Statistical Prediction Based on Ordered Ranked Set Sampling Using Type-II Censored Data from the Rayleigh Distribution under Progressive-Stress Accelerated Life Tests

Hashem, Atef F.; Abdel-Hamid, Alaa H.

doi:https://doi.org/10.1155/2023/5211682

Journal of Mathematics

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 5211682 | https://doi.org/10.1155/2023/5211682

Statistical Prediction Based on Ordered Ranked Set Sampling Using Type-II Censored Data from the Rayleigh Distribution under Progressive-Stress Accelerated Life Tests

Atef F. Hashem^1,2and Alaa H. Abdel-Hamid²

Academic Editor: Ali Sajid

Received23 Dec 2022

Revised01 Feb 2023

Accepted27 Feb 2023

Published30 Mar 2023

Abstract

The objective of ranked set sampling is to gather observations from a population that is more likely to cover the population’s full range of values. In this paper, the ordered ranked set sample is obtained using the idea of order statistics from independent and nonidentically distributed random variables under progressive-stress accelerated life tests. The lifetime of the item tested under normal conditions is suggested to be subject to the Rayleigh distribution with a scale parameter satisfying the inverse power law such that the applied stress is a nonlinear increasing function of time. Considering the type-II censoring scheme, one-sample prediction for censored lifetimes is discussed. Numerous point predictors including the Bayes point predictor, conditional median predictor, and best unbiased predictor for future order statistics are discussed. Additionally, conditional prediction intervals for future order statistics are also studied. The theoretical findings reported in this work are shown by illustrative examples based on simulated data as well as real data sets. The effectiveness of the prediction methods is then evaluated by a Monte Carlo simulation study.

1. Introduction

Modern devices have been designed and engineered to function flawlessly for an extended period of time under regular operating settings thanks to ongoing advancements in manufacturing technology.

As a result, when conducting traditional life research experiments, manufacturers struggle to provide sufficient information about the failure times for their products. Because of this, accelerated life tests (ALTs) or partial ALTs (PALTs) are used to quickly obtain the required information regarding product failure times and establish the relationship between product life and external stress variables. In ALTs, products are checked under situations that are more stressful than usual to discover early failure times, whereas in PALTs, they are checked in both normal and accelerated situations. Stress levels higher than those used during manufacturing are applied to products during ALTs and PALTs. The observed failures of the products collected from such ALTs or PALTs are used to predict how long they will live under normal conditions of use.

Several techniques, including progressive stress, step stress, and constant stress, can be used to apply stress to ALTs. For further explanation on ALTs and PALTs, one can refer to [1–15].

The researcher or experimenter may not be able to obtain complete data on the failure times of the units in the test trial as the units may be broken or excluded from the test prior to failure or when the unit is canceled. Censored data are those obtained from such situations. Censoring may have the significant benefit of reducing the total cost and duration of the experiment. Type-I and type-II censoring are the two methods that are frequently employed. In type-I censoring, the number of observed failures is a random variable (RV), while the experimental time is fixed. In contrast, the experimental time is an RV in type-II censoring, whereas the observed failure rate is fixed. A number of studies, see [16–18], have covered these two CSs in some detail.

Prediction is viewed as a significant problem in statistical inference. It has numerous uses in reliability, quality control, engineering, business, meteorology, medical sciences, and other fields as well. It is the challenge of predicting the values of unobserved (future) observations or functions of such observations from currently accessible (informative) observations. Two frequently used prediction strategies are the one- and two-sample techniques. An interval predictor and a point predictor are both examples of predictors. It has been discussed by a number of authors, including [5, 7, 19–22].

The ranked set sampling method was proposed in [23] as a more accurate way to compute the mean pasture yield. When determining the population mean, a theoretical base for this sampling method was enhanced and developed in [24]. It could be applied to choose sample units more economically for a test or study. It is frequently recommended when ordering sample units is cheap and simple and measuring sample units is very expensive or complicated. It could be used in a variety of disciplines, including agriculture, biology, ecology, engineering, medicine, and social studies [25]. The steps listed below could be used to obtain a ranked set sample (RSS) with size from the provided population:(1)Simple random samples (SRSs), each of the same size , are created by selecting items from the provided population.(2)The items are ranked, according to the variable of interest, for each sample. Several techniques, including expert opinion, readily available information, a person’s professional judgment, and other information, may be used in ranking the items.(3)A single item is measured in each of the ranked samples.(4)A sample is chosen for actual measurement as follows:(i)The smallest item, say , is measured in the first sample, and the other items are not measured.(ii)The second smallest item, say , is measured in the second sample, and the other items are not measured.(iii)This approach is repeated until the greatest item of the latest sample, say , is measured.(5)The procedure described above is referred to as a one cycle RSS with size , and the data obtained are shown by . It is observed that are independent RVs with nonidentical distributions (IRVNIDs).(6)The preceding steps of cycles are repeated to obtain an RSS of size extracted from items. The resulting data are denoted as

The ordered RSS (ORSS) was devised in [26], in which the authors demonstrated how much more effective ORSS is than SRS. It can be achieved by ordering the RSS, , in ascending order of magnitude. This proposal was based on the idea of order statistics from IRVNID.

Several researchers have investigated the estimation and prediction problems on the basis of the SRS and ORSS of various distributions. The distribution-free prediction intervals for record values and future order statistics were constructed in [27]. In [28, 29], it was investigated how to predict unobserved data under a type-II censoring scheme (CS) and how to estimate the parameters of Rayleigh and Pareto distributions using Bayesian methods. Based on type-I CS, the step-stress ALT data were used in [30, 31] to estimate the parameters of Rayleigh and exponential distributions. The Bayesian method was explained in [32] to estimate the parameters taken into consideration using progressive-stress ALT (PRSALT) data that are exponentially distributed.

Due to the importance of predictions, ALTs, and RSSs in many areas as mentioned above, many experimenters and engineers would like to obtain the failure times of some items in a short time. Additionally, they may need to predict future failure times for some items that cannot be obtained in the normal state of the experiment. These requirements and their importance motivate us to consider this article in which we apply the PRSALT, with a nonlinear increasing function of time, to items whose lifetimes under normal condition stress are supposed to follow the Rayleigh distribution (RD). ORSSs are obtained using the idea of order statistics from IRVNID under PRSALTs.

Under type-II censoring, numerous point predictors including the Bayes point predictor (BPPRR) (using squared error (SER), linear-exponential (LEX), and general entropy (GEN) loss functions), conditional median predictor (CMPR), and best unbiased predictor (BUPR) for future order statistics are discussed. Furthermore, conditional prediction intervals (CPIs) for future order statistics are also studied.

The remaining sections are arranged as follows: Section 2 discusses the ORSS under the PRSALT. The model and type-II censoring are explained in Section 3. Section 4 discusses some point predictors and CPIs of future order statistics. In Section 5, representative examples are provided. In Sections 6 and 7, respectively, simulation studies and conclusions are presented.

2. Description of the Model under PRSALT

The RD was originally proposed in [33] in the field of acoustics; since its inception, several researchers have applied the distribution in numerous branches of technology and science. It is extensively applied in communication engineering and oceanography to model wave heights. Furthermore, it has a wide range of applications in lifetime data analysis, particularly in survival analysis and reliability theory. The fact that the RD’s failure rate is a linearly increasing function of time at a constant rate makes it a good model for the lifespan of parts and objects that deteriorate quickly over time. As a result, compared to the exponential distribution, the RD’s reliability function deteriorates over time at a significantly faster rate.

Assume that an item’s lifetime under normal use is represented by the RV , which is subject to RD with a scale parameter of . Then, the cumulative distribution function (CDF), , of is represented by

2.1. Progressive-Stress Model Based on the Rayleigh Distribution

Previous studies of the PRSALT have indicated that the imposed stress is expressed as an increasing linear function of time, see [5, 6, 9]. While in some papers such as [7, 11, 34], the authors suggested PRSALTs taking into account that the imposed stress is represented as a nonlinear increasing function of time. The PRSALT is performed under the following fundamental assumptions.

2.1.1. Assumptions

(1)The lifetime of an item under design stress is governed by RD with CDF (2).(2)The imposed stress is a nonlinear increasing function of time with the form, see Figure 1,(3)The relation between the scale parameter in CDF (2) and the imposed stress is controlled by the inverse power law with two positive parameters and , i.e.,(4)The testing process starts by dividing the testable items into () groups, each of which has items and is administered under PRSALT. Thus,(5)For , the failure times in group , indicated by , , , (with realizations , , , ), are statistically independent RVs.(6)The items’ failure mechanisms remain unchanged under any level of stress.(7)Cumulative exposure model [1] links the distribution under accelerated stress to that under normal stress.

Based on CDF (2) and according to Assumptions 2, 3, and 7, the cumulative exposure model, , can be expressed as

The CDF under PRSALT, , takes the formwhere the function is the assumed CDF with .

Cumulative exposure model (6), according to Assumptions 3 and 4, becomes

Using CDFs (2) and (7), the CDF for an item presented in group under PRSALT takes the form

One can notice that CDF (9) concerns a Weibull distribution with

The corresponding probability density function (PDF), , and the hazard rate function (HRF), , of (9) are given, respectively, by

PDF (11) and HRF (12) are plotted in Figure 2 for , and different values of and . It can be noticed that PDF (11) is always unimodal, while HRF (12) is always increasing since .

(a)

(b)

2.2. Ranked Set Sampling with Accelerated Life Tests under Progressive Stress

The next algorithm can be applied to obtain an RSS with size under PRSALT with () levels of stress:(1)Fixed values for , , and are assigned, such that .(2) items are chosen from the provided population, and they are divided into SRSs, all of the same size .(3) is set.(4)The items to be examined are divided into () groups, as is previously indicated in Section 1. Each group is an SRS consisting of items and is performed under PRSALT with stress levels .(5)The SRSs in all groups are ordered without practical measurement.(6)In the -th ordered SRS, , a single item is measured.(7)In group , the -th smallest item, say , is measured.(8) is set. If , then the previous steps are halted, and it is suggested that we proceed to Step 10. If not, the smallest item in group , say , is measured.(9)Steps 4–8 are iterated.(10)An RSS of size is now generated under PRSALT as follows:(11)The method described in the previous steps is called a one-cycle RSS of size under PRSALT, and the outcomes are shown by For instance, denotes the fifth smallest item in the fifth sample presented in the third group. Additionally, the elements of the RSS are IRVNID.(12)Steps 2–9 of cycles are iterated to obtain an RSS of size . The obtained data are shown by

We presume that is a one-cycle RSS from a given population under PRSALT with CDF (9) and PDF (11). The CDF and PDF of , denoted by and , are then the CDF and PDF of the -th order statistic of group , respectively. They can be written as [35, 36]where and are given by (9) and (11), respectively.

It is possible to rewrite CDF (16) and PDF (17) aswhere

3. The Model with Type-II Censoring

Type-II CS can be imposed to the ordered one cycle RSS under PRSALT as follows: having determined the RSS for group , , , we order and determine the first statistics in it, say . The data collected from this procedure are known as one-cycle type-II censored ORSS and are represented by . Based on the idea of order statistics from IRVNID which was proposed in [37], it is possible to write the likelihood function for one-cycle ORSS with type-II CS aswhere denote the total of all permutations of , and , .

Likelihood function (21) can be rewritten as follows:where denotes the permanent of a square real matrix of size .

Substitute CDF (18) and PDF (19) into (21), the likelihood function can be expressed as follows:

Considering equations (9) and (11) and the next relations, we obtainand it is possible to express the likelihood function as follows:where , , , and , and

The likelihood function can be modified using the relationships provided in (25) as follows:where and are as given in (10), , , , , , and , andwhere is as defined in (27).

For -cycle ORSS, the likelihood function under type-II CS can be expressed as

The likelihood function can be rewritten using the relations provided in (25) aswhere and are as given in (10),and , , , , , , , , , , and .

3.1. Formulating the Prior and Posterior Density Functions

It is appropriate to select and to be dependent since they are merged, as shown in (10). We presume that and are distributed according to the Lomax distribution. The following is a possible representation of the joint prior density of and :where

Using (36) and (37), joint prior density (35) takes the following form:

The hyperparameter values can be specified in such a way that the prior means become the approximate expected value of the corresponding parameters.

Using (33) and (38), it is possible to write the joint posterior density function of and as follows:where

3.2. Loss Functions

Both Bayes analysis and statistical decision inference rely heavily on the loss function. Its choice must be taken into account for calculating the Bayes estimators for and and any function of them. Due to its equal weighting of overestimation and underestimation, the SER loss function is one of the most widely used symmetric loss functions for evaluating estimator performance in practice. The following is a formulation of the SER loss function:where indicates the estimator of .

Considering the SER loss function, the Bayes estimate (BE) of is provided by

In some circumstances, overestimating or underestimating might have different effects. Engineering, medicinal, and biomedical sciences frequently encounter such circumstances. Overestimation is typically more harmful than underestimation. For instance, when we estimate the average dependable working life of components, an asymmetric loss function may be more suitable in this case. There are many asymmetric loss functions proposed for use, including the LEX and GEN loss functions.

The following formula for the LEX loss function was provided in [38]:where and is the LEX estimator of .

Considering the LEX loss function, the BE of is provided by

The following formula for the GEN loss function was provided in [39]:

Considering the GEN loss function, the BE of is provided by

The methods for obtaining point predictors and prediction intervals for future order statistics are covered in the section that follows.

4. One-Sample Prediction Procedure

The following is how a one-sample prediction scheme is carried out: Suppose that, for and , is an informative type-II -cycle ORSS of size taken from a sample of size . Suppose that be the unobserved future order statistics from the same sample, which is yet to observe. Let . Predicting the remaining order statistics is our current goal.

The conditional PDF of , with realization , can take the following form [35, 36, 40]:where and , andwhere .

4.1. Bayesian Prediction by a Point

In the following manner, based on different loss functions, the BPPRs of the -th order statistic, , in the future sample, will be obtained.

The predictive PDF of can be formulated as follows:

Considering the SER, LEX, and GEN loss functions, the BPPRs of are given, respectively, by

4.2. Best Unbiased Predictors, Conditional Median Predictors, and Conditional Prediction Intervals

A median unbiased predictor is defined according to the concept of median unbiasedness. Several characteristics of the median unbiased predictor, in the context of traditional type-II CS, were investigated by Takada [41]. The CMPR was introduced in [42] as a specific kind of median unbiased predictor. If the statistic is the median of the conditional distribution of , it is named the CMPR of .

The conditional CDF of that corresponds to PDF (47) takes the following form:where and are given, respectively, by (48) and (49).

Replacing by their BEs , the CMPR of can be achieved by solving the next equation with respect to :

The following two equations should be simultaneously solved to calculate the bounds of CPI of :Here, and are given, respectively, by (48) and (49).

The predictor is called BUPR of if the predictor error has a mean of zero and a variance that is smaller than or equal to that of any other unbiased predictors of .

The following integral gives the BUPR of :where is given by (47).

If the parameters are unknown, then they can be replaced by their BEs . Further explanation about BUPR, CMPR, and CPI can be found in [20, 22] and [43].

5. Illustrative Examples

Simulated data as well as real data sets are used in this section to demonstrate the point predictor methods described in this article.

5.1. Simulated Data Set

The hyperparameter values (, and ) are selected to produce the population parameter values ( and ). Under PRSALT with two groups, we generate five SRSs, each of size 10, and divide each SRS into two groups, each of size 5, see the third column of Table 1. In each SRS, the first and second groups are generated using CDF (9) with , and . We apply the technique of RSS to these SRSs to obtain a one-cycle RSS and then arrange it to obtain the ORSS. This is shown in the last two columns of Table 1. We apply the type-II censoring procedure to the values of the ORSS, listed in the last two columns of Table 1, by selecting the first values of them.

Based on the ORSS under PRSALT listed in Table 1, the BPPRs, CMPRs, BUPRs, and 95% CPIs for are computed and presented in Table 2.

5.2. Application to Real Data

Our goal now is to demonstrate the point predictor methods discussed in this article using a real data set examined in [44]. The data represent the failure times (in hours) of electrolytic capacitors with a size of 32 volts and 22 microfarads put under two groups of PRSALT. There are 30 units in each testing group. The failure times (in hours) are as follows: First group : 7.21, 10.24, 10.26, 10.37, 10.51, 10.56, 11.25, 11.28, 11.29, 11.35, 12.23, 12.25, 12.36, 12.57, 13.03, 13.04, 13.05, 13.27, 13.46, 13.49, 14.23, 14.45, 15.00, 15.43, 15.47, 16.55, 17.07, 17.21, 17.23, 18.49. Second group : 7.36, 7.55, 7.57, 8.00, 8.23, 8.46, 9.02, 9.03, 9.04, 9.22, 9.32, 9.34, 9.49, 10.28, 10.53, 11.33, 11.34, 11.54, 12.16, 12.53, 12.55, 13.20, 14.06, 14.21, 14.21, 14.21, 16.24, 16.41, 17.53, 21.26.

Before moving on, the statistical test of Kolmogorov–Smirnov (K–S) and its accompanying value are used for each group to determine whether Weibull distribution with CDF (9) is valid for fitting the aforementioned data. It can be shown from the aforementioned data and CDF (9) that the estimates ( and ) maximize the likelihood function of and . The K–S test statistic and the value are given, respectively, by First group: K–S statistic = 0.18113, and value = 0.27858 Second group: K–S statistic = 0.21773 and value = 0.11632

As can be seen, the Weibull distribution with CDF (9) matches the provided real data set well because all of the values are higher than 0.050. This is further demonstrated by depicting the empirical CDF of the provided real data set along with CDF (4) for each group, as shown in Figure 3. We select the hyperparameter values (, and ) to produce the population parameter values ( and ) using (21) and (22).

Under PRSALT with two groups, we choose five SRSs of size 10 each and divide each SRS into two groups of size 5 each, see the third column of Table 3. In each SRS, the first and second groups are drawn, respectively, from the above data under the first and second levels of stress. The technique of RSS is applied to these SRSs to obtain a one-cycle RSS and then order it to obtain the ORSS presented in the last two columns of Table 3. The type-II censoring procedure is applied to the values of the ORSS, listed in the last two columns of Table 3, by selecting the first values of them. Based on the ORSS under PRSALT listed in Table 3, the BPPRs, CMPRs, BUPRs, and 95% CPIs for are computed and presented in Table 4.

6. Simulation Study

A Monte Carlo simulation study is executed in this section to determine BPPRs, CMPRs, BUPRs, and CPIs for the -th order statistic in group , . The next steps can be followed to perform a Monte Carlo simulation:(1)Using (36) and (37), we choose the hyperparameter values (, and ) to produce the population parameter values ( and ). The hyperparameter values were determined to meet the unbiasedness requirements [3, 45] as follows: where E denotes the expectation.(2)Assume (two groups), we generate five SRSs of size 10 each and divide each SRS into two groups of size 5 each. In each SRS, the two groups are generated using CDF (9) with , and , respectively.(3)As described in Subsection 2.2, we apply the RSS method to the SRSs generated in Step 2 to get a one-cycle RSS, which is then ordered to get the ORSS.(4)We apply the type-II censoring procedure to the ORSS values that are acquired in Step 3 by selecting the first values of them.(5)We iterate Steps 2–4 times to obtain -cycle type-II censored ORSSs.(6)The BPPRs, CMPRs, BUPRs, and 95% CPIs for are computed, as indicated earlier in Section 4.(7)If is a prediction of , then the mean squared prediction errors (MSPERs) and biases of are given by(8)We iterate the above steps 1000 times.(9)The coverage probabilities (CPRs) of the CPIs are computed according to the following relation:(10)We compute the average of BPPRs (ABPPR), CMPRs (ACMPR), and BUPRs (ABUPR).

Tables 5–10 present the obtained numerical results.

6.1. Simulation Results

The results presented in Tables 5–10 indicate the following:(1)Through MSPERs, BUPRs are the most accurate point predictors.(2)Through bias and MSPERs, the BPPRs based on LEX (at ) and GEN (at ) loss functions perform better than the BPPRs based on SER loss functions.(3)The MSPERs and bias of BPPR, BUPR, and CMPR of increase as the index increases for .(4)By increasing the stress level, the MSPERs (bias) of BPPR, BUPR, and CMPR of decrease (increases).(5)The CPRs of the CPIs are right near the actual confidence levels by increasing .(6)By increasing the stress level, the AILs decrease since by increasing the stress level, the failure times decrease.(7)The AILs of CPI of increase as the index increases for .

Except for a few unusual cases, the results above are accurate, and this could be because of data fluctuations.

7. Conclusion

The ORSS method has received increasing attention in recent years due to its effectiveness in estimation. This fact has been demonstrated in this article since it has been observed that the estimates calculated under ORSSs are more effective than those calculated under SRSs. Based on type-II censoring, the ORSS method under PRSALTs has been applied to items to be tested. The lifetime of an item under normal use was supposed to follow RD with a scale parameter satisfying the inverse power law such that the imposed stress is expressed by a nonlinear increasing function of time. A one-sample prediction procedure for the unobserved failure times under type-II censoring has been investigated. Some point predictors such as the BPPR, CMPR, and BUPR as well as CPI for future order statistics have been discussed. The performance and effectiveness of the prediction methods described in the article have been demonstrated through Monte Carlo simulations as well as real data. The numerical results have shown that the prediction methods have perfect performance.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to extend their appreciation to the Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University (IMSIU), Saudi Arabia, for funding this research work through Grant No. 221412055.

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Copyright © 2023 Atef F. Hashem and Alaa H. Abdel-Hamid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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