International Journal of Analysis and Applications

Volume 19, Number 1 (2021), 153-164

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-19-2021-153

PREDICTION INTERVALS FOR THE FIRST AND LAST POINT IN FUTURE

SAMPLE HAVING FROM A NEW BATHTUB SHAPE FAILURE RATE LIFE TIME

MODEL IN THE PRESENCE OF OUTLIERS

AYED R.A. ALANZI∗

Department of Mathematics, College of Science and Human Studies at Hotat Sudair, Majmaah University,

Majmaah 11952, Saudi Arabia

∗Corresponding author: a.alanzi@mu.edu.sa

Abstract. Acquiring Bayesian prediction intervals for the first and final points of observation with the

bathtub-shape distribution of the failure rate life-time type under the conditions of available outliers is the

focus of the research. These bounds of predication acquired on the basis of the right Type-II censored

sample. The procedure is presented with the help of a wide range of illustrative example.

1. Introduction

The researchers have to apply the same type of distribution in the context of numerous statistical problems

to use the previously obtained data in order to predict the future data. This need has been the subject of

a number of academic researches and studies with the analysis of the corresponding practical application

( [1]; [2]; [3]). Simultaneously, the research which offered the most significant applications was conducted and

further analyzed [4]. More particularly, the researcher devoted his work to the increasing function of failure

rate or two parameters of the bathtub-shape life-time distribution. Chen states that the distribution has λ

and β as parameters; in that case, the following equations denote the functions of cumulative distribution

and probability density:

Received November 5th, 2020; accepted November 30th, 2020; published January 7th, 2021.

2010 Mathematics Subject Classification. 62E17.

Key words and phrases. bathub-shaped model; Bayesian prediction; censored samples; outliers; bivariate prior.

©2021 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

153

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-153


Int. J. Anal. Appl. 19 (1) (2021) 154

f(x; λ, β) = λβ xβ−1ex
β −λ (ex

β
−1), x > 0, λ > 0, β > 0,(1.1)

and

F(x; λ, β) = 1 − e−λ (e
xβ−1).(1.2)

According to the calculation of Selim [5], the Bayesian estimations are based on the bathbut-shape life-

time distribution with two parameters founded on the record values. In the studies of Niazi and Abd-

Elrahman [6], Bayesian prediction boundaries are acquired for bathbut-shape life-time distribution with two

parameters and a failure rate function. Derivation of Bayesian prediction bounds is done for new model

of life-time bathbut-shape failure rate with the doubly right Type-II censored samples [7]. The theoretical

and practical value of the research was huge, considering the life-time distribution function based on the

Bayesian prediction intervals [8]. There was also an investigation devoted to the Chen distribution in the

e-Bayesian assessment on the basis of the censoring scheme type-I [9]. Th present research deals with model

of Chen(λ, β) in the framework of two different sampling plans to obtain Bayesian intervals for prediction

needed in the future studies: Provided that x1, x2, · · · , xn is an order random sample from model (1.2)

with size n and xr, xr+1, · · · , xn, the sample has (n − r) as the largest for the sample observations. The

statistical analysis uses merely the ordered observations that remained, i.e. x = (x1, · · · , xr). It is clear

that this sampling type includes a complete ordered sample r = n as a special case.

In the first case, a censored sample Type-II is x1 < · · · < xr with r < n defined as an observed sample,

while the unobserved sample is identified with xr+1 < xr+2 < · · ·xn as the remaining values. The study of

all the rest values of n−r is done with the use of an observed sample.

In the second case, the sample of type-II, identical to that in the previous case, is presented with x1 <

· · · < xr while z1 < z2 < · · · < zm is viewed as a future sample of unobserved type with the use of the same

population. In terms of using the selected configurations of sampling, it is essential to specify the prediction

intervals, which apply previous observations with the aim to identify the future observations.

2. Function of Likelihood

Provided that a random sample with the derivation from the population with specified probabilities in

(1.1) and (1.2) is x1 < x2 < · · · < xn, it is time to assign x1,x2, · · · ,xn to the life test. Recording of

the failure times is done merely beginning from the failure timeframe rth with r < n. The analyzed study

outputs are identified in the research as the censored data Type-II with the estimated function of likelihood

presented below:



Int. J. Anal. Appl. 19 (1) (2021) 155

L(λ, β; x) ∝ [1 −FX(x(r); λ, β)]n−r
r∏
i=1

[fX(x(i); λ, β)]

= (λβ)r exp
{ r∑
i=1

(β ln x(i) + x
β
(i)

) −λT1(β; x)
}
, x(k) > 0

(2.1)

where

x = (x(1), · · · ,x(r)),

T1(β; x) =

r∑
i=1

(e
x
β
(i) − 1) + (n−r)(ex

β
(r) − 1).(2.2)

3. Density Functions of Prior and Posterior Types

In the present work, it is taken that the researcher applies a simple prior density function for making

proper measurement. According to the previous assumptions for the λ and β parameters, they are presented

as:

π(λ, β) = π1(λ) π2(β)(3.1)

where π1(λ) is a conjugate prior given by

π1(λ) =
ba11

Γ(a1)
λa1−1 e−b1 λ, λ > 0 (a1, b1 > 0).(3.2)

and

π2(β) =
ba22

Γ(a2)
βa2−1 e−b2 β, β > 0 (a2, b2 > 0).(3.3)

According to Sarhan et all [10], the function analyzed in the points (3.2) and (3.3) should be applied with

the corresponding two parameters to identify the bathtub-shaped distribution. Further summarization of

joint posterior density function of λ and β parameters is given below with the use of the joint prior density

function estimated in (4.1) and likelihood function estimated in (5.4):

π∗2 (λ, β, |x) ∝ λ
r+a1−1βr +a2−1 exp

{ r∑
i=1

(β ln x(i) + x
β
(i)

) − b2 β − λ [T1(β; x) + b1]
}
,(3.4)

Therefor, the λ and β posterior density function can presented as

π∗(λ, β|x) ∝ g1(λ|data) g2(β|data) g3(λ,β|data),(3.5)



Int. J. Anal. Appl. 19 (1) (2021) 156

with g1(λ|β,data) as Gamma density under the constraints of r shape and T1(β; x) scale, while a proper

density function of g2(β|data) is presented below as:

g2(β|data) ∝
1

[T1(β; x)]r
βr−1 exp

{ r∑
i=1

β(ln x(i) − b2)
}

(3.6)

while g3(λ,β|data)) can be presented as follows:

g3(λ, β|data) = λa1 βa2 e−λb1+x
β

.(3.7)

Thereby, taking into consideration the squared error loss function, all functions of λ and β can be interpreted

through the Bayesian estimation as follows:

ĝ(λ, β) =

∫∞
0

∫∞
0
g(λ, β)g1(λ|data) g2(β|λ, data)g3(λ, β|data)dλdβ∫∞

0

∫∞
0
g1(λ|data) g2(β|λ, data) g3(λ, β|data)dλdβ

.(3.8)

Analysis of the equation (5.6) makes it possible to claim that its conversion into a simple closed form is

impossible. Thus, estimation of the Bayesian predications in terms of λ and β as the inputs denoted above

is not possible in this form either. Consequently, one of the assumptions covers the potential efficiency of

applying the technique of importance sampling. It should be done in accordance with the idea suggested

by Chen and Shao [11], which implies approximation of (5.6) to find a solution to the restrictions related to

simple closed forms.

3.1. Technique of Importance Sampling. The methodology of importance sampling technique is applied

to compute and validate the λ, β Bayes estimates as well as a number of constructed relevant functions, for

instance g(λ, β). Algorithm presents the process of approximating the function of posterior density.

Algorithm:

(1) Using g1(β|data) for the estimation of β.

(2) Using g2(λ|β, data) for the estimation of λ.

(3) Repeating the stage 1 and 2 consecutively for generation of (λ1, β1), (λ2, β2), · · · , (λM, βM ).

The following equation presents the process of approximating in the context of the procedure of importance

sampling under the restrictions of Bayesian estimates for g(λ, β) along with the relevant control for squared

error loss:

ĝBS(λ, β) =

M∑
i=1

g(λi, βi) g3(λi, βi|data)∑M
i=M0

g3(λi, βi|data)
,(3.9)



Int. J. Anal. Appl. 19 (1) (2021) 157

4. Prediction with Outliers Presence

In the analysis of the process of predicting the future observations with the outliers presence, it is appro-

priate to use the formal definitions given for (1.1). Besides, I apply the random sample of x1, x2, · · · , xn

created on the basis of Chen (λ, β) on the basis of the given function of population density. The following

stage is using the independent unobserved sample as y1, y2, · · · , ym as a result of using the same data to

form a future sample. The next stage covers further testing of the boundaries of Bayesian prediction for

sth with a single outlier in the range of future estimates for ys, s = 1, 2, · · · , m. The following equation

presents the ys density function for a provided θ under the conditions described above:

h(ys|θ) = D(s) [(s− 1)Fs−2(1 −F)m−sF?f + Fs−1(1 −F)m−sf?

+(m−s)Fs−1(1 −F)m−s−1(1 −F?)f],(4.1)

with

D(s) =

(
m− 1
s− 1

)
(4.2)

The function of density is presented as f = f(y|θ) and the function of cumulative distribution is given

as F = F(y|θ) for all ys that can’t be defined as outliers. Balakrishnan and Ambagabpitiya [12] refer to

f∗ = f∗(y|θ) and F∗ = F∗(y|θ) as to outliers. Acquiring the f∗ and F∗ functions is done for the Chen (λ, β)

model via using a different parameter λ by λ λ0, or λ + λ0 according to the classification of the outliers.

The study of Alanzi and Niazi [8] is devoted the analysis of prediction interval on the basis of using doubly

Type-II censored sample for the future to have λ replaced by λ λ0, or λ + λ0. Furthermore, the research

conducted by Alanzi [13] was devoted to study of a right Type-II censored sample to have λ replaced with

λ λ0 with further calculation of the interval for prediction aimed at first and final future observation. The

present study implies replacement of λ with λ + λ0 as well as calculating the interval for predication needed

for the first and final observation that involve using the right Type-II censored samples.

5. Prediction of The First Observation

The case of prediction of the first implies having distribution in the first y1 in the m-size future sample

via adding s = 1 in (4.1) with only one outlier presence (type λ + λ0); it is done as follow:

h(y1|θ) = (1 −F)m−1f? + (m− 1)(1 −F)m−2(1 −F?)f,(5.1)

It is possible to acquire Y1 density function with a single type λ + λ0 outlier presence in the case of Chen(λ, β)

via changing of (1.1) for f and (1.2) for F in (5.1). On replacement of λ for λ + λ0. f
∗ and F∗ have the same



Int. J. Anal. Appl. 19 (1) (2021) 158

values as they do in (1.1) and (1.2). It is possible to present a density function in the following simplified

form:

h1(y1|λ, β) = f(y1; (λm + λ0), β),(5.2)

with cdf of y1 presented as follows:

H1(y1|λ, β) = F(y1; (λm + λ0), β).(5.3)

Estimation of the predictive density of Y = y1, with x, (λm + λ0) and β is as follows:

h∗1(y|x) =
∫ ∞
0

∫ ∞
0

h1(y1|λ, β) π∗(λ, β|x) dλdβ,(5.4)

Estimation of the Y = y1, predictive distribution function with x, λ and β is as follows:

H∗1 (y |x) =
∫ ∞
0

∫ ∞
0

H1(y1|x, λ, β) π∗(λ, β|x) dλdβ,(5.5)

{(λi, βi); i = 1, 2, · · · ,M} are assumed to be MCMC samples obtained after generation from π∗(λ, β|x)

and corresponding parameters of estimation to ensure consistency of h∗1(y1|x, λ, β) and H∗(y1|x, λ, β).

Thus,

ĥ∗1(y |x) =
M∑
i=1

h1(y1|λi, βi) hi(5.6)

and

Ĥ∗1 (y |x) =
M∑
i=1

H1(y1|λi, βi) hi(5.7)

with

gi =
g3(λi, βi|data)
M∑
i=1

g3(λi, βi|data)
; i = 1, 2, · · · , M.(5.8)

On the basis of the above-mentioned analysis, the Bayesian estimation for Y1,(1−τ) 100 % implies having

P [L(x) ≤ Y1 ≤ U(x)] = 1 − τ, with L(x) as the highest limit for y1 and U(x) as the lowest one. The

following estimation is made on the basis of prior estimates for (5.7), 1 − τ
2

and τ
2
, thus:

P[Y ≥ L(x)|x] = 1 −
τ

2
⇒ Ĥ∗1 (L(x)|x) =

τ

2
(5.9)

and

P [Y ≤ U(x)|x] =
τ

2
⇒ Ĥ∗1 (U(x)|x) = 1 −

τ

2
.(5.10)

Calculation of the prediction limits of y1 is done using the equations (5.9) and (5.10).



Int. J. Anal. Appl. 19 (1) (2021) 159

6. Prediction of The Last Observation

Distribution of the last in a m-size sample with only a single outlier presence is ensured when s = m

is added in (4.1). It is possible to present the Ym density function for a provided θ with a single outlier

presence as follows:

h2(ym|θ) = (m− 1)Fm−2F?f + Fm−1f?,(6.1)

Provided that a single outlier of type λ + λ0 is presence, it is possible to obtain the Ym density function

in the case Chen (λ, β) via replacing (1.1) for f and (1.2) for F in (6.1). The research uses the f∗ value

from (1.1) and F∗ value from (1.2) after λ is replaced with λ + λ0. It is possible to present the mentioned

density function in the following simplified form:

h2(ym|λ, β) =
[
(λ + λ0)

m−1∑
j=0

B1j(ym) + λ(m− 1)
m−2∑
j=0

B2j(ym)
]
, ym > 0,

(6.2)

with

B1j(ym) = a1j(m)f(ym; λ (j + 1) + λ0, β),

B2j(ym) = a2j(m)
[
f(ym; λ (j + 1), β) −f(ym; λ (j + 2) + λ0, β),(6.3)

with ` = 1, 2,

a`j(m) = (−1)j
(
m− `
j

)
,(6.4)

The cdf that is related to pdf h2(ym|λ, β) is as follows:

H2(ym|λ, β) = D(s)
[
(λ + λ0)

m−1∑
j=0

B∗1j(ym) + β(m− 1)
m−2∑
j=0

B∗2j(ym)
]
,(6.5)

with

B∗1j(ym) =
a1j(m)

λ (j + 1) + λ0
F(ym; λ (j + 1) + λ0, β),

B∗2j(ym) =
a2j(m)

λ (j + 1)
F(ym; λ (j + 1), β)

−
a2j(m)

λ (j + 2) + λ0
F(ym; λ (j + 2) + λ0, β),(6.6)

with F(ym; λ (j+1)+λ0, β) presented via (1.2). The ym predictive density provided that there are constraints

for x from the outlier λ + λ0 can be estimated via using the equations (6.2) in (5.4) and the algorithm.



Int. J. Anal. Appl. 19 (1) (2021) 160

h∗2(ym|x) =
∫ ∞
0

∫ ∞
0

h2(ym|λ, β) π∗(λ, β|x) dλdβ,(6.7)

With the predictive cdf of ym, G
∗
2(ym|x) can be defined as follows:

H∗2 (ym|x) =
∫ ∞
0

∫ ∞
0

H2(ym|λ, β) π∗(λ, β|x) dλdβ,(6.8)

with π∗(λ, β|x) presented in (6.5) and H2(ym|λ, β) presented in (3.5). Clearly, it is not possible to present

either (6.7) or (6.8) in a closed form. Consequently, evaluation can’t be done with the use of analytical

approch. On the basis of the MCMC samples {(λi, βi), i = 1, 2, · · · , M}, along with a consistent estimator

used for G∗2(ym|x) and g∗2 (ym|x) simulation of data, it is reasonable to state:

ĥ∗2(ym|x) =
M∑
i=1

h2(ym|λi, βi) hi,(6.9)

and

Ĥ∗2 (ym|x) =
M∑
i=1

H2(ym|λi, βi) hi,(6.10)

with estimation of hi from (5.8). Moreover, it is possible to estimate Ĝ
∗
2(ym|x) and ĝ∗2 (ym|x) for all ym with

the use of MCMC samples {(λi, βi), i = 1, 2, · · · , M} that bear certain similarity to them. Furthermore,

the Bayesian prediction boundaries of a (1−τ)100% type for ym implies having P [L(x) ≤ ym ≤ U(x)] = 1−τ

with L(x) as the lowest Bayesian prediction limit for ym and U(x) is the highest. It is possible to acquire

them and identify as L(x) and U(x), which can be show via non-linear equations solutions as follows for ym

by non-linear equations solutions.

P[Y ≥ L(x)|x] = 1 −
τ

2
⇒ Ĥ∗2 (L(x)|x) =

τ

2
(6.11)

and

P [Y ≤ U(x)|x] =
τ

2
⇒ Ĥ∗2 (U(x)|x) = 1 −

τ

2
.(6.12)

Iterative statistical methodologies could apply the equations (6.11) and (6.12) mentioned above to ensure

advanced regression that implies also implemented control for ym multicellularity or backward analysis and

other factors in other cases.

For instance, generating λ = 0.983884 for the prior parameters a1 = 1.3 and b1 = 2.1 using the

equation (3.2) for the prior density. Also, generating β = 3.90261 for the prior parameters a2 = 3.2 and

b2 = 1.4 using the equation (3.3) for the prior density. Further, Chen distribution with λ = 0.983884 and

β = 1, 3.90261 on the basis of using a different value of r allows generating a random n = 30 size sample.



Int. J. Anal. Appl. 19 (1) (2021) 161

For an illustration of the example, I can make an assumption that there is different m = 10 size sample with

a single outlier λ + λ0 presence. There is a set goal to obtain prediction limits Y1 and Y15 estimated as 95%

for the provided λ0 value in terms of the first and last of future sample. Table 1,2,3 demonstrate the limits

with the provided λ + λ0 values.

Table 1. Bayesian prediction intervals 95 % for y1 and y15 with a single λ + λ0 outlier

presence, n = 30,r = 20.

λ0 Observations y1 y15

0 Lower and Upper limits (0.199913, 0.690383) 0.991503, 1.20199)

Length 0.490471 0.210491

Percentage of Coverage 95.09 % 94.84 %

1 Lower and Upper limits (0.196435, 0.679618) (0.986197, 1.20059)

Length 0.483183 0.214388

Percentage of Coverage 94.57 % 95.24 %

2 Lower and Upper limits (0.193236, 0.669636) (0.985184, 1.20058)

Length 0.4764 0.2154

Percentage of Coverage 94.06 % 95.33 %

3 Lower and Upper limits (0.190279 , 0.660341) (0.984974 , 1.20058)

Length 0.470062 0.21561

Percentage of Coverage 93.41% 95.33%

4 Lower and Upper limits (0.187532 , 0.651651) (0.98493, 1.20058)

Length 0.46412 0.215654

Percentage of Coverage 92.93% 95.33%



Int. J. Anal. Appl. 19 (1) (2021) 162

Table 2. Bayesian prediction intervals 95 % for y1 and y15 with a single λ + λ0, outlier

presence n = 30, r = 25.

λ0 Observations y1 y15

0 Lower and Upper limits (0.193992, 0.679774) (0.983196, 1.19774)

Length 0.485782 0.214544

Percentage of Coverage 94.72 % 95.29%

1 Lower and Upper limits (0.190741, 0.669518) (0.977914, 1.1963)

Length 0.478776 0.218387

Percentage of Coverage 94.18 % 95.62%

2 Lower and Upper limits (0.187741, 0.659979) (0.976825, 1.1963)

Length 0.472238 0.219473

Percentage of Coverage 93.57% 95.74%

3 Lower and Upper limits (0.184957, 0.651072) (0.976582, 1.1963)

Length 0.466115 0.219716

Percentage of Coverage 93.01 % 95.7 %

4 Lower and Upper limits (0.182364, 0.642726) (0.976527, 1.1963)

Length 0.460362 0.219771

Percentage of Coverage 92.02 % 95.7%



Int. J. Anal. Appl. 19 (1) (2021) 163

Table 3. Bayesian prediction intervals 95 % for y1 and y15 with a single λ + λ0 outlier

presence, n = r = 30.

λ0 Observations y1 y15

0 Lower and Upper limits (0.197556, 0.685817) (0.987722, 1.1998)

Length 0.488261 0.212081

Percentage of Coverage 94.99 % 95.06%

1 Lower and Upper limits (0.194182, 0.675296) (0.982433, 1.19838)

Length 0.481114 0.521755

Percentage of Coverage 94.43% 95.44%

2 Lower and Upper limits (0.191073, 0.665526) 0.981386, 1.19838)

Length 0.474453 0.21595

Percentage of Coverage 93.9% 95.59%

3 Lower and Upper limits (0.188195, 0.656417) (0.981161, 1.19838)

Length 0.468223 0.216995

Percentage of Coverage 93.27 % 95.59%

4 Lower and Upper limits (0.185517, 0.647892) (0.981112, 1.19838)

Length 0.462375 0.21722

Percentage of Coverage 92.57% 95.6%



Int. J. Anal. Appl. 19 (1) (2021) 164

7. Conclusion on The Findings

The research analyzes a single λ + λ0 outlier as multiple outliers can be studied only on the basis of a

more profound analysis. It is possible to acquire the Bayesian prediction limits of the first y1 and last y15

in the homogeneous case for the future observation with no outliers by λ + λ0 setting in (5.7) and (6.2)

equation. Table 1, 2 and 3 show the potential impact of λ value on the future observation boundaries and

restrictions.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

References

[1] I. R. Dunsmor, The Bayesian Predictive Distribution in Life Testing Models, Technometrics. 16 (1975), 455-460.

[2] A. Aitchison, I.R. Dunsmore, Statistical Prediction Analysis, Cambridge University Press, (1975).

[3] S. Geisser, Predictive Inference: An Introduction, Chapman and Hall, London, (1993).

[4] Z. Chen, A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Stat. Probab.

Lett. 49 (2000), 155-162.

[5] M. A. Selim, Bayesian Estimations from the Two-Parameter Bathtub-Shaped Lifetime Distribution Based on Record

Values. Pak J. Stat. Oper. Res. 8(2) (2012), 155-165.

[6] S. F. Niazi, A. M. Abd-Elrahman, Bayesian prediction bounds for a two-parameter lifetime distribution with bathtub-

shaped failure rate function. SYLWAN 159(6) (2015), 34-50.

[7] S. F. Niazi, A. M. Abd-Elrahman, Bayesian prediction bounds of doubly type-ii censored samples for a new bathtub shape

failure rate life time model. J. Appl. Math. Stat. Sci. 1 (2016), 21-30.

[8] A. R. Alanzi, S. F. Niazi, Bayesian Prediction Intervals For A New Bathtub Shape Failure Rate Lifetime Model In The

Presence Of Outliers. Adv. Appl. Stat. 63 (2020), 1-22.

[9] A. Algarni, A. M. Almarashi, H. Okasha, H. K. T. Ng, E-Bayesian Estimation of Chen Distribution Based on Type-I

Censoring Scheme. Entropy, 22(6) (2020), 636.

[10] A. M. Sarhan, D. C. Hamliton, B. Smith, Parameter estimation for a twoparameter bathtub-shaped lifetime distribution,

Appl. Math. Model. 36 (2012), 5380-5392.

[11] M.-H. Chen, Q.-M. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graph. Stat. 8(1)

(1999), 2327-2341.

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[13] A. R. Alanzi, Bathtub-Shape Failure Rate Life Time Model in a New Version with the Use of Bayesian Prediction Bounds

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	1. Introduction
	2. Function of Likelihood 
	3. Density Functions of Prior and Posterior Types
	3.1. Technique of Importance Sampling

	4. Prediction with Outliers Presence
	5. Prediction of The First Observation
	6. Prediction of The Last Observation
	7. Conclusion on The Findings
	References