Computational Statistics and Data Analysis 53 (2009) 3560–3570
Contents lists available at ScienceDirect
Computational Statistics and Data Analysis
journal homepage: www.elsevier.com/locate/csda
Expected values of the number of failures for two populations under
joint Type-II progressive censoring
Safar Parsi a , Ismihan Bairamov b,∗
a
Department of Statistics, University of Mohaghegh Ardabili, Ardabil, Iran
b
Department of Mathematics, Izmir University of Economics, Izmir, Turkey
article
info
Article history:
Received 27 November 2008
Received in revised form 19 March 2009
Accepted 20 March 2009
Available online 7 April 2009
a b s t r a c t
The model in which joint Type-II progressive censoring is implemented on two samples
from different populations in a combined manner is considered and the probabilities of
failures are discussed. This model may have relevance for practical applications, when
an experimenter need to know the expected values of the number of failures for each
population. This knowledge plays an important role in the selection of appropriate
sampling plans and the construction of efficient estimators for parameters. The formula
allowing numerical computation of the expected value of the number of failures for each
of the two populations is given. Also, a detailed numerical study of this expected value is
carried out for different parametric families of distributions.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
Recently, Balakrishnan and Rasouli (2008) have introduced a joint censoring scheme which is important in performing
comparative life-tests of two identical products manufactured in different units under the same conditions. To describe this
scheme, suppose products are being manufactured by two lines under the same conditions and two samples of products of
sizes m and n are selected from these two lines and are placed simultaneously on a life-testing experiment. Then, based on
cost considerations and time restrictions for completion of the test, suppose the researcher chooses to terminate the lifetesting experiment as soon as a certain number of failures occur. In this situation, either point or interval estimation of the
mean lifetimes of products manufactured by two lines may be of interest. Under joint Type-II censoring, specimens of two
products under study are placed on a life-test simultaneously, the successive failure times and the corresponding product
types are recorded, and the experiment is terminated as soon as a specified total number of failures (say, k) occurs.
Suppose X1 , . . . , Xm , the lifetimes of m specimens of product A, are independent and identically distributed (i.i.d.) random
variables, with cumulative distribution function (c.d.f.) F (x) and probability density function (p.d.f.) f (x). Let Y1 , . . . , Yn be
i.i.d. random variables with c.d.f. G(x) and p.d.f. g (x), representing the lifetime of n specimens of product B. Further, suppose
W1 ≤ · · · ≤ WN denote the order statistics of the N = m + n random variables {X1 , . . . , Xm ; Y1 , . . . , Yn }. Then, under the
joint Type-II censoring scheme, the observable data consists of (Z, W), where W = (W1 , . . . , Wk ), with k (1 ≤ k < N) being
a pre-fixed integer, and Z = (Z1 , . . . , Zk ) with Zi = 1 or 0 accordingly as Wi is from an X - or Y -failure.
A joint censoring scheme has been previously considered in the literature. For example, Basu (1968), discussed a
generalized Savage statistic, while Johnson and Mehrotra (1972) studied the locally most powerful rank tests. Bhattacharyya
and Mehrotra (1981) considered the problem of testing the equality of two distributions, under the assumption of
exponentiality. The developments under this sampling scheme have been reviewed by Bhattacharyya (1995) in Chapter 7
∗
Corresponding author. Tel.: +90 232 4888131; fax: +90 232 4888339.
E-mail addresses: [email protected] (S. Parsi), [email protected] (I. Bairamov).
0167-9473/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.csda.2009.03.023
S. Parsi, I. Bairamov / Computational Statistics and Data Analysis 53 (2009) 3560–3570
3561
of Balakrishnan and Basu (1995) and all research has focused on nonparametric and parametric tests of hypotheses.
Balakrishnan and Rasouli (2008) have studied the exact likelihood inference for two exponential populations under joint
Type-II censoring.
If a researcher desires to remove live units at points other than the termination point of the life-test, the above described
scheme will be of no value because the joint Type-II censoring does not allow for units to be lost or removed from the test
at points other than the final termination point. More general censoring schemes are therefore required.
Rasouli and Balakrishnan (in press) consider the Joint Type-II Progressive Censoring (JPC) scheme. Under the JPC scheme,
m units of product A with c.d.f. F (x) and p.d.f. f (x) and n units of product B with c.d.f. G(y) and p.d.f. g (y) are placed on
test at time zero (thus, we have N = m + n units on test), and k failures from both product A and product B are going
to be observed. Immediately following the first failure, r1 of the surviving units from both of product A and product B are
randomly selected and removed (we can partition r1 into r10 and r100 such that r10 and r100 are the number of failures of product
A and the number of failures of product B respectively (r1 = r10 + r100 )). Then, immediately following the second observed
0
failure, r2 of the surviving units from both product A and product B are removed (r2 = r2 + r200 ), and so on. This experiment
terminates at the time when the kth failure is observed and the remaining rk = N − k − r1 − r2 − · · · − rk−1 surviving units
from two populations are all removed. If r1 = r2 = · · · = rk−1 = 0, then rk = N − k, which corresponds to the joint Type-II
censoring. If r1 = r2 = · · · = rk = 0, then N = k, which corresponds to the complete sample.
Several authors have addressed inferential issues based on progressive Type-II censored samples (for example, see
Gibbons and Vance (1983), Balakrishnan et al. (1987), Balakrishnan and Sandhu (1995) and Bairamov and Eryilmaz (2006)).
Viveros and Balakrishnan (1994) discussed interval estimation of life parameters based on progressive Type-II censored
data. Kuş and Kaya (2007) examined a simultaneous confidence interval for parameters of Pareto distribution based on
progressive type-II censored data. (For more details see also Balakrishnan and Aggarwala (2000) and Balakrishnan (2007).)
Rasouli and Balakrishnan (in press) developed exact inferential methods based on MLEs and compared their performance
with those based on approximate, Bayesian and bootstrap methods, based on the JPC scheme under the assumption of
exponentiality for both samples. For recent works on progressive censoring see also Balakrishnan and Rasouli (2008), Huang
and Wu (2008), Wu (2008), Sanjel and Balakrishnan (2008), Soliman (2008) and Balakrishnan et al. (2008).
In practical applications, a researcher may need to know the expected values of the number of failures for each of the
two populations. This information is important for the selection of an appropriate sampling plan, because in a statistical
inference for parameters, the number of failures for a population are directly related to the estimator efficiency. In this
study, we consider a joint Type-II progressive censoring scheme for two populations when the censoring is implemented
on the two samples in a combined manner. Section 2 presents the details of the proposed model.
The structure of probability of failures is discussed in Section 3. The formula which is most valid for the computation of
expected values of the number of failures for each of the two populations is given in Section 4. Finally, a detailed numerical
study of the expected values are carried out for various classes of life distributions such as Gamma, Pareto and Weibull.
It is important to examine how well the proposed method works for approximation of the expected values (A.E.) of the
number of failures. To study the performance of this approach 1,000, JPC samples from different distributions and censoring
schemes are simulated and the simulated expected values of the number of failures are obtained. The simulation results are
also shown in Tables 1–3 in the Appendix. It can be seen that the approximate expected values of the number of failures are
close to the simulated expected values of the number of failures for different distributions and censoring schemes. Hence,
the proposed method is reliable for approximating the expected values of the number of failures.
2. The model
Suppose m and n independent units are placed in a test with the corresponding lifetimes being identically distributed with
p.d.f.’s f (x; θ 1 ) and g (y; θ 2 ) and c.d.f.’s F (x; θ 1 ) and G(y; θ 2 ), θ 1 = (θ11 , θ12 , . . . , θ1d1 ), θ 2 = (θ21 , θ22 , . . . , θ2d2 ), θ 1 ∈ Rd1 ,
θ 2 ∈ Rd2 , respectively. Under the JPC with censoring scheme R = (R1 = r1 , . . . , Rk = rk ), the observable data consists of
0
(WR , Z, R ) where WR = (W1R:k:N , W2R:k:N , . . . , WkR:k:N ) and Z = (Z1 , . . . , Zk ) with Zi = 1 if WiR is from X (product A) failure
0
0
and Zi = 0 if WiR is from Y (product B) failure and R0 = (R1 = r10 , . . . , R0k−1 = rk−1 ).
Pk
R
R
Let Mk =
i=1 Zi denote the number of X -failures in W and Nk = k − Mk (i.e., the number of Y -failures in W ). The
R
0
likelihood function of (W , Z, R ) is derived by Rasouli and Balakrishnan (in press) as:
L(θ 1 , θ 2 ; wR , z, r0 ) = C
k
Y
r 0 r 00
{f (wi ; θ 1 )}zi {g (wi ; θ 2 )}1−zi F̄ (wi ; θ 1 ) i Ḡ(wi ; θ 2 ) i ,
(1)
i=1
where w1 < w2 < · · · < wk , and m0 ≡ n0 ≡ 0 and F̄ = 1 − F and Ḡ = 1 − G are the survival functions of the two
0
populations. Note that rk = N − k − r1 − r2 − · · · − rk−1 , rk0 = m − mk − r10 − r2 − · · · − rk0 −1 . For i = 1, 2, . . . , k, ri00 = ri − ri0
and C = D1 D2 with
D1 =
k
Y
j =1
"
m−
j −1
X
i=1
zi −
j −1
X
i=1
!
0
ri
!
#
j −1
j −1
X
X
00
zj + n −
(1 − zi ) −
ri (1 − zj ) ,
i=1
i=1
3562
S. Parsi, I. Bairamov / Computational Statistics and Data Analysis 53 (2009) 3560–3570
  j−1

  jP
−1
jP
−1 0
jP
−1
P


m−
zi −
ri
n−
(1−zi )− ri00


  i=1 i=1   i=1

i=1




0
0

rj
rj −rj
k−1 


Y
D2 =
.
!
jP
−1



j=1 
m+n−j−
ri




i=1




rj


From (1), we have the joint density of (WR , Z, R0 ) as
fWR ,Z,R0 (w, z, r0 ; θ 1 , θ 2 ) = C
k
Y
{f (wi ; θ 1 )}zi {g (wi ; θ 2 )}1−zi
ri0 F̄ (wi ; θ 1 )
ri00
Ḡ(wi ; θ 2 )
i=1
0 < w1 < w2 < · · · < wk < ∞.
In the following theorem, the expressions for the probability of failures are given.
Theorem 3.1. The joint density of (WR , Z, R0 ) is
0
fWR ,Z,R0 (w, z, r0 ; θ 1 , θ 2 ) = fWR |Z=z,R0 =r0 (w; θ 1 , θ 2 |Z = z, R0 = r0 )P (Z = z, R0 = r ),
where
0
P (Z = z, R0 = r ) = αk
k−1
Y
αi βi ,
(2)
i=1
and
mz1 + n (1 − z1 )
α1 = P (Z1 = z1 ) =
mp + nq
0
β1 = P (R1 = r1 |Z1 = z1 ) =
0
m−m1
0
r1
pz1 q1−z1 ,
n−n1
r1 −r10
N −1
r1
z1 = 0 or 1,
(3)
,
r10 = max(0, r1 + n1 − n), . . . , min(r1 , m − m1 ),
(4)
and for i = 2, . . . , k, zi = 0 or 1,
αi = P (Zi = zi |Z1 = z1 , . . . Zi−1 = zi−1 , R01 = r10 , . . . R0i−1 = ri0−1 )
!
!
iP
−1
iP
−1
0
00
m − mi−1 −
rj zi + n − ni−1 −
rj (1 − zi )
j =1
=
j =1
i −1
m − mi−1 −
P
!
0
rj
i −1
p+
n − ni−1 −
j=1
P
pzi q1−zi .
!
00
rj
(5)
q
j =1
For i = 2, . . . , k − 1, with r00 ≡ r000 ≡ 0,
0
βi = P (R0i = ri0 |Z1 = z1 , . . . Zi = zi , R01 = r10 , . . . , Ri−1 = ri0−1 )



i−1
i−1
m−mi −

P 0
j=1
0
ri
P 00
r
n−ni −
rj
j=1

=
N−
iP
−1
r j −i
0
ri −ri
!
j

,
j=1
ri
ri0 = max(0, ri + ni +
i−1
X
rj00 − n), . . . , min(ri , m − mi − ri0−1 − · · · − r10 ),
j =1
where p = P (Z = 1) and q = 1 − p.
(6)
S. Parsi, I. Bairamov / Computational Statistics and Data Analysis 53 (2009) 3560–3570
3563
3. Expected value of Mk
Pk
R
With the knowledge of the expected value of Mk =
i=1 Zi , the number of X -failures in W presents data of interest.
In this section, by using Theorem 3.1, we will be able to approximate the expected value of Mk . We will need the Taylor
expansion of the function in several variables (see, Adams (2007)).
For the d-dimensional function T (x1 , . . . , xd ) we have
T (x1 , . . . , xd ) = T (a1 , . . . , ad ) +
d
X
(xi − ai )Txi (a1 , . . . , ad ) + R2 ,
i =1
where Txi denotes the partial derivative with respect to xi and R2 is the remainder, which reduces to the tangent plane
approximation
≈
T (x1 , . . . , xd ) ≈ T (x1 , . . . , xd ) = T (a1 , . . . , ad ) +
d
X
(xi − ai )Txi (a1 , . . . , ad ),
i =1
≈
where T (x1 , . . . , xd ) is the Taylor polynomial of T (x1 , . . . , xd ) of degree 1 near the point (a1 , . . . , ad ).
In the special case, when d = 2 and the function depends on two variables, x1 and x2 , the Taylor polynomial near the point
(a1 , a2 ) is:
T (x1 , x2 ) = T (a1 , a2 ) + (x1 − a1 )Tx1 (a1 , a2 ) + (x2 − a2 )Tx2 (a1 , a2 ) + R2
and
≈
T (x1 , x2 ) ≈ T (x1 , x2 ) = T (a1 , a2 ) + (x1 − a1 )Tx1 (a1 , a2 ) + (x2 − a2 )Tx2 (a1 , a2 ).
(7)
From (2)–(6), we have
µZ1 ≡ E (Z 1) =
m
mp + nq
p
(8)
0
r (m − m )
1
1
µR01 |Z1? ≡ E R1 |Z1 = z1 =
.
(N − 1)
(9)
Conditioning with respect to Z1 and Z2 gives
µZ
?
0?
2 |Z1 ;R1
≡ E Z2 |Z1 = z1 , R01 = r10
m − m1 − r10
=
p,
(10)
m − m1 − r10 p + n − n1 − r100 q
0
µR0 |Z ? ;R0 ? ≡ E R02 |Z1 = z1 , Z2 = z2 , R1 = r10
2 2 1
r2 m − m2 − r10
.
=
N − r10 − 2
(11)
Continuing this process we obtain
µZ | Z ?
i
0
;R ?
i−1 i−1
0
≡ E Zi |Z1 = z1 , . . . , Zi−1 = zi−1 , R01 = r1 , . . . , R0i−1 = ri0−1
!
iP
−1
0
m − mi−1 −
rj
j =1
=
i−1
m − mi−1 −
P
!
0
rj
i−1
p+
n − ni−1 −
j =1
P
! p,
00
rj
i = 3, . . . , k
(12)
q
j =1
and
0
µR0 |Z ? ;R0 ? ≡ E R0i |Z1 = z1 , . . . , Zi = zi , R01 = r10 , . . . , Ri−1 = ri0−1
i i
i−1
!
iP
−1
0
ri m − mi −
rj
j=1
=
i −1
N−
P
j =1
0
! ,
rj − i
i = 3, . . . , k.
(13)
3564
S. Parsi, I. Bairamov / Computational Statistics and Data Analysis 53 (2009) 3560–3570
From (8) and (9), it can be seen immediately that
m
µZ1 = E (Z 1) =
p
mp + nq
h
0
µR01 = EZ1 ER1 R1 |Z1 = z1
i
r1 m −
=
m
p
mp+nq
.
(N − 1)
However, it can be observed from (10)–(13), that it is not easy to get exact expressions for µZi and µR0 for i = 2, 3, . . . , k. By
i
using the Taylor expansion of the function of several variables, we can approximate them. For approximating µZ2 , we define
≈
≈
µZ1 = µZ1 , µR0 = µR0 and
1
1
Ψ (z1 , r10 ) = µZ
2
0
= E Z2 |Z1 = z1 , R01 = r1
0
|Z ? ; R ?
1
1
m − m1 − r10
=
p.
0
m − m1 − r1 p + n − n1 − r100 q
The function Ψ (z1 , r10 ) depends on two variables, z1 and r10 . We use the Taylor expansion for the function Ψ (z1 , r10 )
≈
≈
substituting (a1 , a2 ) = (µZ1 , µR0 ) in (7). Thus, we have
1
≈
≈
µ
µ
m − Z1 − R0
1
≈
p
Ψ (z1 , r10 ) = ≈
≈
≈
≈
m − µZ1 − µR0 p + n − (1 − µZ1 ) − (r1 − µR0 ) q
1
1
≈
≈
≈
≈
≈
≈
+ (z1 − µZ1 )Ψz1 (µZ1 , µR0 ) + (r10 − µR01 )Ψr10 (µZ1 , µR01 ).
1
By applying a repeated conditional expectation formula, we have
≈
≈
µZ2 = EZ1 [E 0 (Ψ (z1 , r10 ))]
R
1
≈
≈
m − µZ 1 − µR 0
1
= ≈
≈
m − µZ1 − µR0
p+
1
≈
p.
≈
n − (1 − µZ1 ) − (r1 − µR0 ) q
1
Since the expected values after 2 terms vanish, when approximating µR0 this method yields
2
2 ≈
r2
m−
≈
P
j =1
µR 0 =
2
!
≈
µZ j − µR 0
1
.
≈
µ
N − R0 − 2
1
Thus, by using similar considerations, we get the approximate values of µZi and µR0 for i = 3, 4, . . . , k as follows:
i
m−
≈
µZi =
m−
i−1 ≈
i−1 ≈
j =1
j =1
P
µZ j −
P
µ
i −1 ≈
i −1 ≈
j=1
j =1
P
µZj −
!
i −1
p+
0
Rj
P
n−
P
µ
!
0
Rj
≈
(1 − µZj ) −
j=1
i−1
P
≈
! p i = 3, . . . , k.
(rj − µR0j ) q
j =1
And
ri
≈
i ≈
iP
−1 ≈
P
µZj − µ 0
m−
R
j=1
µR 0 =
i
j =1
i −1 ≈
N−
P
j =1
!
j
!
i = 3, . . . , k.
µR 0 − i
j
Therefore, the approximate value for the expected value of the number of failures from X ’s is
E (Mk ) ≈
k
X
≈
µZ i .
i =1
(14)
S. Parsi, I. Bairamov / Computational Statistics and Data Analysis 53 (2009) 3560–3570
3565
3.1. Some special distributions
To evaluate the expected value of Mk for special distributions such as Gamma, Pareto and Weibull, we need to compute
the value of p given in (15) for these distributions.
Gamma distribution. Let X and Y be independent random variables with Gamma p.d.f.’s Gamma(α1 , β1 ) and
Gamma(α2 , β2 ), respectively. The p.d.f.’s of X and Y are
f (x; α1 , β1 ) =
xα1 −1
x
I (x > 0)
α
exp −
α
x
exp −
I (x > 0),
β2
Γ (α1 )β1 1
β1
and
g (x; α2 , β2 ) =
xα2 −1
Γ (α2 )β2 2
where I is the indicator function. It is not difficult to verify that, in this case,
I
Pg = P (Z = 1) = P (X < Y ) =
β2
β1 +β2
(α1 , α2 )
B(α1 , α2 )
,
where B(a, b) is the beta function and Ix (a, b) is the incomplete beta function.
The expected value of Mk , in the case where X ’s have Gamma(α1 , β1 ) and Y ’s have Gamma(α2 , β2 ) distributions,
respectively, under the setup of the JPC scheme, can be calculated using (14) by substituting p = Pg .
Pareto distribution. Let X and Y be independent random variables with Pareto p.d.f.’s Pareto(λ1 , µ1 ), and Pareto(λ2 , µ2 ),
respectively. The p.d.f.’s of X and Y are
λ
f (x; λ1 , µ1 ) =
λ1 µ1 1
I (x > µ1 )
xλ1 +1
and
λ
g (x; λ2 , µ2 ) =
λ2 µ2 2
xλ2 +1
I (x > µ2 ).
In this case, we have
λ1
µ1
λ2
,
λ1 +λ2 µ2
Pp = P (Z = 1) = P (X < Y ) =

λ1
µ2 λ2


,
λ1 + λ2 µ1



1 −
if µ1 < µ2
if µ1 > µ2 .
The expected value of Mk for two Pareto distributions under the setup of the JPC scheme can be evaluated from (14) by
substituting p = Pp .
Weibull distribution. Let X and Y be independent random variables with corresponding Weibull p.d.f.’s, Weibull(β1 , α1 )
and Weibull(β2 , α2 ), where αi , i = 1, 2, are the scale parameters and βi , i = 1, 2, are the shape parameters. The p.d.f.’s of
X and Y are given as
β1
f (x; β1 , α1 ) =
α1
β2
g (x; β2 , α2 ) =
α2
x
β 1 −1
" #
β1
x
exp −
I (x > 0)
α1
β2 −1
" #
β2
x
exp −
I (x > 0).
α2
α1
and
x
α2
In this case, we have

∞
X
(−1)j α2 jβ1
β1


1
−
Γ
j
+
1
,


j!
α1
β2
j=0
Pw = P (Z = 1) = P (X < Y ) =
∞
X

(−1)j α2 jβ1
β1


Γ
j+1 ,

j!
α1
β2
j=0
if β1 < β2
if β1 > β2 .
The expected value of Mk for these two Weibull distributions under the JPC scheme can be found by substituting p = Pw
in (14).
3566
S. Parsi, I. Bairamov / Computational Statistics and Data Analysis 53 (2009) 3560–3570
Fig. 1. The approximate expected values of the number of failures.
4. Computation results and simulation study
In this section, some numerical results on calculation of the expected value of Mk are presented. We have considered
different sample sizes for the two populations from different distributions and censoring schemes. The expected values
for Mk , calculated from the formula (14) for Gamma, Weibull and Pareto distributions mentioned above, are presented in
Tables 1–3 in the Appendix.
Table 1 consists of the approximate values of E (Mk )− the expected number of failures from X ’s in the JPC scheme, for
observations X1 , X2 , . . . , Xm and Y1 , Y2 , . . . , Yn from populations having Gamma(α1 , β1 ) and Gamma(α2 , β2 ) distributions
respectively, with the censoring scheme R = (r1 , . . . , rk ). Similarly, in Table 2, the numerical values of Mk for two
Pareto(λ1 , µ1 ) and Pareto(λ2 , µ2 ) distributions and in Table 3 for two Weibull(β1 , α1 ) and Weibull(β2 , α2 ) distributions
are given.
It is important to examine the effectiveness of the proposed method in approximating the expected values of the number
of failures. To study the performance of this approach, we also simulate 1,000 JPC samples from different distributions and
different combinations of N, k, m, n, p and censoring schemes R and obtain the simulated expected values of the number of
failures. The simulation results are also shown in Tables 1–3. The approximate expected values of the number of failures are
close to the simulated expected values of the number of failures, for different distributions and censoring cases. Hence, this
method is reliable where it is necessary to obtain the approximate values of the expected values of the number of failures.
In practical applications, it is very important for the researcher to have information about the expected values of the number of failures for the two populations before the experiment. Because in statistical inference procedures for parameters the
number of failures for a population is directly related to estimators efficiency, information about the expected values of the
number of failures for the two populations before the experiment is extremely important in the selection of a sampling plan.
From Table 1, for example, for X and Y having Gamma distributions with parameters (α1 , β1 ) = (4, 12), (α2 , β2 ) = (6, 4),
respectively, and p = 0.166, k = 10, m = 15, n = 15, if we choose the censoring scheme (0, . . . , 0, 20), then we expect
that approximately 2 failure times will be from the X sample. In the case of the censoring scheme (4, 0, 4, 0, 4, 0, 4, 0, 4, 0),
we will expect approximately 4 observations of failure times from the X sample.
From Table 2, for X and Y with Pareto distributions with parameters (λ1 , µ1 ) = (3, 6) and (λ2 , µ2 ) = (4, 4), respectively,
for m = 50, n = 50, p = 0.085, k = 50, if the ordinary joint Type-II scheme is selected as (0, . . . , 0, 50), then approximately
8 observations of failure times are expected from the X sample, whereas for the JPC scheme (50, 0, . . . , 0), this number
increases to 25. Therefore, for inference procedures for the parameters of X sample, we should prefer this scheme.
From Table 3, with X having Weibull distribution with parameters (β1 , α1 ) = (5, 3) and Y having Weibull distribution
with parameters (β2 , α2 ) = (5, 5) we observe that for m = 10, n = 20, k = 10, p = 0.928 in an ordinary joint Type-II
censoring scheme (0, 0, . . . , 0, 20), approximately 8 observations are expected from the X sample and 2 from the Y sample.
However, if we choose the JPC scheme (10, 10, 0, . . . , 0) we would expect 5 observations of failure times from the X sample
and 5 from the Y sample. This means that if m and n are small and q less than p, and we employ an ordinary Type-II censoring
scheme, then it is quite likely that we will observe very few failures from the second population in our censored sample, thus
resulting in poor inference for the parameters of the second population. A JPC scheme balances this situation and results
in failures from both populations, thus yielding more accurate inference for both parameters of the two populations. For
example, Fig. 1, below, shows the approximate expected values of the number of failures with respect to p ∈ (0, 1) for
S. Parsi, I. Bairamov / Computational Statistics and Data Analysis 53 (2009) 3560–3570
3567
Table 1
The Approximation of the Expected (A.E.) and Simulated Expected (S.E.) values of Mk for some small, moderate and large values of m, n, k and Pg for two
Gamma populations when joint Type-II progressive censoring is implemented.
Scheme no.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
526
53
54
55
56
57
N
30
k
Scheme(R)
(m, n)
(α1 , α2 , β1 , β2 ) =
(4, 6, 12, 4), p = 0.166
A.E . Mk
S .E . Mk
(α1 , α2 , β1 , β2 ) =
(2, 5, 7, 3), p = 0.580
A.E . Mk
S .E . Mk
(α1 , α2 , β1 , β2 ) =
(3, 5, 2, 3), p = 0.904
A.E . Mk
S .E . Mk
10
(0, . . ., 0, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(30, 20)
(25, 25)
(20, 30)
(30, 20)
(25, 25)
(20, 30)
(30, 20)
(25, 25)
(20, 30)
(30, 20)
(25, 25)
(20, 30)
(70, 30)
(50, 50)
(30, 70)
(70, 30)
(50, 50)
(30, 70)
(70, 30)
(50, 50)
(30, 70)
(70, 30)
(50, 50)
(30, 70)
3.568
2.038
1.083
6.389
4.710
3.023
7.246
5.711
3.940
3.567
2.037
1.083
5.917
4.861
3.419
5.413
3.550
2.287
6.316
3.584
1.865
9.802
7.313
4.813
7.898
4.805
2.427
10.184
5.965
3.018
13.202
9.882
6.560
6.143
4.358
3.014
11.773
9.792
7.813
8.530
6.030
4.137
14.811
12.329
9.849
7.235
3.737
1.741
13.691
9.727
5.763
23.637
12.142
5.433
34.807
24.831
14.888
7.238
5.665
3.944
6.794
5.125
3.432
7.860
6.256
4.375
7.237
5.664
3.944
8.267
6.706
4.771
7.831
6.440
4.846
10.747
8.356
5.774
10.115
7.607
5.080
10.778
8.378
5.780
14.144
10.910
7.471
13.436
10.089
6.731
13.178
11.247
9.217
12.120
10.111
8.096
16.342
13.913
11.371
15.110
12.600
10.086
15.150
11.440
7.273
14.131
10.127
6.092
37.307
27.806
17.402
35.109
25.092
15.061
9.369
8.725
7.456
7.020
5.414
3.728
8.309
6.743
4.828
9.368
8.724
7.456
9.565
8.083
6.329
9.495
8.919
7.123
13.863
12.532
9.652
10.272
7.720
5.265
13.556
11.178
7.843
17.972
14.884
9.999
13.510
10.160
6.836
18.159
17.175
15.662
12.232
10.269
8.289
22.364
20.822
18.354
15.187
12.712
10.236
19.014
17.747
15.074
14.270
10.347
6.406
46.723
41.546
29.080
35.130
25.204
15.253
(20, 0, . . ., 0)
(10, 10, 0, . . ., 0)
(0, . . ., 0, 10, 10)
(0, 0, 0, 0, 10, 10, 0, 0, 0, 0)
(4, 0, 4, 0, 4, 0, 4, 0, 4, 0)
15
(0, . . ., 0, 15)
(15, 0, . . ., 0)
(5, 0, . . ., 0, 5, 0, . . . 0, 5)
20
(0, . . ., 0, 10)
(10, 0, . . ., 0)
50
20
(0, . . ., 0, 30)
(0, . . ., 0, 30)
25
(0, . . ., 0, 25)
(25, 0, . . ., 0)
100
20
(0, . . ., 0, 80)
(80, 0, . . ., 0)
50
(0, . . ., 0, 50)
(50, 0, . . ., 0)
3.562
2.020
1.080
6.433
4.753
3.123
6.161
4.567
3.001
3.586
2.103
1.095
4.958
3.534
2.189
4.529
2.928
1.664
6.432
3.716
1.828
9.870
7.346
4.886
7.603
4.697
2.354
10.312
6.146
3.063
13.133
9.854
6.537
6.178
4.371
3.044
11.861
9.840
7.940
8.619
6.130
4.143
14.774
12.302
9.906
7.174
3.736
1.754
13.617
9.673
5.897
23.629
12.149
5.440
34.849
24.741
14.796
7.246
5.676
3.903
6.639
5.071
3.357
6.724
5.031
3.437
7.228
5.607
3.944
6.911
5.332
3.701
7.069
5.490
3.754
10.764
8.381
5.784
10.013
7.632
5.063
10.488
8.122
5.579
14.104
10.885
7.470
13.364
10.063
6.745
13.215
11.250
9.320
12.151
10.009
8.061
16.273
13.840
11.417
15.014
12.474
10.138
15.133
11.491
7.318
13.969
10.027
6.205
37.252
27.684
17.349
35.104
24.981
15.038
9.365
8.719
7.438
6.953
5.219
3.715
7.036
5.454
3.950
9.308
8.722
7.374
8.029
6.930
5.560
8.713
7.668
6.072
13.849
12.490
9.467
10.124
7.702
5.270
13.187
10.729
7.684
17.941
14.647
9.953
13.392
10.109
6.854
18.132
17.229
15.637
12.245
10.342
8.341
22.378
20.741
18.225
15.157
12.769
10.192
18.970
17.711
15.068
14.225
10.348
6.397
46.748
41.423
28.902
35.225
25.092
15.260
N = 30, k = 10, m = 20, n = 10 for two cases R = (0, . . . , 0, 20) (joint Type-II censoring) and R = (20, 0, . . . , 0) (JPC)
schemes. It is easy to see that the JPC scheme balances the situation.
3568
S. Parsi, I. Bairamov / Computational Statistics and Data Analysis 53 (2009) 3560–3570
Table 2
The Approximation of the Expected (A.E.) and Simulated Expected (S.E.) values of Mk for some small, moderate and large values of m, n, k and Pp for two
Pareto populations when joint Type-II progressive censoring is implemented.
Scheme no.
N
k
Scheme(R)
(m, n)
(λ1 , λ2 , µ1 , µ2 ) =
(3, 4, 6, 4), p = 0.085
A.E . Mk
S .E . Mk
(λ1 , λ2 , µ1 , µ2 ) =
(4, 2, 7, 5), p = 0.340
A.E . Mk
S .E . Mk
(λ1 , λ2 , µ1 , µ2 ) =
(3, 7, 3, 5), p = 0.849
A.E . Mk
S .E . Mk
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
30
10
(0, . . ., 0, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(30, 20)
(25, 25)
(20, 30)
(30, 20)
(25, 25)
(20, 30)
(30, 20)
(25, 25)
(20, 30)
(30, 20)
(25, 25)
(20, 30)
(70, 30)
(50, 50)
(30, 70)
(70, 30)
(50, 50)
(30, 70)
(70, 30)
(50, 50)
(30, 70)
(70, 30)
(50, 50)
(30, 70)
2.352
1.145
0.559
6.293
4.553
2.922
7.044
5.510
3.902
2.352
1.145
0.559
5.140
4.328
2.966
4.643
2.929
1.729
5.212
2.266
1.016
9.722
7.218
4.682
7.594
4.423
1.598
10.000
5.035
1.852
13.156
9.845
6.463
4.005
2.557
1.646
11.705
9.727
7.720
6.317
3.843
2.378
14.755
12.282
9.794
4.498
2.009
0.870
13.577
9.641
5.755
20.583
7.785
2.924
34.736
24.790
14.865
5.387
3.656
2.218
6.536
4.821
3.104
7.512
5.880
4.005
5.386
3.656
2.218
7.142
5.639
3.839
6.467
4.823
3.354
8.404
5.763
3.517
9.907
7.375
4.819
8.979
6.307
3.911
11.727
8.160
5.025
13.272
9.916
6.529
9.492
7.473
5.655
11.887
9.873
7.850
12.191
9.632
7.308
14.907
12.397
9.878
11.291
7.100
3.794
13.870
9.841
5.791
30.042
19.304
10.432
34.921
24.911
14.889
9.013
8.118
6.629
6.962
5.298
3.632
8.075
6.511
4.664
9.013
8.118
6.629
9.334
8.100
6.105
9.214
8.427
6.817
13.286
11.636
8.891
10.215
7.698
5.211
12.997
10.838
7.656
17.180
14.233
9.892
13.438
10.124
6.806
17.229
15.950
14.209
12.200
10.218
8.239
21.167
19.336
16.845
15.152
12.679
10.199
18.424
16.572
13.212
14.227
10.286
6.329
45.018
38.585
26.970
35.117
25.176
15.205
(20, 0, . . ., 0)
(10, 10, 0, . . ., 0)
(0, . . ., 0, 10, 10)
(0, 0, 0, 0, 10, 10, 0, 0, 0, 0)
(4, 0, 4, 0, 4, 0, 4, 0, 4, 0)
15
(0, . . ., 0, 15)
(15, 0, . . ., 0)
(5, 0, . . ., 0, 5, 0, . . . 0, 5)
20
(0, . . ., 0, 10)
(10, 0, . . ., 0)
50
20
(0, . . ., 0, 30)
(0, . . ., 0, 30)
25
(0, . . ., 0, 25)
(25, 0, . . ., 0)
100
20
(0, . . ., 0, 80)
(80, 0, . . ., 0)
50
(0, . . ., 0, 50)
(50, 0, . . ., 0)
2.366
1.117
0.547
6.337
4.698
3.158
6.025
4.546
2.992
2.454
1.157
0.551
4.360
2.993
1.991
3.836
2.267
1.266
5.411
2.349
0.966
9.714
7.292
4.884
7.224
4.130
1.610
10.025
5.247
1.869
13.090
9.871
6.563
4.064
2.563
1.635
11.880
9.756
7.842
6.514
3.908
2.376
14.812
12.297
9.878
4.434
2.051
0.851
13.500
9.640
5.891
20.709
7.849
2.901
34.829
24.708
14.803
5.345
3.672
2.228
6.558
4.918
3.204
6.521
4.808
3.190
5.451
3.672
2.186
6.041
4.319
2.737
5.670
4.061
2.494
8.371
5.762
3.585
9.858
7.508
4.949
8.864
6.236
3.788
11.756
8.256
5.057
13.242
9.941
6.716
9.572
7.490
5.653
11.909
9.907
7.885
12.079
9.709
7.344
14.903
12.441
9.888
11.259
7.143
3.847
13.913
9.916
5.897
30.001
19.353
10.417
34.937
25.015
14.904
9.009
8.084
6.572
6.792
5.247
3.635
7.047
5.427
3.855
9.001
8.118
6.578
7.843
6.608
5.076
8.462
7.273
5.560
13.275
11.636
8.777
10.204
7.649
5.217
12.660
10.412
7.430
17.093
14.096
9.770
13.451
10.129
6.771
17.258
15.937
14.233
12.288
10.274
8.393
21.097
19.249
16.800
15.180
12.628
10.258
18.456
16.599
13.283
14.116
10.323
6.349
44.993
38.529
26.857
35.106
25.120
15.324
Acknowledgements
The authors are grateful to two anonymous referees, the Co-Editor and Associate editor for the careful reading and
constructive suggestions, which were helpful in improving the presentation.
S. Parsi, I. Bairamov / Computational Statistics and Data Analysis 53 (2009) 3560–3570
3569
Table 3
The Approximation of the Expected (A.E.) and Simulated Expected (S.E.) values of Mk for some small, moderate and large values of m, n, k and Pw for two
Weibull populations when joint Type-II progressive censoring is implemented.
Scheme no.
N
k
Scheme(R)
(m, n)
(β1 , β2 , α1 , α2 ) =
(4, 5, 3, 2), p = 0.160
A.E . Mk
S .E . Mk
(β1 , β2 , α1 , α2 ) =
(6, 2, 3, 3), p = 0.431
A.E . Mk
S .E . Mk
(β1 , β2 , α1 , α2 ) =
(5, 5, 3, 5), p = 0.928
A.E . Mk
S .E . Mk
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
30
10
(0, . . ., 0, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(20, 10)
(15, 15)
(10, 20)
(30, 20)
(25, 25)
(20, 30)
(30, 20)
(25, 25)
(20, 30)
(30, 20)
(25, 25)
(20, 30)
(30, 20)
(25, 25)
(20, 30)
(70, 30)
(50, 50)
(30, 70)
(70, 30)
(50, 50)
(30, 70)
(70, 30)
(50, 50)
(30, 70)
(70, 30)
(50, 50)
(30, 70)
3.490
1.976
1.045
6.379
4.704
3.025
7.236
5.692
3.919
3.490
1.975
1.044
5.866
4.823
3.387
5.382
3.503
2.247
6.234
3.497
1.805
9.797
7.306
4.799
7.875
4.761
2.370
10.151
5.886
2.939
13.198
9.882
6.571
6.004
4.236
2.917
11.768
9.791
7.805
8.382
5.885
4.016
14.807
12.326
9.849
7.059
3.614
1.675
13.683
9.724
5.762
23.394
11.855
5.253
34.802
24.828
14.886
6.143
4.427
2.839
6.578
4.890
3.224
7.605
5.962
4.107
6.142
4.427
2.838
7.651
6.006
4.126
7.008
5.442
3.901
9.335
6.762
4.355
9.927
7.405
4.900
9.656
7.095
4.625
12.640
9.216
5.968
13.274
9.919
6.577
10.954
8.925
6.983
11.912
9.900
7.894
13.821
11.283
8.842
14.921
12.409
9.902
12.911
8.759
5.002
13.917
9.887
5.886
32.936
22.583
13.007
34.942
24.915
14.904
9.525
9.012
7.891
7.147
5.450
3.762
8.423
6.844
4.857
9.525
9.011
7.890
9.669
8.285
6.633
9.619
9.146
6.940
14.128
12.985
9.919
10.335
7.818
5.316
13.829
11.285
7.942
18.365
15.006
10.000
13.476
10.196
6.856
18.588
17.773
16.414
12.268
10.307
8.319
22.938
21.575
19.078
15.210
12.725
10.258
19.265
18.283
16.038
14.349
10.386
6.444
47.499
43.050
29.728
35.141
25.216
15.276
(20, 0, . . ., 0)
(10, 10, 0, . . ., 0)
(0, . . ., 0, 10, 10)
(0, 0, 0, 0, 10, 10, 0, 0, 0, 0)
(4, 0, 4, 0, 4, 0, 4, 0, 4, 0)
15
(0, . . ., 0, 15)
(15, 0, . . ., 0)
(5, 0, . . ., 0, 5, 0, . . . 0, 5)
20
(0, . . ., 0, 10)
(10, 0, . . ., 0)
50
20
(0, . . ., 0, 30)
(0, . . ., 0, 30)
25
(0, . . ., 0, 25)
(25, 0, . . ., 0)
100
20
(0, . . ., 0, 80)
(80, 0, . . ., 0)
50
(0, . . ., 0, 50)
(50, 0, . . ., 0)
3.543
1.996
1.057
6.393
4.788
3.150
6.171
4.597
3.021
3.549
2.020
1.048
4.992
3.449
2.186
4.466
2.816
1.607
6.371
3.563
1.854
9.790
7.327
4.859
7.721
4.665
2.307
10.263
6.035
2.960
13.214
9.945
6.656
6.067
4.313
3.006
11.642
9.696
7.803
8.473
5.949
4.010
14.800
12.363
9.863
6.959
3.500
1.645
13.771
9.682
5.820
23.529
11.884
5.291
34.686
24.787
14.862
6.179
4.458
2.794
6.707
4.923
3.396
6.583
4.916
3.225
6.157
4.483
2.790
6.340
4.675
3.065
6.284
4.527
2.963
9.305
6.736
4.348
9.950
7.442
4.941
9.469
7.000
4.508
12.602
9.202
5.968
13.324
9.943
6.567
10.921
8.885
7.013
11.988
9.909
7.944
13.803
11.208
8.830
14.970
12.524
9.924
12.874
8.771
4.960
13.982
9.996
5.942
32.920
22.490
13.057
35.071
24.928
14.946
9.544
9.033
7.873
6.846
5.316
3.676
7.043
5.523
3.942
9.571
9.019
7.737
8.120
7.039
5.785
8.910
7.889
6.302
14.070
13.015
9.739
10.122
7.698
5.294
13.396
10.940
7.871
18.302
14.856
9.986
13.446
10.171
6.830
18.599
17.704
16.307
12.216
10.247
8.206
22.935
21.588
18.896
15.172
12.682
10.214
19.254
18.327
16.058
14.053
10.350
6.476
47.552
42.966
29.564
35.126
25.110
15.289
Appendix
Proof of Theorem 3.1. We can obtain αi from the fact that in a population containing two members (one X and one Y ), the
probability of observing X first is
p = P (Z = 1) = P (X < Y ) =
Z
0
∞
Ḡ(x; θ 2 )dF (x; θ 1 ).
(15)
3570
S. Parsi, I. Bairamov / Computational Statistics and Data Analysis 53 (2009) 3560–3570
Now for the m observations of X ’s, n observations of Y ’s and k times censoring using Bayes’ theorem with prior probabilities
p/(p + q) for z = 1 and q/(p + q) for z = 0 and with mass probabilities a/(a + b) for z = 1 and b/(a + b) for z = 0 for
each step, we have the posterior mass probabilities ap/(ap + bq) for z = 1 and bp/(ap + bq) for z = 0. Here in the jth step
for derivation of probability
0
P (Zj = zj |Z1 = z1 , . . . Zj−1 = zj−1 , R1 = r10 , . . . , R0j−1 = rj0−1 ),
the numbers a and b are taken as aj = m − mj−1 −
Pj−1 0 i=1 rj , bj = n − nj−1 −
Pj−1
00
i =1 r j .
Also, we can obtain βi from the fact that in the ith step, R0i has a hypergeometric distribution with parameters m − mi −
Pi−1
0
j =1 r j ,
n − ni −
Pi−1
00
j =1 r j
and ri .
Note that in the literature the probability p = P (X < Y ) is called stress–strength reliability. In recent years the problem
of estimating p has been given much attention in reliability and related fields. For a comprehensive review of results on
estimating stress–strength reliability we refer Kotz et al. (2003).
References
Adams, R.A., 2007. Calculus. Addison Wesley, Longman.
Bairamov, I., Eryilmaz, S., 2006. Spacings, exceedances and concomitants in progressive type censoring scheme. J. Statist. Planning Inference 136, 527–536.
Balakrishnan, N., 2007. Progressive censoring methodology: An appraisal. Test 16, 211–259.
Balakrishnan, N., Aggarwala, R., 2000. Progressive Censoring: Theory, Methods and Applications. Birkhauser, Boston.
Balakrishnan, N., Basu, A.P. (Eds.), 1995. The Exponential Distribution: Theory, Methods and Applications. Gordon and Breach, Newark, NJ.
Balakrishnan, N., Kannan, R.A., Lin, C.T., Ng, H.K.T., 1987. A point and interval estimation for Gaussian distribution based on progressively type-II censored
samples. IEEE Trans. Reliab. 52 (1), 90–95.
Balakrishnan, N., Rasouli, A., 2008. Exact likelihood inference for two exponential populations under joint Type-II censoring. Comput. Statist. Data Anal. 52,
2725–2738.
Balakrishnan, N., Sandhu, R.A., 1995. A simple simulation algorithm for generating progressively type-II censored sample. Amer. Statistician 49, 229–230.
Balakrishnan, N., Burkschat, M., Cramer, E., Hofmann, G., 2008. Fisher information based progressive censoring plans. Comput. Statist. and Data Anal. 53
(2), 366–380.
Basu, A.P., 1968. On a generalized Savage statistic with applications to life testing. Ann. Math. Statist. 39, 1591–1604.
Bhattacharyya, G.K., 1995. Inferences under two-sample and multi-sample situations. In: Balakrishnan, N., Basu, A.P. (Eds.), The Exponential Distribution:
Theory, Methods and Applications. Gordon and Breach, Newark, NJ, pp. 93–118 (Chapter 7).
Bhattacharyya, G.K., Mehrotra, K.G., 1981. On testing equality of two exponential distributions under combined Type-II censoring. J. Amer. Statist. Assoc.
76, 886–894.
Gibbons, D.I., Vance, L.C., 1983. Estimators for the 2-parameter Weibull distribution with progressively censored samples. IEEE Trans. Reliab. R-32, 95–99.
Huang, S.R., Wu, S.J., 2008. Reliability sampling plans under progressive type-I interval censoring using cost functions. IEEE Trans. Reliab. 57 (3), 445–451.
Johnson, R.A., Mehrotra, K.G., 1972. Locally most powerful rank tests for the two-sample problem with censored data. Ann. Math. Statist. 43, 823–831.
Kuş, C., Kaya, M.F., 2007. Estimation for the parameters of the Pareto distribution under progressive censoring. Commun. Statist. Theor. Meth. 36, 1359–1365.
Kotz, S., Lumelskii, Y., Pensky, M., 2003. The Stress-Strength Model and its Generalizations. Theory and Applications. World Scientific.
Rasouli, A., Balakrishnan, N., 2009. Exact likelihood inference for two exponential populations under joint progressive type-II censoring. Commun. Statist.
Theory Methods (in press).
Sanjel, D., Balakrishnan, N., 2008. A Laguerre polynomial approximation for a goodness-of-fit test for exponential distribution based on progressively
censored data. J. Stat. Comput. and Simul. 78 (6), 503–513.
Soliman, A.A., 2008. Estimations for pareto model using general progressive censored data and asymmetric loss. Commun. Statist. Theory Methods 37 (9),
1353–1370.
Viveros, R., Balakrishnan, N., 1994. Interval estimation of parameters of life from progressively censored data. Technometrics 36, 84–91.
Wu, S.J., 2008. Estimation of the two-parameter bathtub-shaped lifetime distribution with progressive censoring. J. Appl. Statist. 35 (10), 1139–1150.