Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2018 / Article

Research Article | Open Access

Volume 2018 |Article ID 3465909 | https://doi.org/10.1155/2018/3465909

Jyun-You Chiang, Shuai Wang, Tzong-Ru Tsai, Ting Li, "Model Selection Approaches for Predicting Future Order Statistics from Type II Censored Data", Mathematical Problems in Engineering, vol. 2018, Article ID 3465909, 29 pages, 2018. https://doi.org/10.1155/2018/3465909

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

Academic Editor: Mohammed Nouari
Received07 Apr 2018
Revised04 Jul 2018
Accepted14 Aug 2018
Published08 Oct 2018

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.

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, [16] 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 where identical components are placed in the test simultaneously. Assuming that component fails, the experiment is terminated. Thus the last 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 item is equivalent to the life length of a (n-s+1)-out-of-n system that was made up of identical components with independent life lengths. When s = 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 Figure 1.

The rest of this paper is organized as follows. Section 2 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

Let denote the failure time of item and , which follows a location-scale family, having the probability density function (PDF) and cumulative distribution function (CDF): andrespectively, where is location parameter and is scale parameter. and are the PDF and CDF of a member, respectively, in the location-scale family. Denote the sample size by , and denote type II censored sample with failures by , which are the realizations of , where . Our goal is to predict for . Let and here and after to simplify the notations. Kaminsky and Rhodin [9] considered prediction of having observed , The predictive likelihood functions (PLF) of , and isPlease note that the capital notation in is unknown and can be predicted based on the sample . Based on the proposed method by Raqab [13], the PLF of , and in (3) can be represented as a product of two likelihood functions, the PLF of and (i.e., which is denoted as ) and the PLF of (i.e., which is denoted as ). Both likelihood functions are presented, respectively, byand In practice, we can obtain the MLEs of and , denoted by and , respectively, through maximizing in (4). Then use and to replace and as the plug-in parameters in (5) to predict . Let for , for and , then we can rewrite (4) and (5) byandwhere and . After straightforward computations, the MLEs of , and respectively can be obtained as the solutions ofandwhere andBecause of no analytic presentation for and , one needs to use numerical gradient computation methods, for example, the Newton-Raphson method, for obtaining and via by equating (8) and (9). To obtain proper initial solutions for implementing gradient computation methods, we consider using the approximate MLEs (AMLE) of and from Hossain and Willan [25] as their initial solutions in this study.

2.2. Approximate Maximum Likelihood Predictors

When we obtain the MLEs and , we can predict by using two approximation methods, the expected value prediction method and Taylor series prediction method. The resulting predictors of based on the expected prediction method is denoted by , and the resulting predictors of based on the Taylor series prediction method is denoted by . The two approximate methods mainly use two different methods to get the approximates of and . Mehrotra and Nanda [26] proposed approximate maximum likelihood estimators for the ND and gamma distribution by replacing and 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 and at the points to obtain modified MLEs of the parameters of the ND and Rayleigh distribution, where for . 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 .

Based on the expected value prediction method, replacing with , and replacing and by their respective expected values in (10). According to Raqab [13], the expected value of , and can be presented, respectively, byand

Based on the Taylor series prediction method, replacing with and replacing and with their Taylor series approximations at points and (), respectively, in (10). In this study, we denote the and of under the candidate distribution by and , 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 predicting 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 byThrough using (17), we can obtain . The MLEs of normal distribution parameters are denoted by and . Replacing and with and in (6), we can represent (6) bywhere is the CDF of the standard ND. According to (15) and (16), and can be replaced with their respective expected values in (10). Equation (10) can be rewritten asThe values of are available and have been tabulated by Teichroew [28]. Hence, of for ND can be derived as Because is a necessary condition, we modify (20) byand use in (21) to protect for .

Based on the Taylor series prediction method, the functions and are expanded by using the Taylor series around points and (), respectively. According to Raqab [13], we can approximate and byandThe values of and are given in Appendix A. Equation (10) can be rewritten byThe of can be obtained bywhere

If the underlying distribution is SEV, the PDF of the SEV is given by

Based on the expected value prediction method, . Using (8) and (9), the MLEs of and are denoted by and , respectively. Replacing and with and in (6), (6) can be represented bywhere is the CDF of the standard SEV. Then and are replaced with their respective expected values in Eq. (10). Equation (10) can be rewritten asThe of can be obtained asfor and .

Based on the Taylor series prediction method, expanding and by using the Taylor series at the points and (), respectively. We obtainandThe values of and are given in Appendix B. Equation (10) can be rewritten asThe of can be derived asfor

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, modification approach (shorted as approach), and modification D approach (shorted as the D approach), to obtain the predictor of . It is noticed that the idea of the approach and D approach is based on goodness-of-fit test methods. All these three approaches can be implemented to obtain the predictor of via using Algorithms 13.

Algorithm 1 (the RRML approach).
Step 1. Collect a type II censored sample, which has size and observed failure times; we consider candidate distributions.
Step 2. Obtain () and for the candidate distribution . Obtain under the candidate distribution and label it by for , and or 2.
Step 3. Let denote the predicted value of for or 2. Based on the method proposed by Dumonceaux and Antle [16], we can obtain , which can provide the largest maximum likelihood information by If the candidate distributions are ND and SEV, Steps 2 and 3 in Algorithm 1 can be reduced to Step 2’ and Step 3’ as the following, respectively:
Step 2’. Obtain (, ), (, ), and . Obtain under the ND () and obtain under the SEV () for and or 2.
Step 3’. Let denote the predicted value of . Then

Algorithm 2 (the approach).
Step 1. Collect a type II censored sample, which has size and observed failure times.
Step 2. Obtain () for , and then obtain for , and or 2.
Step 3. Based on the method proposed by Castro-Kuriss et al. [29], the modification of with censored observations can be presented bywhere . The definition of is the same as that of (2), it represents the CDF of the assumed distribution in model selection. Evaluate the value of through using the candidate distribution for .
Step 4. Let be the predicted value of for or 2, then can be obtained with the smallest . That is, is the value corresponding to , which is defined byIf the candidate distributions are ND and SEV, Steps 2, 3, and 4 in Algorithm 2 can be reduced to Step 2’ and Step 3’ as the following, respectively:
Step 2’. Obtain () and (). Obtain the under the ND and obtain the under the SEV for and or 2.
Step 3’. The modification of with censored observations can be presented bywhere . The definition of is the same as that of (2); it represents the CDF of the assumed distribution in model selection. Evaluate the values of through using the ND and SEV and denot them by and , respectively.
Step 4’. Let denote the predicted value of , then can be obtained by

Algorithm 3 (the approach).
Step 1. Collect a type II censored sample, which has size and observed failure times.
Step 2. Obtain () for , and then obtain for , and or 2.
Step 3. Based on the method proposed by Castro-Kuriss et al. [29], the modification of with censored observations can be presented bywhere .
Step 4. Let be the predicted value of for or 2, then can be obtained with the smallest . That is, is the value corresponding to , which is defined byIf the candidate distributions are ND and SEV, Steps 2, 3, and 4 in Algorithm 3 can be reduced to Step 2’ and Step 3’ as the following, respectively:
Step 2’. Obtain () and (). Obtain under the ND and obtain under the SEV for and or 2.
Step 3’. The modification of with censored observations can be presented bywhere . Evaluate the value of by using the ND and SEV and denote them by and .
Step 4’. Let denote the predicted value of , then can be obtained by

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, , used in the simulation were randomly generated from the ND and SEV with location parameter and scale parameter . Then, the order statistic is predicted and denoted by for for the sample sizes and 60. For the purpose of comparison, the values of the bias and mean square error (MSE) of are evaluated using Monte Carlo runs:andwhere is the predicted value of that is obtained in the iteration of simulation for . 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 and . As or 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.


Assumed Distribution
Normal distributionExtreme Value distribution
biasMSEbiasMSEbiasMSEbiasMSE

1089-0.11890.1295-0.34300.2140-0.19700.1406-0.34300.2140
78-0.11930.0949-0.28320.1509-0.15970.0995-0.28320.1509
79-0.15690.2159-0.33080.2753-0.30580.2619-0.40070.3196
67-0.12060.0738-0.25600.1170-0.14020.0752-0.25600.1170
68-0.30090.1652-0.15700.2134-0.34710.1947-0.25920.2399
69-0.19640.3199-0.29800.3498-0.41470.4169-0.46780.4566
56-0.12220.0747-0.24480.1142-0.12980.0749-0.24480.1142
57-0.16900.1570-0.29130.1968-0.24520.1773-0.32660.2152
58-0.22040.2661-0.29790.2877-0.38420.3310-0.43040.3606
59-0.27020.4625-0.34290.4841-0.55570.6235-0.59140.6567

201618-0.06750.0802-0.23430.1178-0.20740.1067-0.28320.1414
1416-0.06260.0514-0.19640.0780-0.14670.0622-0.22080.0875
1418-0.08020.1259-0.16070.1372-0.29950.1939-0.33380.2143
1214-0.05890.0411-0.17530.0640-0.11620.0477-0.19030.0693
1216-0.08630.0936-0.14690.1019-0.23810.1317-0.27030.1467
1218-0.11190.1857-0.16940.1942-0.40850.3102-0.43330.3296
1012-0.06640.0376-0.17070.0579-0.10750.0420-0.18120.0614
1014-0.08840.0818-0.13850.0886-0.20720.1084-0.23920.1211
1016-0.11510.1417-0.15520.1474-0.33760.2210-0.35880.2343
1018-0.14970.2628-0.19600.2701-0.52580.4627-0.54600.4823

3024270.02950.0763-0.11610.0584-0.25440.1226-0.23960.1020
2124-0.00590.0500-0.10780.0386-0.19520.0725-0.18190.0591
2127-0.02380.0933-0.09910.0827-0.33170.1889-0.26870.1366
1821-0.04820.0343-0.10870.0325-0.18510.0574-0.17000.0487
1824-0.07480.0650-0.11110.0605-0.31070.1451-0.24090.1030
1827-0.09610.1203-0.13280.1210-0.42700.2816-0.37170.2418
1518-0.06540.0267-0.10960.0310-0.13540.0388-0.15790.0430
1521-0.08620.0540-0.10520.0553-0.25720.1096-0.23310.0943
1524-0.11380.0911-0.13100.0941-0.38360.2094-0.35740.1924
1527-0.14640.1578-0.17220.1648-0.56140.4196-0.54630.4115

4032360.06530.0634-0.06690.0404-0.26960.1229-0.21450.0840
28320.05640.0426-0.05360.0225-0.19610.0685-0.14680.0418
28360.06220.0661-0.02680.0495-0.32240.1660-0.24420.1073
24280.05030.0349-0.04700.0173-0.16730.0514-0.11750.0294
24320.05670.0476-0.01880.0307-0.24640.1060-0.15880.0554
24360.06790.0697-0.00300.0560-0.32020.1666-0.25750.1199
20240.04020.0318-0.04590.0152-0.16290.0464-0.10490.0244
20280.03880.0405-0.02080.0261-0.22360.0887-0.12030.0394
20320.04980.0511-0.00350.0402-0.25030.1088-0.16660.0637
20360.05480.07690.00150.0674-0.32150.1688-0.26580.1294

5040450.06600.0504-0.04470.0310-0.28130.1201-0.19920.0700
35400.05640.0352-0.03870.0175-0.20430.0676-0.13300.0342
35450.06710.0551-0.01380.0407-0.32230.1587-0.23720.0970
30350.05060.0293-0.03210.0131-0.17420.0503-0.10160.0228
30400.05800.0371-0.00740.0239-0.24790.0992-0.15000.0467
30450.06760.05640.00760.0457-0.32450.1580-0.25470.1092
25300.04090.0260-0.03130.0110-0.17240.0446-0.09030.0180
25350.04960.0313-0.00650.0193-0.22350.0822-0.11080.0319
25400.05670.03930.00810.0299-0.25300.1017-0.16400.0559
25450.07020.05770.02210.0492-0.32450.1573-0.26460.1164

6048540.06390.0445-0.03500.0266-0.29010.1196-0.19250.0630
42480.05540.0309-0.02750.0144-0.21010.0643-0.12270.0285
42540.06440.0457-0.00460.0341-0.32640.1523-0.23250.0886
36420.05000.0247-0.02400.0106-0.18030.0491-0.09290.0186
36480.05730.0325-0.00240.0209-0.25050.0969-0.14650.0429
36540.06780.04880.01480.0396-0.32680.1542-0.25260.1040
30360.04160.0215-0.02260.0092-0.17860.0466-0.07980.0154
30420.04520.0260-0.00300.0162-0.23040.0786-0.10820.0276
30480.05730.03330.01580.0257-0.25320.0983-0.15880.0512
30540.06740.04940.02330.0429-0.32900.1568-0.26680.1139


Assumed Distribution
Normal distributionExtreme Value distribution
biasMSEbiasMSEbiasMSEbiasMSE

10890.12250.1563-0.29650.1583-0.06690.0868-0.29690.1589
780.04170.1109-0.26850.1365-0.07390.0793-0.26920.1376
790.16590.2742-0.12330.1783-0.11310.1742-0.25390.2066
67-0.01550.0969-0.26870.1355-0.08640.0797-0.27020.1364
680.07100.2206-0.17250.1852-0.12870.1759-0.26350.2118
690.20750.46280.03040.3333-0.17410.2954-0.25710.3099
56-0.05320.1017-0.27670.1497-0.09130.0909-0.2780.1516
57-0.00370.2267-0.21440.2142-0.15480.202-0.28480.2367
580.08460.4086-0.05440.3325-0.21040.3158-0.28580.3299
590.21960.72790.09130.5969-0.27170.4812-0.32970.4914

2016180.31210.2071-0.07260.0498-0.04410.0543-0.17450.0747
14160.20530.1345-0.10280.0466-0.04550.0431-0.16580.0639
14180.33060.23890.10390.0949-0.07020.0961-0.13270.1031
12140.14240.1072-0.11980.0481-0.04450.0404-0.16460.0605
12160.22340.17020.04400.0791-0.07450.0868-0.1330.0925
12180.36850.31540.20980.1901-0.09490.155-0.14020.1594
10120.09670.0948-0.13600.0561-0.04960.0454-0.17180.0667
10140.16240.14580.01560.0853-0.07890.0949-0.1360.1015
10160.26670.25530.14670.1775-0.10320.1629-0.14170.1668
10180.43830.49680.31990.3774-0.13350.2516-0.16910.2546

3024270.54820.35430.07880.03300.08460.0522-0.05520.0298
21240.43510.24720.02700.02120.07170.0477-0.05730.0238
21270.54480.35620.20680.08210.0820.0574-0.00620.0386
18210.37430.19800.00030.01880.05940.0452-0.06340.0228
18240.42150.24140.11350.04780.05880.0523-0.02220.0343
18270.52500.34200.26120.11980.0650.0640.00120.0518
15180.31460.1777-0.02320.02210.03080.0473-0.08210.0289
15210.32570.18570.05750.03920.00690.0597-0.05780.0447
15240.36880.21900.14090.0693-0.00920.0766-0.05880.0667
15270.47530.32790.27590.1478-0.01850.1067-0.05980.0998

4032360.55360.34860.11410.03420.07150.0397-0.03580.0221
28320.44710.23460.05640.01830.06750.0352-0.03460.0166
28360.55510.34850.23450.08470.07420.04220.00380.0289
24280.39230.19460.02930.01440.06110.0344-0.03610.0148
24320.44710.23340.14340.04360.06660.0374-0.00060.0234
24360.55840.35370.30260.12570.080.04540.02790.0349
20240.37320.18310.01420.01490.05750.0362-0.03840.0163
20280.39290.19210.09320.03270.05880.0364-0.00590.0229
20320.44550.23720.19580.06740.06780.03920.0210.0289
20360.55260.35220.33740.15360.07620.04660.03620.0386

5040450.56050.34440.13860.03570.06860.0324-0.02040.018
35400.44840.22910.07170.01740.06110.028-0.02380.0127
35450.56000.34320.24910.08640.06980.03340.00860.0237
30350.39250.18590.04030.01270.05380.0282-0.02650.0117
30400.44690.22890.15600.04260.06170.03040.00690.0189
30450.55940.34930.31340.12710.07220.03630.02770.0283
25300.37100.17360.02420.01250.04990.0287-0.02760.0126
25350.39380.18690.10680.02960.05660.03030.00470.0184
25400.44410.23240.20150.06610.06120.03230.02170.0245
25450.56230.34680.35580.15550.07730.03770.04360.0318

6048540.56170.34380.15020.03740.06310.0267-0.01510.0148
42480.44750.22470.08010.01700.05620.024-0.01780.0107
42540.56290.34690.26190.08980.06660.02940.01390.0205
36420.39430.18020.05060.01190.05290.0232-0.0170.0097
36480.44780.22240.16300.04180.05860.02460.01030.0159
36540.56200.34440.32430.12750.06890.03120.03060.0246
30360.36900.16600.03120.01090.04480.0238-0.02020.0102
30420.39040.18050.11030.02780.05030.02510.00590.0155
30480.44720.22390.21190.06340.06060.02660.02760.0204
30540.56100.34560.36060.15710.07010.03230.04050.0273

To evaluate the performance of the three proposed model selection approaches for MLP, Tables 35 report the simulation results for three model selection approaches from the ND. Tables 68 respectively report the simulation results for three model selection approaches from the SEV. The column “correct (%)” presented in Tables 38 is the correct model selection rate in all simulation runs. From Tables 38 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 approach or 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 or 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.


RML approach
biasMSEbiasMSECorrect (%)

1089-0.17960.1372-0.34300.21400.6430
78-0.15800.1001-0.28320.15090.6263
79-0.24590.2408-0.37320.30230.6263
67-0.14660.0772-0.25600.11700.5959
68-0.33160.1836-0.22260.23130.5959
69-0.31500.3745-0.38670.40980.5959
56-0.13990.0774-0.24480.11420.5908
57-0.22060.1701-0.31640.20980.5908
58-0.31440.3048-0.37090.33030.5908
59-0.41890.5512-0.46950.57940.5908

201618-0.13010.0875-0.25420.12750.7195
1416-0.10870.0556-0.20930.08310.6955
1418-0.17410.1534-0.22810.17020.6955
1214-0.09440.0443-0.18500.06740.6657
1216-0.15970.1112-0.20080.12290.6657
1218-0.23820.2419-0.27640.25610.6657
1012-0.09480.0404-0.17870.06060.6477
1014-0.14890.0949-0.18490.10460.6477
1016-0.21640.1806-0.24320.19010.6477
1018-0.31040.3566-0.34190.36990.6477

3024270.00920.0718-0.12240.06040.9508
2124-0.02350.0472-0.11300.04020.9345
2127-0.04930.0950-0.11350.08880.9345
1821-0.07000.0348-0.11720.03510.8478
1824-0.11140.0752-0.13570.07250.8478
1827-0.15310.1561-0.17880.15810.8478
1518-0.09200.0295-0.12340.03440.7104
1521-0.13640.0676-0.14510.07000.7104
1524-0.19850.1293-0.20610.13340.7104
1527-0.27890.2534-0.29450.26210.7104

4032360.05330.0603-0.07080.04130.9717
28320.04760.0408-0.05600.02290.9754
28360.05140.0651-0.03240.05070.9754
24280.04450.0334-0.04840.01750.9777
24320.04970.0464-0.02180.03110.9777
24360.05930.0691-0.00830.05720.9777
20240.03520.0308-0.04690.01540.9817
20280.03310.0399-0.02280.02630.9817
20320.04340.0508-0.00690.04070.9817
20360.04670.0768-0.00420.06830.9817

5040450.05890.0488-0.04720.03130.9820
35400.05050.0341-0.04030.01770.9847
35450.05950.0542-0.01780.04110.9847
30350.04730.0286-0.03290.01310.9876
30400.05390.0366-0.00930.02410.9876
30450.06240.05590.00420.04630.9876
25300.03810.0257-0.03190.01110.9878
25350.04660.0310-0.00760.01940.9878
25400.05270.03890.00600.03000.9878
25450.06600.05770.01910.04970.9878

6048540.05860.0435-0.03680.02700.9913
42480.05220.0300-0.02840.01440.9919
42540.06110.0453-0.00640.03450.9919
36420.04770.0243-0.02450.01070.9933
36480.05430.0322-0.00380.02100.9933
36540.06440.04870.01260.04000.9933
30360.04010.0212-0.02290.00920.9928
30420.04330.0257-0.00370.01630.9928
30480.05550.03330.01480.02580.9928
30540.06490.04920.02150.04290.9928


approach
biasMSEbiasMSEcorrect (%)

1089-0.14890.1300-0.34300.21400.6694
78-0.12850.0932-0.28320.15090.6654
79-0.20160.2273-0.35170.28800.6654
67-0.11810.0716-0.25600.11700.6616
68-0.31120.1713-0.18090.21900.6616
69-0.25600.3466-0.34610.37980.6616
56-0.11380.0725-0.24480.11420.6305
57-0.18200.1603-0.29650.19940.6305
58-0.26210.2818-0.33390.30610.6305
59-0.35290.5053-0.41700.53060.6305

201618-0.11590.0843-0.24970.12520.7314
1416-0.08920.0529-0.20380.08100.7198
1418-0.14560.1450-0.21090.16100.7198
1214-0.07480.0420-0.17950.06550.6935
1216-0.13080.1049-0.18350.11590.6935
1218-0.19890.2247-0.24750.23690.6935
1012-0.07410.0379-0.17240.05840.6781
1014-0.11930.0886-0.16680.09760.6781
1016-0.17910.1639-0.21560.17210.6781
1018-0.25800.3209-0.29850.33180.6781

302427-0.06100.0618-0.14220.06660.7082
2124-0.06370.0390-0.12110.04150.7488
2127-0.09980.0901-0.13550.09350.7488
1821-0.07550.0314-0.11740.03480.8027
1824-0.11770.0693-0.13690.07020.8027
1827-0.16150.1472-0.18260.15340.8027
1518-0.07780.0277-0.11840.03330.6998
1521-0.12060.0629-0.13650.06620.6998
1524-0.17960.1204-0.19350.12540.6998
1527-0.25660.2319-0.27740.24160.6998

403236-0.04160.0511-0.10130.04930.7000
2832-0.02870.0289-0.07420.02590.6958
2836-0.04670.0643-0.08000.06290.6958
2428-0.02250.0230-0.06090.01930.7010
2432-0.02780.0389-0.05050.03530.7010
2436-0.03900.0710-0.06500.07020.7010
2024-0.02070.0203-0.05540.01630.7535
2028-0.02320.0321-0.03820.02770.7535
2032-0.02180.0468-0.03620.04450.7535
2036-0.03460.0793-0.05560.07840.7535

504045-0.03140.0421-0.07800.03830.7221
3540-0.02440.0245-0.05900.02030.7185
3545-0.03760.0552-0.06700.05280.7185
3035-0.01710.0190-0.04510.01450.7377
3040-0.01930.0314-0.03750.02780.7377
3045-0.03030.0596-0.05110.05830.7377
2530-0.01660.0168-0.04010.01180.7656
2535-0.01120.0252-0.02430.02100.7656
2540-0.01160.0373-0.02400.03420.7656
2545-0.01790.0630-0.03570.06110.7656

604854-0.02540.0383-0.06640.03310.7520
4248-0.01790.0219-0.04630.01680.7444
4254-0.02760.0485-0.05240.04530.7444
3642-0.01370.0164-0.03640.01180.7635
3648-0.01410.0283-0.03060.02440.7635
3654-0.01920.0537-0.03830.05170.7635
3036-0.00980.0146-0.03040.00980.7864
3042-0.00680.0217-0.01840.01770.7864
3048-0.00280.0321-0.01310.02910.7864
3054-0.01370.0565-0.03070.05470.7864


approach
biasMSEbiasMSEcorrect (%)

1089-0.14910.1300-0.34300.21400.6687
78-0.12860.0932-0.28320.15090.6648
79-0.20160.2274-0.35170.28810.6648
67-0.11810.0716-0.25600.11700.6615
68-0.31130.1713-0.18090.21900.6615
69-0.25600.3466-0.34610.37980.6615
56-0.11380.0725-0.24480.11420.6306
57-0.18200.1603-0.29650.19940.6306
58-0.26210.2818-0.33390.30610.6306
59-0.35290.5054-0.41700.53070.6306

201618-0.11630.0844-0.24980.12530.7293
1416-0.08930.0529-0.20390.08100.7190
1418-0.14580.1451-0.21090.16110.7190
1214-0.07480.0420-0.17950.06550.6933
1216-0.13080.1049-0.18350.11590.6933
1218-0.19900.2248-0.24760.23700.6933
1012-0.07410.0379-0.17240.05840.6781
1014-0.11930.0886-0.16680.09760.6781
1016-0.17910.1639-0.21560.17210.6781
1018-0.25800.3209-0.29840.33180.6781

302427-0.06400.0618-0.14330.06690.7006
2124-0.06460.0390-0.12130.04160.7454
2127-0.10090.0903-0.13610.09370.7454
1821-0.07570.0314-0.11750.03480.8016
1824-0.11780.0693-0.13690.07020.8016
1827-0.16200.1472-0.18290.15340.8016
1518-0.07780.0276-0.11840.03330.6998
1521-0.12060.0629-0.13650.06620.6998
1524-0.17960.1204-0.19350.12540.6998
1527-0.25650.2319-0.27730.24160.6998

403236-0.04580.0514-0.10280.04970.6912
2832-0.03000.0290-0.07450.02600.6916
2836-0.04860.0642-0.08110.06300.6916
2428-0.02300.0230-0.06100.01930.6990
2432-0.02860.0389-0.05080.03540.6990
2436-0.03980.0711-0.06550.07030.6990
2024-0.02060.0203-0.05540.01630.7536
2028-0.02310.0321-0.03820.02770.7536
2032-0.02170.0468-0.03620.04450.7536
2036-0.03450.0792-0.05540.07840.7536

504045-0.03490.0424-0.07930.03870.7099
3540-0.02590.0246-0.05940.02040.7138
3545-0.03970.0555-0.06820.05310.7138
3035-0.01790.0190-0.04530.01460.7353
3040-0.01990.0313-0.03780.02780.7353
3045-0.03100.0597-0.05150.05840.7353
2530-0.01650.0168-0.04010.01180.7659
2535-0.01110.0252-0.02420.02100.7659
2540-0.01150.0373-0.02400.03420.7659
2545-0.01790.0629-0.03570.06100.7659

604854-0.02940.0387-0.06790.03350.7393
4248-0.01990.0220-0.04690.01690.7384
4254-0.03020.0487-0.05380.04550.7384
3642-0.01460.0164-0.03660.01180.7617
3648-0.01490.0283-0.03090.02440.7617
3654-0.02000.0539-0.03880.05190.7617
3036-0.00970.0146-0.03040.00980.7867
3042-0.00670.0217-0.01840.01770.7867
3048-0.00280.0320-0.01300.02910.7867
3054-0.01350.0565-0.03060.05470.7867


RML approach
biasMSEbiasMSEcorrect(%)

1089-0.04610.0865-0.29690.15890.5938
78-0.06790.0798-0.26920.13760.5587
79-0.05200.1779-0.22580.19770.5587
67-0.08970.0814-0.27020.13640.5445
68-0.08660.1746-0.24540.20470.5445
69-0.06820.3140-0.17160.30480.5445
56-0.10100.0935-0.27800.15160.5242
57-0.12340.2004-0.27150.23140.5242
58-0.12280.3226-0.21190.32140.5242
59-0.10700.5253-0.18350.50580.5242

201618-0.00890.0544-0.16260.07130.7451
1416-0.02220.0422-0.15920.06180.7003
1418-0.00120.1000-0.07640.09880.7003
1214-0.02720.0398-0.16030.05920.6548
1216-0.01830.0883-0.08460.08820.6548
12180.02260.1753-0.03320.16700.6548
1012-0.03710.0447-0.16920.06580.6058
1014-0.02770.0946-0.08950.09600.6058
10160.00280.1753-0.04150.17070.6058
10180.05120.30560.00520.29090.6058

3024270.11430.0639-0.04540.02990.9188
21240.09910.0541-0.04950.02350.9066
21270.11710.07170.01210.04230.9066
18210.07760.0471-0.05800.02230.9046
18240.08120.0569-0.01100.03510.9046
18270.09780.07420.02410.05620.9046
15180.03600.0465-0.07830.02810.8915
15210.02220.0569-0.04530.04230.8915
15240.01810.0730-0.03420.06260.8915
15270.02420.1017-0.02000.09330.8915

4032360.09300.0472-0.02820.02260.9441
28320.08690.0402-0.02920.01670.9393
28360.09940.05420.01740.03250.9393
24280.08090.0391-0.03120.01470.9346
24320.08960.04540.00960.02500.9346
24360.11090.05990.04780.04130.9346
20240.07600.0406-0.03440.01610.9227
20280.08040.04230.00210.02390.9227
20320.09280.04850.03470.03220.9227
20360.11080.06310.06050.04760.9227

5040450.08150.0378-0.01570.01850.9673
35400.07350.0317-0.02010.01280.9593
35450.08530.04080.01720.02620.9593
30350.06790.0311-0.02300.01170.9557
30400.07710.03560.01390.02020.9557
30450.09290.04650.04150.03320.9557
25300.06290.0317-0.02480.01260.9475
25350.07070.03430.01000.01900.9475
25400.07890.03890.03150.02690.9475
25450.09880.05060.05930.03910.9475

6048540.07180.0302-0.01180.01540.9808
42480.06490.0266-0.01520.01090.9731
42540.07740.03530.02010.02260.9731
36420.06180.0256-0.01490.00980.9707
36480.06950.02850.01550.01690.9707
36540.08450.03840.04110.02810.9707
30360.05420.0256-0.01830.01020.9633
30420.06060.02790.00990.01600.9633
30480.07270.03070.03440.02210.9633
30540.08550.04200.05180.03290.9633


approach
biasMSEbiasMSEcorrect (%)

1089-0.00130.0904-0.29690.15890.4340
78-0.02220.0817-0.26920.13760.3635
790.03920.2032-0.18200.18690.3635
67-0.04180.0830-0.27020.13640.3208
680.00590.1959-0.20140.19230.3208
690.08790.3923-0.05790.31990.3208
56-0.05390.0936-0.27800.15160.3094
57-0.03120.2201-0.22530.21820.3094
580.01990.3925-0.10530.33730.3094
590.10070.6842-0.01080.58370.3094

2016180.02210.0565-0.15210.06870.6408
14160.00440.0437-0.15150.05970.5695
14180.05030.1109-0.04170.09910.5695
12140.00110.0405-0.15270.05690.5011
12160.03540.0953-0.04740.08580.5011
12180.11100.20440.03960.17800.5011
1012-0.00170.0460-0.15910.06280.4051
10140.04030.1047-0.04030.09430.4051
10160.10410.20170.04310.17910.4051
10180.21560.39250.15150.34470.4051

3024270.13460.0585-0.03640.02980.8704
21240.11910.0495-0.04150.02300.8433
21270.14710.07430.03050.04630.8433
18210.10520.0451-0.04910.02160.8042
18240.11680.05900.00690.03680.8042
18270.14240.08800.05480.06580.8042
15180.07030.0455-0.06780.02670.7363
15210.06760.0589-0.02490.04260.7363
15240.06890.0790-0.00240.06490.7363
15270.09930.12330.03740.10670.7363

4032360.11300.0465-0.01960.02340.9068
28320.10610.0375-0.02210.01680.8795
28360.12580.05800.03370.03710.8795
24280.09850.0351-0.02510.01470.8582
24320.11300.04460.02230.02720.8582
24360.14020.06940.06920.05070.8582
20240.09450.0358-0.02850.01590.8438
20280.09940.04030.01120.02490.8438
20320.11910.05170.05190.03690.8438
20360.14130.07590.08520.06000.8438

5040450.09960.0382-0.00820.01940.9309
35400.09190.0304-0.01370.01310.9069
35450.10940.04770.03180.03130.9069
30350.08540.0287-0.01760.01180.8834
30400.10100.03680.02640.02230.8834
30450.12380.05710.06350.04220.8834
25300.08140.0283-0.01980.01250.8760
25350.09250.03390.01990.02020.8760
25400.10220.04230.04650.03080.8760
25450.13440.06430.08700.05170.8760

6048540.08740.0319-0.00540.01640.9491
42480.08040.0264-0.00990.01140.9272
42540.10030.04080.03380.02670.9272
36420.07760.0242-0.01020.01000.9116
36480.09120.03040.02660.01890.9116
36540.11100.04720.06010.03540.9116
30360.07250.0239-0.01360.01030.8991
30420.08020.02850.01850.01720.8991
30480.09670.03580.04940.02620.8991
30540.11620.04200.07570.04300.8991


approach
biasMSEbiasMSEcorrect (%)

1089-0.00180.0902-0.29690.15890.4362
78-0.02230.0817-0.26920.13760.364
790.03890.2031-0.18210.18700.364
67-0.04190.0830-0.27020.13640.321
680.00580.1959-0.20150.19230.321
690.08770.3923-0.05800.31990.321
56-0.05390.0936-0.27800.15160.3093
57-0.03120.2201-0.22530.21820.3093
580.01990.3925-0.10530.33730.3093
590.10070.6842-0.01080.58370.3093

2016180.02120.0564-0.15240.06880.6441
14160.00410.0436-0.15160.05970.5715
14180.04990.1106-0.04190.09900.5715
12140.00110.0405-0.15280.05690.5014
12160.03540.0954-0.04740.08580.5014
12180.11100.20440.03950.17800.5014
1012-0.00170.0460-0.15910.06280.405
10140.04040.1047-0.04030.09430.405
10160.10410.20170.04310.17910.405
10180.21570.39250.15160.34470.405

3024270.13100.0578-0.03770.02980.8771
21240.11810.0493-0.04180.02300.8474
21270.14520.07360.02950.04610.8474
18210.10460.0450-0.04930.02160.8059
18240.11640.05890.00670.03680.8059
18270.14160.08760.05440.06560.8059
15180.07040.0455-0.06770.02670.7359
15210.06780.0589-0.02480.04260.7359
15240.06900.0790-0.00230.06490.7359
15270.09940.12340.03740.10670.7359

4032360.11030.0454-0.02060.02320.9126
28320.10410.0374-0.02270.01680.8831
28360.12440.05750.03290.03690.8831
24280.09790.0350-0.02520.01470.8598
24320.11250.04450.02210.02710.8598
24360.13930.06900.06870.05050.8598
20240.09460.0358-0.02840.01590.8433
20280.09950.04030.01120.02490.8433
20320.11920.05180.05190.03690.8433
20360.14140.07590.08530.06010.8433

5040450.09730.0373-0.00910.01920.9364
35400.09040.0301-0.01420.01310.9113
35450.10740.04690.03060.03100.9113
30350.08480.0287-0.01770.01180.8852
30400.10030.03670.02610.02220.8852
30450.12330.05680.06320.04210.8852
25300.08140.0283-0.01970.01250.876
25350.09260.03390.01990.02020.876
25400.10220.04230.04650.03080.876
25450.13450.06430.08710.05170.876

6048540.08450.0310-0.00650.01620.9554
42480.07940.0261-0.01020.01130.9301
42540.09920.04020.03320.02650.9301
36420.07720.0241-0.01030.01000.9127
36480.09050.03020.02630.01880.9127
36540.10990.04700.05950.03530.9127
30360.07250.0239-0.01360.01030.899
30420.08030.02850.01860.01720.899
30480.09680.03580.04940.02620.899
30540.11630.05320.07580.04300.899

5. Illustrative Examples

In this section, three numerical examples are presented to illustrate the proposed approaches in Sections 24.

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 were:   0.22, 0.50, 0.88, 1.00, 1.32, 1.33, 1.54, 1.76, 2.50, 3.00.

Let be the logs of the ten observations, i.e., . 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 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.

Through using Newton-Raphson algorithm, we obtained the MLEs of and as and for the ND and SEV, respectively.

The values via using ND and SE