# Journal of Statistical Theory and Applications

Volume 17, Issue 1, March 2018, Pages 146 - 157

# Bayesian Inference Based on Multiply Type-II Censored Sample from a General Class of Distributions

Authors
A.R. Shafaya_shafay2013@yahoo.com
Nature Science Department, Community College of Riyadh, King Saud University, P.O. Box 28095, Riyadh 11437, Saudi Arabia, Department of Mathematics, Faculty of Science, Fayoum University, Fayoum, Egypt
M.M. Mohie El-Dinmmmmoheeldin@yahoo.com
Department of Mathematics, Faculty of Science, Al-Azhar University, Egypt
Y. Abdel-Atyyahia1970@yahoo.com
Department of Mathematics, Faculty of Science, Al-Azhar University, Egypt
Received 18 October 2016, Accepted 21 December 2017, Available Online 31 March 2018.
DOI
https://doi.org/10.2991/jsta.2018.17.1.11How to use a DOI?
Keywords
Order statistics; Multiply Type-II censored sample; Bayesian prediction; Bayesian estimation; The inverse Weibull distribution; The inverse exponential distribution; The inverse Rayleigh distribution
Abstract

In this paper, we consider a general form for the underlying distribution and a general conjugate prior, and develop a general procedure for Bayesian estimation based on an observed multiply Type-II censored sample. The problem of predicting the order statistics from a future sample are also discussed from a Bayesian view-point. For the illustration of the developed results, the inverse Weibull distribution is used as example. Finally, two numerical examples are presented for illustrating all the inferential procedures developed here.

Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

## 1. Introduction

In reliability analysis, experiments often get terminated before all units on test fail due to cost and time considerations. In such cases, failure information is available only on part of the sample, and only partial information on all units that had not failed. Such data are called censored data. There are several forms of censored data. One of the most common forms of censoring is Type-II right censoring which can be described as follows: Consider n identical units under observation in a life-testing experiment and suppose only the first rn failure times X1:n, ⋯, Xr:n are observed and the rest of the data are only known to be larger than Xr:n. A generalization of Type-II censoring scheme is multiple Type-II censoring scheme. Under this scheme, we observe only the j1th,j2th,,jrth failure times Xj1:n, Xj2:n,...,Xjr:n, where 1 ≤ j1 < j2 < ⋯ < jrn, and the rest of the data are not available. Particular applications of such censoring are found in reliability theory and survival analysis. Surveys regarding censored data can be found in Nelson , Balakrishnan and Cohen , Cohen , Balakrishnan and Aggarwala , and McCool . For a survey on multiple Type-II censoring, one may refer to Kong .

Let X1:n < X2:n < ⋯ < Xn:n be the order statistics from a random sample of size n from an absolutely continuous cumulative distribution function (CDF) F(x) ≡ F(x|θ) with probability density function (PDF) f (x) ≡ f (x|θ), where the parameter θ ∈ Θ may be a real vector. These order statistics have been used in a wide range of problems, including robust statistical estimation, detection of outliers, characterization of probability distributions, goodness-of-fit tests, entropy estimation, analysis of censored samples, reliability analysis, quality control and strength of materials; for more details, see Arnold et al. , David and Nagaraja , Balakrishnan and Rao [7, 8], and the references contained therein. The joint density of multiply Type-II censored order statistics X_=(Xj1:n,Xj2:n,,Xjr:n) is given by

fX_(x_)=n!(njr)![1F(xjr)](njr)i=1r1(jiji11)![F(xji)F(xji1)](jiji11)f(xji),
where x_=(xj1,,xjr) is a vector of realizations and j0 = 0.

We consider here the general inverse exponential form for the underlying distribution, suggested by Mohie El-Din et al. , that is described as follows; Motivated by the fact that the CDF can be written in the form

F(x|θ)=exp[λ(x;θ)],
where λ (x; θ) = − lnF(x|θ). Of course, some conditions need to be imposed so that F(x|θ) is a valid CDF. These conditions are: λ (x; θ) is continuous, monotone decreasing and differentiable function, with λ (x; θ) → ∞ as x → −∞ and λ (x; θ) → 0 as x → ∞. The PDF corresponding to (1.2) is given by
f(x|θ)=λ(x;θ)exp[λ(x;θ)],
where λ′(x; θ) is the derivative of λ (x; θ) with respect to x. With an appropriate choice of λ (x; θ), several distributions that are used in reliability studies can be obtained as special cases such as the inverse exponential, inverse Rayleigh, inverse Weibull, inverse Pareto, negative exponential, negative Weibull, negative Pareto, negative power, Gumbel, exponentiated-Weibull, loglogistic, Burr X, inverse Burr XII and inverse paralogistic distributions.

In many practical problems, one may wish to use past data to predict an observation from a future sample from the same population. As in the case of estimation, a predictor can be either a point or an interval predictor. Several researchers have considered Bayesian prediction for future observations based on Type-II censored data; see Dunsmore , Nigm and Hamdy , Nigm [20, 21], AL-Hussaini and Jaheen , AL-Hussaini , and Raqab and Madi . Bayesian prediction bounds for future observations based on multiply Type-II censored data have been discussed by several authors, including Abdel-Aty et al. , Schenk et al. , Mohie El-Din et al. , and Shafay et al. . Recently, Mohie El-Din et al.  have considered the general inverse exponential form for the underlying distribution, given in (1.2), and a general conjugate prior and developed a general procedure for determining the two-sample Bayesian prediction for future lifetimes based on a right Type-II censored sample. In this paper, we discuss the same problem based on a multiply Type-II censored data which involves some additional complications.

The rest of this paper is organized as follows. In Section 2, we present the structure of the posterior distribution and derive the Bayesian estimation for the unknown parameters. In Section 3, the problem of predicting the order statistics from a future sample is then discussed when the observed sample is a multiply Type-II censored sample from the same parent distribution. The results for the inverse Weibull distribution are presented in Section 4. Finally, in Section 5, we present some numerical results for illustrating all the inferential methods developed here.

## 2. The Posterior Distribution and Bayesian Estimation

In this section, we use the general inverse exponential form given in (1.2) and a general conjugate prior to develop general procedure for determining the Bayesian estimator of the unknown parameter θ.

Upon using (1.2) and (1.3) in (1.1), we obtain the likelihood function of the multiply Type-II censored sample X_=(Xj1:n,Xj2:n,,Xjr:n) as

L(θ;x_)k1=1j1j0kr=1jrjr1w=1njr+1Ck1,,kr,wA(θ;x_)exp[B(θ;x_)],
where Ck1,,kr,w=(i=1rcki(jiji1))cw(njr+1), ci(j)=(1)ji(i1)!(ji)! for i = 1,..., j,
A(θ;x_)=i=1(λ(xji;θ)),
and
B(θ;x_)=(njrw+1)λ(xjr;θ)+i=1r(kiλ(xji;θ)+(jiji1ki)λ(xji1;θ)).

From the Bayesian viewpoint, the unknown parameter is regarded as a realization of a random variable, which has some prior distribution. We consider here a general conjugate prior, suggested by AL-Hussaini , that is given by

π(θ;δ)C(θ;δ)exp[D(θ;δ)],
where θ ∈ Θ is the vector of parameters of the distribution in (1.2) and δ is the vector of prior parameters. The prior family in (2.2) includes several priors used in the literature as special cases.

Upon combining (2.1) and (2.2), the posterior density function of θ, given the multiply Type-II censored data, is obtained as

π*(θ|x_)=L(θ;x_)π(θ;δ)/θΘL(θ;x_)π(θ;δ)dθ=I1k1=1j1j0kr=1jrjr1w=1njr+1Ck1,,kr,wη(θ;x_)exp[ζ(θ;x_)],
where
η(θ;x_)=C(θ;δ)A(θ;x_),ζ(θ;x_)=B(θ;x_)+D(θ;δ),
and
I=k1=1j1j0kr=1jrjr1w=1njr+1Ck1,,kr,wθΘη(θ;x_)exp[ζ(θ;x_)]dθ.
The Bayesian estimator of θ under the squared error loss function is the mean of the posterior density function, given by
θ^=Eπ*(θ|x_)[θ]=I1k1=1j1j0kr=1jrjr1w=1njr+1Ck1,,kr,wθΘθη(θ;x_)exp[ζ(θ;x_)]dθ.

## 3. Bayesian Predication

Let Y1:mY2:m ≤ ... ≤ Ym:m be the order statistics from a future random sample of size m from the same population. We use here the general inverse exponential form given in (1.2) and the general conjugate prior given in (2.2) to develop a general procedure for deriving the point and interval predictions for Ys:m, 1 ≤ sm, based on the observed multiply Type-II censored sample.

It is well known that the marginal density function of the sth order statistic from a sample of size m from a continuous distribution with CDF F(x) and PDF f (x) is given, see Arnold et al. , by

fYs:m(y|θ)=m!(s1)!(ms)![F(y)]s1[1F(y)]msf(y),y>0.
Upon substituting (1.2) and (1.3) in (3.1), the marginal density function of Ys:m in (3.1) becomes
fYs:m(y|θ)=m!(s1)!q=1ms+1cq(ms+1)(λ(y;θ))exp[(mq+1)λ(y;θ)],
where cq(ms+1)=(1)(msq+1)(q1)!(msq+1)! for q = 1,...,ms + 1.

Upon combining (2.3) and (3.2), the Bayesian predictive density function of Ys:m, given X = x, is then

fYs:m*(y|x)=θΘπ*(θ|x_)fYs:m(y|θ)dθ=m!I1(s1)!k1=1j1j0kr=1jrjr1w=1njr+1q=1ms+1Ck1,,kr,wcq(ms+1)×θΘ(λ(y;θ))η(θ;x_)exp[(ζ(θ;x_)+(mq+1)λ(y;θ))]dθ.
From (3.3), we simply obtain the predictive cumulative distribution function FYs:m*(t|x_), for t ≥ 0, as
FYs:m*(t|x)=0tfYs:m*(y|x_)dy=m!I1(s1)!k1=1j1j0kr=1jrjr1w=1njr+1q=1ms+1Ck1,,kr,wcq(ms+1)mq+1×θΘη(θ;x_)exp[{ζ(θ;x_)+(mq+1)λ(t;θ)}]dθ.
The Bayesian point predictor of Ys:m under the squared error loss function is the mean of the predictive density, given by
Y^s:m=0yfYs:m*(y|x_)dy.
The Bayesian predictive bounds of a two-sided equi-tailed 100(1 − γ)% interval for Ys:m, 1 ≤ sm, can be obtained by solving the following two equations:
FYs:m*(LYs:m|x_)=γ2andFYs:m*(UYs:m|x)=1γ2,
where LYs:m and UYs:m denote the lower and upper bounds, respectively.

### Remark 3.1.

In the case when the observed sample is right Type-II censored (i.e., ji = i, 1 ≤ ir), the predictive density and cumulative distribution functions of Ys:m given in (3.3) and (3.4) reduce to expressions (2.13) and (2.14) of Mohie El-Din et al. , respectively.

## 4. The Inverse Weibull Distribution

Several distributions that are used in reliability studies can be obtained as special cases from the general inverse exponential form given in (1.2). We consider in this section the inverse Weibull distribution as illustrative example. The inverse Weibull model has been derived as a suitable model for describing the degradation phenomena of mechanical components, such as the dynamic components of diesel engines, see for example Murthy et al. . The physical failure process given by Erto and Rapone  also leads to the inverse Weibull model. Erto and Rapone  showed that the inverse Weibull model provides a good fit to survival data such as the times to breakdown of an insulating fluid subject to the action of constant tension, see also Nelson . Calabria and Pulcini  provided an interpretation of the inverse Weibull distribution in the context of a load-strength relationship for a component.

The distribution function of the inverse Weibull distribution is given by

F(x|θ)=exp[α/xβ],x>0,
where α > 0 and β > 0 are the scale and shape parameters, respectively, and so we have
λ(x;α,β)=α/xβandλ(x;α,β)=αβ/xβ+1.
We provide here the Bayesian inference for the inverse Weibull distribution, when the scale parameter is unknown and the shape parameter is known. It is assumed that the scale parameter has a gamma prior distribution with the shape and scale parameters as c and d, respectively and it has the density function
π(α;δ)αc1exp[dα],α>0,
where c and d can be chosen to suit the prior belief of the experimenter in terms of location and variability of the prior distribution. Moreover, Jeffrey’s prior can be obtained as a special case of (4.2) by substituting c = d = 0. So, we have
C(α;δ)=αc1andD(α;δ)=dα,
where δ = (c, d).

Hence, the posterior density function of α, given the multiply Type-II censored data, become

π*(α|x_)=I1k1=1j1j0kr=1jrjr1w=1njr+1Ck1,,kr,wαr+c1×exp[α((njrw+1)xjrβ+k1xj1β+i=1r(kixjiβ+(jiji1ki)xji1β)+d)],
where
I=Γ(r+c)k1=1j1j0kr=1jrjr1w=1njr+1Ck1,,kr,w[(njrw+1)xjrβ+k1xj1β+i=2r(kixjiβ+(jiji1ki)xji1β)+d](r+c).
The Bayesian estimator of α is then given by
α^=Γ(r+c+1)I1k1=1j1j0kr=1jrjr1w=1njr+1Ck1,,kr,w×[(njrw+1)xjrβ+k1xj1β+i=2r(kixjiβ+(jiji1ki)xji1β)+d](r+c+1).
The predictive density and cumulative distribution functions of Ys:m are given, respectively, as
fYs:m*(y|x_)=m!Γ(r+c+1)I1(s1)!k1=1j1j0kr=1jrjr1w=1njr+1q=1ms+1βCk1,,kr,wcq(ms+1)yβ+1×[(njrw+1)xjrβ+k1xj1β+i=2r(kixjiβ+(jiji1ki)xji1β)+(mq+1)yβ+d](r+c+1)
and
FYs:m*(t|x_)=m!Γ(r+c)I1(s1)!k1=1j1j0kr=1jrjr1w=1njr+1q=1ms+1Ck1,,kr,wcq(ms+1)mq+1×[(njrw+1)xjrβ+k1xj1β+i=2r(kixjiβ+(jiji1ki)xji1β)+(mq+1)tβ+d](r+c).
The Bayesian point predictor of Ys:m under the squared error loss function is then obtained as the mean of the Bayesian predictive density function in (4.4), and this would require numerical integration. The Bayesian predictive bounds of a two-sided equi-tailed 100(1 − τ)% interval for Ys:m can be obtained by solving the two equations in (3.6) where FYs:m*(t|x_) as in (4.5).

### Remark 4.1.

In the case when the observed sample is right Type-II censored (i.e., ji = i, 1 ≤ ir), the posterior density function of α becomes

π*(α|x_)=I1w=1nr+1cw(nr+1)αr+c1exp[α((nrw+1)xrβ+i=1r(1xiβ)+d)],
where
I=Γ(r+c)w=1nr+1cw(nr+1)[(nrw+1)xrβ+i=1r(1xiβ)+d](r+c),
and so the Bayesian estimator of α becomes
α^=Γ(r+c+1)I1w=1nr+1cw(nr+1)[(nrw+1)xrβ+i=1r(1xiβ)+d](r+c+1).
The predictive density and cumulative distribution functions of Ys:m in this case become
fYs:m*(y|x_)=m!Γ(r+c+1)I1(s1)!w=1nr+1q=1ms+1βcw(nr+1)cq(ms+1)yβ+1×[(nrw+1)xrβ+i=1r(1xiβ)+(mq+1)yβ+d](r+c+1)
and
FYs:m*(t|x_)=m!Γ(r+c)I1(s1)!w=1nr+1q=1ms+1cw(nr+1)cq(ms+1)mq+1×[(nrw+1)xrβ+i=1r(1xiβ)+(mq+1)tβ+d](r+c).

### Remark 4.2.

The inverse Weibull distribution contains many of important special cases such as the inverse exponential and inverse Rayleigh distributions. The corresponding results of the inverse exponential and inverse Rayleigh distributions can be obtained by setting β = 1 and 2, respectively.

## 5. Numerical Results

To illustrate the inferential procedures developed in the preceding sections, we present in this section a numerical study for the inverse exponential and inverse Rayleigh distributions as special cases of the inverse Weibull distribution when β = 1 and β = 2, respectively.

## 5.1. Numerical results for the inverse exponential distribution

To illustrate the prediction results for the inverse exponential distribution, we generated order statistics from a sample of size n = 10 from the inverse exponential distribution with α = 5. The generated order statistics are listed as follows: 1.31662, 3.04073, 5.25839, 5.39344, 5.64310, 5.88457, 13.72085, 22.43066, 29.19947, 1393.06058.

We then used these data to consider two different multiply Type-II censoring schemes:

1. (1)

Scheme 1: r = 6 with j1 = 1, j2 = 3, j3 = 5, j4 = 6, j5 = 7 and j6 = 8. Under this censoring scheme, we obtained the following data: 1.31662, 5.25839, 5.64310, 5.88457, 13.72085 and 22.43066;

2. (2)

Scheme 2: r = 8 with j1 = 1, j2 = 3, j3 = 4, j4 = 5, j5 = 6, j6 = 7, j7 = 9 and j8 = 10. Under this censoring scheme, we obtained the following data: 1.31662, 5.25839, 5.39344, 5.64310, 5.88457, 13.72085, 29.19947 and 1393.06058.

We assume these data to have come from the inverse exponential distribution, where the parameter α is unknown. Based on the above two multiply Type-II censoring schemes, we used the results presented earlier in Section 4 (when β = 1) to compute the Bayesian estimate of α. In addition, suppose we want to predict the order statistics from a future unobserved sample with size m = 10. In this case, the point predictors and the 95% two-sample Bayesian prediction intervals for the order statistics Ys:10, for s = 1,...,10, are all obtained for three different choices of the prior, namely,

1. (1)

Informative gamma prior 1 (GP 1) : E(α) = 5 and Var(α) = 2.5. So, we have c = 10 and d = 2.

2. (2)

Informative gamma prior (GP 2): E(α) = 5 and Var(α) = 5. So, we have c = 5 and d = 1.

3. (3)

Jeffreys’ prior (JP): π (α) ∝ 1/α. So, we have c = 0 and d = 0.

The Bayesian estimates of α based on the three choices of the prior and the two multiply Type-II censoring schemes, are presented in Table 1. Table 2 shows the point predictors and the 95% two-sample Bayesian prediction intervals for the order statistics Ys:10, for s = 1,...,10, based on the three choices of the prior and the two multiply Type-II censoring schemes.

GP 1 GP 2 JP
Scheme 1 5.009 5.014 5.028
Scheme 2 5.006 5.010 5.021
Table 1.

The Bayesian estimates of α based on the three choices of the prior and the two multiply Type-II censoring schemes.

prior
s
GP 1 GP 2 JP

LYs:10 UYs:10 Ŷs:10 LYs:10 UYs:10 Ŷs:10 LYs:10 UYs:10 Ŷs:10
1 Scheme 1 0.721 4.619 2.094 0.686 4.737 2.462 0.620 4.971 3.017
Scheme 2 0.722 4.616 1.986 0.687 4.733 2.153 0.621 4.964 2.681
2 Scheme 1 1.153 6.715 3.045 1.092 6.889 3.538 0.980 7.233 4.916
Scheme 2 1.154 6.710 2.952 1.093 6.883 3.242 0.983 7.223 4.142
3 Scheme 1 1.563 9.230 4.172 1.479 9.460 4.627 1.326 9.919 5.788
Scheme 2 1.565 9.224 4.089 1.480 9.453 4.383 1.329 9.905 5.094
4 Scheme 1 2.012 12.611 5.490 1.906 12.907 5.837 1.710 13.494 6.980
Scheme 2 2.014 12.603 5.243 1.909 12.895 5.462 1.714 13.477 6.103
5 Scheme 1 2.543 17.563 7.287 2.411 17.940 7.798 2.170 18.695 9.358
Scheme 2 2.546 17.555 7.074 2.416 17.927 7.359 2.176 18.671 8.869
6 Scheme 1 3.209 25.583 9.996 3.067 26.067 10.350 2.755 27.043 11.907
Scheme 2 3.210 25.569 9.526 3.104 26.040 10.042 2.758 27.010 11.005
7 Scheme 1 4.101 40.475 14.542 3.908 41.115 14.991 3.548 42.423 16.007
Scheme 2 4.109 40.450 14.282 3.915 41.083 14.638 3.554 42.368 15.215
8 Scheme 1 5.400 75.022 22.780 5.165 75.931 23.081 4.721 77.798 24.457
Scheme 2 5.439 74.978 22.361 5.171 75.769 22.584 4.726 77.699 23.762
9 Scheme 1 7.563 199.92 46.646 7.269 201.38 47.159 6.705 204.44 48.724
Scheme 2 7.669 199.01 45.127 7.274 201.22 46.827 6.714 204.19 47.951
10 Scheme 1 12.340 1977.3 1071.8 11.95 1978.8 1074.5 11.172 1983.3 1085.7
Scheme 2 12.433 1976.6 1069.1 11.99 1977.3 1072.9 11.188 1980.7 1080.3
Table 2.

The Bayesian prediction for Ys:10 (s = 1,2,...,10) based on the three choices of the prior and the two multiply Type-II censoring schemes.

## 5.2. Numerical results of the inverse Rayleigh distribution

To illustrate the prediction results for the inverse Rayleigh distribution, we generated order statistics from a sample of size n = 10 from the inverse exponential distribution with α = 5. The generated order statistics are listed as follows: 1.35854, 1.80942, 2.04633, 2.08176, 2.34668, 2.94300, 4.81843, 5.26941, 8.16025, 59.60427.

We then used these data to consider two different multiply Type-II censoring schemes:

1. (1)

Scheme 1: r = 6 with j1 = 1, j2 = 3, j3 = 5, j4 = 6, j5 = 7 and j6 = 8. Under this censoring scheme, we obtained the following data: 1.35854, 2.04633, 2.34668, 2.94300, 4.81843 and 5.26941;

2. (2)

Scheme 2: r = 8 with j1 = 1, j2 = 3, j3 = 4, j4 = 5, j5 = 6, j6 = 7, j7 = 9 and j8 = 10. Under this censoring scheme, we obtained the following data: 1.35854, 2.04633, 2.08176, 2.34668, 2.94300, 4.81843, 8.16025 and 59.60427.

We assume these data to have come from the inverse Rayleigh distribution, where the parameter α is unknown. Based on the above two multiply Type-II censoring schemes, we used the results presented earlier in Section 4 (when β = 2) to compute the Bayesian estimate of α. In addition, suppose we want to predict the order statistics from a future unobserved sample with size m = 10. In this case, the point predictors and the 95% two-sample Bayesian prediction intervals for the order statistics Ys:10, for s = 1,...,10, are all obtained for three different choices of the hyper-parameters c and d, namely,

1. (1)

Informative gamma prior 1 (GP 1): E(α) = 5 and Var(α) = 2.5. So, we have c = 10 and d = 2.

2. (2)

Informative gamma prior 2 (GP 2): E(α) = 5 and Var(α) = 5. So, we have c = 5 and d = 1.

3. (3)

Jeffreys’ prior (JP): π (α) ∝ 1/α. So, we have c = 0 and d = 0.

The Bayesian estimates of α based on the three choices of the prior and the two multiply Type-II censoring schemes, are presented in Table 3. Table 4 shows the point predictors and the 95% two-sample Bayesian prediction intervals for the order statistics Ys:10, for s = 1,...,10, based on the three choices of the prior and the two multiply Type-II censoring schemes.

GP 1 GP 2 JP
Scheme 1 5.334 5.458 5.730
Scheme 2 5.240 5.367 5.645
Table 3.

The Bayesian estimates of α based on the three choices of the prior and the two multiply Type-II censoring schemes.

prior
s
GP 1 GP 2 JP

LYs:10 UYs:10 Ŷs:10 LYs:10 UYs:10 Ŷs:10 LYs:10 UYs:10 Ŷs:10
1 Scheme 1 0.877 2.217 1.415 0.865 2.269 1.735 0.843 2.377 1.945
Scheme 2 0.878 2.216 1.392 0.866 2.268 1.682 0.844 2.376 1.874
2 Scheme 1 1.109 2.672 1.713 1.092 2.736 1.941 1.060 2.867 2.105
Scheme 2 1.110 2.270 1.699 1.093 2.734 1.887 1.062 2.865 2.002
3 Scheme 1 1.291 3.133 2.000 1.271 3.206 2.402 1.233 3.357 2.784
Scheme 2 1.292 3.131 1.979 1.273 3.203 2.038 1.235 3.354 2.572
4 Scheme 1 1.465 3.663 2.291 1.442 3.745 2.696 1.401 3.917 2.903
Scheme 2 1.466 3.660 2.287 1.444 3.740 2.473 1.402 3.913 2.711
5 Scheme 1 1.647 4.323 2.632 1.623 4.416 3.091 1.577 4.609 3.334
Scheme 2 1.648 4.322 2.622 1.624 4.412 2.806 1.580 4.605 3.049
6 Scheme 1 1.850 5.217 3.060 1.825 5.323 3.547 1.777 5.545 3.908
Scheme 2 1.852 5.210 3.019 1.828 5.320 3.176 1.780 5.542 3.631
7 Scheme 1 2.091 6.563 3.646 2.065 6.686 4.007 2.017 6.947 4.817
Scheme 2 2.093 6.561 3.573 2.067 6.682 3.824 2.020 6.941 4.230
8 Scheme 1 2.410 8.935 4.498 2.374 9.088 5.626 2.326 9.410 6.715
Scheme 2 2.411 8.931 4.466 2.377 9.081 5.415 2.329 9.402 5.988
9 Scheme 1 2.840 14.588 6.130 2.816 14.802 7.474 2.771 15.258 8.970
Scheme 2 2.842 14.497 6.072 2.822 14.714 7.278 2.775 15.218 8.002
10 Scheme 1 3.627 45.883 14.136 3.610 46.410 16.857 3.577 47.542 18.545
Scheme 2 3.639 44.912 12.612 3.623 45.645 14.849 3.582 46.604 17.013
Table 4.

The Bayesian prediction for Ys:10 (s = 1,2,...,10) based on the three choices of the prior and the two multiply Type-II censoring schemes.

## 5.3. Concluding Remarks

From the results in Tables 14, the following points can be observed:

1. (1)

The results obtained based on the multiply Type-II censoring scheme 2 (with r = 8) is more precise than the corresponding ones based on the multiply Type-II censoring scheme 2 (with r = 6), as we would expect;

2. (2)

A comparison of the results for the informative gamma priors with the corresponding ones for the Jeffreys’ prior reveals that the former produce more precise results, as we would expect;

3. (3)

The results obtained based on informative gamma prior 1 (with Var(α) = 2.5) is more precise than the corresponding ones based on informative gamma prior 2 (with Var(α) = 5), as we would expect;

4. (4)

The width of the Bayesian prediction intervals decreases with increasing β.