated
life testing as keywords, produced the frequency table (Table 1.4) and the
histogram for the publication record⁴ (see Fig. 1.1).
Table 1.4
Period No. of Publications
3 1954–1963
4 1964–1973
26 1974–1983
45 1984–1993
47 1994–1998
31 1999–2003
61 2004–2008
131 2009–2013
227 2014–2018
103 2019–2022
33

Figure 1.1 Histogram of publication record data on hybrid censoring for
the period 1954–2022 for given time intervals given in Table 1.4.
34

1.6 Scope of the book
The primary objectives in preparing this book have been to produce an
up-to-date volume, with emphasis on both theory and applications that will
form as a reference guide for practitioners involved in the design of a life-
testing experiment as well as in the analysis of lifetime data observed from
such experiments. The models and methods described in the book would
provide the reader a know-how regarding the designs and implementations
of various hybrid censoring schemes and their merits and demerits. A cen-
tral aspect in our presentation and analysis of the different hybrid censoring
schemes is to identify the key shared features as well as the structural ele-
ments. In particular, this enables a structured and efficient approach to
complex hybrid censoring models and is intended to support both the de-
sign and analysis of new, possibly even more complex models and the
development of associated statistical procedures. A complete treatment of
basic theory, including the derivation of all associated characteristics, prop-
erties and inferential results, is presented for all the different hybrid cen-
soring schemes considered in the literature. To facilitate a better under-
standing and appreciation of these developments, a comprehensive review
of all pertinent results on the conventional censoring schemes, viz., Type-II,
Type-I, and progressive censoring, is also given.
Throughout the book, many numerical examples as well as examples
based on real-life data sets have been presented in order to demonstrate
practicability and usefulness of the theoretical results discussed. In doing
so, due recognition has been given to all the published works on hybrid cen-
soring methodology by citing them appropriately. Thus, the Bibliography at
the end of the book contains an exhaustive list of publications on the sub-
ject to-date and it should be valuable for any researcher interested in work-
ing on this topic of research.
The coverage of the rest of the book, intended to give a detailed overview
of all pertinent developments, is as follows. Chapter 2 presents some pre-
liminary results on order statistics, progressively Type-II right censored
order statistics, generalized order statistics and sequential order statistics,
and they get used repeatedly in the ensuing chapters. In Chapter 3, pertinent
inferential methods and results are described for the classical censoring
schemes, viz., Type-II, Type-I, and progressive censoring, and they become
foundational for the corresponding results for hybrid censoring schemes
35

developed subsequently. Chapter 4 discusses various models and distri-
butional properties of hybrid censoring designs and inter-relationships be-
tween different forms of hybrid censoring. While Chapter 5 presents infer-
ential results for the case of exponential lifetime distribution, Chapter 6
deals with inferential results for some other important lifetime distributions
such as Weibull, log-normal, Birnbaum-Saunders, generalized exponential,
(log-)Laplace, and uniform. The generalization to progressive hybrid cen-
soring situation is handled in Chapter 7. Chapter 8 deals with the derivation
of information measures for different forms of hybrid censored data. These
include Fisher information, entropy, Kullback-Leibler information, and Pit-
man closeness. Next, applications of hybrid censoring to reliability prob-
lems are discussed in Chapters 9 and 10 with Chapter 9 focussing on step-
stress accelerated life-tests and Chapter 10 handling competing risks anal-
ysis, stress-strength analysis, acceptance sampling plans, and optimal de-
sign problems. Chapters 11 and 12 deal with model validation methods and
prediction (both point and interval prediction) issues, respectively. Finally,
adaptive progressive hybrid censoring schemes are addressed in Chapter 13.
Even though the coverage of the topic this way is quite elaborate and de-
tailed, some gaps that exist in the literature and also some problems that
will be worthy of further study are pointed out at different places in the
book. These could serve as convenient starting points for any one interested
in engaging in this area of research.
36

1.7 Notation
The following (incomplete) list provides notation that will be used through-
out the book. In selected cases, page numbers are provided in order to refer
to pertinent definition.
Notation Explanation Page
Random variables
random variables X,Y,Z
vector of random variables
X ,…,X
X,X
i-th order statistic in a sample
of size n
X
i-th uniform order statistic in
a sample of size n
U
i-th normalized spacing in a
sample of size n
S
i-th progressively Type-II
censored order statistic
based on censoring plan
i-th uniform progressively
Type-II censored order
statistic based on censoring
plan
vector of progressively
Type-II censored order
statistics
random counter
(for order statistics) or
(for
progressively Type-II
censored order statistics).
D
random counter for
corresponding hybrid
censoring scheme
,
Censoring plans, coefficients, constants
1 n
n
i:n
i:n
i,n
37

censoring plan ,
etc.
, 1 ≤ i ≤ m,
for a censoring plan
c
, 1 ≤ j ≤ r a
,
k + 1 ≤ j ≤ k
Density functions, cumulative distribution functions
density function/cumulative
distribution function (of a
random variable X)
f,f ,F,F
quantile function of F F
survival/reliability function
hazard rate
h = f/(1 − F) of F,F
h,h
cumulative distribution
function of a (standard)
exponential distribution (with
mean ϑ)
density function, cumulative
distribution function, and
quantile function of a
standard normal distribution
N(0,1)
φ,Φ,Φ
distribution function of a
χ²-distribution with d
degrees of freedom
distribution function of a
-distribution with scale
parameter ϑ and shape
parameter β
r−1
j,r
1 2
X X
←
X
X
−1
38

distribution function of X
f ,F
density function of
cumulative distribution
function of
joint density
function/cumulative
distribution function of
X ,…,X
f ,F
joint density
function/cumulative
distribution function of
f ,F
likelihood function for
parameter
θ = (θ ,…,θ ) given
some data or x ,d
log-likelihood function for
parameter
θ = (θ ,…,θ ) given
some data or x ,d
Fisher information about
θ = (θ ,…,θ ) (in X)
Distributions
uniform distribution 354 Uniform(a,b)
beta distribution 354 Beta(α,β)
power distribution 354 Power(α)
reflected power distribution 354 RPower(β)
exponential distribution 354 Exp(μ,ϑ), Exp(ϑ)
Weibull distribution 355 Weibull(ϑ,β)
gamma distribution 355
normal distribution 356 N(μ,σ²)
χ²-distribution 355
F-distribution 355 F
j:n
j:n j:n
1:n m:n
1,…,m:n 1,…,m:n
1,…,m:m:n 1,…,m:m:n
1 p
d
1 p
d
1 p
n,m
39

Pareto distribution 355 Pareto(α)
Laplace distribution 356 Laplace(μ,ϑ)
binomial distribution 356 bin(n,p)
Special functions
exponential function
natural logarithm
gamma function Γ(α)
incomplete gamma function
ratio defined as
, t ≥ 0
beta function
incomplete beta function
ratio defined as
,
0 < t < 1
I (α,β)
n factorial defined by ,
, where
n!
binomial coefficient ,
, k ≤ n
multinomial coefficient ,
,
univariate B-Spline B of
degree k with knots a ,…,a
357 B (⋅|a ,…,a )
indicator function on the set
A
positive part of x
negative part of x
largest integer k satisfying
k ≤ x
⌊x⌋
sign of x sgn(x)
divided differences of order
ν − j at
x > … > x for
358 [x ,…,x ]h
t
k
k 1
k k 1
ν j
j ν
40

function h
Sets
integers {1,2,3,…}
real numbers
n-fold Cartesian product of
set of (k × n)-matrices
set of all permutations of
(1,…,n)
measurable space with
σ-algebra
probability space with
σ-algebra and probability
measure P
Symbols
X is distributed according to
a cumulative distribution
function F
X ∼ F
independent and identically
distributed
iid
X ,…,X are iid random
variables from a cumulative
distribution function F
equality in distribution
convergence in distribution
convergence almost
everywhere
left endpoint of the support
of F
α(F)
right endpoint of the support
of F
ω(F)
1 n
41

p-th quantile of F ξ = F (p),
p ∈ (0,1)
p-th quantile of a standard
normal distribution
z
p-th quantile of a
χ²-distribution with ν
degrees of freedom
r-dimensional Lebesgue
measure
one-point distribution in
x/Dirac measure in x
δ
trace of a matrix A tr(A)
determinant of a matrix A
median of cumulative
distribution function F
med(F)
a
(a )
1 = (1,…,1)=(1 ) 1
g(t−)
g(t+)
represents all additive terms
of a function which do not
contain the variable of the
function
const
Operations
, x∧y
, x∨y
, x∧y
, x∨y
for a
transposed vector of vector x
inverse matrix of Σ
p
←
p
x
k
⁎k
⁎k
•r
T
−1
42

Abbreviations
almost surely a.s.
almost everywhere a.e.
for example / exempli gratia e.g.
id est i.e.
with respect to w.r.t.
increasing/decreasing failure
rate
IFR/DFR
best linear unbiased
estimator
BLUE
best linear invariant estimator BLIE
maximum likelihood
estimator
MLE
approximate maximum
likelihood estimator
AMLE
uniformly minimum variance
unbiased estimator
UMVUE
best unbiased predictor BUP
maximum likelihood
predictor
MLP
approximate maximum
likelihood predictor
AMLP
median unbiased predictor MUP
conditional median predictor CMP
best linear unbiased predictor BLUP
best linear equivariant
predictor
BLEP
predictive likelihood function PLF
highest probability density HPD
mean squared error MSE
43

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cation and Production Acceptance Tests: Exponential Distribution.
Washington, DC: U.S. Government Printing Office; 1977.
Nelson, 1982 W. Nelson, Applied Life Data Analysis. New York:
Wiley; 1982.
Rényi, 1953 A. Rényi, On the theory of order statistics, Acta
Mathematica Academiae Scientiarum Hungaricae 1953;4:191–
231.
Schneider, 1986 H. Schneider, Truncated and Censored Samples
from Normal Populations. New York: Marcel Dekker; 1986.
Sukhatme, 1937 P.V. Sukhatme, Test of significance for sam-
ples of the -population with two degrees of freedom, Annual
of Eugenics 1937;8:52–56.
¹ “The uniform distribution is not a suitable model for lifetimes, but is
used here for illustrating the idea behind the two censoring schemes, and
their primary difference. The uniform distribution, however, can be thought
of as a nonparametric model, obtained after performing a probability
46

integral transformation on the observed lifetimes, as will be explained later
on in the book!”
² “Keep in mind that the notation used here is a preliminary one, and an
unified notation that will be used throughout the book will be presented
later at the end of this chapter.”
³ “The condition may sometimes require the number of complete failures
observed to be even more than one, depending on the assumed model and
the number of parameters involved in it!”
⁴ “Retrieved on June 1, 2022 from http://zbmath.org/.”
47

Abstract
In this chapter, some essential distributional properties and results are
presented for order statistics and progressively censored order statis-
tics. These include marginal, joint and conditional distributions,
Markov dependence, conditional block independence and mixture re-
sults. The uniform, exponential and Weibull distributions are used as
examples. Finally, some general results for generalized order statistics
and sequential order statistics are presented. All the results described
here will be useful in subsequent chapters when inferential results are
developed for various forms of hybrid censoring.
Keywords
Order statistics; Progressively Type-II censored order statistics; Progres-
sively Type-I censored order statistics; Sequential order statistics; Gener-
alized order statistics; Markov dependence; Mixture property; Block inde-
pendence property
Chapter Outline
2.1 Introduction
2.2 Order statistics
• 2.2.1 Joint and marginal distributions
• 2.2.2 Conditional distributions
• 2.2.3 Markov dependence
• 2.2.4 Conditional block independence
• 2.2.5 Results for uniform distribution
• 2.2.6 Results for exponential distribution
• 2.2.7 Results for Weibull distribution
• 2.2.8 Results for symmetric distributions
2.3 Progressively Type-II right censored order statistics
• 2.3.1 Joint and marginal distributions
• 2.3.2 Conditional distributions and dependence structure
• 2.3.3 Conditional block independence
49

• 2.3.4 Mixture representation
• 2.3.5 General progressive Type-II censoring
• 2.3.6 Results for uniform distribution
• 2.3.7 Results for exponential distribution
• 2.3.8 Progressive Type-I right censoring
2.4 Generalized order statistics
2.5 Sequential order statistics
50

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having once been refused, I dread a second attempt. “A burnt child
fears the fire;” and a singed lover trembles before the blazing eyes
of the object of his adoration. I have yet a short time before the
expiration of my hour of trial, and the character of “Sir Thomas
Clifford” from which to borrow courage. (Enter Stella, c.)
Stella. Well, mysterious “Festus,” what new fancy is agitating
your fertile brain?
Festus. Madam, to tell you the truth, I was—thinking—of you.
Stella. Of me, or of your future salary?
Festus. Both.
Stella. What of me?
Festus. (Very awkward and confused.) That I think—I think—
that you—you—are—are—
Stella. Well, what am I?
Festus. (Abruptly.) A very fine reader.
Stella. Oh! is that all?
Festus. All worth mentioning.
Stella. Sir!
Festus. That is all I am at liberty to mention.
Stella. What if I should grant you liberty to say more?
Festus. Oh! then—then I should say—I should say—
Stella. Well, what would you say?
Festus. It’s your turn to read.
Stella. (Aside.) Stupid! (Aloud.) Well, sir, what shall I read?
Festus. Oh! oblige me by making your own selection.
Stella. There’s “The Bells,” by Poe. Do you like that?
Festus. Oh, exceedingly!

Stella. But I don’t know how to read it: it’s very difficult.
Festus. Perhaps I can assist you. (Aside.) I’ll provoke her a bit;
see if she has a temper.
Stella. Well, you are very kind. (Aside.) I’ll see if I can make
him talk.
Festus. Well, then, you take the book, and read. (Hands her copy
of Poe.) When I think you need correcting, I will speak.
Stella. Very well. (They sit, c. Stella reads in a very tragic tone,
emphasizing the words in Italics.)
“Hear the sledges with the bells,
Silver bells!”
Festus. Oh, stop, stop, stop! Dear me! that’s not the way to read.
There’s no silver in your bells. Listen:—
Sil-ver bells!”
Very silvery, don’t you see?
Stella. Oh, yes! excuse me. (Reads in a very silly tone.)
Sil———verbells!”
Festus. Oh, no, no! that’s too silly.
Stella. Sir!
Festus. I mean, there’s too much of the sil in silver. (Repeats
his reading. She imitates it.)
Festus. Ah! that’s better. Thank you: you are charming. (She
looks at him.) That is, a charming reader. Go on.

Stella. (Reads.)
“What a world of merriment their melody foretells!
How they tinkle”—
Festus. (Interrupting.) I beg your pardon: “twinkle.”
Stella. No, sir: “tinkle.”
Festus. But I am sure it is “twinkle.”
Stella. Can’t I believe my own eyes?
Festus. Not unless they “twinkle.”
Stella. Look for yourself. (Shows him the book.)
Festus. My stars! it is “tinkle.” I beg your pardon. Go on.
Stella.
“How they tinkle, tinkle, tinkle,
In the icy air”—
Festus. No, no: frosty,—frosty air.
Stella. No, sir: it’s icy air.
Festus. You are mistaken: “frosty.”
Stella. Am I? Look for yourself.
Festus. Well, I declare! It is, I see, icy. I beg your pardon. Go on.
Stella. I see, I see. You are bent on interrupting me. What do
you mean, sir?
Festus. What can you expect, if you don’t know how to read?
Stella. Sir, this is provoking. I don’t know how to read?
Festus. Not “The Bells,” I know.

Stella. Oh! do you? Well, sir, I know you are no gentleman; and I
know, if you want “The Bells” read (starts up, and throws book at
him), read it yourself.
Festus. Madam, what am I to understand by this?
Stella. That your presence is no longer agreeable to me.
Festus. Oh, very well, very well! I understand you wish me to go.
(Stella stands, r., with her back to him.) You wish me to go. I will
intrude no longer. (Very loud.) Since you—wish—me—to—go—
(Aside.) Confound it, I believe she does! (Aloud.) Very well,
madam, very well. Good-evening. (Exit, l.)
Stella. He’ll be back in three minutes. (Enter Festus, l.)
Festus. I forgot my hat. You’ll excuse me if I take my—(Aside.)
Confound it, she won’t speak! (Stands irresolute a moment, and
then approaches her.) Madam,—Stella,—I was wrong. You can read
“The Bells” divinely. I hear them ringing in my ears now. I beg your
pardon. Read “The Bells” in any manner you please: I shall be
delighted to listen.
Stella. Oh, very well! Since you have returned, I will read.
Reading. “The Bells,” Poe. Stella.
Festus. Splendid, splendid!
Stella. Now, sir, I shall be happy to listen to you once more.
Festus. Your “Bells” have stirred the fires of patriotism within my
heart; and I will give you, as my selection, “Sheridan’s Ride.”
Reading. “Sheridan’s Ride,” Reid. Festus.
Stella. Excellent! Mr. Festus, you are a very spirited rider,—I mean
reader. Now, suppose, for variety, we have another scene.
Festus. With all my heart. What shall it be?
Stella. Oh! you select. Pray. Mr. Festus, did you have any design
in selecting the scene from “The Marble Heart”?

Festus. Well, I like that. You selected it yourself.
Stella. But the play was your selection; and you were very perfect
in the part of “Raphael.”
Festus. Well, I selected what I thought I should most excel in.
Stella. You excel in love-making! That’s good. But I must say, you
act it well.
Festus. Yes—that is—I think that circumstances—occurring—
which would make—circumstances—perfectly—that is, I mean to say
that—circumstances—indeed—what were you saying?
Stella. Ha, ha, ha! O mighty Festus! you’ve lost your place; but,
as you have a partiality for love-scenes, what is your next?
Festus. What say you to a scene from “The Hunchback”? “The
secretary of my lord”? You know the scene,—“Julia” and “Sir Thomas
Clifford.”
Stella. Oh, yes! I am familiar with it; but I think, as an applicant
for a situation, you are making me perform more than my share of
work.
Festus. Oh! if you object—
Stella. Oh! but I don’t object. Proceed. (Sits, l. of table. Festus
exits, l.)
SCENE FROM “THE HUNCHBACK.”
(Arranged for this piece.)
Julia, Stella.
Sir Thomas Clifford, Festus.

Jul. (Alone.) A wedded bride?
Is it a dream?
Oh, would it were a dream!
How would I bless the sun that waked me from it!
I am wrecked
By mine own act! What! no escape? no hope?
None! I must e’en abide these hated nuptials!
Hated!—ay, own it, and then curse thyself
That mad’st the bane thou loathest for the love
Thou bear’st to one who never can be thine!
Yes, love! Deceive thyself no longer. False
To say ’tis pity for his fall,—respect
Engendered by a hollow world’s disdain,
Which hoots whom fickle fortune cheers no more!
’Tis none of these: ’tis love, and, if not love,
Why, then, idolatry! Ay, that’s the name
To speak the broadest, deepest, strongest passion
That ever woman’s heart was borne away by!
He comes! Thoud’st play the lady,—play it now!
(Enter Clifford, l.)
Speaks he not?
Or does he wait for orders to unfold
His business? Stopped his business till I spoke,
I’d hold my peace forever!
(Clifford kneels, presenting a letter.)
Does he kneel?
A lady am I to my heart’s content!
Could he unmake me that which claims his knee,
I’d kneel to him,—I would, I would! Your will?
Clif. This letter from my lord.
Jul. Oh, fate! who speaks?
Clif. The secretary of my lord. (Rises.)

Jul. I breathe!
I could have sworn ’twas he!
(Makes an effort to look at him, but is unable.)
So like the voice!—
I dare not look lest there the form should stand.
How came he by that voice? ’Tis Clifford’s voice
If ever Clifford spoke! My fears come back.
Clifford, the secretary of my lord!
Fortune hath freaks, but none so mad as that.
It cannot be!—it should not be! A look,
And all were set at rest.
(Tries to look at him again, but cannot.)
So strong my fears,
Dread to confirm them takes away the power
To try and end them. Come the worst, I’ll look.
(She tries again, and is again unequal to the task.)
I’d sink before him if I met his eye!
Clif. Wilt please your ladyship to take the letter?
Jul. There, Clifford speaks again! Not Clifford’s breath
Could more make Clifford’s voice: not Clifford’s tongue
And lips more frame it into Clifford’s speech.
A question, and ’tis over! Know I you?
Clif. Reverse of fortune, lady, changes friends:
It turns them into strangers. What I am
I have not always been.
Jul. Could I not name you?
Clif. If your disdain for one, perhaps too bold
When hollow fortune called him favorite,
Now by her fickleness perforce reduced
To take an humble tone, would suffer you—
Jul. I might?

Ju g t
Clif. You might.
Jul. O Clifford! is it you?
Clif. Your answer to my lord. (Gives the letter.)
Jul. Your lord!
Clif. Wilt write it?
Or, will it please you send a verbal one?
I’ll bear it faithfully.
Jul. You’ll bear it?
Clif. Madam,
Your pardon: but my haste is somewhat urgent.
My lord’s impatient, and to use despatch
Were his repeated orders.
Jul. Orders? Well (takes letter),
I’ll read the letter, sir. ’Tis right you mind
His lordship’s orders. They are paramount.
Nothing should supersede them. Stand beside them!
They merit all your care, and have it! Fit,
Most fit, they should. Give me the letter, sir.
Clif. You have it, madam.
Jul. So! How poor a thing
I look! so lost while he is all himself!
Have I no pride?
If he can freeze, ’tis time that I grow cold.
I’ll read the letter.
(Opens it, and holds it as about to read it.)
Mind his orders! So!
Quickly he fits his habits to his fortunes!
He serves my lord with all his will! His heart’s

He serves my lord with all his will! His heart s
In his vocation. So! Is this the letter?
’Tis upside down, and here I’m poring on’t!
Most fit I let him see me play the fool!
Shame! Let me be myself!
(She sits a while at table, vacantly gazing on the letter,
then looks at Clifford.)
How plainly shows his humble suit!
It fits not him that wears it. I have wronged him!
He can’t he happy—does not look it—is not!
That eye which reads the ground is argument
Enough. He loves me. There I let him stand,
And I am sitting! (Rises, and points to a chair.)
Pray you, take a chair.
(He bows as acknowledging and declining the honor.
She looks at him a while.)
Clifford, why don’t you speak to me? (Weeps.)
Clif. I trust
You’re happy.
Jul. Happy? Very, very happy!
You see I weep I am so happy. Tears
Are signs, you know, of naught but happiness.
When first I saw you, little did I look
To be so happy. Clifford!
Clif. Madam?
Jul. Madam!
I call thee Clifford, and thou call’st me madam!
Clif. Such the address my duty stints me to.
Thou art the wife elect of a proud earl
Whose humble secretary sole am I.
Jul. Most right! I had forgot! I thank you, sir,
For so reminding me and give you joy

For so reminding me, and give you joy
That what, I see, had been a burthen to you
Is fairly off your hands.
Clif. A burthen to me?
Mean you yourself? Are you that burthen, Julia?
Say that the sun’s a burthen to the earth!
Say that the blood’s a burthen to the heart!
Say health’s a burthen, peace, contentment, joy,
Fame, riches, honors, everything that man
Desires, and gives the name of blessing to!—
E’en such a burthen Julia were to me
Had fortune let me wear her.
Jul. (Aside.) On the brink
Of what a precipice I’m standing! Back,
Back! while the faculty remains to do’t!
A minute longer, not the whirlpool’s self
More sure to suck thee down! One effort! (Sits.) There!
(Recovers her self-possession, takes up the letter, and reads.)
To wed to-morrow night! Wed whom? A man
Whom I can never love! I should before
Have thought of that. To-morrow night. This hour
To-morrow,—how I tremble!
At what means
Will not the desperate snatch! What’s honor’s price?
Nor friends, nor lovers,—no, nor life itself!
Clifford, this moment leave me!
(Clifford retires up the stage out of her sight.)
Is he gone?
Oh, docile lover! Do his mistress’ wish
That went against his own! Do it so soon,
Ere well ’twas uttered! No good-by to her!
No word, no look! ’Twas best that so he went.
Alas the strait of her who owns that best
Which last she’d wish were done! What’s left me now?
To weep, to weep!

o eep, to eep
(Leans her head upon her arm, which rests upon the table, her
other arm hanging listless at her side. Clifford comes
down the stage, looks a moment at her, approaches her,
and, kneeling, takes her hand.)
Clif. My Julia!
Jul. Here again?
Up, up! By all thy hopes of heaven go hence!
To stay’s perdition to me! Look you, Clifford!
Were there a grave where thou art kneeling now,
I’d walk into’t and be inearthed alive
Ere taint should touch my name! Should some one come
And see thee kneeling thus! Let go my hand!—
Remember, Clifford, I’m a promised bride—
And take thy arm away! It has no right
To clasp my waist! Judge you so poorly of me
As think I’ll suffer this? My honor, sir!
(She breaks from him, quitting her seat.)
I’m glad you’ve forced me to respect myself:
You’ll find that I can do so.
Clif. There was a time I held your hand unchid;
There was a time I might have clasped your waist:
I had forgot that time was past and gone.
I pray you, pardon me.
Jul. (Softened.) I do so, Clifford.
Clif. I shall no more offend.
Jul. Make sure of that.
No longer is it fit thou keep’st thy post
In’s lordship’s household. Give it up! A day,
An hour, remain not in it.

Clif. Wherefore?
Jul. Live
In the same house with me, and I another’s?
Put miles, put leagues, between us! The same land
Should not contain us.
O Clifford, Clifford!
Rash was the act, so light that gave me up,
That stung a woman’s pride, and drove her mad,
Till in her frenzy she destroyed her peace!
Oh, it was rashly done! Had you reproved,
Expostulated, had you reasoned with me,
Tried to find out what was indeed my heart,
I would have shown it, you’d have seen it, all
Had been as naught can ever be again.
Clif. Lov’st thou me, Julia?
Jul. Dost thou ask me, Clifford?
Clif. These nuptials may be shunned—
Jul. With honor?
Clif. Yes.
Jul. Then take me! Hold!—hear me, and take me, then!
Let not thy passion be my counsellor;
Deal with me, Clifford, as my brother. Be
The jealous guardian of my spotless name.
Scan thou my cause as ’twere thy sister’s. Let
Thy scrutiny o’erlook no point of it,
And turn it o’er not once, but many a time,
That flaw, speck, yea, the shade of one,—a soil
So slight not one out of a thousand eyes
Could find it out,—may not escape thee; then
Say if these nuptials can be shunned with honor!

Clif. They can.
Jul. Then take me, Clifford—
Festus. Stop one moment. (Looks at watch.) Time’s up.
Stella. So soon?
Festus. The tone of your voice expresses regret. What is your
decision?
Stella. My decision?
Festus. Upon my application for the situation of reader. Shall I
have it?
Stella. Perhaps the terms will not suit.
Festus. Madam, I am willing to serve you on any terms. Allow me
to throw off the mask of “Festus,” which of course you have seen
through, and offer myself for a situation under the name of—
Stella. Stop: you are not going to pronounce that name before all
these good people?
Festus. Of course not. But what shall I do? Stella, I feel that
“Raphael” and “Sir Thomas Clifford” have inspired me to attempt
love-making on my own account. Grant me the opportunity to make
application for the situation made vacant by my unceremonious exit
the other night. Let “Festus” apply once more.
Stella. What shall I say? (To audience.) Would you? He seems
to have found his tongue; and who knows but what he may make an
agreeable beau? I think he had better call again; for to have a lover
who can make love by borrowing, is, at least,—under the
circumstances—under the circumstances—what is it, Festus?
Festus. Circumstances? Why, under the circumstances, I should
say it was “An Original Idea.”

George M. Baker.
Note. The “Readings” and “Scenes” maybe
varied to suit the taste of the performers. “The
Garden Scene” in “Romeo and Juliet,” scenes
from “Ingomar,” “The School for Scandal,” etc.,
have been used with good effect.
[2] Or the evening of the performance.

A. W. Pinero’s Plays
Price, 50 Cents Each

THE AMAZONS Farce in Three Acts. Seven
males, five females. Costumes, modern; scenery, not
difficult. Plays a full evening.
THE CABINET MINISTER Farce in Four Acts.
Ten males, nine females. Costumes, modern society;
scenery, three interiors. Plays a full evening.
DANDY DICK Farce in Three Acts. Seven males,
four females. Costumes, modern; scenery, two
interiors. Plays two hours and a half.
THE GAY LORD QUEX Comedy in Four Acts. Four
males, ten females. Costumes, modern; scenery, two
interiors and an exterior. Plays a full evening.
HIS HOUSE IN ORDER Comedy in Four Acts. Nine
males, four females. Costumes, modern; scenery, three
interiors. Plays a full evening.
THE HOBBY HORSE Comedy in Three Acts. Ten
males, five females. Costumes, modern; scenery easy.
Plays two hours and a half.
IRIS Drama in Five Acts. Seven males, seven
females. Costumes, modern; scenery, three interiors.
Plays a full evening.
LADY BOUNTIFUL Play in Four Acts. Eight males,
seven females. Costumes, modern; scenery, four
interiors, not easy. Plays a full evening.
LETTY Drama in Four Acts and an Epilogue.
Ten males, five females. Costumes, modern; scenery
complicated. Plays a full evening.
Sent p epaid on eceipt of p ice b

Sent prepaid on receipt of price by
Walter H. Baker & Company
No. 5 Hamilton Place, Boston, Massachusetts

THE MAGISTRATE Farce in Three Acts. Twelve
males, four females. Costumes, modern; scenery, all
interior. Plays two hours and a half.
THE NOTORIOUS MRS. EBBSMITH Drama in Four Acts.
Eight males, five females. Costumes, modern; scenery,
all interiors. Plays a full evening.
THE PROFLIGATE Play in Four Acts. Seven males,
five females. Scenery, three interiors, rather
elaborate; costumes, modern. Plays a full evening.
THE SCHOOLMISTRESS Farce in Three Acts. Nine
males, seven females. Costumes, modern; scenery, three
THE SECOND MRS. TANQUERAY Play in Four Acts.
Eight males, five females. Costumes, modern; scenery,
three interiors. Plays a full evening.
SWEET LAVENDER Comedy in Three Acts. Seven
males, four females. Scene, a single interior;
costumes, modern. Plays a full evening.
THE TIMES Comedy in Four Acts. Six males, seven
females. Scene, a single interior; costumes, modern.
Plays a full evening.
THE WEAKER SEX Comedy in Three Acts. Eight
males, eight females. Costumes, modern; scenery, two
A WIFE WITHOUT A SMILE Comedy in Three Acts.
Five males, four females. Costumes, modern; scene, a
single interior. Plays a full evening.

THE AWAKENING Play in Four Acts. By C. H. Chambers.
Four males, six females. Scenery, not difficult, chiefly
interiors; costumes, modern. Plays a full evening.
Price, 50 Cents.
THE FRUITS OF ENLIGHTENMENT Comedy in Four
Acts. By L. Tolstoi. Twenty-one males, eleven females.
Scenery, characteristic interiors; costumes, modern.
Plays a full evening. Recommended for reading clubs.
Price, 25 Cents.
HIS EXCELLENCY THE GOVERNOR Farce in
Three Acts. By R. Marshall. Ten males, three females.
Costumes, modern; scenery, one interior.
Acting rights reserved. Time, a full evening.
Price, 50 Cents.
AN IDEAL HUSBAND Comedy in Four Acts.
By Oscar Wilde. Nine males, six females. Costumes,
modern; scenery, three interiors. Plays a full evening.
Acting rights reserved. Sold for reading.
Price, 50 Cents.
THE IMPORTANCE OF BEING EARNEST
Farce in Three Acts. By Oscar Wilde. Five males, four
females. Costumes, modern; scenes, two interiors and
an exterior. Plays a full evening. Acting rights reserved.
Price, 50 Cents.
LADY WINDERMERE’S FAN Comedy in Four
Acts. By Oscar Wilde. Seven males, nine females.
Costumes, modern; scenery, three interiors.
Plays a full evening. Acting rights reserved.
Price, 50 Cents.
NATHAN HALE Play in Four Acts.

By Clyde Fitch. Fifteen males, four females.
Costumes of the eighteenth century in America.
Scenery, four interiors and two exteriors.
Acting rights reserved. Plays a full evening.
Price, 50 Cents.
THE OTHER FELLOW Comedy in Three Acts.
By M. B. Horne. Six males, four females.
Scenery, two interiors; costumes, modern.
Professional stage rights reserved. Plays a full evening.
Price, 50 Cents.
THE TYRANNY OF TEARS Comedy in Four Acts.
By C. H. Chambers. Four males, three females.
Scenery, an interior and an exterior; costumes,
modern. Acting rights reserved. Plays a full evening.
Price, 50 Cents.
A WOMAN OF NO IMPORTANCE Comedy in Four Acts.
By Oscar Wilde. Eight males, seven females. Costumes,
modern; scenery, three interiors and an exterior. Plays a
full evening. Stage rights reserved. Offered for reading only.
Price, 50 Cents.
Walter H. Baker & Company
No. 5 Hamilton Place, Boston, Massachusetts
S. J. PARKHILL & CO., PRINTERS, BOSTON, U.S.A.
Transcriber's Notes:
The cover image is in the public domain.

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