Prediction of Future Lifetimes for a Simple Step-Stress Model with
Type-II Censoring and Rayleigh Distribution
MOHAMMAD A. AMLEH
Department of Mathematics, Faculty of Science
Zarqa University, Zarqa, 13110
JORDAN
Abstract: - In this paper, we discuss the prediction problem of the lifetimes to failure of units from Rayleigh
distribution with Type-II censoring for a simple step-stress setup under cumulative exposure model. We consider
several methods of point prediction, including maximum likelihood predictor, conditional median predictor, and best
unbiased predictor. In addition, we discuss the prediction intervals for future lifetimes of the censored units using
pivotal quantity, highest conditional density, and shortest-length based methods. Monte Carlo simulation is conducted
to compare the proposed prediction methods. Further, a real data set is analyzed for illustrative purposes.
Keywords: - Conditional median predictor; Cumulative exposure model; Maximum likelihood prediction;
Pivot quantity; Prediction intervals; Rayleigh distribution; Step-stress accelerated life test.
Received: June 8, 2021. Revised: February 3, 2022. Accepted: February 20, 2022. Published: March 24, 2022.
1 Introduction
Accelerated life tests (ALTs) are commonly used to
test components operated at higher than usual levels of
stress. The failure data obtained from such tests are
then transformed to estimate the distribution of failures
under specified conditions, which improves component
designs and makes better component selections.
In ALT, the model is chosen according to the
relationship between the parameters of the lifetime
distribution and the conditions of the accelerated stress.
If we use a constant stress level and some selected
stress levels are very low, then we get many non-failed
units during the testing time, which reduces the
effectiveness of the test. To overcome this problem,
step stress accelerated life test (SSALT) can be used.
For more details on ALTs, one may refer to Nelson
[20] and Kundu and Ganguly [17].
In the SSALT, the level of stress in the test will be
increased in steps at different intermediate stages of the
experiment. Accordingly, a test unit is subjected to a
specified level of stress for a prefixed period of time, if
it does not fail during that period of time, then the stress
level is changed for future prefixed time. This process
continues until the test units fail or some termination
conditions will be used. If there are two levels of stress,
the SSALT is known as simple SSALT. In order to
analyze the data under SSALT, there is more than one
model that connects the lifetime's distribution under
various stress levels to the failure times under the step
stress test. The most popular model is known as the
cumulative exposure model (CEM), which was
proposed by Nelson [19]. In this model, it is assumed
that the remaining lifetime of the experiment units is
dependent only on the cumulative exposure the units
have experienced, with no memory on how the
exposure was accumulated. Statistical inferences of
step-stress test under CEM were discussed by many
authors. Estimation of the parameters in a simple step-
stress test under CEM for Weibull and exponential
distributions are addressed by Bai and Kim [4] and
Xiong [22], respectively. Balakrishnan et al. [5]
presented a simple step-stress model under Type-I
censoring and lognormally distributed lifetimes. Mitra
et al. [18] discussed a simple step-stress model for two-
parameter exponential distribution with Type-II
censoring.
For Rayleigh distribution, Ebrahem and Al-Masri [11]
discussed the estimation problem of the parameters for
a simple step-stress model of Rayleigh distribution with
log-linear link function. Chandra and Khan [10]
presented the estimation problem of the parameters for
simple step-stress model under Rayleigh distribution
with Type-I and Type-II censoring. Kumar et al [16]
considered the Bayesian inference for Rayleigh
distribution under step-stress partially accelerated test
with progressive Type-II censoring and binomial
removal. Kotb and El-Din [15] presented a parametric
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inference for step-stress tests from Rayleigh
distribution under ordered ranked set sampling.
It may worth mentioning that no attention has been
paid to the problem of prediction of new lifetimes of
Rayleigh distribution under CEM. In fact, the
prediction problem has not been discussed extensively
for step-stress model in the literature. Basak [6], and
Basak and Balakrishnan ([7], [8]) considered the
problem of predicting failure times of censored units
for a simple step-stress model from exponential
distribution with Type-I censoring and Type-II
censoring, respectively. Recently, Amleh and Raqab
([2], [3]) discussed the prediction problem for step-
stress plan for Lomax distribution under CEM, and for
Weibull distribution under Khamis-Higgens model,
respectively.
In this paper, the simple SSALT for the Rayleigh
distribution based on CEM is considered. It is assumed
that failures occur according to Type-II censoring
scheme, in which the experiment is terminated as soon
as the failure occurs. Specifically, the aim of the
paper is predicting future order statistics based on
Type-II censored units under simple step-stress setup
with Rayleigh CEM using point prediction and
prediction intervals.
The rest of the paper is organized as follows. The CEM
under Rayleigh distribution, basic model assumptions
and maximum likelihood estimation of the original
parameters based on the observed data are discussed in
Section 2. Point predictors including maximum
likelihood predictor, conditional median predictor, and
the best unbiased predictor are presented in Section 3.
In Section 4, we develop different methods for
obtaining prediction limits of the censored lifetimes. To
assess the effectiveness of the prediction procedures,
we perform a simulation study and real data analysis in
Section 5. Finally, we conclude the paper in Section 6.
2 Model description and maximum
likelihood estimation
Rayleigh distribution was introduced in 1880 as part of
a problem in the field of acoustics. Over the following
years, significant work has taken place in the
distribution in different fields of science and
technology. Rayleigh distribution is related to other
known distributions such as Weibull, chi-square and
extreme values distributions. An important feature of
the Rayleigh distribution is that its hazard rate function
is an increasing function of time. This means that if the
failure times have Rayleigh distribution, an intense
aging item occurs. For more details on Rayleigh
distribution one may refer to Johnson et al. [13]. The
probability density function (pdf) of the Rayleigh
distribution is given by
with cumulative distribution function (cdf)
where is the scale parameter. The hazard rate
function of the Rayleigh distribution is increasing in
and given by
So, Rayleigh distribution may describe the lifetime
of an increasing failure rate items.
The simple step-stress test under Type-Ⅱ censoring is
performed as follows. All units are initially put on the
lower stress and run until time . Then, the stress is
increased to high level , and the test continues until a
pre-determined number of failures are observed. Let
denotes the random number of failures before , and
, denotes the number of failures after . If
, then the test is terminated at the first level.
Otherwise, the stress level is accelerated to the next
step, and the test continues until the required failures.
The following are the basic assumptions that specify
our model:
1- Units are tested at two levels of stress
;
2- The lifetimes of the units for both stress levels
follow Rayleigh distribution;
3- The scale parameters for the lifetime
distribution are , corresponding to
stress level ,
4- Failures follow the CEM.
According to the above assumptions, the ordered
lifetimes that are observed, which are denoted by the
vector data , have the following form
Here, represents the observed values of the variable
, which denotes
the Type-II censored order statistics. The CEM for
simple step-stress test is given by
where the equivalent shifting time, , is a solution of
the equation . By solving the above
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equation for , we have
. As a result of that,
the Rayleigh CEM for simple step-stress test is
distributed as
with the corresponding pdf
The likelihood function of the parameters
based on the observed Type-II censored data is given
by
Based on the likelihood function given in (7 a), (7 b)
and (7 c), it is observed that the maximum likelihood
estimators (MLEs) of the parameters exist
only if Therefore, according to the
step-stress setup, the likelihood function in (2.7 c) is
given by
Using Eq.'s (5) and (6), we have
which can be simplified as:
Consequently, the log-likelihood function
is given by
So, the likelihood equations are given by
.
The estimation procedure, through equations (12) and
(13), does not result in closed form. Therefore, Eq.’s
(12) and (13) can be solved simultaneously using a
numerical technique as Newton-Raphson method, or
similar methods, see [1]. The algorithm used for
generating the data and computing the MLEs of the
parameters is performed according to the
following algorithm:
Step 1: Generate a random sample of size following
standard uniform distribution, and
obtain the order statistics:
;
Step 2: Find the random variable such that
, where T represents
the failure time, so we have:
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Step 3: Generate the data based on the order statistics
as follows:
Step 4: Compute the MLEs of using Eq.'s
(12) and (13) based on the censored data
as in (14).
3 Prediction of future order statistics
In this section, we discuss the problem of predicting
new failure times based on some observed Rayleigh
failure times under the CEM. The problem can be
described as follows. Let
denote the observed ordered lifetime units, which is
known as informative sample, and let
, be the unobserved future lifetime taken from
the same sample. The prediction problem concerns on
how we can predict the future lifetimes , given the
observed ordered statistics ,.
Based on the Markovian property of censored order
statistics, it is known that the conditional distribution of
givenwhere:
is equivalent to the distribution of given
. Therefore, the density of given is
the same as the density of the order statistic
out of units from the population with left
truncated density
, where
and are given in Section 2 as in Eq. (5) and
(6), respectively. Therefore, the density of
given can be expressed as:
where
.
3.1 Maximum likelihood predictor
The maximum likelihood prediction method was
suggested by Kaminsky and Rhodin [12]. This method
includes the prediction of future order statistics in
addition to estimating the unknown parameters in the
proposed model. The predictive likelihood function
(PLF) of is given by
where is the conditional
density ofgiven the observed value of , as
in Eq. (15), and is the density of . In
fact, the PLF of can be formed as
Taking the case when , we obtain
So, the log PLF can be written as
Using (19), the predictive likelihood equations (PLEs)
forare obtained and presented as follows:
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Since Eq.'s (20)-(22) cannot be solved explicitly,
numerical techniques will be used to solve them
simultaneously, which leads to find the maximum
likelihood predictor (MLP) of and the predictive
maximum likelihood estimators (PMLEs) of and .
The resulting MLP of is denoted by.
3.2 Conditional median predictor
Raqab and Nagaraja [19] proposed a point predictor
based on the conditional distribution of given,
known as conditional median predictor (CMP). A
predictor is called the CMP of , if it is the median
of the conditional distribution of given, that is
Using the conditional distribution of given, we
can obtain
It can be shown that, given, the distribution of
is a Beta distribution with parameters
, denoted by . So, if is
a random , and represents
the median of, the CMP of can be obtained as
The CMP of Y can be computed approximately by
replacing in Eq. (23) by their corresponding
MLEs.
3.3 Best Unbiased Predictor
A point predictor of is called a best
unbiased predictor (BUP) of, if the mean of its
prediction error, is zero and the variance
of its prediction error, is less than or
equal to that of any other unbiased predictor of .
Using the conditional density of given , as in
Eq. (15), the BUP of is given by
Using the binomial expansion:
we obtain
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The BUP of can be approximated by substituting the
MLEs of the unknown parameters and in Eq.
(24).
4 Prediction intervals
Another aspect of prediction problem is to predict the
future unobserved lifetimes by constructing prediction
intervals (PIs) for based on
the Type-II censored sample .
The pivotal, highest conditional density, and shortest-
length based methods are considered in this section.
4.1 Pivotal-based PIs
Let us consider the random variable
Since the distribution of given, is a Beta
distribution with parameters ,
then can be considered as a pivotal quantity for
obtaining the PI of . By considering
, as a prediction coefficient and using Eq. (25),
we obtain
where is the percentile of the distribution
. Therefore, a
pivotal PI of is, where
The prediction limits can be
evaluated approximately by replacing and by
their corresponding MLEs.
4.2 Highest conditional density PIs
As described above, the conditional distribution of
given is . Therefore,
the conditional pdf of is:
The density in Eq. (26) is unimodal function in. An
interval is called highest conditional density
(HCD) PI of content if
, where
for some . Now, if, then is
a unimodal function, and it attains its maximum value
at
. Following Theorem 9.3.2 of
Casella and Berger [9], the HCD PI can be obtained by
finding two percentiles and such that
, with , satisfying
and,
Eq.'s (27) and (28) can be simplified as
and
where
is the incomplete beta function and is the gamma
function. Consequently, a HCD PI of
is given by , with
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For the special case when ,
, which is
a decreasing function in with and
. Therefore, the PI for is of the form
such that . This concludes
that
,
When , is uniform
. Here are taken such that
and . So, we have
and
Finally, when , the density
, is increasing
function with and . In this
case, we select the PI for Y to be of the form
such that
which implies that
So, a HCD PI of is given
by
4.3 Shortest-Length based Method
Based on the fact that the conditional distribution of
given is
a , we select the constants c
and d that satisfy the equation:
Here, the constants and are chosen to minimize the
length of the PI .
The optimization problem for figuring out the shortest-
length (SL) PI can be expressed as:
Minimize
Subject to
The SL PI can be constructed by
minimizing the Lagrangian function:
where is the Lagrange multiplier. By differentiating
with respect to , respectively, we have:
where represents the density of the
distribution . The above equations can be
formed equivalently as:
and,
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The constants and are obtained by solving (31) and
(32) numerically. Hence, a PI of,
based on this technique, is given by,
such that:
5 Simulation study and data analysis
In this section, we conduct a simulation study for
computations of the prediction methods that are
presented in the above sections and comparing their
performance. A real data set is considered to illustrate
the different techniques suggested in this paper.
5.1 Simulation study
In this subsection, we perform an intensive Monte
Carlo simulation study for performance evaluation of
the suggested predictors; which were presented in
Section 3. The performances are measured in terms of
the biases and the mean square prediction errors
(MSPEs) of the predictors. The bias and MSPE of a
predictor of , are defined as
and,
respectively. In addition, we compare the PIs, that are
discussed in Section 4, in terms of their estimated
average lengths (ALs) and coverage probabilities
(CPs).
Consequently, a Monte Carlo simulation is conducted
based on different censoring schemes and sample sizes
from the Rayleigh distribution under CEM. For
particular values of , we generate Type-II
censored samples as described in Section 2 based on
the following schemes:
Scheme 1:,, and
Scheme 2:,, and .
In both cases, we find the value of the point
predictors; MLP, CMP and BUP. Moreover, we
compute 95% PIs based on pivotal quantity, HCD
and SL methods. Type-II censored samples from
Rayleigh distribution were randomly generated
under these two different schemes with 2000
replications of the simulation process. Using these
random samples, prediction biases and MSPEs of
the predictors are computed. The so obtained results
are presented in Table 1. In Table 2, we have
presented the ALs and CPs of the PIs.
Based on these tables, we observe the following
remarks
1. For fixed values of , biases and the MSPEs
of the point predictors increase as increases, which is
due to the variation of the lifetime to be predicted as
gets large.
2. The prediction biases of the BUP are smaller than
those of the CMP and the MLP for all the
considered cases. However, the biases resulted from
the CMP are close to the biases of the BUP and
smaller than the biases of the MLP. By considering
the MSPE as an optimality criterion, it is observed
that, the CMP outperforms both the BUP and the MLP.
Further, it is noticed that the MSPEs of the three
predictors are close to each other's, especially when is
close to , which can be explained by observing the
closeness of the MLEs of the parameters and the
corresponding PMLEs in most of the considered cases.
3. The PIs obtained using the SL method is more
efficient than other methods based on the AL
criterion. Its performance tends to be higher when
gets large. On the other hand, it is observed that the
HCD PIs outperforms the pivot PIs in the sense of
ALs when tends to be close to . As
approaches, the pivot PIs are competeive. Based
on the CP criterion, the HCD PIs are superior to the
PIs obtained by SL and pivot methods. The CPs of
SL and pivot PIs are very close. It is evident that the
CPs of all obtained PIs increase when increases.
In this sense, the worse CP occurs when the lifetime
to be predicted is immediately after the last
observed lifetime.
As a summary, for point prediction aspect, the CMP is
the best predictor as it is computationally attractive and
has good performances in terms of the biases and
MSPE criteria. For prediction interval part, the SL
method produces efficient PIs over other methods
based on the AL and CP criteria.
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Table 1: Biases and MSPEs of the point predictors for the censored lifetimes
Scheme 1:,, and .
(n, r)
s
MLP
CMP
BUP
(30, 20)
Bias
MSPE
Bias
MSPE
Bias
MSPE
22
-0.0615
0.0245
-0.0218
0.0233
-0.0071
0.0237
24
-0.0673
0.0326
-0.0175
0.0315
-0.0033
0.0323
26
-0.1041
0.0509
-0.0420
0.0473
-0.0279
0.0478
28
-0.1177
0.0682
-0.0344
0.0640
-0.0174
0.0651
30
-0.2057
0.1478
-0.0501
0.1360
-0.0106
0.1411
(40, 25)
27
-0.0433
0.0167
-0.0157
0.0161
-0.0050
0.0164
30
-0.0560
0.0242
-0.0201
0.0233
-0.0101
0.0236
35
-0.0825
0.0469
-0.0275
0.0456
-0.0174
0.0461
38
-0.1187
0.0829
-0.0381
0.0808
-0.0237
0.0820
40
-0.1924
0.1502
-0.0416
0.1465
-0.0038
0.1523
(50, 30)
32
-0.0294
0.0123
-0.0080
0.0121
0.0005
0.0124
35
-0.0413
0.0153
-0.0151
0.0148
-0.0073
0.0150
40
-0.0568
0.0248
-0.0189
0.0238
-0.0115
0.0240
45
-0.0819
0.0472
-0.0270
0.0459
-0.0184
0.0463
50
-0.1971
0.1443
-0.0574
0.1332
-0.0211
0.1365
Scheme 2:,, and .
(n, r)
s
MLP
CMP
BUP
(30, 20)
Bias
MSPE
Bias
MSPE
Bias
MSPE
22
-0.1077
0.0878
-0.0320
0.0843
-0.0044
0.0866
24
-0.1203
0.1143
-0.0244
0.1135
0.0018
0.1173
26
-0.1927
0.1760
-0.0702
0.1646
-0.0439
0.1671
28
-0.2593
0.2920
-0.0919
0.2761
-0.0605
0.2803
30
-0.4489
0.6233
-0.1366
0.5729
-0.0635
0.5903
(40, 25)
27
-0.0930
0.0635
-0.0408
0.0607
-0.0210
0.0613
30
-0.0959
0.0871
-0.0249
0.0856
-0.0062
0.0874
35
-0.1524
0.1682
-0.0394
0.1675
-0.0204
0.1703
38
-0.2459
0.3144
-0.0790
0.3043
-0.0522
0.3086
40
-0.4218
0.6127
-0.1264
0.5870
-0.0571
0.6042
(50, 30)
32
-0.0634
0.0463
-0.0232
0.0453
-0.0074
0.0459
35
-0.0807
0.0587
-0.0282
0.0563
-0.0134
0.0571
40
-0.1070
0.0986
-0.0289
0.0958
-0.0150
0.0971
45
-0.1756
0.1728
-0.0613
0.1659
-0.0455
0.1673
50
-0.4225
0.5844
-0.1366
0.5251
-0.0686
0.5354
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Table 2: ALs and CPs of 95% PIs of the censored lifetimes
Scheme 1:,, and .
(n, r)
s
Pivotal Method
HCD Method
SL Method
(30, 20)
AL
CP
AL
CP
AL
CP
22
0.3579
0.654
0.3289
0.653
0.3288
0.656
24
0.5233
0.831
0.5079
0.831
0.5034
0.832
26
0.7059
0.883
0.7142
0.885
0.6899
0.881
28
0.9489
0.918
1.0519
0.933
0.9323
0.907
30
1.7211
0.951
0.966
1.6782
0.942
(40, 25)
27
0.2517
0.593
0.2306
0.574
0.2305
0.575
30
0.4062
0.806
0.3954
0.805
0.3937
0.809
35
0.6763
0.892
0.6918
0.886
0.6678
0.891
38
0.9532
0.914
1.0842
0.935
0.9416
0.901
40
1.6706
0.955
0.960
1.6323
0.945
(50, 30)
32
0.2046
0.594
0.1871
0.569
0.1871
0.569
35
0.3209
0.761
0.3111
0.760
0.3105
0.763
40
0.4776
0.848
0.4764
0.847
0.4715
0.850
45
0.6874
0.884
0.7123
0.887
0.6816
0.883
50
1.6841
0.954
0.984
1.6464
0.952
Scheme 2:,, and .
(n, r)
s
Pivotal Method
HCD Method
SL Method
(30, 20)
AL
CP
AL
CP
AL
CP
22
0.6719
0.661
0.6175
0.640
0.6173
0.640
24
0.9939
0.802
0.9646
0.799
0.9560
0.797
26
1.3014
0.860
1.3167
0.866
1.2719
0.852
28
1.8087
0.914
2.0051
0.928
1.777
0.907
30
3.2288
0.952
0.962
3.1485
0.945
(40, 25)
27
0.4768
0.607
0.4368
0.578
0.4367
0.579
30
0.7790
0.777
0.7583
0.780
0.7551
0.786
35
1.2733
0.876
1.3026
0.886
1.2574
0.867
38
1.7843
0.879
2.0295
0.916
1.7626
0.870
40
3.1547
0.937
0.953
3.0825
0.931
(50, 30)
32
0.3734
0.600
0.3414
0.563
0.3414
0.564
35
0.6075
0.756
0.5888
0.760
0.5878
0.762
40
0.9015
0.828
0.8993
0.827
0.8900
0.831
45
1.2881
0.850
1.3349
0.863
1.2773
0.851
50
3.1285
0.948
0.952
3.0586
0.945
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5.2 Data analysis
To clarify the prediction methods proposed in this
paper, we perform a real data analysis. The dataset is
taken from Han and Kundu [12], it represents a total of
31 failure times (in hundred hours) from a sample of
35 prototypes of a solar lighting device with two
dominant failure modes, controller failure and
capacitor failure. Here, temperature is the stress factor
whose level was changed during the test in the range
of 293K to 353K with the normal operating
temperature at 293K, and stress change time at 500
hours. These data have been used previously by Kotb
and El-Din [13]. The data are recorded in Table 3.
To visualize the accuracy of our model, i.e., Rayleigh
CEM, the true cdf of the lifetimes is plotted in Fig. 1,
along with the corresponding cdf based on the
maximum likelihood estimate. However, it was shown
by Kotb and El-Din [15] that Rayleigh distribution is
appropriate for analyzing this data set.
Suppose the life test is terminated when the
lifetime is observed, i.e., we observe a Type-II
censored sample with , . Our purpose
is to obtain point predictors of the unobserved
lifetimes, and the
associated PIs.
First we compute the MLEs of by solving
Eq.'s (12) and (13) simultaneously, it is found
that and. For predicting the
future censored lifetimes, point predictors and PIs are
reported in Table 4.
It can be observed that the point predictors are close to
the true values, with advantage to the CMP. Moreover,
the point predictors obtained are lying within all
considered PIs. It can be observed that all PIs obtained
contain the true values of the future order statistics. It
is also noticed that the PIs become wider when gets
large, the reason is that the fluctuation of
tends to be high as moves away from the observed
failures times. Although all PIs are close in the sense
of AL criterion, the PIs constructed by SL method
have shortest lengths.
Table 4: Point predictors and PIs for future lifetimes
of .
Point predictors of
s
True value
MLP
CMP
BUP
28
5.408
5.379
5.412
5.425
30
5.483
5.475
5.518
5.531
31
5.717
5.533
5.582
5.594
33
-----
5.682
5.750
5.765
35
-----
5.967
6.095
6.128
95% PIs of
s
True
value
Pivotal PI
HCD PI
SL PI
28
5.408
(5.348,
5.571)
(5.340,
5.541)
(5.339,
5.540)
30
5.483
(5.393,
5.737)
(5.388,
5.723)
(5.377,
5.707)
31
5.717
(5.426,
5.833)
(5.427,
5.832)
(5.409,
5.803)
33
-----
(5.522,
6.093)
(5.540,
6.166)
(5.500,
6.057)
35
-----
(5.703,
6.740)
(5.753,
)
(5.657,
6.663)
Fig. 1: The empirical cdf (dots); and the estimated cdf of
Rayleigh CEM model based on MLE (solid line).
Table 3: Lifetimes of prototypes of a solar lighting
device on a simple step-stress test
Temperature
Level
Recorded data
0.140 0.783 1.324
1.582 1.716 1.794
1.883 2.293 2.660
2.674 2.725 3.085
3.924 4.396 4.612
4.892
5.002 5.022 5.082
5.112 5.147 5.238
5.244 5.247 5.305
5.337 5.407 5.408
5.445 5.483 5.717
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6 Conclusions
In this paper, we have addressed the prediction of
future lifetimes of a simple step stress test of Rayleigh
distribution under CEM when the data are Type-II
censored. Several point predictors are proposed
including, maximum likelihood, conditional median,
and best unbiased predictors. We have also discussed
another aspect of prediction, which is constructing
prediction intervals for the future lifetimes. We have
compared the performance of the predictors obtained
by extensive Monte Carlo simulation study by
considering the biases and MSPEs of the suggested
predictors. Prediction intervals were also assessed in
terms of the average lengths and coverage
probabilities. It is observed that the CMP has the best
performance among all point predictors. In the context
of interval prediction, it is observed that the SL based
method is the most suitable method for obtaining PIs
of future lifetimes.
It is worth mentioning that the results of this paper
were mainly obtained for Type-II censored scheme,
but our techniques can be performed for other
censoring schemes, as Type-I, hybrid or progressive
censoring.
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