On Designing Constant-Stress Partially Accelerated Life Tests under Time-Censoring

It is not easy to obtain more failure data from products with high quality and long life at normal (use) condition. Thus, accelerated tests are needed in this respect. This paper considers the constant-stress partially accelerated life tests with type-I censoring under Weibull distribution. The maxi...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2014
1. Verfasser:	Ismail, A.A.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут проблем міцності ім. Г.С. Писаренко НАН України 2014
Schriftenreihe:	Проблемы прочности
Schlagworte:	Научно-технический раздел
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/112695
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	On Designing Constant-Stress Partially Accelerated Life Tests under Time-Censoring / A.A. Ismail // Проблемы прочности. — 2014. — № 1. — С. 162-170. — Бібліогр.: 8 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-112695
record_format	dspace
spelling	irk-123456789-1126952020-11-24T18:27:21Z On Designing Constant-Stress Partially Accelerated Life Tests under Time-Censoring Ismail, A.A. Научно-технический раздел It is not easy to obtain more failure data from products with high quality and long life at normal (use) condition. Thus, accelerated tests are needed in this respect. This paper considers the constant-stress partially accelerated life tests with type-I censoring under Weibull distribution. The maximum likelihood estimators of the model parameters are derived. Partially accelerated life tests plans are developed such that the generalized asymptotic variance of the maximum likelihood estimators of the model parameters is minimized. The plan is to specify the proportion of test units that should be allocated to run under use condition. Simulation studies are made for illustrative purposes. Отримання даних щодо руйнування високоякісних деталей з великою довговічністю в нормальних умовах експлуатації є досить трудомістким процесом, тому виникає необхідність використання прискорених випробувань. Розглядаються частково прискорені ресурсні випробування за постійних навантажень і цензурування в часі типу І для розподілу Вейбулла. Планування таких випробувань проводиться таким чином, щоб мінімізувати загальну асимптотичну дисперсію оціночних функцій максимальної ймовірності параметрів моделі. План експерименту дозволяє визначити, яку частину ресурсних випробувань об’єктів необхідно проводити за нормальних експлуатаційних умов. Ефективність даного підходу проілюстровано на деяких прикладах числових розрахунків. Получение данных по разрушению высококачественных деталей с большой долговечностью в нормальных условиях эксплуатации является весьма трудоемким процессом, поэтому возникает необходимость использования ускоренных испытаний. Рассматриваются частично ускоренные ресурсные испытания при постоянных нагрузках и цензурировании по времени типа I для распределения Вейбулла. Планирование таких испытаний проводится таким образом, чтобы минимизировать обобщенную асимптотическую дисперсию оценочных функций максимальной вероятности параметров модели. План эксперимента позволяет определить, какую именно часть ресурсных испытаний объектов следует проводить при нормальных эксплуатационных условиях. Эффективность данного подхода проиллюстрирована на некоторых примерах численных расчетов. 2014 Article On Designing Constant-Stress Partially Accelerated Life Tests under Time-Censoring / A.A. Ismail // Проблемы прочности. — 2014. — № 1. — С. 162-170. — Бібліогр.: 8 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/112695 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Научно-технический раздел Научно-технический раздел
spellingShingle	Научно-технический раздел Научно-технический раздел Ismail, A.A. On Designing Constant-Stress Partially Accelerated Life Tests under Time-Censoring Проблемы прочности
description	It is not easy to obtain more failure data from products with high quality and long life at normal (use) condition. Thus, accelerated tests are needed in this respect. This paper considers the constant-stress partially accelerated life tests with type-I censoring under Weibull distribution. The maximum likelihood estimators of the model parameters are derived. Partially accelerated life tests plans are developed such that the generalized asymptotic variance of the maximum likelihood estimators of the model parameters is minimized. The plan is to specify the proportion of test units that should be allocated to run under use condition. Simulation studies are made for illustrative purposes.
format	Article
author	Ismail, A.A.
author_facet	Ismail, A.A.
author_sort	Ismail, A.A.
title	On Designing Constant-Stress Partially Accelerated Life Tests under Time-Censoring
title_short	On Designing Constant-Stress Partially Accelerated Life Tests under Time-Censoring
title_full	On Designing Constant-Stress Partially Accelerated Life Tests under Time-Censoring
title_fullStr	On Designing Constant-Stress Partially Accelerated Life Tests under Time-Censoring
title_full_unstemmed	On Designing Constant-Stress Partially Accelerated Life Tests under Time-Censoring
title_sort	on designing constant-stress partially accelerated life tests under time-censoring
publisher	Інститут проблем міцності ім. Г.С. Писаренко НАН України
publishDate	2014
topic_facet	Научно-технический раздел
url	http://dspace.nbuv.gov.ua/handle/123456789/112695
citation_txt	On Designing Constant-Stress Partially Accelerated Life Tests under Time-Censoring / A.A. Ismail // Проблемы прочности. — 2014. — № 1. — С. 162-170. — Бібліогр.: 8 назв. — англ.
series	Проблемы прочности
work_keys_str_mv	AT ismailaa ondesigningconstantstresspartiallyacceleratedlifetestsundertimecensoring
first_indexed	2025-07-08T04:25:14Z
last_indexed	2025-07-08T04:25:14Z
_version_	1837051375444295680
fulltext	UDC 539.4 On Designing Constant-Stress Partially Accelerated Life Tests under Time- Censoring A. A. Ismail King Saud University, Riyadh, Saudi Arabia; Cairo University, Giza, Egypt ÓÄÊ 539.4 Î ïëàíèðîâàíèè ÷àñòè÷íî óñêîðåííûõ ðåñóðñíûõ èñïûòàíèé ïðè ïîñòîÿííûõ íàãðóçêàõ è öåíçóðèðîâàíèè ïî âðåìåíè À. À. Èñìàèë Óíèâåðñèòåò èì. êîðîëÿ Ñàóäà, Ýð-Ðèàä, Ñàóäîâñêàÿ Àðàâèÿ; Êàèðñêèé óíèâåðñèòåò, Ãèçà, Åãèïåò Ïîëó÷åíèå äàííûõ ïî ðàçðóøåíèþ âûñîêîêà÷åñòâåííûõ äåòàëåé ñ áîëüøîé äîëãîâå÷íîñòüþ â íîðìàëüíûõ óñëîâèÿõ ýêñïëóàòàöèè ÿâëÿåòñÿ âåñüìà òðóäîåìêèì ïðîöåññîì, ïîýòîìó âîçíè- êàåò íåîáõîäèìîñòü èñïîëüçîâàíèÿ óñêîðåííûõ èñïûòàíèé. Ðàññìàòðèâàþòñÿ ÷àñòè÷íî óñêîðåííûå ðåñóðñíûå èñïûòàíèÿ ïðè ïîñòîÿííûõ íàãðóçêàõ è öåíçóðèðîâàíèè ïî âðåìåíè òèïà I äëÿ ðàñïðåäåëåíèÿ Âåéáóëëà. Ïëàíèðîâàíèå òàêèõ èñïûòàíèé ïðîâîäèòñÿ òàêèì îáðàçîì, ÷òîáû ìèíèìèçèðîâàòü îáîáùåííóþ àñèìïòîòè÷åñêóþ äèñïåðñèþ îöåíî÷íûõ ôóíê- öèé ìàêñèìàëüíîé âåðîÿòíîñòè ïàðàìåòðîâ ìîäåëè. Ïëàí ýêñïåðèìåíòà ïîçâîëÿåò îïðåäå- ëèòü, êàêóþ èìåííî ÷àñòü ðåñóðñíûõ èñïûòàíèé îáúåêòîâ ñëåäóåò ïðîâîäèòü ïðè íîðìàëü- íûõ ýêñïëóàòàöèîííûõ óñëîâèÿõ. Ýôôåêòèâíîñòü äàííîãî ïîäõîäà ïðîèëëþñòðèðîâàíà íà íåêîòîðûõ ïðèìåðàõ ÷èñëåííûõ ðàñ÷åòîâ. Êëþ÷åâûå ñëîâà: ÷àñòè÷íî óñêîðåííûå ðåñóðñíûå èñïûòàíèÿ, ïîñòîÿííîå íàïðÿæå- íèå, ðàñïðåäåëåíèå Âåéáóëëà, îáîáùåííàÿ àñèìïòîòè÷åñêàÿ äèñïåðñèÿ, ïëàí ýêñïå- ðèìåíòà. N o t a t i o n ALT – accelerated life test PALT – partially accelerated life test n – total number of test items in PALTs T – time of censoring X – lifetime of an item at normal use condition Y – lifetime of an item at accelerated use condition � – acceleration factor (��1) (� )� – denotes maximum likelihood estimate � – Weibull scale parameter � – Weibull shape parameter xi – observed lifetime of item i tested at normal use condition y j – observed lifetime of item j tested at accelerated use condition � �ui aj, – indicator functions: �ui iI X T� ( ), �aj jI Y T� ( ) – proportion of sample units allocated to accelerated condition * – optimum proportion of sample units allocated to accelerated condition © A. A. ISMAIL, 2014 162 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 1 n nu a, – numbers of items failed at normal use and accelerated use conditions, respectively x x Tnu( ) ( )...1 – ordered failure times at normal use condition y y Tna( ) ( )...1 – ordered failure times at accelerated use condition Introduction. It is not easy to obtain more failure data from products with high quality and long life at use condition. Hence, in order to assure rapid failure and then to shorten the testing period, all or some of test units may be subjected to stress conditions more severe than normal ones. Such accelerated life tests (ALTs) or partially accelerated life tests (PALTs) result in shorter lives than would be observed under use condition. In ALTs test items are run only at accelerated conditions, while in PALTs they are run at both use and accelerated conditions. As Nelson [1] indicates, the stress can be applied in various ways, commonly used methods are step-stress and constant-stress. Under step-stress PALTs, a test item is first run at use condition and, if it does not fail for a specified time, then it is run at accelerated condition until failure occurs or the observation is censored. But the constant-stress PALTs run each item at either use condition, or accelerated condition only, i.e., each unit is run at a constant-stress level until the test is terminated. Accelerated test stresses involve higher than usual temperature, voltage, pressure, load, humidity, ..., etc., or some combination of them. The objective of PALTs is to collect more failure data in a limited time without necessarily using high stresses for all test units. In practice, PALTs are easier to implement and have many advantages, which include: (1) Time saving: PALTs can substantially shorten the duration of the test without affecting the accuracy of lifetime distribution estimates. (2) Economical: Testing units under PALTs can reduce the costs of experiments because not all test units are run at higher stresses. (3) Adaptable: PALTs provide a flexible test strategy, especially for new products when one presumably has little information regarding appropriate test stresses. In such situations, it may not be easy for the experimenter to determine suitable test stress levels. Moreover, the constant-stress PALTs approach is simple and has several advantages: firstly, it is easier to maintain a constant stress level in most tests. Secondly, accelerated test models are better developed. Thirdly, data analysis for reliability estimation is well- developed and computerized [1]. For an overview of constant-stress PALTs, there are few studies in the literature on designing constant-stress PALTs: Bai and Chung [2] used the maximum likelihood method to estimate the scale parameter and the acceleration factor for exponentially distributed lifetimes under type-I censoring. They also considered the problem of optimally designing constant-stress PALTs that terminates at a predetermined time. Ismail et al. [3] considered the constant-stress PALTs plans under Pareto distribution of the second kind with type-I censoring. Abdel-Ghani [4] considered only the estimation problem in constant-stress PALTs for the Weibull distribution parameters, the present investigation extends this work in which PALTs plans with two levels of stress are developed under type-I censoring. The rest of this paper is organized as follows: In Section 1 the Weibull distribution is introduced as a failure time model and the test method is also described. Section 2 presents the maximum likelihood (ML) estimates of the model parameters. In Section 3 optimum test plans of simple constant-stress PALTs are developed. To illustrate the theoretical results, simulation studies are carried out in Section 4. 1. The Model and Test Method. This section introduces the assumed model for product life and also fully describes the test method. 1.1. The Weibull Distribution: a Failure Time Model. The lifetimes of the test items are assumed to follow a two-parameter Weibull distribution. The Weibull distribution is one On Designing Constant-Stress Partially Accelerated Life Tests ... ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 1 163 of the most commonly used distributions in reliability engineering because of the many shapes it attains for various values of �. It can therefore model a great variety of data and life characteristics; see Dimitri [5]. So, it is a statistical distribution frequently used in life data analysis. The probability density function (pdf) of a two-parameter Weibull distribution is given by f t t tT ( ; , ) exp{ ( ) },� � � � � � � �� 1 t�0, ��0, ��0. (1) The Weibull reliability function takes the form: R t t( ) exp{ ( ) },� � � � and the corresponding failure rate function is given by h t t ( ) .� � �� 1 1.2. Constant-Stress PALTs. The test procedure of the constant-stress PALTs and its assumptions are described as follows: Test Procedure. In a constant-stress PALTs, the total sample size n of test units is subdivided into two parts such that: 1. n items randomly chosen among n test items sampled are allocated to accelerated condition and the remaining ones are allocated to use condition. 2. Each test item is run until the censoring time is reached or the item fails and the test condition is not changed. Assumptions. 1. The lifetimes X i , i n� �1 1, ... , ( ) of items allocated to use condition, are i.i.d. r.v.’s. 2. The lifetimes Y j , j n�1, ... , of items allocated to accelerated condition, are i.i.d. r.v.’s. 3. The lifetimes X i and Y j are mutually statistically-independent. 2. Maximum Likelihood Parameter Estimation. In a simple constant-stress PALTs, the test item is run either at use condition, or at accelerated condition only. Since the lifetimes of the test items follow the Weibull distribution, the probability density function of an item tested at use condition is given as in (1). For an item tested at accelerated condition, the pdf is given by f y y yY ( ; , , ) exp{ ( )},� � � �� 1 y�0, ��1, ��0, ��0, where Y X� �� 1 . Since the test in type-I censoring terminates when a predetermined time is reached, the observed lifetimes x x Tnu ( ) ( )...1 and y y Tna ( ) ( )...1 are ordered failure times at normal and accelerated conditions, respectively, where T is the time at which the experiment is terminated, nu and na are the numbers of items failed at use and accelerated conditions, respectively. A. A. Ismail 164 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 1 Let us define the indicator functions: �ui iI X T� ( ) and �aj jI Y T� ( ). Then the total likelihood for ( ; , ... , ; , ; , ... , ; )( ) ( )x x y yu n un a n an1 1 1 1 1 1� � � � � � is given by L x y L x L yui i ui aj j aj j n ( , \| , , ) ( , \| , ) ( , \| , , )� � � � � � � � � �� 1 �� i n 1 1( ) � � �� x x i i i n ui1 1 1 exp{ ( ) } ( ) [exp{ ( ) }]� �T ui� � � � � �� y y j j i n aj1 1 exp{ ( ) } [exp{ ( ) }] ,� � � � � T aj where Lui and Laj denote the contributions of the items i and j to the total likelihood function under use and accelerated conditions, respectively, and � �ui ui� �1 and �aj � � �1 �aj . The ML estimate of � can be obtained by � , / � � � � � � � � � ! "n nu a 1 (2) where � � � � � � � � � �� # ui i i n aj j u a j x y T n n T n n 1 1 ( ) ( ) ( ) n # and � �1 . Therefore, two ML non-linear equations can be expressed as follows: n n n y T n n a u a aj j a j � � ( )� � ( ) � � �� 1 � # � � � � � � � � � 1 0 n , (3) n n x y n n n n u a ui i aj j u a u aj � � � � � � � �� ln ln ( ) ln � � � � 1 n i n � ## � � 1 1/ � � � �� n n n a u a ln � � . / � � � � �1 0 (4) From Eq. (3), the ML estimate of � can be easily derived from � ( ) ( � � � � � � � � � � � � � � � � � � � � � � � #n x T n n n y T a ui i u i n u aj j 1 n na j n � � � � � � � � � � � � $ $$ � $ $ $ ! $ $$ " $ $ $� # ) . / � 1 1 (5) ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 1 165 On Designing Constant-Stress Partially Accelerated Life Tests ... Then Eq. (4) after substitution of � can be rewritten as follows: n n x y n x x T n n u a ui i aj j u ui i i u � � � � � � � ln ln ln ( ) ln � � � � � � � � T x T n n i n ui i u i n j n i n � � �� # # ## � � �1 1 11 � � � � � ( ) � � � � � � # n y y T n n T y T n n a aj j j n j a aj j � � � � � � � � � � ln ( ) ln ( 1 a j n ) . � # � 1 0 (6) Obviously, it is very difficult to obtain a closed-form solution for the non-linear equation (6). Thus, an iterative procedure must be used to solve this equation numerically. The Newton–Raphson method is used to obtain the ML estimate of �. Thus, once the value of �� is determined, the ML estimates of � and � are easily obtained from Eqs. (2) and (5), respectively. Concerning the asymptotic variance-covariance matrix of the maximum likelihood estimators (MLE) of the parameters, it can be obtained by numerically inverting the asymptotic Fisher information matrix F. The asymptotic (large sample) matrix F is composed of the negative second derivatives of the natural logarithm of the likelihood function evaluated at the ML estimates. Unfortunately, the exact expressions of the mathematical expectations for the second derivatives given in (7) are very difficult to obtain. Therefore, the observed Fisher information matrix can be written asymptotically by dropping the expectations as follows (see Cohen [6]): F L L L L L � � � � � � % %� % %�%� % %�%� % %�%� % %� 2 2 2 2 2 2 2 ln ln ln ln ln � � � � � � � � � � � � � � � % %�%� % %�%� % %�%� % %� 2 2 2 2 2 ln ln ln ln L L L L � � � � � � � &( � , � , � ).� � � (7) Consequently, the MLEs of �, �, and � have an asymptotic variance-covariance matrix defined by inverting the Fisher information matrix F indicated above. 3. Optimum Test Plans. Now, for the optimal design stage of the test, a new experiment – with test units different from those tested in the stage of parameter estimation – is conducted. The current aim is to obtain the optimal proportion of sample units * allocated to accelerated condition based on the outputs of the stage of parameter estimation that are at the same time considered inputs to the optimal design stage of the test. It is worth noting that the proportion of sample units allocated to accelerated condition is pre-specified for the stage of parameter estimation. But for the optimal design stage of the test is considered a division parameter that has to be optimally determined according to a certain optimality criterion. This section considers the problem of designing simple constant-stress PALTs, which are terminated at a pre-specified time. The optimum test plan for products having a 166 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 1 A. A. Ismail two-parameter Weibull lifetime distribution is developed. The optimality criterion is to find the optimal proportion of sample units * allocated to accelerated condition such that the generalized asymptotic variance (GAV) of the MLE of the model parameters under use condition is minimized. Most of the test plans are equally-spaced test stresses, i.e., the same number of test units are allocated to each stress. Such test plans are usually inefficient for estimating the mean life at design stress [7]. In this section, statistically optimum test plans are developed to determine the optimal sample proportion allocated for each stress. Therefore, to determine the optimal sample proportion * allocated for accelerated condition, is chosen such that the GAV of the MLEs of the model parameters is minimized. The GAV of the MLEs of the model parameters as an optimality criterion is commonly used and defined below as the reciprocal of the determinant of the Fisher information matrix F (see Bai et al. [8].) That is GAV F ( � , � , � ) \| \| .� � � � 1 The minimization of the GAV over solves the following equation: % % GAV � 0. (8) In general, the solution to (8) is not in a closed form and therefore it requires usage of a numerical method, such as the Newton–Raphson method, which is applied to obtain * which minimizes the GAV. Accordingly, the corresponding expected optimal numbers of items failed at use and accelerated conditions can be obtained numerically, as it will be shown in the next section. 4. Simulation Studies. In this section, several data sets generated from Weibull distribution under type-I censored data are considered with sample sizes 25, 30, 50, 75, and 100 using 1000 replications for each sample size. The true parameter values of �, �, and � used in this simulation study are (3, 5, 1.5) and (2, 4, 0.7). The Newton–Raphson method and programs written in the Pascal language are used for obtaining the MLEs of �, �, and �. Tables 1 and 2 summarize the results of the ML estimates of the parameters and the estimated variances of the MLEs. Results of simulation studies provide insight into the sampling behavior of the estimators. The numerical results indicate that the ML estimates approximate the true values of the parameters as the sample size n increases. Also, as shown from the numerical results, the asymptotic variances of the estimators are decreasing as the sample size n is attaining large values. Tables 3 and 4 depict the results of the test design. That is, the optimal sample- proportion * allocated to accelerated condition, the expected fraction failing at each stress, represented by nu * and na * , and the optimal GAV of the MLEs of the model parameters are obtained numerically for each sample size. The test plans developed here are statistically optimum plans because they are more efficient than standard plans for estimating the life distribution at design stress. Standard plans usually involve equally- spaced test stresses, each with the same number of test units, and they are not the optimum test plans. It can be observed from the numerical results, via * , presented in Tables 3 and 4, that the optimum test plans do not allocate the same number of test units to each stress. In practice, the optimum test plans are important for improving precision in parameter estimation and thus improving the quality of the inference. Also, these tables present the ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 1 167 On Designing Constant-Stress Partially Accelerated Life Tests ... optimal GAV of the MLEs of the model parameters which is obtained numerically with * in place of for different sized samples. As anticipated, the optimal GAV decreases as the sample size n increases. 168 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 1 A. A. Ismail T a b l e 1 ML Estimates and Estimated Asymptotic Variances of MLEs for the Set of Parameters (�, �, �) at (3, 5, 1.5), Respectively, Given � 0.50 and T �10 for Different-Sized Samples under Type-I Censoring in Constant-Stress PALTs n Parameter Estimate Variance 25 � � � 3.4721 5.6325 1.8362 2.3527 2.7104 1.2473 30 � � � 3.4192 5.4571 1.6920 1.5521 2.2205 1.0978 50 � � � 3.3255 5.2689 1.5523 1.0358 1.6571 1.0266 75 � � � 3.2030 5.1276 1.5244 0.8659 0.7945 0.8571 100 � � � 3.0324 5.0764 1.4923 0.4655 0.5106 0.3749 T a b l e 2 ML Estimates and Estimated Asymptotic Variances of MLEs for the Set of Parameters (�, �, �) at (2, 4, 0.7), Respectively, Given � 0.50 and T �10 for Different-Sized Samples under Type-I Censoring in Constant-Stress PALTs n Parameter Estimate Variance 25 � � � 2.9425 5.1162 1.0284 0.2711 0.2265 0.1025 30 � � � 2.6782 4.6471 0.9652 0.2341 0.1843 0.0617 50 � � � 2.2783 4.4458 0.8361 0.1456 0.1355 0.0268 75 � � � 2.1845 4.1239 0.7632 0.0519 0.0813 0.0112 100 � � � 2.0322 3.9852 0.7103 0.0343 0.0212 0.0041 Conclusions. This paper discusses the problem of designing simple constant-stress PALTs plans for the Weibull distribution under type-I censoring, in which the dependence between allocations of observations and the actual values of the probability model parameters were investigated numerically. In a constant-stress test, a unit is subjected to a constant level of stress until failure occurs or the observation is censored. The ML estimates of the model parameters were numerically obtained. Optimum constant-stress PALTs plans were also developed. The minimization of the GAV of the MLEs of the model parameters was used as an optimality criterion. In the constant-stress PALTs, the common test plans are usually standard plans which use equally-spaced test stresses, each with the same number of test units. The design problem is to determine the numbers of test units that should be allocated to each stress level to estimate the life distribution accurately based on the values of the model parameters. The test plan provides the most accurate estimate of the model parameters for a given test time and number of test units. Thus, the optimal design of the life tests can be considered as a technique to improve the quality of the inference. As shown from the results, via * , the optimum test plans are not standard plans because the value of * is different from 50% for lower or higher than unity values of shape parameter. These results coincide with finding of Yang [7] concerning the standard plans. Since the value of was obtained to minimize the GAV of the MLEs of the model parameters, the developed test plans are statistically optimum. It is noteworthy that when ��1, h t( ) becomes suitable for representing the failure rate of units exhibiting wear-out type failures. Consequently, the majority of these units tend to fail under the accelerated conditions, which is also confirmed by the results obtained. On the other hand, when �'1, h t( ) becomes suitable for representing the failure rate of units exhibiting early-type failures. Therefore, in this case, the majority of test units tend to fail under use condition, which is also corroborated by the numerical results. ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 1 169 On Designing Constant-Stress Partially Accelerated Life Tests ... T a b l e 3 Results of Optimal Design of the Life Test for Different-Sized Samples under Type-I Censoring in Constant-Stress PALTs, Based on the Parameters (�, �, �) Set at (3, 5, 1.5), Respectively n * nu * na * Optimal GAV 25 30 50 75 100 0.5738 0.6177 0.7182 0.7617 0.8215 22 41 48 54 63 36 84 173 262 312 0.2237 0.0833 0.0261 0.0072 0.0024 T a b l e 4 Results of Optimal Design of the Life Test for Different-Sized Samples under Type-I Censoring in Constant-Stress PALTs, Based on the Parameters (�, �, �) Set at (2, 4, 0.7), Respectively n * nu * na * Optimal GAV 25 30 50 75 100 0.4273 0.4972 0.5481 0.6152 0.6507 31 74 105 122 138 19 37 46 62 71 0.3681 0.1924 0.1137 0.0491 0.0134 Therefore, the optimal design results are quite lucrative for using the new optimal proportions, in order to run futher experiments. In fact, the test is called an optimally designed test because the test is based on the optimal values of the allocating or dividing proportions. Finally, since the developed test plans are statistically optimum plans, the usefulness of the optimal design lies in the fact that it can serve as a benchmark for comparison with other designs. As a future work, the problem of designing PALTs under the context of Bayesian approach will be considered for the same probability distribution. Acknowledgements. This project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center. Ð å ç þ ì å Îòðèìàííÿ äàíèõ ùîäî ðóéíóâàííÿ âèñîêîÿê³ñíèõ äåòàëåé ç âåëèêîþ äîâãîâ³÷í³ñòþ â íîðìàëüíèõ óìîâàõ åêñïëóàòàö³¿ º äîñèòü òðóäîì³ñòêèì ïðîöåñîì, òîìó âèíèêàº íåîáõ³äí³ñòü âèêîðèñòàííÿ ïðèñêîðåíèõ âèïðîáóâàíü. Ðîçãëÿäàþòüñÿ ÷àñòêîâî ïðè- ñêîðåí³ ðåñóðñí³ âèïðîáóâàííÿ çà ïîñò³éíèõ íàâàíòàæåíü ³ öåíçóðóâàííÿ â ÷àñ³ òèïó ² äëÿ ðîçïîä³ëó Âåéáóëëà. Ïëàíóâàííÿ òàêèõ âèïðîáóâàíü ïðîâîäèòüñÿ òàêèì ÷èíîì, ùîá ì³í³ì³çóâàòè çàãàëüíó àñèìïòîòè÷íó äèñïåðñ³þ îö³íî÷íèõ ôóíêö³é ìàêñèìàëü- íî¿ éìîâ³ðíîñò³ ïàðàìåòð³â ìîäåë³. Ïëàí åêñïåðèìåíòó äîçâîëÿº âèçíà÷èòè, ÿêó ÷àñòèíó ðåñóðñíèõ âèïðîáóâàíü îá’ºêò³â íåîáõ³äíî ïðîâîäèòè çà íîðìàëüíèõ åêñï- ëóàòàö³éíèõ óìîâ. Åôåêòèâí³ñòü äàíîãî ï³äõîäó ïðî³ëþñòðîâàíî íà äåÿêèõ ïðèêëà- äàõ ÷èñëîâèõ ðîçðàõóíê³â. 1. W. Nelson, Accelerated Life Testing: Statistical Models, Data Analysis, and Test Plans, Wiley, New York (1990). 2. D. S. Bai and S. W. Chung, “Optimal design of partially accelerated life tests for the exponential distribution under type-I censoring,” IEEE Trans. Reliab., 41, No. 3, 400–406 (1992). 3. A. A. Ismail, A. A. Abdel-Ghaly, and E. H. El-Khodary, “Optimum constant-stress life test plans for Pareto distribution under type-I censoring,” J. Stat. Comput. Simul., 81, No. 12, 1835–1845 (2011). 4. M. M. Abdel-Ghani, Investigation of Some Lifetime Models under Partially Accelerated Life Tests, Ph.D. Thesis, Cairo University (1998). 5. K. Dimitri, Reliability Engineering Handbook, Vol. 1, Prentice Hall, Englewood Cliffs, NJ (1991). 6. A. C. Cohen, “Maximum likelihood estimation in the Weibull distribution based on complete and on censored samples,” Technometrics, 7, No. 4, 579–588 (1965). 7. G. B. Yang, “Optimum constant-stress accelerated life-test plans,” IEEE Trans. Reliab., 43, No. 4, 575–581 (1994). 8. D. S. Bai, J. G. Kim, and Y. R. Chun, “Design of failure-censored accelerated life-test sampling plans for lognormal and Weibull distributions,” Eng. Opt., 21, 197–212 (1993). Received 23. 01. 2013 170 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 1 A. A. Ismail << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /All /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Warning /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJDFFile false /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /LeaveColorUnchanged /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments false /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /Description << /CHS <FEFF4f7f75288fd94e9b8bbe5b9a521b5efa7684002000500044004600206587686353ef901a8fc7684c976262535370673a548c002000700072006f006f00660065007200208fdb884c9ad88d2891cf62535370300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c676562535f00521b5efa768400200050004400460020658768633002> /CHT <FEFF4f7f752890194e9b8a2d7f6e5efa7acb7684002000410064006f006200650020005000440046002065874ef653ef5728684c9762537088686a5f548c002000700072006f006f00660065007200204e0a73725f979ad854c18cea7684521753706548679c300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c4f86958b555f5df25efa7acb76840020005000440046002065874ef63002> /DAN <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> /DEU <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> /ESP <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> /FRA <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> /ITA <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> /JPN <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> /KOR <FEFFc7740020c124c815c7440020c0acc6a9d558c5ec0020b370c2a4d06cd0d10020d504b9b0d1300020bc0f0020ad50c815ae30c5d0c11c0020ace0d488c9c8b85c0020c778c1c4d560002000410064006f0062006500200050004400460020bb38c11cb97c0020c791c131d569b2c8b2e4002e0020c774b807ac8c0020c791c131b41c00200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000410064006f00620065002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002e> /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken voor kwaliteitsafdrukken op desktopprinters en proofers. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.) /NOR <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> /PTB <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> /SUO <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> /SVE <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> /ENU (Use these settings to create Adobe PDF documents for quality printing on desktop printers and proofers. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.) >> /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ << /AsReaderSpreads false /CropImagesToFrames true /ErrorControl /WarnAndContinue /FlattenerIgnoreSpreadOverrides false /IncludeGuidesGrids false /IncludeNonPrinting false /IncludeSlug false /Namespace [ (Adobe) (InDesign) (4.0) ] /OmitPlacedBitmaps false /OmitPlacedEPS false /OmitPlacedPDF false /SimulateOverprint /Legacy >> << /AddBleedMarks false /AddColorBars false /AddCropMarks false /AddPageInfo false /AddRegMarks false /ConvertColors /NoConversion /DestinationProfileName () /DestinationProfileSelector /NA /Downsample16BitImages true /FlattenerPreset << /PresetSelector /MediumResolution >> /FormElements false /GenerateStructure true /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles true /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /NA /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /LeaveUntagged /UseDocumentBleed false >> ] >> setdistillerparams << /HWResolution [2400 2400] /PageSize [612.000 792.000] >> setpagedevice

On Designing Constant-Stress Partially Accelerated Life Tests under Time-Censoring

Institution

Ähnliche Einträge