# Model Selection Approaches for Predicting Future Order Statistics from Type II Censored Data

Authors: Jyun-You Chiang; Shuai Wang; Tzong-Ru Tsai; Ting Li
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper studies a discriminant problem of location-scale family in case of prediction from type II censored samples. Three model selection approaches and two types of predictors are, respectively, proposed to predict the future order statistics from censored data when the best underlying distribution is not clear with several candidates. Two members in the location-scale family, the normal distribution and smallest extreme value distribution, are used as candidates to illustrate the best model competition for the underlying distribution via using the proposed prediction methods. The performance of correct and incorrect selections under correct specification and misspecification is evaluated via using Monte Carlo simulations. Simulation results show that model misspecification has impact on the prediction precision and the proposed three model selection approaches perform well when more than one candidate distributions are competing for the best underlying distribution. Finally, the proposed approaches are applied to three data sets.


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## Body

## 1. Introduction

For saving testing time and sample resource, censoring schemes often are considered to implement life tests. Type I censoring scheme and type II censoring scheme are two popular censoring schemes based on the criteria of test time censoring and failure number censoring. Plenty studies can be found for evaluating the reliability of lifetime components via using type I censoring test or type II censoring test. See examples like, [1–6] etc.In this study, we mainly restrict our attention to using type II censoring scheme for predicting the censored sample for reliability evaluation when a discriminant problem is considered. In the type II censoring scheme, we consider an experiment wheren identical components are placed in the test simultaneously. Assuming that rth component fails, the experiment is terminated. Thus the last (n-r) components are censored. In many engineering applications, censored data are not allowed for implementing statistical methods to obtain information. For example, if we like to conduct a factorial design or fractional factorial design based on the experimental design methods, most experimental design methods cannot be implemented with censored data. In such situation, a reliable procedure for predicting censored or unobserved observations is required. Moreover, if we can predict the unobserved observations and transform a censored data set into a complete data set, the parameter estimation problem becomes easy especially for dealing with the cases, which have no analytic solutions of the parameter estimators can be obtained. The purpose of predicting life length of the sth(r<s≤n) item is equivalent to the life length of a (n-s+1)-out-of-n system that was made up of n identical components with independent life lengths. Whens = n, it is better known as the parallel system. For this issue, various methods have been developed to predict the censored data. Kaminsky and Nelson [7] provided interval and point prediction of order statistics. Fertig et al. [8] provided Monte Carlo estimates of the distribution percentiles to construct prediction intervals for samples from a Weibull or smallest extreme value distribution (SEV). Kaminsky and Rhodin [9] provided the maximum likelihood predictor (MLP) to predict the future order statistics and then estimate the unknown parameters. Wu et al. [10] proposed five new pivotal quantities to obtain prediction intervals of future order statistics from the Pareto distribution. Kundu and Raqab [11] describes the Bayesian inference and prediction of the two-parameter Weibull distribution. Panahi and Sayyareh [12] proposed parameter estimation and prediction of order statistics for the Burr type XII distribution. Some of these predictions are complex, or they need to construct complex statistical models. Therefore, these existing methods are not easy to apply.In order to solve this problem, Raqab [13] modified the MLP method and proposed four modified MLPs (MMLPs) to predict the future order statistics for the normal distribution (ND). In order to simplify the estimation function, they considered four types of modification to approximate the terms of hazard rate and extended hazard rate functions form a ND, which has unknown mean and known standard deviation. Yang and Tong [14] used MMLP method to predict type II censored data from factorial experiments. They derived the simple explicit solutions for parameters for a ND, which has unknown mean and unknown standard deviation. Chiang [15] used another three MMLP procedures to predict type II censored data under the Weibull distribution. In his procedures, it is difficult to find the only root solution to the parameter estimation. However, the parameter estimation of MMLP method can be obtained via simple parameter explicit solution only in the ND. For other commonly used distributions, the likelihood equations of MMLP may be nonlinear and does not admit explicit solutions. Hence the parameter estimation of MMLP loses the advantage for other commonly used distributions.Another important problem in life testing experiments is the model selection based on the existing sample. In practical applications, many statistical distributions are much alike, especially in censored data, and the underlying distribution of product quality characteristics is usually unknown. They may fit the data well in practical applications. However, their predictions may lead to a significant difference. Therefore, correctly identifying the underlying distribution is an important issue and it has long been studied. Dumonceaux and Antle [16] applied ratio of maximized likelihood (RML) to discriminating between the lognormal and Weibull distributions. Kundu and Manglick [17] proposed statistical methods to discriminate between the lognormal and gamma distributions. Kundu and Raqab [18] proposed a selection to discriminate between the generalized Rayleigh and lognormal distribution. Yu [19] provided a misspecification analysis method to discriminate between the ND and SEV for the design of experiment. Dey and Kundu [20] studied the discrimination problem between the lognormal and log-logistic distributions. Elsherpieny et al [21] considered the discrimination problem between the Weibull and log-logistic distributions. Ashour and Hashish [22] provided a numerical comparison study for using RML-procedure, S-procedure, and F-procedure in failure model discrimination. Pakyari [23] presented diagnostic tools based on the likelihood ratio test and the minimum Kolmogorov distance method to discriminate between the generalized exponential, geometric extreme exponential, and Weibull distributions. Elsherpieny et al. [24] provided a method to discriminate the gamma and log-logistic distributions based on progressive type II censored data. Although the inference methods in the aforementioned studies are valuable, the impacts of model misspecification on predicting the future order statistics have not been well studied.Among the model discrimination problems, due to the well-developed theory and inferential procedures for the location-scale family of distributions, the model discrimination within the location-scale family of distributions is particularly important and it has received much attention. The main purpose of this paper is to address these issues and provide satisfactory estimators of parameters and predictors of future order statistics when the underlying distribution is unknown but it is a member in the location-scale family. Specifically, for lifetime analysis, the essence of this study is to predict the future order statistics for type II censored data when the underlying distribution is unknown but is a member of the location-scale family. The major contributions of this study for censored data prediction are presented in Figure1.Figure 1
The flow chart of the major contribution of this study.The rest of this paper is organized as follows. Section2 presents materials and methods. In this section, statistical methods to obtain approximate predictors for type II right censored variables are studied and two prediction methods are proposed to predict the type II right-censored variables based on the AMLEs. The ND and SEV are considered as the candidate distributions to compete the best distribution for obtaining the predictors of type II right-censored variables. In Section 3, we provide three algorithms to implement the three proposed model selection approaches to deal with the discrimination problem when obtaining the predictors of type II right-censored variables based on the proposed methods. An intensive simulation study is conducted in Section 4 to evaluate the performance of the proposed approaches. Then, three examples are used to demonstrate the applications of the proposed methodologies in Section 5. Some concluding remarks are provided in Section 6.


## 2. Methods for Approximate Predictors

### 2.1. Approximate Maximum Likelihood Estimation

LetYi denote the failure time of ith item and Xi=log⁡⁡(Yi), which follows a location-scale family, having the probability density function (PDF) and cumulative distribution function (CDF): (1)fx;μ,σ=1σgx-μσ,and(2)Fx;μ,σ=Gx-μσ,-∞<μ<∞,σ>0,-∞<x<∞,respectively, where μ is location parameter and σ is scale parameter. g(·) and G(·) are the PDF and CDF of a member, respectively, in the location-scale family. Denote the sample size by n, and denote type II censored sample with r failures by x1:n≤x2:n≤⋯≤xr:n, which are the realizations of X1:n≤X2:n≤⋯≤Xr:n, where 1≤r<s≤n. Our goal is to predict xs:n for r<s≤n. Let f(x)≡fx;μ,σ and F(x)≡Fx;μ,σ here and after to simplify the notations. Kaminsky and Rhodin [9] considered prediction of xs:n having observed x=x1:n,x2:n,…,xr:n, The predictive likelihood functions (PLF) of Xs:n, μ and σ is(3)LXs:n,μ,σ;x≡fx,Xs:n;μ,σ=n!s-r-1!n-s!∏j=1rfxj:nFXs:n-Fxr:ns-r-1fXs:n1-FXs:nn-s.Please note that the capital notation Xs:n in FXs:n is unknown and can be predicted based on the sample x. Based on the proposed method by Raqab [13], the PLF of Xs:n, μ and σ in (3) can be represented as a product of two likelihood functions, the PLF of μ and σ (i.e., which is denoted as L1) and the PLF of Xs:n (i.e., which is denoted as L2). Both likelihood functions are presented, respectively, by(4)L1μ,σ;x=n!n-r!∏j=1rfxj:n1-Fxr:nn-r,and (5)L2Xs:n;μ,σ,x=n-r!s-r-1!n-s!FXs:n-Fxr:ns-r-11-Fxr:nn-r×1-FXs:nn-sfXs:n.In practice, we can obtain the MLEs of μ and σ, denoted by μ^ and σ^, respectively, through maximizing L1μ,σ;x in (4). Then use μ^ and σ^ to replace μ and σ as the plug-in parameters in (5) to predict Xs:n. Let zj:n=(xj:n-μ)/σ for j=1,…,r, Zs:n=(Xs:n-μ)/σ for s=r+1,…,n and z=(z1:n,z2:n,…,zr:n), then we can rewrite (4) and (5) by(6)L1≡L1μ,σ;z=C1∏j=1rσ-1fzj:n1-Fzr:nn-rand(7)L2≡L2Zs:n;μ^,σ^,z=C2σ-1FZs:n-Fzr:ns-r-11-Fzr:nn-r1-FZs:nn-sfZs:n,where C1=n!/n-r! and C2=n-r!/[s-r-1!n-s!]. After straightforward computations, the MLEs of μ, σ and Zs:n respectively can be obtained as the solutions of(8)∂log⁡L1⁡∂μ=1σ∑j=1rΨzj:n+n-rhzr:n=0(9)∂log⁡L1∂σ=1σ-r+∑j=1rΨzj:nzj:n+n-rhzr:nzr:n=0and(10)∂log⁡L2∂Zs:n=s-r-1h1zr:n,Zs:n-ΨZs:n-n-shZs:n=0,where (11)ΨZj:n=-f′Zj:nfZj:n,j=1,…,n,whereZj:n=zj:nifj≤r,(12)hZj:n=fZj:n1-FZj:n,j=1,…,n,whereZj:n=zj:nifj≤r,and(13)h1zr:n,Zs:n=fZs:nFZs:n-Fzr:n.Because of no analytic presentation for μ^ and σ^, one needs to use numerical gradient computation methods, for example, the Newton-Raphson method, for obtaining μ^ and σ^ via by equating (8) and (9). To obtain proper initial solutions for implementing gradient computation methods, we consider using the approximate MLEs (AMLE) of μ and σ from Hossain and Willan [25] as their initial solutions in this study.


### 2.2. Approximate Maximum Likelihood Predictors

When we obtain the MLEsμ^ and σ^, we can predict Xs:n by using two approximation methods, the expected value prediction method and Taylor series prediction method. The resulting predictors of Xs:n based on the expected prediction method is denoted by MLPE, and the resulting predictors of Xs:n based on the Taylor series prediction method is denoted by MLPT. The two approximate methods mainly use two different methods to get the approximates of h1zr:n,Zs:n and hZs:n. Mehrotra and Nanda [26] proposed approximate maximum likelihood estimators for the ND and gamma distribution by replacing hx and xhx by their respective expected values and efficiencies compared to those for the best linear unbiased estimators for these distributions. Balakrishnan and Cohen [27] used the Taylor series expansion of hx and fx/F(x) at the points F-1ps to obtain modified MLEs of the parameters of the ND and Rayleigh distribution, where pi=i/(n+1) for i=1,2,…,n. The main point of their approach is that likelihood equations involve complicated terms and it is not possible to obtain an explicit form for MLE. So we follow their ideas and find an explicit form for the predictor of Xs:n.Based on the expected value prediction method, replacingμ,σ with (μ^,σ^), and replacing h1zr:n,Zs:n and hZs:n by their respective expected values in (10). According to Raqab [13], the expected value of fZj:n, h1zr:n,Zs:n and hZs:n can be presented, respectively, by(14)EfZj:n=1n+1∑k=j+1n+1EΨZk:n+1,j≤nandZj:n=zj:nifj≤r,(15)EhZj:n=1n-j∑k=j+1nEΨZk:n,j≤n-1andZj:n=zj:nifj≤r,and(16)Eh1Zi:n,Zj:n=1j-i-1∑k=jnEΨZk:n,j-i≥2,andZj:n=zj:nifj≤r.Based on the Taylor series prediction method, replacingμ,σ with (μ^,σ^) and replacing hZs:n and h1Zr:n,Zs:n with their Taylor series approximations at points F-1ps and (F-1pr,F-1ps), respectively, in (10). In this study, we denote the MLPE and MLPT of Xs:n under the candidate distribution M by X^s:nM,1 and X^s:nM,2, respectively.There are many common distributions in location-scale family of distributions. The widely used members including the ND, SEV, logistic distribution, etc. It is impossible to list all inference formulas for predictingXs:n under all widely used members in the location-scale family. In this study, we use ND and SEV as candidates to illustrating the applications of the proposed methods. But the suggested algorithms in this study can be applied for the cases with more than two candidate members. The reason to select the ND and SEV as candidates is due to the fact that the Weibull distribution and lognormal distribution are two widely used distributions for life testing applications. The Weibull and lognormal distributions can be respectively transformed into the SEV and ND by taking log-transformation.If the underlying distribution is normal, the PDF of normal distribution is given by(17)gZ=ϕz=12πe-z2/2.Through using (17), we can obtain Ψz=-ϕ′(z)/ϕ(z)=z. The MLEs of normal distribution parameters are denoted by μ^N and σ^N. Replacing μ and σ with μ^N and σ^N in (6), we can represent (6) by(18)L^Nμ^N,σ^N=C1∏j=1rσ^N-1ϕzj:n1-Φzr:nn-r,where Φ(·) is the CDF of the standard ND. According to (15) and (16), h1zr:n,Zs:n and hZs:n can be replaced with their respective expected values in (10). Equation (10) can be rewritten as(19)EZs:n-Z^s:n=0.The values of E(Zj:n) are available and have been tabulated by Teichroew [28]. Hence, MLPE of Xs:n for ND can be derived as (20)X^s:nN,1=μ^N+σ^NEZs:n.Because EZs:n≥zr:n is a necessary condition, we modify (20) by(21)X^s:nN,1=max⁡μ^N+σ^NEZs:n,xr:nand use X^s:nN,1 in (21) to protect Xs:n for r+1≤s≤n.Based on the Taylor series prediction method, the functionshZs:n and h1zr:n,Zs:n are expanded by using the Taylor series around points F-1ps and (F-1pr,F-1ps), respectively. According to Raqab [13], we can approximate hZs:n and h1zr:n,Zs:n by(22)hZs:n=fZs:n1-FZs:n≈α+βZs:n,and(23)h1zr:n,Zs:n=fZs:nFZs:n-Fzr:n≈γ+ρzr:n-vsZs:n.The values of α,β,γ,ρ and vs are given in Appendix A. Equation (10) can be rewritten by(24)s-r-1γ+ρzr:n-vsZs:n-zs:n-n-sα+βZs:n=0.The MLPT of Xs:n can be obtained by(25)X^s:nN,2=max⁡s-r-1ρxr:ns-r-1vs+1+n-sβ+1-s-r-1ρs-r-1vs+1+n-sβμ^N+s-r-1γ-n-sαs-r-1vs+1+n-sβσ^N,xr:n,where r+1≤s≤n.If the underlying distribution is SEV, the PDF of the SEV is given by(26)gz=ϕsevz=ez-ez.Based on the expected value prediction method,Ψz=-ϕ′sev(z)/ϕsev(z)=ez-1. Using (8) and (9), the MLEs of μ and σ are denoted by μ^S and σ^S, respectively. Replacing μ and σ with μ^S and σ^S in (6), (6) can be represented by(27)L^Sμ^S,σ^S=C1∏j=1rσ^S-1ϕsevzj:n1-Φsevzr:nn-r,where Φsevz=1-exp⁡[-exp⁡(z)] is the CDF of the standard SEV. Then h1zr:n,Zs:n and hZs:n are replaced with their respective expected values in Eq. (10). Equation (10) can be rewritten as(28)s-r-1Eh1zr:n,Zs:n-eZ^s:n-1-n-sEhZs:n=0.The MLPE of Xs:n can be obtained as(29)X^s:nSEV,1=max⁡μ^S+σ^Sln⁡EΨZs:n+1,xr:nfor r+1≤s≤n and EΨZs:n=E(eZs:n-1).Based on the Taylor series prediction method, expandinghZs:n and h1zr:n,Zs:n by using the Taylor series at the points F-1ps and (F-1pr,F-1ps), respectively. We obtain(30)hZs:n=fZs:n1-FZs:n≈1-αs-βsZs:n,and(31)h1zr:n,Zs:n=fZs:nFZs:n-Fzr:n≈γE+ρEzr:n+vEZs:n.The values of αs,βs,γE,ρE and vE are given in Appendix B. Equation (10) can be rewritten as(32)s-r-1γE+ρEzr:n+vEZs:n-eZs:n-1-n-s1-αs-βsZs:n=0The MLPT of Xs:n can be derived as(33)X^s:nSEV,2=max⁡-s-r-1vExr:ns-r-1ρE+βs+n-sβs+1+s-r-1vEs-r-1ρE+βs+n-sβsμ^S-s-r-1γE+αs-n-s+n-sαss-r-1ρE+βs+n-sβsσ^S,xr:n,for r+1≤s≤n.


## 2.1. Approximate Maximum Likelihood Estimation

LetYi denote the failure time of ith item and Xi=log⁡⁡(Yi), which follows a location-scale family, having the probability density function (PDF) and cumulative distribution function (CDF): (1)fx;μ,σ=1σgx-μσ,and(2)Fx;μ,σ=Gx-μσ,-∞<μ<∞,σ>0,-∞<x<∞,respectively, where μ is location parameter and σ is scale parameter. g(·) and G(·) are the PDF and CDF of a member, respectively, in the location-scale family. Denote the sample size by n, and denote type II censored sample with r failures by x1:n≤x2:n≤⋯≤xr:n, which are the realizations of X1:n≤X2:n≤⋯≤Xr:n, where 1≤r<s≤n. Our goal is to predict xs:n for r<s≤n. Let f(x)≡fx;μ,σ and F(x)≡Fx;μ,σ here and after to simplify the notations. Kaminsky and Rhodin [9] considered prediction of xs:n having observed x=x1:n,x2:n,…,xr:n, The predictive likelihood functions (PLF) of Xs:n, μ and σ is(3)LXs:n,μ,σ;x≡fx,Xs:n;μ,σ=n!s-r-1!n-s!∏j=1rfxj:nFXs:n-Fxr:ns-r-1fXs:n1-FXs:nn-s.Please note that the capital notation Xs:n in FXs:n is unknown and can be predicted based on the sample x. Based on the proposed method by Raqab [13], the PLF of Xs:n, μ and σ in (3) can be represented as a product of two likelihood functions, the PLF of μ and σ (i.e., which is denoted as L1) and the PLF of Xs:n (i.e., which is denoted as L2). Both likelihood functions are presented, respectively, by(4)L1μ,σ;x=n!n-r!∏j=1rfxj:n1-Fxr:nn-r,and (5)L2Xs:n;μ,σ,x=n-r!s-r-1!n-s!FXs:n-Fxr:ns-r-11-Fxr:nn-r×1-FXs:nn-sfXs:n.In practice, we can obtain the MLEs of μ and σ, denoted by μ^ and σ^, respectively, through maximizing L1μ,σ;x in (4). Then use μ^ and σ^ to replace μ and σ as the plug-in parameters in (5) to predict Xs:n. Let zj:n=(xj:n-μ)/σ for j=1,…,r, Zs:n=(Xs:n-μ)/σ for s=r+1,…,n and z=(z1:n,z2:n,…,zr:n), then we can rewrite (4) and (5) by(6)L1≡L1μ,σ;z=C1∏j=1rσ-1fzj:n1-Fzr:nn-rand(7)L2≡L2Zs:n;μ^,σ^,z=C2σ-1FZs:n-Fzr:ns-r-11-Fzr:nn-r1-FZs:nn-sfZs:n,where C1=n!/n-r! and C2=n-r!/[s-r-1!n-s!]. After straightforward computations, the MLEs of μ, σ and Zs:n respectively can be obtained as the solutions of(8)∂log⁡L1⁡∂μ=1σ∑j=1rΨzj:n+n-rhzr:n=0(9)∂log⁡L1∂σ=1σ-r+∑j=1rΨzj:nzj:n+n-rhzr:nzr:n=0and(10)∂log⁡L2∂Zs:n=s-r-1h1zr:n,Zs:n-ΨZs:n-n-shZs:n=0,where (11)ΨZj:n=-f′Zj:nfZj:n,j=1,…,n,whereZj:n=zj:nifj≤r,(12)hZj:n=fZj:n1-FZj:n,j=1,…,n,whereZj:n=zj:nifj≤r,and(13)h1zr:n,Zs:n=fZs:nFZs:n-Fzr:n.Because of no analytic presentation for μ^ and σ^, one needs to use numerical gradient computation methods, for example, the Newton-Raphson method, for obtaining μ^ and σ^ via by equating (8) and (9). To obtain proper initial solutions for implementing gradient computation methods, we consider using the approximate MLEs (AMLE) of μ and σ from Hossain and Willan [25] as their initial solutions in this study.


## 2.2. Approximate Maximum Likelihood Predictors

When we obtain the MLEsμ^ and σ^, we can predict Xs:n by using two approximation methods, the expected value prediction method and Taylor series prediction method. The resulting predictors of Xs:n based on the expected prediction method is denoted by MLPE, and the resulting predictors of Xs:n based on the Taylor series prediction method is denoted by MLPT. The two approximate methods mainly use two different methods to get the approximates of h1zr:n,Zs:n and hZs:n. Mehrotra and Nanda [26] proposed approximate maximum likelihood estimators for the ND and gamma distribution by replacing hx and xhx by their respective expected values and efficiencies compared to those for the best linear unbiased estimators for these distributions. Balakrishnan and Cohen [27] used the Taylor series expansion of hx and fx/F(x) at the points F-1ps to obtain modified MLEs of the parameters of the ND and Rayleigh distribution, where pi=i/(n+1) for i=1,2,…,n. The main point of their approach is that likelihood equations involve complicated terms and it is not possible to obtain an explicit form for MLE. So we follow their ideas and find an explicit form for the predictor of Xs:n.Based on the expected value prediction method, replacingμ,σ with (μ^,σ^), and replacing h1zr:n,Zs:n and hZs:n by their respective expected values in (10). According to Raqab [13], the expected value of fZj:n, h1zr:n,Zs:n and hZs:n can be presented, respectively, by(14)EfZj:n=1n+1∑k=j+1n+1EΨZk:n+1,j≤nandZj:n=zj:nifj≤r,(15)EhZj:n=1n-j∑k=j+1nEΨZk:n,j≤n-1andZj:n=zj:nifj≤r,and(16)Eh1Zi:n,Zj:n=1j-i-1∑k=jnEΨZk:n,j-i≥2,andZj:n=zj:nifj≤r.Based on the Taylor series prediction method, replacingμ,σ with (μ^,σ^) and replacing hZs:n and h1Zr:n,Zs:n with their Taylor series approximations at points F-1ps and (F-1pr,F-1ps), respectively, in (10). In this study, we denote the MLPE and MLPT of Xs:n under the candidate distribution M by X^s:nM,1 and X^s:nM,2, respectively.There are many common distributions in location-scale family of distributions. The widely used members including the ND, SEV, logistic distribution, etc. It is impossible to list all inference formulas for predictingXs:n under all widely used members in the location-scale family. In this study, we use ND and SEV as candidates to illustrating the applications of the proposed methods. But the suggested algorithms in this study can be applied for the cases with more than two candidate members. The reason to select the ND and SEV as candidates is due to the fact that the Weibull distribution and lognormal distribution are two widely used distributions for life testing applications. The Weibull and lognormal distributions can be respectively transformed into the SEV and ND by taking log-transformation.If the underlying distribution is normal, the PDF of normal distribution is given by(17)gZ=ϕz=12πe-z2/2.Through using (17), we can obtain Ψz=-ϕ′(z)/ϕ(z)=z. The MLEs of normal distribution parameters are denoted by μ^N and σ^N. Replacing μ and σ with μ^N and σ^N in (6), we can represent (6) by(18)L^Nμ^N,σ^N=C1∏j=1rσ^N-1ϕzj:n1-Φzr:nn-r,where Φ(·) is the CDF of the standard ND. According to (15) and (16), h1zr:n,Zs:n and hZs:n can be replaced with their respective expected values in (10). Equation (10) can be rewritten as(19)EZs:n-Z^s:n=0.The values of E(Zj:n) are available and have been tabulated by Teichroew [28]. Hence, MLPE of Xs:n for ND can be derived as (20)X^s:nN,1=μ^N+σ^NEZs:n.Because EZs:n≥zr:n is a necessary condition, we modify (20) by(21)X^s:nN,1=max⁡μ^N+σ^NEZs:n,xr:nand use X^s:nN,1 in (21) to protect Xs:n for r+1≤s≤n.Based on the Taylor series prediction method, the functionshZs:n and h1zr:n,Zs:n are expanded by using the Taylor series around points F-1ps and (F-1pr,F-1ps), respectively. According to Raqab [13], we can approximate hZs:n and h1zr:n,Zs:n by(22)hZs:n=fZs:n1-FZs:n≈α+βZs:n,and(23)h1zr:n,Zs:n=fZs:nFZs:n-Fzr:n≈γ+ρzr:n-vsZs:n.The values of α,β,γ,ρ and vs are given in Appendix A. Equation (10) can be rewritten by(24)s-r-1γ+ρzr:n-vsZs:n-zs:n-n-sα+βZs:n=0.The MLPT of Xs:n can be obtained by(25)X^s:nN,2=max⁡s-r-1ρxr:ns-r-1vs+1+n-sβ+1-s-r-1ρs-r-1vs+1+n-sβμ^N+s-r-1γ-n-sαs-r-1vs+1+n-sβσ^N,xr:n,where r+1≤s≤n.If the underlying distribution is SEV, the PDF of the SEV is given by(26)gz=ϕsevz=ez-ez.Based on the expected value prediction method,Ψz=-ϕ′sev(z)/ϕsev(z)=ez-1. Using (8) and (9), the MLEs of μ and σ are denoted by μ^S and σ^S, respectively. Replacing μ and σ with μ^S and σ^S in (6), (6) can be represented by(27)L^Sμ^S,σ^S=C1∏j=1rσ^S-1ϕsevzj:n1-Φsevzr:nn-r,where Φsevz=1-exp⁡[-exp⁡(z)] is the CDF of the standard SEV. Then h1zr:n,Zs:n and hZs:n are replaced with their respective expected values in Eq. (10). Equation (10) can be rewritten as(28)s-r-1Eh1zr:n,Zs:n-eZ^s:n-1-n-sEhZs:n=0.The MLPE of Xs:n can be obtained as(29)X^s:nSEV,1=max⁡μ^S+σ^Sln⁡EΨZs:n+1,xr:nfor r+1≤s≤n and EΨZs:n=E(eZs:n-1).Based on the Taylor series prediction method, expandinghZs:n and h1zr:n,Zs:n by using the Taylor series at the points F-1ps and (F-1pr,F-1ps), respectively. We obtain(30)hZs:n=fZs:n1-FZs:n≈1-αs-βsZs:n,and(31)h1zr:n,Zs:n=fZs:nFZs:n-Fzr:n≈γE+ρEzr:n+vEZs:n.The values of αs,βs,γE,ρE and vE are given in Appendix B. Equation (10) can be rewritten as(32)s-r-1γE+ρEzr:n+vEZs:n-eZs:n-1-n-s1-αs-βsZs:n=0The MLPT of Xs:n can be derived as(33)X^s:nSEV,2=max⁡-s-r-1vExr:ns-r-1ρE+βs+n-sβs+1+s-r-1vEs-r-1ρE+βs+n-sβsμ^S-s-r-1γE+αs-n-s+n-sαss-r-1ρE+βs+n-sβsσ^S,xr:n,for r+1≤s≤n.


## 3. Three Model Selection Approaches

When several candidate distributions are competing for the best underlying distribution and the users cannot identify which one distribution is the best, we suggest three approaches to discriminate the candidate distributions, the ratio of the maximized likelihood (RRML) approach, modificationDSP approach (shorted as DSP approach), and modificationD approach (shorted as theD approach), to obtain the predictor of X^s:n. It is noticed that the idea of the DSP approach andD approach is based on goodness-of-fit test methods. All these three approaches can be implemented to obtain the predictor of Xs:n via using Algorithms 1–3.Algorithm 1 (the RRML approach).
Step 1. Collect a type II censored sample, which has size n and r observed failure times; we consider k candidate distributions.
Step 2. Obtain (μ^Mi,σ^Mi) and L^Miμ^Mi,σ^Mi for the candidate distribution Mi,i=1,2,…,k. Obtain Xs:n under the candidate distribution Mi and label it by X^s:nMi,j for s=r+1,…,n, i=1,2,…,k and j=1 or 2.
Step 3. Let X^s:nA1,j denote the predicted value of Xs:n for j=1 or 2. Based on the method proposed by Dumonceaux and Antle [16], we can obtain X^s:nA1,j, which can provide the largest maximum likelihood information by (34)L^A1μ^A1,σ^A1=max⁡L^M1μ^M1,σ^M1,L^M2μ^M2,σ^M2,…,L^Mkμ^Mk,σ^Mk.
If the candidate distributions are ND and SEV, Steps 2 and 3 in Algorithm1 can be reduced to Step 2’ and Step 3’ as the following, respectively:
Step 2’. Obtain (μ^N, σ^N), (μ^S, σ^S), L^Nμ^N,σ^N and L^Sμ^S,σ^S. Obtain Xs:n under the ND (X^s:nN,j) and obtain Xs:n under the SEV (X^s:nSEV,j) for s=r+1,…,n and j=1 or 2.
Step 3’. Let X^s:nA1 denote the predicted value of Xs:n. Then(35)X^s:nA1=X^s:nN,jifL^Nμ^N,σ^N>L^Sμ^S,σ^SX^s:nSEV,j,otherwise.fors=r+1,…,nandj=1or2.Algorithm 2 (theDSP approach).
Step 1. Collect a type II censored sample, which has size n and r observed failure times.
Step 2. Obtain (μ^Mi,σ^Mi) for i=1,2,…,k, and then obtain X^s:nMi,j for s=r+1,…,n, i=1,2,…,k and j=1 or 2.
Step 3. Based on the method proposed by Castro-Kuriss et al. [29], the modification of DSP with censored observations can be presented by(36)DSPμ,σ=max1≤i≤r⁡2πarcsin⁡i-0.5n-arcsin⁡Ui:n,where Ui:n=Gxi:n-μ/σ. The definition of G∙ is the same as that of (2), it represents the CDF of the assumed distribution in model selection. Evaluate the value of DSP through using the candidate distribution Mi for i=1,2,…,k.
Step 4. Let X^s:nA2,j be the predicted value of Xs:n for j=1 or 2, then X^s:nA2,j can be obtained with the smallest D^SP. That is, X^s:nA2,j is the value corresponding to D^SPA2μ^A2,σ^A2, which is defined by(37)D^SPA2μ^A2,σ^A2=min⁡D^SPμ^M1,σ^M1,D^SPμ^M2,σ^M2,…,D^SPμ^Mk,σ^Mk.
If the candidate distributions are ND and SEV, Steps 2, 3, and 4 in Algorithm2 can be reduced to Step 2’ and Step 3’ as the following, respectively:
Step 2’. Obtain (μ^N,σ^N) and (μ^S,σ^S). Obtain the X^s:nN,j under the ND and obtain the X^s:nSEV,j under the SEV for s=r+1,…,n and j=1 or 2.
Step 3’. The modification of DSP with censored observations can be presented by(38)DSPμ,σ=max1≤i≤r⁡2πarcsin⁡i-0.5n-arcsin⁡Ui:n,where Ui:n=Gxi:n-μ/σ. The definition of G∙ is the same as that of (2); it represents the CDF of the assumed distribution in model selection. Evaluate the values of DSP through using the ND and SEV and denot them by D^SPNμ^N,σ^N and D^SPSEVμ^S,σ^S, respectively.
Step 4’. Let X^s:nA2,j denote the predicted value of Xs:n, then X^s:nA2,j can be obtained by(39)X^s:nA2=X^s:nN,j,ifD^SPNμ^N,σ^N<D^SPSEVμ^S,σ^SX^s:nSEV,j,ifD^SPNμ^N,σ^N≥D^SPSEVμ^S,σ^Sfors=r+1,…,nandj=1or2.Algorithm 3 (theD approach).
Step 1. Collect a type II censored sample, which has size n and r observed failure times.
Step 2. Obtain (μ^Mi,σ^Mi) for i=1,2,…,k, and then obtain X^s:nMi,j for s=r+1,…,n, i=1,2,…,k and j=1 or 2.
Step 3. Based on the method proposed by Castro-Kuriss et al. [29], the modification of D(μ,σ) with censored observations can be presented by(40)Dμ,σ=max1≤i≤r⁡2πi-0.5n-Ui:n+0.5n,where Ui:n=Gxi:n-μ/σ.
Step 4. Let X^s:nA3,j be the predicted value of Xs:n for j=1 or 2, then X^s:nA3,j can be obtained with the smallest D^(μ^Mi,σ^Mi). That is, X^s:nA3,j is the value corresponding to D^A3μ^A3,σ^A3, which is defined by(41)D^A3μ^A3,σ^A3=min⁡D^μ^M1,σ^M1,D^μ^M2,σ^M2,…,D^μ^Mk,σ^Mk.
If the candidate distributions are ND and SEV, Steps 2, 3, and 4 in Algorithm3 can be reduced to Step 2’ and Step 3’ as the following, respectively:
Step 2’. Obtain (μ^N,σ^N) and (μ^S,σ^S). Obtain X^s:nN,j under the ND and obtain X^s:nSEV,j under the SEV for s=r+1,…,n and j=1 or 2.
Step 3’. The modification of D(μ,σ) with censored observations can be presented by(42)Dμ,σ=max1≤i≤r⁡2πi-0.5n-Ui:n+0.5n,where Ui:n=Gxi:n-μ/σ. Evaluate the value of D(μ,σ) by using the ND and SEV and denote them by D^Nμ^N,σ^N and D^SEVμ^S,σ^S.
Step 4’. Let X^s:nA3,j denote the predicted value of Xs:n, then X^s:nA3,j can be obtained by(43)X^s:nA3,j=X^s:nN,j,ifD^Nμ^N,σ^N<D^SEVμ^S,σ^SX^s:nSEV,j,ifD^Nμ^N,σ^N≥D^SEVμ^S,σ^Sfors=r+1,…,nandj=1or2.


## 4. Monte Carlo Simulations

A Monte Carlo simulation study was conducted in this section, by using R language, to evaluate the performance of the proposed three approaches with two predicting methods. We consider the ND and SEV as the candidate distributions for competing the best lifetime model in the simulation study. The data sets of type II censoring sample,x1:n,…,xr:n, used in the simulation were randomly generated from the ND and SEV with location parameter μ=0 and scale parameter σ=1. Then, the sth order statistic is predicted and denoted by X^s:n for s=r+1,r+2,…,n for the sample sizes n=20,30,40,50 and 60. For the purpose of comparison, the values of the bias and mean square error (MSE) of X^s:n are evaluated using N=10000 Monte Carlo runs:(44)bais=1N∑i=1NX^s:n,i-Xs:nand(45)MSE=1N∑i=1NX^s:n,i-Xs:n2,where X^s:n,i is the predicted value of Xs:n that is obtained in the ith iteration of simulation for i=1,…,N. All simulation results are displayed in Tables 1 and 2 with the candidate distributions of ND and SEV. From Tables 1 and 2, we notice that the bias and MSE are large when the misspecification model is used. The impact of misspecification depends on the values of r and s. As n or r increases, the simulated bias and MSE are decreased. We also find that the MSE based on using the Taylor series prediction method is smaller than that based on using the expected values prediction method when the sample size is or larger than 30.Table 1
The corresponding bias and MSEs for different settings with model misspecification when true distribution is ND.
Assumed Distribution Normal distribution Extreme Value distribution X ^ s : n N , 1 X ^ s : n N , 2 X ^ s : n S E V , 1 X ^ s : n S E V , 2 n r s bias MSE bias MSE bias MSE bias MSE 10 8 9 -0.1189 0.1295 -0.3430 0.2140 -0.1970 0.1406 -0.3430 0.2140 7 8 -0.1193 0.0949 -0.2832 0.1509 -0.1597 0.0995 -0.2832 0.1509 7 9 -0.1569 0.2159 -0.3308 0.2753 -0.3058 0.2619 -0.4007 0.3196 6 7 -0.1206 0.0738 -0.2560 0.1170 -0.1402 0.0752 -0.2560 0.1170 6 8 -0.3009 0.1652 -0.1570 0.2134 -0.3471 0.1947 -0.2592 0.2399 6 9 -0.1964 0.3199 -0.2980 0.3498 -0.4147 0.4169 -0.4678 0.4566 5 6 -0.1222 0.0747 -0.2448 0.1142 -0.1298 0.0749 -0.2448 0.1142 5 7 -0.1690 0.1570 -0.2913 0.1968 -0.2452 0.1773 -0.3266 0.2152 5 8 -0.2204 0.2661 -0.2979 0.2877 -0.3842 0.3310 -0.4304 0.3606 5 9 -0.2702 0.4625 -0.3429 0.4841 -0.5557 0.6235 -0.5914 0.6567  20 16 18 -0.0675 0.0802 -0.2343 0.1178 -0.2074 0.1067 -0.2832 0.1414 14 16 -0.0626 0.0514 -0.1964 0.0780 -0.1467 0.0622 -0.2208 0.0875 14 18 -0.0802 0.1259 -0.1607 0.1372 -0.2995 0.1939 -0.3338 0.2143 12 14 -0.0589 0.0411 -0.1753 0.0640 -0.1162 0.0477 -0.1903 0.0693 12 16 -0.0863 0.0936 -0.1469 0.1019 -0.2381 0.1317 -0.2703 0.1467 12 18 -0.1119 0.1857 -0.1694 0.1942 -0.4085 0.3102 -0.4333 0.3296 10 12 -0.0664 0.0376 -0.1707 0.0579 -0.1075 0.0420 -0.1812 0.0614 10 14 -0.0884 0.0818 -0.1385 0.0886 -0.2072 0.1084 -0.2392 0.1211 10 16 -0.1151 0.1417 -0.1552 0.1474 -0.3376 0.2210 -0.3588 0.2343 10 18 -0.1497 0.2628 -0.1960 0.2701 -0.5258 0.4627 -0.5460 0.4823  30 24 27 0.0295 0.0763 -0.1161 0.0584 -0.2544 0.1226 -0.2396 0.1020 21 24 -0.0059 0.0500 -0.1078 0.0386 -0.1952 0.0725 -0.1819 0.0591 21 27 -0.0238 0.0933 -0.0991 0.0827 -0.3317 0.1889 -0.2687 0.1366 18 21 -0.0482 0.0343 -0.1087 0.0325 -0.1851 0.0574 -0.1700 0.0487 18 24 -0.0748 0.0650 -0.1111 0.0605 -0.3107 0.1451 -0.2409 0.1030 18 27 -0.0961 0.1203 -0.1328 0.1210 -0.4270 0.2816 -0.3717 0.2418 15 18 -0.0654 0.0267 -0.1096 0.0310 -0.1354 0.0388 -0.1579 0.0430 15 21 -0.0862 0.0540 -0.1052 0.0553 -0.2572 0.1096 -0.2331 0.0943 15 24 -0.1138 0.0911 -0.1310 0.0941 -0.3836 0.2094 -0.3574 0.1924 15 27 -0.1464 0.1578 -0.1722 0.1648 -0.5614 0.4196 -0.5463 0.4115  40 32 36 0.0653 0.0634 -0.0669 0.0404 -0.2696 0.1229 -0.2145 0.0840 28 32 0.0564 0.0426 -0.0536 0.0225 -0.1961 0.0685 -0.1468 0.0418 28 36 0.0622 0.0661 -0.0268 0.0495 -0.3224 0.1660 -0.2442 0.1073 24 28 0.0503 0.0349 -0.0470 0.0173 -0.1673 0.0514 -0.1175 0.0294 24 32 0.0567 0.0476 -0.0188 0.0307 -0.2464 0.1060 -0.1588 0.0554 24 36 0.0679 0.0697 -0.0030 0.0560 -0.3202 0.1666 -0.2575 0.1199 20 24 0.0402 0.0318 -0.0459 0.0152 -0.1629 0.0464 -0.1049 0.0244 20 28 0.0388 0.0405 -0.0208 0.0261 -0.2236 0.0887 -0.1203 0.0394 20 32 0.0498 0.0511 -0.0035 0.0402 -0.2503 0.1088 -0.1666 0.0637 20 36 0.0548 0.0769 0.0015 0.0674 -0.3215 0.1688 -0.2658 0.1294  50 40 45 0.0660 0.0504 -0.0447 0.0310 -0.2813 0.1201 -0.1992 0.0700 35 40 0.0564 0.0352 -0.0387 0.0175 -0.2043 0.0676 -0.1330 0.0342 35 45 0.0671 0.0551 -0.0138 0.0407 -0.3223 0.1587 -0.2372 0.0970 30 35 0.0506 0.0293 -0.0321 0.0131 -0.1742 0.0503 -0.1016 0.0228 30 40 0.0580 0.0371 -0.0074 0.0239 -0.2479 0.0992 -0.1500 0.0467 30 45 0.0676 0.0564 0.0076 0.0457 -0.3245 0.1580 -0.2547 0.1092 25 30 0.0409 0.0260 -0.0313 0.0110 -0.1724 0.0446 -0.0903 0.0180 25 35 0.0496 0.0313 -0.0065 0.0193 -0.2235 0.0822 -0.1108 0.0319 25 40 0.0567 0.0393 0.0081 0.0299 -0.2530 0.1017 -0.1640 0.0559 25 45 0.0702 0.0577 0.0221 0.0492 -0.3245 0.1573 -0.2646 0.1164  60 48 54 0.0639 0.0445 -0.0350 0.0266 -0.2901 0.1196 -0.1925 0.0630 42 48 0.0554 0.0309 -0.0275 0.0144 -0.2101 0.0643 -0.1227 0.0285 42 54 0.0644 0.0457 -0.0046 0.0341 -0.3264 0.1523 -0.2325 0.0886 36 42 0.0500 0.0247 -0.0240 0.0106 -0.1803 0.0491 -0.0929 0.0186 36 48 0.0573 0.0325 -0.0024 0.0209 -0.2505 0.0969 -0.1465 0.0429 36 54 0.0678 0.0488 0.0148 0.0396 -0.3268 0.1542 -0.2526 0.1040 30 36 0.0416 0.0215 -0.0226 0.0092 -0.1786 0.0466 -0.0798 0.0154 30 42 0.0452 0.0260 -0.0030 0.0162 -0.2304 0.0786 -0.1082 0.0276 30 48 0.0573 0.0333 0.0158 0.0257 -0.2532 0.0983 -0.1588 0.0512 30 54 0.0674 0.0494 0.0233 0.0429 -0.3290 0.1568 -0.2668 0.1139Table 2
The corresponding bias and MSEs for different settings with model misspecification when the true distribution is SEV.
Assumed Distribution Normal distribution Extreme Value distribution X ^ s : n N , 1 X ^ s : n N , 2 X ^ s : n S E V , 1 X ^ s : n S E V , 2 n r s bias MSE bias MSE bias MSE bias MSE 10 8 9 0.1225 0.1563 -0.2965 0.1583 -0.0669 0.0868 -0.2969 0.1589 7 8 0.0417 0.1109 -0.2685 0.1365 -0.0739 0.0793 -0.2692 0.1376 7 9 0.1659 0.2742 -0.1233 0.1783 -0.1131 0.1742 -0.2539 0.2066 6 7 -0.0155 0.0969 -0.2687 0.1355 -0.0864 0.0797 -0.2702 0.1364 6 8 0.0710 0.2206 -0.1725 0.1852 -0.1287 0.1759 -0.2635 0.2118 6 9 0.2075 0.4628 0.0304 0.3333 -0.1741 0.2954 -0.2571 0.3099 5 6 -0.0532 0.1017 -0.2767 0.1497 -0.0913 0.0909 -0.278 0.1516 5 7 -0.0037 0.2267 -0.2144 0.2142 -0.1548 0.202 -0.2848 0.2367 5 8 0.0846 0.4086 -0.0544 0.3325 -0.2104 0.3158 -0.2858 0.3299 5 9 0.2196 0.7279 0.0913 0.5969 -0.2717 0.4812 -0.3297 0.4914  20 16 18 0.3121 0.2071 -0.0726 0.0498 -0.0441 0.0543 -0.1745 0.0747 14 16 0.2053 0.1345 -0.1028 0.0466 -0.0455 0.0431 -0.1658 0.0639 14 18 0.3306 0.2389 0.1039 0.0949 -0.0702 0.0961 -0.1327 0.1031 12 14 0.1424 0.1072 -0.1198 0.0481 -0.0445 0.0404 -0.1646 0.0605 12 16 0.2234 0.1702 0.0440 0.0791 -0.0745 0.0868 -0.133 0.0925 12 18 0.3685 0.3154 0.2098 0.1901 -0.0949 0.155 -0.1402 0.1594 10 12 0.0967 0.0948 -0.1360 0.0561 -0.0496 0.0454 -0.1718 0.0667 10 14 0.1624 0.1458 0.0156 0.0853 -0.0789 0.0949 -0.136 0.1015 10 16 0.2667 0.2553 0.1467 0.1775 -0.1032 0.1629 -0.1417 0.1668 10 18 0.4383 0.4968 0.3199 0.3774 -0.1335 0.2516 -0.1691 0.2546  30 24 27 0.5482 0.3543 0.0788 0.0330 0.0846 0.0522 -0.0552 0.0298 21 24 0.4351 0.2472 0.0270 0.0212 0.0717 0.0477 -0.0573 0.0238 21 27 0.5448 0.3562 0.2068 0.0821 0.082 0.0574 -0.0062 0.0386 18 21 0.3743 0.1980 0.0003 0.0188 0.0594 0.0452 -0.0634 0.0228 18 24 0.4215 0.2414 0.1135 0.0478 0.0588 0.0523 -0.0222 0.0343 18 27 0.5250 0.3420 0.2612 0.1198 0.065 0.064 0.0012 0.0518 15 18 0.3146 0.1777 -0.0232 0.0221 0.0308 0.0473 -0.0821 0.0289 15 21 0.3257 0.1857 0.0575 0.0392 0.0069 0.0597 -0.0578 0.0447 15 24 0.3688 0.2190 0.1409 0.0693 -0.0092 0.0766 -0.0588 0.0667 15 27 0.4753 0.3279 0.2759 0.1478 -0.0185 0.1067 -0.0598 0.0998  40 32 36 0.5536 0.3486 0.1141 0.0342 0.0715 0.0397 -0.0358 0.0221 28 32 0.4471 0.2346 0.0564 0.0183 0.0675 0.0352 -0.0346 0.0166 28 36 0.5551 0.3485 0.2345 0.0847 0.0742 0.0422 0.0038 0.0289 24 28 0.3923 0.1946 0.0293 0.0144 0.0611 0.0344 -0.0361 0.0148 24 32 0.4471 0.2334 0.1434 0.0436 0.0666 0.0374 -0.0006 0.0234 24 36 0.5584 0.3537 0.3026 0.1257 0.08 0.0454 0.0279 0.0349 20 24 0.3732 0.1831 0.0142 0.0149 0.0575 0.0362 -0.0384 0.0163 20 28 0.3929 0.1921 0.0932 0.0327 0.0588 0.0364 -0.0059 0.0229 20 32 0.4455 0.2372 0.1958 0.0674 0.0678 0.0392 0.021 0.0289 20 36 0.5526 0.3522 0.3374 0.1536 0.0762 0.0466 0.0362 0.0386  50 40 45 0.5605 0.3444 0.1386 0.0357 0.0686 0.0324 -0.0204 0.018 35 40 0.4484 0.2291 0.0717 0.0174 0.0611 0.028 -0.0238 0.0127 35 45 0.5600 0.3432 0.2491 0.0864 0.0698 0.0334 0.0086 0.0237 30 35 0.3925 0.1859 0.0403 0.0127 0.0538 0.0282 -0.0265 0.0117 30 40 0.4469 0.2289 0.1560 0.0426 0.0617 0.0304 0.0069 0.0189 30 45 0.5594 0.3493 0.3134 0.1271 0.0722 0.0363 0.0277 0.0283 25 30 0.3710 0.1736 0.0242 0.0125 0.0499 0.0287 -0.0276 0.0126 25 35 0.3938 0.1869 0.1068 0.0296 0.0566 0.0303 0.0047 0.0184 25 40 0.4441 0.2324 0.2015 0.0661 0.0612 0.0323 0.0217 0.0245 25 45 0.5623 0.3468 0.3558 0.1555 0.0773 0.0377 0.0436 0.0318  60 48 54 0.5617 0.3438 0.1502 0.0374 0.0631 0.0267 -0.0151 0.0148 42 48 0.4475 0.2247 0.0801 0.0170 0.0562 0.024 -0.0178 0.0107 42 54 0.5629 0.3469 0.2619 0.0898 0.0666 0.0294 0.0139 0.0205 36 42 0.3943 0.1802 0.0506 0.0119 0.0529 0.0232 -0.017 0.0097 36 48 0.4478 0.2224 0.1630 0.0418 0.0586 0.0246 0.0103 0.0159 36 54 0.5620 0.3444 0.3243 0.1275 0.0689 0.0312 0.0306 0.0246 30 36 0.3690 0.1660 0.0312 0.0109 0.0448 0.0238 -0.0202 0.0102 30 42 0.3904 0.1805 0.1103 0.0278 0.0503 0.0251 0.0059 0.0155 30 48 0.4472 0.2239 0.2119 0.0634 0.0606 0.0266 0.0276 0.0204 30 54 0.5610 0.3456 0.3606 0.1571 0.0701 0.0323 0.0405 0.0273To evaluate the performance of the three proposed model selection approaches for MLP, Tables3–5 report the simulation results for three model selection approaches from the ND. Tables 6–8 respectively report the simulation results for three model selection approaches from the SEV. The column “correct (%)” presented in Tables 3–8 is the correct model selection rate in all simulation runs. From Tables 3–8 we find that the three model selection approaches have good ability to identify the correct underlying distribution with a high probability. Moreover, the MSEs of these three approaches are close to those simulated MSEs of the cases by using the real underlying distribution. Overall, the correct model selection rates through using DSP approach or D approach are higher than that of using the RRML approach when the sample size is smaller than 30. When the sample size grows to or over 30, the performance of the RRML approach is improved and the correct model selection rate of the RRML approach is higher than that are obtained by using the DSP or D approach. To compare the performance of using two different MLPs, the MSEs of using the expected values prediction method are smaller than that using the Taylor series prediction method when the sample size is smaller than 30. The proposed approaches can perform well under large sample size cases.Table 3
The corresponding bias and MSEs for different settings of RML approach when the true distribution is ND.
RML approach X ^ s : n A 1,1 X ^ s : n A 1 , 2 n r s bias MSE bias MSE Correct (%) 10 8 9 -0.1796 0.1372 -0.3430 0.2140 0.6430 7 8 -0.1580 0.1001 -0.2832 0.1509 0.6263 7 9 -0.2459 0.2408 -0.3732 0.3023 0.6263 6 7 -0.1466 0.0772 -0.2560 0.1170 0.5959 6 8 -0.3316 0.1836 -0.2226 0.2313 0.5959 6 9 -0.3150 0.3745 -0.3867 0.4098 0.5959 5 6 -0.1399 0.0774 -0.2448 0.1142 0.5908 5 7 -0.2206 0.1701 -0.3164 0.2098 0.5908 5 8 -0.3144 0.3048 -0.3709 0.3303 0.5908 5 9 -0.4189 0.5512 -0.4695 0.5794 0.5908  20 16 18 -0.1301 0.0875 -0.2542 0.1275 0.7195 14 16 -0.1087 0.0556 -0.2093 0.0831 0.6955 14 18 -0.1741 0.1534 -0.2281 0.1702 0.6955 12 14 -0.0944 0.0443 -0.1850 0.0674 0.6657 12 16 -0.1597 0.1112 -0.2008 0.1229 0.6657 12 18 -0.2382 0.2419 -0.2764 0.2561 0.6657 10 12 -0.0948 0.0404 -0.1787 0.0606 0.6477 10 14 -0.1489 0.0949 -0.1849 0.1046 0.6477 10 16 -0.2164 0.1806 -0.2432 0.1901 0.6477 10 18 -0.3104 0.3566 -0.3419 0.3699 0.6477  30 24 27 0.0092 0.0718 -0.1224 0.0604 0.9508 21 24 -0.0235 0.0472 -0.1130 0.0402 0.9345 21 27 -0.0493 0.0950 -0.1135 0.0888 0.9345 18 21 -0.0700 0.0348 -0.1172 0.0351 0.8478 18 24 -0.1114 0.0752 -0.1357 0.0725 0.8478 18 27 -0.1531 0.1561 -0.1788 0.1581 0.8478 15 18 -0.0920 0.0295 -0.1234 0.0344 0.7104 15 21 -0.1364 0.0676 -0.1451 0.0700 0.7104 15 24 -0.1985 0.1293 -0.2061 0.1334 0.7104 15 27 -0.2789 0.2534 -0.2945 0.2621 0.7104  40 32 36 0.0533 0.0603 -0.0708 0.0413 0.9717 28 32 0.0476 0.0408 -0.0560 0.0229 0.9754 28 36 0.0514 0.0651 -0.0324 0.0507 0.9754 24 28 0.0445 0.0334 -0.0484 0.0175 0.9777 24 32 0.0497 0.0464 -0.0218 0.0311 0.9777 24 36 0.0593 0.0691 -0.0083 0.0572 0.9777 20 24 0.0352 0.0308 -0.0469 0.0154 0.9817 20 28 0.0331 0.0399 -0.0228 0.0263 0.9817 20 32 0.0434 0.0508 -0.0069 0.0407 0.9817 20 36 0.0467 0.0768 -0.0042 0.0683 0.9817  50 40 45 0.0589 0.0488 -0.0472 0.0313 0.9820 35 40 0.0505 0.0341 -0.0403 0.0177 0.9847 35 45 0.0595 0.0542 -0.0178 0.0411 0.9847 30 35 0.0473 0.0286 -0.0329 0.0131 0.9876 30 40 0.0539 0.0366 -0.0093 0.0241 0.9876 30 45 0.0624 0.0559 0.0042 0.0463 0.9876 25 30 0.0381 0.0257 -0.0319 0.0111 0.9878 25 35 0.0466 0.0310 -0.0076 0.0194 0.9878 25 40 0.0527 0.0389 0.0060 0.0300 0.9878 25 45 0.0660 0.0577 0.0191 0.0497 0.9878  60 48 54 0.0586 0.0435 -0.0368 0.0270 0.9913 42 48 0.0522 0.0300 -0.0284 0.0144 0.9919 42 54 0.0611 0.0453 -0.0064 0.0345 0.9919 36 42 0.0477 0.0243 -0.0245 0.0107 0.9933 36 48 0.0543 0.0322 -0.0038 0.0210 0.9933 36 54 0.0644 0.0487 0.0126 0.0400 0.9933 30 36 0.0401 0.0212 -0.0229 0.0092 0.9928 30 42 0.0433 0.0257 -0.0037 0.0163 0.9928 30 48 0.0555 0.0333 0.0148 0.0258 0.9928 30 54 0.0649 0.0492 0.0215 0.0429 0.9928Table 4
The corresponding bias and MSEs for different settings ofDsp approach when the true distribution is ND.
D s p approach X ^ s : n A 2,1 X ^ s : n A 2 , 2 n r s bias MSE bias MSE correct (%) 10 8 9 -0.1489 0.1300 -0.3430 0.2140 0.6694 7 8 -0.1285 0.0932 -0.2832 0.1509 0.6654 7 9 -0.2016 0.2273 -0.3517 0.2880 0.6654 6 7 -0.1181 0.0716 -0.2560 0.1170 0.6616 6 8 -0.3112 0.1713 -0.1809 0.2190 0.6616 6 9 -0.2560 0.3466 -0.3461 0.3798 0.6616 5 6 -0.1138 0.0725 -0.2448 0.1142 0.6305 5 7 -0.1820 0.1603 -0.2965 0.1994 0.6305 5 8 -0.2621 0.2818 -0.3339 0.3061 0.6305 5 9 -0.3529 0.5053 -0.4170 0.5306 0.6305  20 16 18 -0.1159 0.0843 -0.2497 0.1252 0.7314 14 16 -0.0892 0.0529 -0.2038 0.0810 0.7198 14 18 -0.1456 0.1450 -0.2109 0.1610 0.7198 12 14 -0.0748 0.0420 -0.1795 0.0655 0.6935 12 16 -0.1308 0.1049 -0.1835 0.1159 0.6935 12 18 -0.1989 0.2247 -0.2475 0.2369 0.6935 10 12 -0.0741 0.0379 -0.1724 0.0584 0.6781 10 14 -0.1193 0.0886 -0.1668 0.0976 0.6781 10 16 -0.1791 0.1639 -0.2156 0.1721 0.6781 10 18 -0.2580 0.3209 -0.2985 0.3318 0.6781  30 24 27 -0.0610 0.0618 -0.1422 0.0666 0.7082 21 24 -0.0637 0.0390 -0.1211 0.0415 0.7488 21 27 -0.0998 0.0901 -0.1355 0.0935 0.7488 18 21 -0.0755 0.0314 -0.1174 0.0348 0.8027 18 24 -0.1177 0.0693 -0.1369 0.0702 0.8027 18 27 -0.1615 0.1472 -0.1826 0.1534 0.8027 15 18 -0.0778 0.0277 -0.1184 0.0333 0.6998 15 21 -0.1206 0.0629 -0.1365 0.0662 0.6998 15 24 -0.1796 0.1204 -0.1935 0.1254 0.6998 15 27 -0.2566 0.2319 -0.2774 0.2416 0.6998  40 32 36 -0.0416 0.0511 -0.1013 0.0493 0.7000 28 32 -0.0287 0.0289 -0.0742 0.0259 0.6958 28 36 -0.0467 0.0643 -0.0800 0.0629 0.6958 24 28 -0.0225 0.0230 -0.0609 0.0193 0.7010 24 32 -0.0278 0.0389 -0.0505 0.0353 0.7010 24 36 -0.0390 0.0710 -0.0650 0.0702 0.7010 20 24 -0.0207 0.0203 -0.0554 0.0163 0.7535 20 28 -0.0232 0.0321 -0.0382 0.0277 0.7535 20 32 -0.0218 0.0468 -0.0362 0.0445 0.7535 20 36 -0.0346 0.0793 -0.0556 0.0784 0.7535  50 40 45 -0.0314 0.0421 -0.0780 0.0383 0.7221 35 40 -0.0244 0.0245 -0.0590 0.0203 0.7185 35 45 -0.0376 0.0552 -0.0670 0.0528 0.7185 30 35 -0.0171 0.0190 -0.0451 0.0145 0.7377 30 40 -0.0193 0.0314 -0.0375 0.0278 0.7377 30 45 -0.0303 0.0596 -0.0511 0.0583 0.7377 25 30 -0.0166 0.0168 -0.0401 0.0118 0.7656 25 35 -0.0112 0.0252 -0.0243 0.0210 0.7656 25 40 -0.0116 0.0373 -0.0240 0.0342 0.7656 25 45 -0.0179 0.0630 -0.0357 0.0611 0.7656  60 48 54 -0.0254 0.0383 -0.0664 0.0331 0.7520 42 48 -0.0179 0.0219 -0.0463 0.0168 0.7444 42 54 -0.0276 0.0485 -0.0524 0.0453 0.7444 36 42 -0.0137 0.0164 -0.0364 0.0118 0.7635 36 48 -0.0141 0.0283 -0.0306 0.0244 0.7635 36 54 -0.0192 0.0537 -0.0383 0.0517 0.7635 30 36 -0.0098 0.0146 -0.0304 0.0098 0.7864 30 42 -0.0068 0.0217 -0.0184 0.0177 0.7864 30 48 -0.0028 0.0321 -0.0131 0.0291 0.7864 30 54 -0.0137 0.0565 -0.0307 0.0547 0.7864Table 5
The corresponding bias and MSEs for different settings ofD approach when the true distribution is ND.
D approach X ^ s : n A 3,1 X ^ s : n A 3,2 n r s bias MSE bias MSE correct (%) 10 8 9 -0.1491 0.1300 -0.3430 0.2140 0.6687 7 8 -0.1286 0.0932 -0.2832 0.1509 0.6648 7 9 -0.2016 0.2274 -0.3517 0.2881 0.6648 6 7 -0.1181 0.0716 -0.2560 0.1170 0.6615 6 8 -0.3113 0.1713 -0.1809 0.2190 0.6615 6 9 -0.2560 0.3466 -0.3461 0.3798 0.6615 5 6 -0.1138 0.0725 -0.2448 0.1142 0.6306 5 7 -0.1820 0.1603 -0.2965 0.1994 0.6306 5 8 -0.2621 0.2818 -0.3339 0.3061 0.6306 5 9 -0.3529 0.5054 -0.4170 0.5307 0.6306  20 16 18 -0.1163 0.0844 -0.2498 0.1253 0.7293 14 16 -0.0893 0.0529 -0.2039 0.0810 0.7190 14 18 -0.1458 0.1451 -0.2109 0.1611 0.7190 12 14 -0.0748 0.0420 -0.1795 0.0655 0.6933 12 16 -0.1308 0.1049 -0.1835 0.1159 0.6933 12 18 -0.1990 0.2248 -0.2476 0.2370 0.6933 10 12 -0.0741 0.0379 -0.1724 0.0584 0.6781 10 14 -0.1193 0.0886 -0.1668 0.0976 0.6781 10 16 -0.1791 0.1639 -0.2156 0.1721 0.6781 10 18 -0.2580 0.3209 -0.2984 0.3318 0.6781  30 24 27 -0.0640 0.0618 -0.1433 0.0669 0.7006 21 24 -0.0646 0.0390 -0.1213 0.0416 0.7454 21 27 -0.1009 0.0903 -0.1361 0.0937 0.7454 18 21 -0.0757 0.0314 -0.1175 0.0348 0.8016 18 24 -0.1178 0.0693 -0.1369 0.0702 0.8016 18 27 -0.1620 0.1472 -0.1829 0.1534 0.8016 15 18 -0.0778 0.0276 -0.1184 0.0333 0.6998 15 21 -0.1206 0.0629 -0.1365 0.0662 0.6998 15 24 -0.1796 0.1204 -0.1935 0.1254 0.6998 15 27 -0.2565 0.2319 -0.2773 0.2416 0.6998  40 32 36 -0.0458 0.0514 -0.1028 0.0497 0.6912 28 32 -0.0300 0.0290 -0.0745 0.0260 0.6916 28 36 -0.0486 0.0642 -0.0811 0.0630 0.6916 24 28 -0.0230 0.0230 -0.0610 0.0193 0.6990 24 32 -0.0286 0.0389 -0.0508 0.0354 0.6990 24 36 -0.0398 0.0711 -0.0655 0.0703 0.6990 20 24 -0.0206 0.0203 -0.0554 0.0163 0.7536 20 28 -0.0231 0.0321 -0.0382 0.0277 0.7536 20 32 -0.0217 0.0468 -0.0362 0.0445 0.7536 20 36 -0.0345 0.0792 -0.0554 0.0784 0.7536  50 40 45 -0.0349 0.0424 -0.0793 0.0387 0.7099 35 40 -0.0259 0.0246 -0.0594 0.0204 0.7138 35 45 -0.0397 0.0555 -0.0682 0.0531 0.7138 30 35 -0.0179 0.0190 -0.0453 0.0146 0.7353 30 40 -0.0199 0.0313 -0.0378 0.0278 0.7353 30 45 -0.0310 0.0597 -0.0515 0.0584 0.7353 25 30 -0.0165 0.0168 -0.0401 0.0118 0.7659 25 35 -0.0111 0.0252 -0.0242 0.0210 0.7659 25 40 -0.0115 0.0373 -0.0240 0.0342 0.7659 25 45 -0.0179 0.0629 -0.0357 0.0610 0.7659  60 48 54 -0.0294 0.0387 -0.0679 0.0335 0.7393 42 48 -0.0199 0.0220 -0.0469 0.0169 0.7384 42 54 -0.0302 0.0487 -0.0538 0.0455 0.7384 36 42 -0.0146 0.0164 -0.0366 0.0118 0.7617 36 48 -0.0149 0.0283 -0.0309 0.0244 0.7617 36 54 -0.0200 0.0539 -0.0388 0.0519 0.7617 30 36 -0.0097 0.0146 -0.0304 0.0098 0.7867 30 42 -0.0067 0.0217 -0.0184 0.0177 0.7867 30 48 -0.0028 0.0320 -0.0130 0.0291 0.7867 30 54 -0.0135 0.0565 -0.0306 0.0547 0.7867Table 6
The corresponding bias and MSEs for different settings of RML approach when the true distribution is SEV.
RML approach X ^ s : n A 1,1 X ^ s : n A 1 , 2 n r s bias MSE bias MSE correct(%) 10 8 9 -0.0461 0.0865 -0.2969 0.1589 0.5938 7 8 -0.0679 0.0798 -0.2692 0.1376 0.5587 7 9 -0.0520 0.1779 -0.2258 0.1977 0.5587 6 7 -0.0897 0.0814 -0.2702 0.1364 0.5445 6 8 -0.0866 0.1746 -0.2454 0.2047 0.5445 6 9 -0.0682 0.3140 -0.1716 0.3048 0.5445 5 6 -0.1010 0.0935 -0.2780 0.1516 0.5242 5 7 -0.1234 0.2004 -0.2715 0.2314 0.5242 5 8 -0.1228 0.3226 -0.2119 0.3214 0.5242 5 9 -0.1070 0.5253 -0.1835 0.5058 0.5242  20 16 18 -0.0089 0.0544 -0.1626 0.0713 0.7451 14 16 -0.0222 0.0422 -0.1592 0.0618 0.7003 14 18 -0.0012 0.1000 -0.0764 0.0988 0.7003 12 14 -0.0272 0.0398 -0.1603 0.0592 0.6548 12 16 -0.0183 0.0883 -0.0846 0.0882 0.6548 12 18 0.0226 0.1753 -0.0332 0.1670 0.6548 10 12 -0.0371 0.0447 -0.1692 0.0658 0.6058 10 14 -0.0277 0.0946 -0.0895 0.0960 0.6058 10 16 0.0028 0.1753 -0.0415 0.1707 0.6058 10 18 0.0512 0.3056 0.0052 0.2909 0.6058  30 24 27 0.1143 0.0639 -0.0454 0.0299 0.9188 21 24 0.0991 0.0541 -0.0495 0.0235 0.9066 21 27 0.1171 0.0717 0.0121 0.0423 0.9066 18 21 0.0776 0.0471 -0.0580 0.0223 0.9046 18 24 0.0812 0.0569 -0.0110 0.0351 0.9046 18 27 0.0978 0.0742 0.0241 0.0562 0.9046 15 18 0.0360 0.0465 -0.0783 0.0281 0.8915 15 21 0.0222 0.0569 -0.0453 0.0423 0.8915 15 24 0.0181 0.0730 -0.0342 0.0626 0.8915 15 27 0.0242 0.1017 -0.0200 0.0933 0.8915  40 32 36 0.0930 0.0472 -0.0282 0.0226 0.9441 28 32 0.0869 0.0402 -0.0292 0.0167 0.9393 28 36 0.0994 0.0542 0.0174 0.0325 0.9393 24 28 0.0809 0.0391 -0.0312 0.0147 0.9346 24 32 0.0896 0.0454 0.0096 0.0250 0.9346 24 36 0.1109 0.0599 0.0478 0.0413 0.9346 20 24 0.0760 0.0406 -0.0344 0.0161 0.9227 20 28 0.0804 0.0423 0.0021 0.0239 0.9227 20 32 0.0928 0.0485 0.0347 0.0322 0.9227 20 36 0.1108 0.0631 0.0605 0.0476 0.9227  50 40 45 0.0815 0.0378 -0.0157 0.0185 0.9673 35 40 0.0735 0.0317 -0.0201 0.0128 0.9593 35 45 0.0853 0.0408 0.0172 0.0262 0.9593 30 35 0.0679 0.0311 -0.0230 0.0117 0.9557 30 40 0.0771 0.0356 0.0139 0.0202 0.9557 30 45 0.0929 0.0465 0.0415 0.0332 0.9557 25 30 0.0629 0.0317 -0.0248 0.0126 0.9475 25 35 0.0707 0.0343 0.0100 0.0190 0.9475 25 40 0.0789 0.0389 0.0315 0.0269 0.9475 25 45 0.0988 0.0506 0.0593 0.0391 0.9475  60 48 54 0.0718 0.0302 -0.0118 0.0154 0.9808 42 48 0.0649 0.0266 -0.0152 0.0109 0.9731 42 54 0.0774 0.0353 0.0201 0.0226 0.9731 36 42 0.0618 0.0256 -0.0149 0.0098 0.9707 36 48 0.0695 0.0285 0.0155 0.0169 0.9707 36 54 0.0845 0.0384 0.0411 0.0281 0.9707 30 36 0.0542 0.0256 -0.0183 0.0102 0.9633 30 42 0.0606 0.0279 0.0099 0.0160 0.9633 30 48 0.0727 0.0307 0.0344 0.0221 0.9633 30 54 0.0855 0.0420 0.0518 0.0329 0.9633Table 7
The corresponding bias and MSEs for different settings ofDsp approach when the true distribution is SEV.
D s p approach X ^ s : n A 2,1 X ^ s : n A 2 , 2 n r s bias MSE bias MSE correct (%) 10 8 9 -0.0013 0.0904 -0.2969 0.1589 0.4340 7 8 -0.0222 0.0817 -0.2692 0.1376 0.3635 7 9 0.0392 0.2032 -0.1820 0.1869 0.3635 6 7 -0.0418 0.0830 -0.2702 0.1364 0.3208 6 8 0.0059 0.1959 -0.2014 0.1923 0.3208 6 9 0.0879 0.3923 -0.0579 0.3199 0.3208 5 6 -0.0539 0.0936 -0.2780 0.1516 0.3094 5 7 -0.0312 0.2201 -0.2253 0.2182 0.3094 5 8 0.0199 0.3925 -0.1053 0.3373 0.3094 5 9 0.1007 0.6842 -0.0108 0.5837 0.3094  20 16 18 0.0221 0.0565 -0.1521 0.0687 0.6408 14 16 0.0044 0.0437 -0.1515 0.0597 0.5695 14 18 0.0503 0.1109 -0.0417 0.0991 0.5695 12 14 0.0011 0.0405 -0.1527 0.0569 0.5011 12 16 0.0354 0.0953 -0.0474 0.0858 0.5011 12 18 0.1110 0.2044 0.0396 0.1780 0.5011 10 12 -0.0017 0.0460 -0.1591 0.0628 0.4051 10 14 0.0403 0.1047 -0.0403 0.0943 0.4051 10 16 0.1041 0.2017 0.0431 0.1791 0.4051 10 18 0.2156 0.3925 0.1515 0.3447 0.4051  30 24 27 0.1346 0.0585 -0.0364 0.0298 0.8704 21 24 0.1191 0.0495 -0.0415 0.0230 0.8433 21 27 0.1471 0.0743 0.0305 0.0463 0.8433 18 21 0.1052 0.0451 -0.0491 0.0216 0.8042 18 24 0.1168 0.0590 0.0069 0.0368 0.8042 18 27 0.1424 0.0880 0.0548 0.0658 0.8042 15 18 0.0703 0.0455 -0.0678 0.0267 0.7363 15 21 0.0676 0.0589 -0.0249 0.0426 0.7363 15 24 0.0689 0.0790 -0.0024 0.0649 0.7363 15 27 0.0993 0.1233 0.0374 0.1067 0.7363  40 32 36 0.1130 0.0465 -0.0196 0.0234 0.9068 28 32 0.1061 0.0375 -0.0221 0.0168 0.8795 28 36 0.1258 0.0580 0.0337 0.0371 0.8795 24 28 0.0985 0.0351 -0.0251 0.0147 0.8582 24 32 0.1130 0.0446 0.0223 0.0272 0.8582 24 36 0.1402 0.0694 0.0692 0.0507 0.8582 20 24 0.0945 0.0358 -0.0285 0.0159 0.8438 20 28 0.0994 0.0403 0.0112 0.0249 0.8438 20 32 0.1191 0.0517 0.0519 0.0369 0.8438 20 36 0.1413 0.0759 0.0852 0.0600 0.8438  50 40 45 0.0996 0.0382 -0.0082 0.0194 0.9309 35 40 0.0919 0.0304 -0.0137 0.0131 0.9069 35 45 0.1094 0.0477 0.0318 0.0313 0.9069 30 35 0.0854 0.0287 -0.0176 0.0118 0.8834 30 40 0.1010 0.0368 0.0264 0.0223 0.8834 30 45 0.1238 0.0571 0.0635 0.0422 0.8834 25 30 0.0814 0.0283 -0.0198 0.0125 0.8760 25 35 0.0925 0.0339 0.0199 0.0202 0.8760 25 40 0.1022 0.0423 0.0465 0.0308 0.8760 25 45 0.1344 0.0643 0.0870 0.0517 0.8760  60 48 54 0.0874 0.0319 -0.0054 0.0164 0.9491 42 48 0.0804 0.0264 -0.0099 0.0114 0.9272 42 54 0.1003 0.0408 0.0338 0.0267 0.9272 36 42 0.0776 0.0242 -0.0102 0.0100 0.9116 36 48 0.0912 0.0304 0.0266 0.0189 0.9116 36 54 0.1110 0.0472 0.0601 0.0354 0.9116 30 36 0.0725 0.0239 -0.0136 0.0103 0.8991 30 42 0.0802 0.0285 0.0185 0.0172 0.8991 30 48 0.0967 0.0358 0.0494 0.0262 0.8991 30 54 0.1162 0.0420 0.0757 0.0430 0.8991Table 8
The corresponding bias and MSE for different settings ofD approach when the true distribution is SEV.
D approach X ^ s : n A 3,1 X ^ s : n A 3 , 2 n r s bias MSE bias MSE correct (%) 10 8 9 -0.0018 0.0902 -0.2969 0.1589 0.4362 7 8 -0.0223 0.0817 -0.2692 0.1376 0.364 7 9 0.0389 0.2031 -0.1821 0.1870 0.364 6 7 -0.0419 0.0830 -0.2702 0.1364 0.321 6 8 0.0058 0.1959 -0.2015 0.1923 0.321 6 9 0.0877 0.3923 -0.0580 0.3199 0.321 5 6 -0.0539 0.0936 -0.2780 0.1516 0.3093 5 7 -0.0312 0.2201 -0.2253 0.2182 0.3093 5 8 0.0199 0.3925 -0.1053 0.3373 0.3093 5 9 0.1007 0.6842 -0.0108 0.5837 0.3093  20 16 18 0.0212 0.0564 -0.1524 0.0688 0.6441 14 16 0.0041 0.0436 -0.1516 0.0597 0.5715 14 18 0.0499 0.1106 -0.0419 0.0990 0.5715 12 14 0.0011 0.0405 -0.1528 0.0569 0.5014 12 16 0.0354 0.0954 -0.0474 0.0858 0.5014 12 18 0.1110 0.2044 0.0395 0.1780 0.5014 10 12 -0.0017 0.0460 -0.1591 0.0628 0.405 10 14 0.0404 0.1047 -0.0403 0.0943 0.405 10 16 0.1041 0.2017 0.0431 0.1791 0.405 10 18 0.2157 0.3925 0.1516 0.3447 0.405  30 24 27 0.1310 0.0578 -0.0377 0.0298 0.8771 21 24 0.1181 0.0493 -0.0418 0.0230 0.8474 21 27 0.1452 0.0736 0.0295 0.0461 0.8474 18 21 0.1046 0.0450 -0.0493 0.0216 0.8059 18 24 0.1164 0.0589 0.0067 0.0368 0.8059 18 27 0.1416 0.0876 0.0544 0.0656 0.8059 15 18 0.0704 0.0455 -0.0677 0.0267 0.7359 15 21 0.0678 0.0589 -0.0248 0.0426 0.7359 15 24 0.0690 0.0790 -0.0023 0.0649 0.7359 15 27 0.0994 0.1234 0.0374 0.1067 0.7359  40 32 36 0.1103 0.0454 -0.0206 0.0232 0.9126 28 32 0.1041 0.0374 -0.0227 0.0168 0.8831 28 36 0.1244 0.0575 0.0329 0.0369 0.8831 24 28 0.0979 0.0350 -0.0252 0.0147 0.8598 24 32 0.1125 0.0445 0.0221 0.0271 0.8598 24 36 0.1393 0.0690 0.0687 0.0505 0.8598 20 24 0.0946 0.0358 -0.0284 0.0159 0.8433 20 28 0.0995 0.0403 0.0112 0.0249 0.8433 20 32 0.1192 0.0518 0.0519 0.0369 0.8433 20 36 0.1414 0.0759 0.0853 0.0601 0.8433  50 40 45 0.0973 0.0373 -0.0091 0.0192 0.9364 35 40 0.0904 0.0301 -0.0142 0.0131 0.9113 35 45 0.1074 0.0469 0.0306 0.0310 0.9113 30 35 0.0848 0.0287 -0.0177 0.0118 0.8852 30 40 0.1003 0.0367 0.0261 0.0222 0.8852 30 45 0.1233 0.0568 0.0632 0.0421 0.8852 25 30 0.0814 0.0283 -0.0197 0.0125 0.876 25 35 0.0926 0.0339 0.0199 0.0202 0.876 25 40 0.1022 0.0423 0.0465 0.0308 0.876 25 45 0.1345 0.0643 0.0871 0.0517 0.876  60 48 54 0.0845 0.0310 -0.0065 0.0162 0.9554 42 48 0.0794 0.0261 -0.0102 0.0113 0.9301 42 54 0.0992 0.0402 0.0332 0.0265 0.9301 36 42 0.0772 0.0241 -0.0103 0.0100 0.9127 36 48 0.0905 0.0302 0.0263 0.0188 0.9127 36 54 0.1099 0.0470 0.0595 0.0353 0.9127 30 36 0.0725 0.0239 -0.0136 0.0103 0.899 30 42 0.0803 0.0285 0.0186 0.0172 0.899 30 48 0.0968 0.0358 0.0494 0.0262 0.899 30 54 0.1163 0.0532 0.0758 0.0430 0.899


## 5. Illustrative Examples

In this section, three numerical examples are presented to illustrate the proposed approaches in Sections2–4.


### 5.1. Example 1

A test airplane component’s failure time dataset provided in Mann and Fertig [30], in which 13 components were placed on test, and the test was terminated at the time of the 10th failure. The failure times (in hours) of the 10 components that failed wereD1:   0.22, 0.50, 0.88, 1.00, 1.32, 1.33, 1.54, 1.76, 2.50, 3.00.LetY1 be the logs of the ten observations, i.e., Y1=ln⁡(D1). Figure 2 presents the histogram and the estimated PDFs of the ND and SEV. From Figure 2, we find a difficulty to fully decide the best distribution for lifetime fitting due to the fact that both candidate distributions can provide good fitting for this data set. In this example, we consider using DSP approach to discriminate competing models and apply Taylor series prediction method to predicting the future order statistics, which are censored. The R source codes of Example 1 can be found in Appendix C and other designs can be obtained from the authors upon request.Figure 2
The histogram and the estimated probability density functions of airplane component’s failure time in Example 1.Through using Newton-Raphson algorithm, we obtained the MLEs ofμ and σ as (μ^N,σ^N)=(0.479,0.938) and (μ^S,σ^S)=(0.821,0.705) for the ND and SEV, respectively.TheDSP values via using ND and SEV are 0.223 and 0.212, respectively. Because the DSP value obtained from the SEV is smaller than that obtained from the ND, we claim the best distribution of this data set is SEV. The Taylor series prediction for (Y11:13,Y12:13,Y13:13) under the extreme value distribution with the censored sample can be obtained by Y^11:13A2,2,Y^12:13A2,2,Y^13:13A2,2=(1.098,1.281,1.567).


### 5.2. Example 2

In this example, we consider that the tests on endurance of deep groove ball bearings data, reported by Lieblein and Zelen [31] and further studied by Meeker and Escobar (1998), are used to illustrate the methodologies developed in this paper. The data are the numbers of million revolutions before failure for each of the 23 ball bearings in the life test. Meeker and Escobar [32] pointed out that this data (D2) follows lognormal distribution or Weibull distribution. Hence Y2=ln⁡(D2) follows a ND or SEV. The data is given as follows:D2:  17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.40, 51.84, 51.96, 54.12, 55.56, 67.80, 68.64, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92, 128.04, 173.40.For more information about this carbon fiber breaking strength data set, one can be referred to Meeker and Escobar (1998). In this example, we assume that the censoring proportion is 0.8696 (r=20,n=23). Figure 3 presents the histogram and the estimated PDFs of ND and SEV based on the type II right-censored data set. From Figure 3, it is difficult to decide the best distribution from these two candidate distributions.Figure 3
The histogram and the estimated probability density functions of tests on endurance of deep groove ball bearings in Example 2.We consider usingD approach in Example 2 for model selection and use expected values prediction method to predict the future order statistics, which are censored. The MLEs of μ and σ can be obtained via using Newton-Raphson algorithm, the resulting MLEs are (μ^N,σ^N)=(4.148,0.524) and (μ^S,σ^S)=(4.369,0.425) for the ND and SEV, respectively. The D values based on using the ND and SEV are 0.181 and 0.297, respectively. Because the D value obtained from ND is smaller than that obtained from SEV, we claim the best model is normal. The expected values prediction of (Y21:23,Y22:23,Y23:23) via using ND are Y^21:23A3,1,Y^22:23A3,1,Y^23:23A3,1=(4.784,4.922,5.160). In addition, we compare our prediction results with the MMLP values that proposed by Yang and Tong (2006), in which the MMLP is (Y^21:23,Y^22:23,Y^23:23)=(4.662,4.936,5.175). Our predicted results are close to that proposed by Yang and Tong [14] even we cannot initially assume which one of the ND or SEV is the best distribution.


### 5.3. Example 3

We consider the experiment on the pull-off performance for use in automotive engine components, reported by Byrne and Taguchi [33] and further studied by Yang and Tong [14], is used to illustrate the methodologies developed in this study. An experiment was conducted to find a method to maximize the pull-off force. Four control factors that could influence the assembly’s pull-off force have been identified. Repeat 8 times for each run and record the pull-off force in pounds. Table 9 lists the four control factors with their levels and complete data of this experiment. In this example, we assume that the censoring proportion is 0.75 (r=6,n=8). Please note that censored data cannot support the practitioner to conduct experimental design methods. Predicting the unobserved data and using a pseudo-complete data set for conducting experimental design methods is required.Table 9
Factors with levels of each factor and complete data in the experiments.
Run Factor Pull-off force for replicate A B C D x 1 : 8 x 2 : 8 x 3 : 8 x 4 : 8 x 5 : 8 x 6 : 8 x 7 : 8 x 8 : 8 1 1 1 1 1 9.5 15.6 16.9 19.1 19.6 19.6 19.9 20 2 1 2 2 2 15 16.2 19.4 19.6 19.7 19.8 21.9 24.2 3 1 3 3 3 15.6 16.3 16.7 18.2 19.1 20.4 22.6 23.3 4 2 1 2 3 17.4 18.3 18.6 18.9 18.9 21 23.2 24.7 5 2 2 3 1 18.6 19.4 19.7 21.4 25.1 25.3 25.6 27.5 6 2 3 1 2 14.7 16.2 16.3 19.6 19.8 20 22.5 24.7 7 3 1 3 2 16.4 16.8 18.4 18.6 19.1 21.6 23.6 24.3 8 3 2 1 3 14.2 15.1 15.6 16.8 17.8 19.6 23.2 24.4 9 3 3 2 1 16.1 17.3 19.3 19.9 22.6 22.7 23.1 28.6 Note: Factor A is interference with Low (1), Medium (2) and High (3) levels. Factor B is connector wall thickness with Thin (1), Medium (2) and Thick (3) levels. Factor C is insertion depth with Shallow (1), Medium (2) and Deep (3) levels. Factor D is Percent adhesive in connector pre-dip with Low (1), Medium (2) and High (3) levels.We consider using the RRML approach for model selection and use Taylor series prediction method to predict the future order statistics in this example. After combining the uncensored data and the predicted censored data, the pseudo-complete data are shown in Table10.Table 10
The pseudo-complete data and results of model selection.
Run Pull-off force for replicate Model selection x 1 : 8 x 2 : 8 x 3 : 8 x 4 : 8 x 5 : 8 x 6 : 8 X ^ 7 : 8 A 1,2 X ^ 8 : 8 A 1,2 1 9.5 15.6 16.9 19.1 19.6 19.6 19.6 21.2 SEV 2 15 16.2 19.4 19.6 19.7 19.8 19.8 20.73 SEV 3 15.6 16.3 16.7 18.2 19.1 20.4 20.4 21.6 ND 4 17.4 18.3 18.6 18.9 18.9 21 21 21.8 ND 5 18.6 19.4 19.7 21.4 25.1 25.3 25.3 27.3 ND 6 14.7 16.2 16.3 19.6 19.8 20 20 21.1 SEV 7 16.4 16.8 18.4 18.6 19.1 21.6 21.6 22.8 ND 8 14.2 15.1 15.6 16.8 17.8 19.6 19.6 20.9 ND 9 16.1 17.3 19.3 19.9 22.6 22.7 22.7 24.5 ND


## 5.1. Example 1

A test airplane component’s failure time dataset provided in Mann and Fertig [30], in which 13 components were placed on test, and the test was terminated at the time of the 10th failure. The failure times (in hours) of the 10 components that failed wereD1:   0.22, 0.50, 0.88, 1.00, 1.32, 1.33, 1.54, 1.76, 2.50, 3.00.LetY1 be the logs of the ten observations, i.e., Y1=ln⁡(D1). Figure 2 presents the histogram and the estimated PDFs of the ND and SEV. From Figure 2, we find a difficulty to fully decide the best distribution for lifetime fitting due to the fact that both candidate distributions can provide good fitting for this data set. In this example, we consider using DSP approach to discriminate competing models and apply Taylor series prediction method to predicting the future order statistics, which are censored. The R source codes of Example 1 can be found in Appendix C and other designs can be obtained from the authors upon request.Figure 2
The histogram and the estimated probability density functions of airplane component’s failure time in Example 1.Through using Newton-Raphson algorithm, we obtained the MLEs ofμ and σ as (μ^N,σ^N)=(0.479,0.938) and (μ^S,σ^S)=(0.821,0.705) for the ND and SEV, respectively.TheDSP values via using ND and SEV are 0.223 and 0.212, respectively. Because the DSP value obtained from the SEV is smaller than that obtained from the ND, we claim the best distribution of this data set is SEV. The Taylor series prediction for (Y11:13,Y12:13,Y13:13) under the extreme value distribution with the censored sample can be obtained by Y^11:13A2,2,Y^12:13A2,2,Y^13:13A2,2=(1.098,1.281,1.567).


## 5.2. Example 2

In this example, we consider that the tests on endurance of deep groove ball bearings data, reported by Lieblein and Zelen [31] and further studied by Meeker and Escobar (1998), are used to illustrate the methodologies developed in this paper. The data are the numbers of million revolutions before failure for each of the 23 ball bearings in the life test. Meeker and Escobar [32] pointed out that this data (D2) follows lognormal distribution or Weibull distribution. Hence Y2=ln⁡(D2) follows a ND or SEV. The data is given as follows:D2:  17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.40, 51.84, 51.96, 54.12, 55.56, 67.80, 68.64, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92, 128.04, 173.40.For more information about this carbon fiber breaking strength data set, one can be referred to Meeker and Escobar (1998). In this example, we assume that the censoring proportion is 0.8696 (r=20,n=23). Figure 3 presents the histogram and the estimated PDFs of ND and SEV based on the type II right-censored data set. From Figure 3, it is difficult to decide the best distribution from these two candidate distributions.Figure 3
The histogram and the estimated probability density functions of tests on endurance of deep groove ball bearings in Example 2.We consider usingD approach in Example 2 for model selection and use expected values prediction method to predict the future order statistics, which are censored. The MLEs of μ and σ can be obtained via using Newton-Raphson algorithm, the resulting MLEs are (μ^N,σ^N)=(4.148,0.524) and (μ^S,σ^S)=(4.369,0.425) for the ND and SEV, respectively. The D values based on using the ND and SEV are 0.181 and 0.297, respectively. Because the D value obtained from ND is smaller than that obtained from SEV, we claim the best model is normal. The expected values prediction of (Y21:23,Y22:23,Y23:23) via using ND are Y^21:23A3,1,Y^22:23A3,1,Y^23:23A3,1=(4.784,4.922,5.160). In addition, we compare our prediction results with the MMLP values that proposed by Yang and Tong (2006), in which the MMLP is (Y^21:23,Y^22:23,Y^23:23)=(4.662,4.936,5.175). Our predicted results are close to that proposed by Yang and Tong [14] even we cannot initially assume which one of the ND or SEV is the best distribution.


## 5.3. Example 3

We consider the experiment on the pull-off performance for use in automotive engine components, reported by Byrne and Taguchi [33] and further studied by Yang and Tong [14], is used to illustrate the methodologies developed in this study. An experiment was conducted to find a method to maximize the pull-off force. Four control factors that could influence the assembly’s pull-off force have been identified. Repeat 8 times for each run and record the pull-off force in pounds. Table 9 lists the four control factors with their levels and complete data of this experiment. In this example, we assume that the censoring proportion is 0.75 (r=6,n=8). Please note that censored data cannot support the practitioner to conduct experimental design methods. Predicting the unobserved data and using a pseudo-complete data set for conducting experimental design methods is required.Table 9
Factors with levels of each factor and complete data in the experiments.
Run Factor Pull-off force for replicate A B C D x 1 : 8 x 2 : 8 x 3 : 8 x 4 : 8 x 5 : 8 x 6 : 8 x 7 : 8 x 8 : 8 1 1 1 1 1 9.5 15.6 16.9 19.1 19.6 19.6 19.9 20 2 1 2 2 2 15 16.2 19.4 19.6 19.7 19.8 21.9 24.2 3 1 3 3 3 15.6 16.3 16.7 18.2 19.1 20.4 22.6 23.3 4 2 1 2 3 17.4 18.3 18.6 18.9 18.9 21 23.2 24.7 5 2 2 3 1 18.6 19.4 19.7 21.4 25.1 25.3 25.6 27.5 6 2 3 1 2 14.7 16.2 16.3 19.6 19.8 20 22.5 24.7 7 3 1 3 2 16.4 16.8 18.4 18.6 19.1 21.6 23.6 24.3 8 3 2 1 3 14.2 15.1 15.6 16.8 17.8 19.6 23.2 24.4 9 3 3 2 1 16.1 17.3 19.3 19.9 22.6 22.7 23.1 28.6 Note: Factor A is interference with Low (1), Medium (2) and High (3) levels. Factor B is connector wall thickness with Thin (1), Medium (2) and Thick (3) levels. Factor C is insertion depth with Shallow (1), Medium (2) and Deep (3) levels. Factor D is Percent adhesive in connector pre-dip with Low (1), Medium (2) and High (3) levels.We consider using the RRML approach for model selection and use Taylor series prediction method to predict the future order statistics in this example. After combining the uncensored data and the predicted censored data, the pseudo-complete data are shown in Table10.Table 10
The pseudo-complete data and results of model selection.
Run Pull-off force for replicate Model selection x 1 : 8 x 2 : 8 x 3 : 8 x 4 : 8 x 5 : 8 x 6 : 8 X ^ 7 : 8 A 1,2 X ^ 8 : 8 A 1,2 1 9.5 15.6 16.9 19.1 19.6 19.6 19.6 21.2 SEV 2 15 16.2 19.4 19.6 19.7 19.8 19.8 20.73 SEV 3 15.6 16.3 16.7 18.2 19.1 20.4 20.4 21.6 ND 4 17.4 18.3 18.6 18.9 18.9 21 21 21.8 ND 5 18.6 19.4 19.7 21.4 25.1 25.3 25.3 27.3 ND 6 14.7 16.2 16.3 19.6 19.8 20 20 21.1 SEV 7 16.4 16.8 18.4 18.6 19.1 21.6 21.6 22.8 ND 8 14.2 15.1 15.6 16.8 17.8 19.6 19.6 20.9 ND 9 16.1 17.3 19.3 19.9 22.6 22.7 22.7 24.5 ND


## 6. Conclusions

It could be difficult to discriminate a best model sometimes from several candidate distributions. The sample size, estimation methods, and goodness-of-fit testing methods can affect the final results of model selection. In this study, we focus on providing reliable methods to obtain predicting values of censored data to reduce the impact of model misspecification. In this study, three model selection approaches are proposed for predicting the future order statistics from type II censored data, in which the quality characteristic is assumed to follow a location-scale family. The ND and SEV are considered as the candidate members in the location-scale distribution to compete the best underlying distribution. The ND can be the log transformation from the lognormal distribution and the SEV can be the log transformation from the Weibull distribution. Discrimination between lognormal and Weibull distributions is equivalent to the discrimination between ND and SEV. Hence, both ND and SEV are widely used for practical reliability applications.Through any one of three proposed approaches, the robust predictions can be obtained even under model uncertainty. Three examples are used to illustrate the methodologies. Moreover, the performance of these three proposed approaches are evaluated through using Monte Carlo simulations. Numerical results show that the three proposed model selection approaches are robust and effective in obtaining good predicted values for the future order statistics, which are censored.In comparing these three proposed approaches, we recommend usingDSP approach or D approach for model selection and use expected values prediction method to predict the future order statistics for small sample size cases, that is, the sample cases with a size n is less than 30. For large sample size cases (sample size n larger than 30), we recommend using RRML approach for model selection and use Taylor series prediction method to predict the future order statistics. Simulation results show that the proposed approaches are robust and can highly reduce the impact caused by model uncertainty. The proposed approaches can also work well if more than two candidate distribution are competing for the best distribution.Other model selection methods from the current three proposed approaches could also be competitive. How to employ new model selection methods for the topic of type II censored data prediction can be studied in the future.



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*Source: 3465909-2018-10-08.xml*