Estimating the Generalized Exponential Distribution Parameters and the Acceleration Factor under Constant-Stress Partially Accelerated Life Testing with Type-II Censoring
Accelerated life testing (ALT) and partially accelerated life testing (PALT) are frequently used in modern reliability engineering. ALT and PALT are run to obtain information on the life of the products and materials in a shorter time and at lower cost. The experimental units are subject to st...
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| Date: | 2013 |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2013
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| Cite this: | Estimating the Generalized Exponential Distribution Parameters and the Acceleration Factor under Constant-Stress Partially Accelerated Life Testing with Type-II Censoring / A.A. Ismail // Проблемы прочности. — 2013. — № 6. — С. 82-94. — Бібліогр.: 19 назв. — англ. |
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| author | Ismail, A.A. |
| author_facet | Ismail, A.A. |
| citation_txt | Estimating the Generalized Exponential Distribution Parameters and the Acceleration Factor under Constant-Stress Partially Accelerated Life Testing with Type-II Censoring / A.A. Ismail // Проблемы прочности. — 2013. — № 6. — С. 82-94. — Бібліогр.: 19 назв. — англ. |
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| description | Accelerated life testing (ALT) and partially accelerated
life testing (PALT) are frequently used
in modern reliability engineering. ALT and
PALT are run to obtain information on the life
of the products and materials in a shorter time
and at lower cost. The experimental units are
subject to stress conditions that are more severe
than those encountered in normal use condition
to induce early failures. ALT or PALT can be
carried out using constant, step, progressive, cyclic
and random stress loadings. This paper considers
the problem of estimating the generalized
exponential (GE) distribution parameters and
the acceleration factor under constant-stress
PALT model. The main objective is to derive the
maximum likelihood estimators (MLEs) of the
parameters of the GE distribution and the acceleration
factor when the data are type-II censored
from constant-stress PALT. Also, the performance
of the MLEs is investigated numerically
for different sample sizes and different parameter
values using the mean square error. In addition,
the approximate confidence intervals of the
model parameters are constructed. Moreover,
the likelihood ratio bounds (LRB) method is
used to obtain confidence bounds of the model
parameters when the sample size is small. For
illustration, a simulation study is conducted. It
is observed that the simulation results support
the theoretical findings.
Ускоренные и частично ускоренные ресурсные испытания часто проводятся для обеспечения
надежности современной техники. Цель таких испытаний – за короткий срок и с меньшими
затратами получить информацию о ресурсе изделий и материалов. Испытания можно
проводить при постоянных, ступенчатых, прогрессирующих и циклических нагрузках, а
также при нагрузке случайным напряжением. Рассматривается задача оценки параметров
обобщенного экспоненциального распределения и коэффициента ускорения в условиях частично ускоренного ресурсного испытания с постоянным напряжением. С использованием среднеквадратической погрешности проведено численное исследование эффективности оценки методом максимального правдоподобия для разных размеров образцов и значений параметра.
Для параметров модели построены приблизительные доверительные границы. Чтобы получить доверительные границы для параметров модели в случае образца маленького размера,
использовали метод отношения правдоподобия. В качестве примера проведено исследование с
помощью моделирования. Показано, что результаты моделирования согласуются с данными
расчетов.
Прискорені та частково прискорені ресурсні випробування часто проводяться
для забезпечення надійності сучасної техніки. Метою таких випробувань є за
короткий проміжок часу та з найменшими затратами отримати інформацію
про ресурс виробів і матеріалів. Випробування можна проводити за постійних, східчастих, прогресуючих і циклічних навантажень, а також при навантаженні випадковим напруженням. Розглядається задача оцінки параметрів
загального експоненціального розподілу та коефіцієнта прискорення в умовах частково прискореного ресурсного випробування з постійною напругою.
Із використанням середньоквадратичної похибки проведено чисельне дослідження ефективності оцінки методом максимальної правдоподібності для різних розмірів зразка та значень параметра. Для параметрів моделі побудовано
приблизні довірчі границі. Щоб отримати довірчі границі для параметра
моделі у випадку зразка маленького розміру, використовували метод відношення правдоподібності. Як приклад проведено дослідження за допомогою
моделювання. Показано, що результати моделювання збігаються з даними
розрахунків.
|
| first_indexed | 2025-12-07T16:18:37Z |
| format | Article |
| fulltext |
UDC 539.4
Estimating the Generalized Exponential Distribution Parameters and
the Acceleration Factor under Constant-Stress Partially Accelerated
Life Testing with Type-II Censoring
A. A. Ismail
King Saud University, Riyadh, Saudi Arabia
ÓÄÊ 539.4
Îöåíêà ïàðàìåòðîâ îáîáùåííîãî ýêñïîíåíöèàëüíîãî ðàñïðåäåëå-
íèÿ è êîýôôèöèåíòà óñêîðåíèÿ â óñëîâèÿõ ÷àñòè÷íî óñêîðåííîãî
ðåñóðñíîãî èñïûòàíèÿ ñ ïîñòîÿííûì íàïðÿæåíèåì ïðè öåíçóðèðî-
âàíèè òèïà II
À. À. Èñìàèë
Óíèâåðñèòåò èì. êîðîëÿ Ñàóäà, Ýð-Ðèÿä, Ñàóäîâñêàÿ Àðàâèÿ
Óñêîðåííûå è ÷àñòè÷íî óñêîðåííûå ðåñóðñíûå èñïûòàíèÿ ÷àñòî ïðîâîäÿòñÿ äëÿ îáåñïå÷åíèÿ
íàäåæíîñòè ñîâðåìåííîé òåõíèêè. Öåëü òàêèõ èñïûòàíèé – çà êîðîòêèé ñðîê è ñ ìåíüøèìè
çàòðàòàìè ïîëó÷èòü èíôîðìàöèþ î ðåñóðñå èçäåëèé è ìàòåðèàëîâ. Èñïûòàíèÿ ìîæíî
ïðîâîäèòü ïðè ïîñòîÿííûõ, ñòóïåí÷àòûõ, ïðîãðåññèðóþùèõ è öèêëè÷åñêèõ íàãðóçêàõ, à
òàêæå ïðè íàãðóçêå ñëó÷àéíûì íàïðÿæåíèåì. Ðàññìàòðèâàåòñÿ çàäà÷à îöåíêè ïàðàìåòðîâ
îáîáùåííîãî ýêñïîíåíöèàëüíîãî ðàñïðåäåëåíèÿ è êîýôôèöèåíòà óñêîðåíèÿ â óñëîâèÿõ ÷àñòè÷-
íî óñêîðåííîãî ðåñóðñíîãî èñïûòàíèÿ ñ ïîñòîÿííûì íàïðÿæåíèåì. Ñ èñïîëüçîâàíèåì ñðåäíå-
êâàäðàòè÷åñêîé ïîãðåøíîñòè ïðîâåäåíî ÷èñëåííîå èññëåäîâàíèå ýôôåêòèâíîñòè îöåíêè ìå-
òîäîì ìàêñèìàëüíîãî ïðàâäîïîäîáèÿ äëÿ ðàçíûõ ðàçìåðîâ îáðàçöîâ è çíà÷åíèé ïàðàìåòðà.
Äëÿ ïàðàìåòðîâ ìîäåëè ïîñòðîåíû ïðèáëèçèòåëüíûå äîâåðèòåëüíûå ãðàíèöû. ×òîáû ïîëó-
÷èòü äîâåðèòåëüíûå ãðàíèöû äëÿ ïàðàìåòðîâ ìîäåëè â ñëó÷àå îáðàçöà ìàëåíüêîãî ðàçìåðà,
èñïîëüçîâàëè ìåòîä îòíîøåíèÿ ïðàâäîïîäîáèÿ.  êà÷åñòâå ïðèìåðà ïðîâåäåíî èññëåäîâàíèå ñ
ïîìîùüþ ìîäåëèðîâàíèÿ. Ïîêàçàíî, ÷òî ðåçóëüòàòû ìîäåëèðîâàíèÿ ñîãëàñóþòñÿ ñ äàííûìè
ðàñ÷åòîâ.
Êëþ÷åâûå ñëîâà: òåõíèêà îáåñïå÷åíèÿ íàäåæíîñòè, ÷àñòè÷íî óñêîðåííûå
ðåñóðñíûå èñïûòàíèÿ, ïîñòîÿííîå íàïðÿæåíèå, îáîáùåííîå ýêñïîíåíöèàëü-
íîå ðàñïðåäåëåíèå, îöåíêà ïî ìåòîäó ìàêñèìàëüíîãî ïðàâäîïîäîáèÿ, ãðàíè-
öû îòíîøåíèÿ ïðàâäîïîäîáèÿ, öåíçóðèðîâàíèå òèïà II.
N o t a t i o n
ALT – accelerated life testing
PALT – partially accelerated life testing
CSPALT – constant-stress partially accelerated life testing
GE – generalized exponential
MTTF – mean time to failure
ML – maximum likelihood
© A. A. ISMAIL, 2013
82 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
MLEs – maximum likelihood estimates/estimators
LR – likelihood ratio
LRB – likelihood ratio bounds
MSE – mean square error
CI – confidence interval
1�� – confidence level
IW95 – CI width at 95% level of confidence
IW99 – CI width at 99% level of confidence
n – total number of test items in a PALT (sample size)
y r( ) – the time of the rth failure at which the test is terminated
X – lifetime of an item at normal (use) condition
Y – lifetime of an item at accelerated condition
(� )� – denotes maximum likelihood estimate
� – acceleration factor (��1)
� – GE shape parameter (��0)
� – GE scale parameter (��0)
xi – observed lifetime of item i tested at use condition
y j – observed lifetime of item j tested at accelerated condition
ui aj, – indicator functions: ui i rI X y
�( ),( ) aj j rI Y y
�( )( ) ,
i n�1, ... ,
– proportion of sample units allocated to accelerated condition
n nu a, – numbers of items failed at use and accelerated conditions,
respectively
r – total number of failed units (r n nu a� � )
L xui i ui( , ) – the likelihood for ( , ),xi ui i nu�1, ... ,
L yaj j aj( , ) – the likelihood for ( , ),y j aj j na�1, ... ,
x x yn ru( ) ( ) ( )...1 � � � – ordered failure times at use condition
y y yn ra( ) ( ) ( )...1 � � � – ordered failure times at accelerated condition
Introduction. Accelerated life testing (ALT) is frequently used in modern
reliability engineering. By performing life tests at accelerated (i.e., harsher-than-
use) conditions, failures are quickly obtained, and products’ reliability at normal
(use) conditions can then be estimated via a stress–life relationship (data-
extrapolation). In some cases such relationship is not known or can’t be assumed
and then the ALT methods can’t be applied. Alternatively, another test method
namely partially accelerated life testing (PALT) can be used to estimate and
analyze products’ reliability.
Such PALT results in shorter lives than would be observed under use
condition. In PALT some of test units can be run under use condition and the others
under accelerated condition. (In ALT, test units are run only at accelerated
condition. Interested readers can refer to Nelson [1] and Meeker and Escobar [2]
which are two comprehensible sources for ALT).
As Nelson [1] indicates, the stress can be applied in various ways, commonly
used methods are step-stress and constant-stress. Under step-stress PALT, a test
Estimating the Generalized Exponential Distribution Parameters ...
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 83
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 PALT runs 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 a
PALT is then to collect more failure data in a limited time without necessarily
using high stresses to all test units.
Compared with the step-stress accelerated life test (step-test), the constant-
stress accelerated life test (constant-test) has some merits as follows: simple test
method, ripe theory, and precise test data. The constant-stress accelerated-life-test
is commonly used to test the reliability of electric appliances. For an overview of
constant-stress PALT (CSPALT), there are few studies in the literature in this
respect: Bai and Chung [3] used the maximum likelihood method to estimate the
scale parameter and the acceleration factor for exponentially distributed lifetimes
under type-I censoring. Abdel-Ghani [4] considered the estimation problem for the
Weibull distribution parameters. Ismail [5] used the maximum likelihood approach
for estimating the acceleration factor and the parameters of Pareto distribution of
the second kind. Ismail [6] extended the work of Abdel-Ghani [4] to study the
problem of optimal design of the life test with type-II censored data. Recently,
Ismail et al. [7] developed optimum constant-stress life test plans for Pareto
distribution under type-I censoring.
The objective of this paper is to consider the CSPALT with type-II censoring
for estimating the acceleration factor and the generalized exponential distribution
parameters. A new distribution named as generalized exponential (GE) distribution
or exponentiated exponential distribution was introduced by Gupta and Kundu [8]
and since then it received a considerable attention in the literature. Several
properties of the GE distribution were studied quite extensively; see for example,
Gupta and Kundu [8–11]. The readers may be referred to some of the related
literature on GE distribution by Raqab [12], Raqab and Ahsanullah [13] and Zheng
[14]. The simple mathematical structure of the GE distribution enables it to be used
effectively for modeling various lifetime data types with possible censoring or
grouping Baklizi [15].
The two-parameter GE family has the distribution function
F x e x( ; , ) ( ) ,� � � �� � �1 y �0. (1)
The corresponding density function is
f x e eX
x x( ; , ) ( ) ,� � �� � � �� � � � �1 1 x �0, ��0, ��0, (2)
where � and � are the shape and scale parameters, respectively. When ��1 it
coincides with the exponential distribution with mean 1 �. When ��1 the density
function is strictly decreasing and for ��1 it has a unimodal shape. These
densities are illustrated in Gupta and Kundu [9]. It is clear that the GE density
functions are always right skewed and it is observed that GE distributions can be
used quite effectively to analyze skewed data sets.
A. A. Ismail
84 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
The hazard rate function of the GE distribution is
h x
e e
e
x x
x
( ; , )
( )
( )
,� �
�� � � �
� �
�
�
� �
� � �
�
1
1 1
1
(3)
and the MTTF is
MTTF
i i
i
i
�
�
�
��
�
�
��
� �
�
�
�
�
�
( )
.
1 1
1
(4)
The GE distribution can have increasing and decreasing hazard rates depending
on the shape parameter �. The hazard rate increases from 0 to � if ��1 and if
��1 it decreases from � to �. This property leads to a good ability of using this
distribution in reliability and life testing, Abuammoh and Sarhan [16].
The reminder of the paper is organized as follows: in Section 1 the test
procedure and its assumptions used throughout this paper are presented. In Section 2
two different methods are used to obtain the estimations of the unknown parameters.
The first method is the maximum likelihood (ML) method when the sample size is
large and the second one is the likelihood ratio bounds (LRB) method when the
sample size is small. In Section 3 simulation studies are carried out to illustrate the
theoretical results.
1. Test Procedure and Its Assumption. The test procedure of the CSPALT
and its assumptions are described as follows:
Test Procedure. In a CSPALT, the total sample size n of test units is divided
into two parts such that:
1. n
units randomly chosen among n test units sampled are allocated to
accelerated condition and the remaining are allocated to use condition.
2. Each test unit is run until it fails or the test is terminated.
Assumptions.
1. The lifetimes X i , i n� �1 1, ... , ( )
of units allocated to use condition, are
i.i.d. r.v.’s.
2. The lifetimes Y j , j n�1, ... ,
of units allocated to accelerated condition,
are i.i.d r.v.’s.
3. The lifetimes X i and Y j are mutually statistically-independent.
4. The lifetimes of the test units follow the GE distribution.
2. Parameter Estimation.
2.1. ML Estimation. In a simple constant-stress PALT, the test item is run
either at use condition or at accelerated condition only. Since the lifetimes of the
test items follow the GE distribution, the probability density function (pdf) of an
item tested at use condition is given as in (2). While for an item tested at
accelerated condition, the pdf is given by
f y e eY
y y( ; , , ) ( ) ,� � � ��� �� � ��� � � � �1 1
y �0, ��1, ��0, ��0,
(5)
where Y X� �� 1 .
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 85
Estimating the Generalized Exponential Distribution Parameters ...
Since the test in type-II censoring is terminated at a predetermined r of
failures, the observed lifetimes x x yn ru( ) ( ) ( )...1 � � � and y y yn ra( ) ( ) ( )...1 � � �
are ordered failure times at use and accelerated conditions respectively, where nu
and na are the numbers of items failed at use and accelerated conditions,
respectively.
Let the indicator functions: ui i rI X y
�( )( ) and aj j rI Y y
�( ).( ) 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
i
n
aj j aj( , | , , ) ( , | , ) ( , | , ,� � � � � � � �
�
�
�
1
)�
�
�
j
n
1
� � � �
� � � �
�
�{ ( ) } { ( ) }( )��
� � � � �
1 1 11
1
e e exi xi ui r ui
y
i
n
�
� � � �
� � � �
{ ( ) } { ( ) } ,( )���
�� � �� �� �
1 1 11e e e
y j y j a j y r a j
j�
�
1
n
(6)
where
ui ui� �1 , aj aj� �1 , and
� �1 .
We can write the total likelihood function by another possibility, in terms of
the order statistics indicated earlier, as
L x y e e exi
u
xi
i
n
y
( , | , , ) ) } { (� � � ������
� � � �
� � �
� �
�
� �� 1
1
1 1 ( ) ) }r
ui n
n
�
�
� �
�
1
� � �
� � �
�
�
�
�{ ) } { ( )( )������
�� � �� �� �e e e
y j y j
a
r
j
n
y
j n
1
1
1 1
a
n
�
�
1
}.
The log-likelihood function, ln ,L is
ln ln ln ( ) ln( )L n n e xu u i
j
n
i
n
xi� � � � � � �
�
��
��� � � �
�
1 1
11
� � � � � � �
�
n e n n nu
y
a a a
rln[ ( ) ] ln ln ln( )( )1 1
� � � � �
� � � � � � �
�
� �
�� �( ) ln( ) ln[ (� ��
��
��
1 1 1 1
1 1
e y n e
y j
j
n
j
j
n
a
y r( ) ) ].�
(7)
86 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
A. A. Ismail
The normal equations become
�
�� �
� � �
��
��
ln
( )
L n y e
e
y
a j
j
j
n
i
n y j
y j
ua
� � �
�
� �
�
�
��
��1
1 11
�
�
� �
�
� � �
�
��
�� �� �
�� �
y n e e
e
r a
y y
y
r r
r
( )
( ) ( )
( )
( )
( )
,
1
1 1
0
1
(8)
�
�� �
� ��ln
ln( ) ln( )
L n n
e e
u a
i
n
j
n
xi
u
y j
a
�
�
� � � � �
�
�
�
�
� 1 1
1 1
�
�
� �
� �
�
�
� �
�
�
n e e
e
n eu
y y
y
a
r r
r
( ) ln( )
( )
(( ) ( )
( )
1 1
1 1
1
� � �
� �
�� � ��
�� �
y y
y
r r
r
e
e
( ) ( )
( )
) ln( )
( )
,
1
1 1
0
�
� �
�
�
� (9)
�
�� �
�
�
�
ln
( )
( )
L n n x e
e
u a i
i
n xi
xi
u
�
�
� �
�
�
�
�
�
�1
11
� �
�
� �
� � �
�
�
x
n y e e
e
i
u r
y y
y
j
n r r
r
�
� � �
� �
( ) ( )
( )
( ) ( )
( )
1
1 1
1
1
u
y j
y j
a y e
e
j
i
n
� �� �
�
�
�
�
�
( )
( )
� �
��
��
1
11
� �
�
� �
� � �
�
�
��
�� � ��
��
y
n y e e
e
j
a r
y y
y
r r
r
( ) ( )
( )
( ) ( )
( )
1
1 1
1
�
�
�
� 0
1
.
j
na
(10)
Now, we have a system of three nonlinear equations in three unknowns �, �,
and �. It is clear that a closed form solution is very difficult to obtain. Therefore,
iterative procedure can be used to find a numerical solution of the above system.
Concerning the asymptotic confidence intervals of the model parameters, it is
not possible to derive the exact distributions of the MLE of the parameters because
the likelihood equations have no closed form solutions in the unknown parameters
�, �, and �. Thus, approximate confidence intervals of the parameters are derived
based on the asymptotic distributions of the MLE of the elements of the vector of
unknown parameters � � � �� ( , , ). It is known that the asymptotic distribution of
the MLE of � is given by (see Miller [17])
(( � , � , � , ( , , )),�� � ���� ��� � � � � ��! N I0 1
where I �1( , , )� � � is the variance-covariance matrix of the unknown parameters
� � � �� ( , , ). The elements of the 3 3� matrix I , I ij ( , , ),� � � i j, , , ;�1 2 3 can be
approximated by I ij (
� , � , � ),� � � where
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 87
Estimating the Generalized Exponential Distribution Parameters ...
I
L
ij
i j
( � )
ln ( )
.
�
�
� �
�� �� � �
��
�
2
From Eq. (7), we get the following:
�
�� �
� �
"
"
�� ��
2
2 2
2
2
3
2 2
3
2
1
ln
( )
L n y e y e
a j j j
j
y j y j
�� � �
�
�
� �
j
na
�
�
1
�
� � � �
� �
�� � " " " �
�� �� �2 2
5 5
2
51 1y e n er
y
a
yr r
( )
( ) ( ){[( ) ][ ] "
"
� ��
5
2 1
5
2
( ) ( ) }
,
� �
e
y r
(11)
�
�� �
" " " "
"
� � �
�
2
2 2
2 2
2
2 2
2
2
1
1
ln (ln ) {( ) }
( )
L n n nu u
�� �
� �
�
�
a
�2
�
�
� �na (ln ) { ( ) }
,
" " " "
"
#
�
6
2
5 5
2
5
2
1
(12)
�
�� �
�
"
"
� �2
2 2
2
1
1
2
1
1
ln
( )
( )L n n x e eu a i i
ii
n xi xi
��
�
� �
�
� �
�
u
� �
�
� � � �� � � �
n y e eu r
y yr r� � " " "� � � � �
( ){[( ) ][ ]( ) ( )2
2
2
2
1
21 1 �"
"
� �
2
2 1 2
5
2
( ) ( ) }
� �
�
e
y r
� �
�
�
� �
�
�( )
( )
� �
"
"
$
�� ��
1
2
3
3
2
1
y e ej j
jj
n y j y ja
�
� � �
� � � �
n y ea r
y r�� " � " " �"
�� � � �2 2
5 6
2
6
1
6
2 1
1( )
( )
{ [( ) ]( ) e
y r�2
5
2
��
"
( ) }
, (13)
�
����
�
"
��
2
31
ln L y ej
jj
n y ja
� �
�
�
�
�
� �
� � � �
n y ea r
y r� " �" " " �" "
�� � � �
( )
( )( ) {[ ln ] ln6
2
6
1
6 5 6
2 1
6
5
2
}
,
"
(14)
�
����
�
"
��
"
�� �� ��
2
3
2
3
1
ln
( )
( )L y e y e ej
j
j j
y j y j y j
� � �
�
� � �
"3
2
11 1jj
n
j
n
j
j
naa a
y
�� �
�� �
%
&
'
('
)
*
'
+'
� �
88 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
A. A. Ismail
� � � �
�
� �n y e
y y e
a r
y
r r
r�
"
�� " � ��
��
� ��( )
( ) ( )
( )
{[ ( )
5
2 6
11 1
y r( ) ]" "�
6
2
5
� �
�
� �
�� "
�� �
e
y r( ) ( )
},6
2 1
(15)
�
���� "
�
"
� ��
2
11 31
ln L x e y e
i
ii
n
j
jj
nxiu
y ja
� � �
�
�
�
�
� �
� � �� � � �n
y e y e
u
r
y
r
yr r
"
� " " " "� � � �
4
2 2
1
2 2
1{[ ln ] ]( ) ( )
( ) ( )
4 �
� �� � � �
� " "
"
� �"� � � ��
y e
n
y er
y a
r
yr r
( ) ( )ln } {[( ) (
2
2 1
2
5
2 6
1 ) ln "6 �
� �� � � �
y e y er
y
r
yr r
( ) ( )
( ) ( )] ln },�" " � �" "� �� � ��
6
1
5 6
2 1
6 (16)
where
"
�
1 1i e xi� �
�
, "
�
2 1� �
�
e
y r( ) , "
��
3 1j e
y j� �
�
,
"
� �
4 1 1� � �
�
( ) ,( )e
y r "
�� �
5 1 1� � �
�
( ) ,( )e
y r "
��
6 1� �
�
e
y r( ) .
Thus, the approximate 100(1��)% two sided confidence intervals for �, �,
and � are, respectively, given by
� ( � ) ,/� ��, �Z I2 11
1
� ( � ) ,/� ��, �Z I2 22
1
� ( � ) ,/� ��, �Z I2 33
1
where Z� /2 is the upper 100(� 2)th percentile of a standard normal distribution.
2.2. LRB Estimation. The ML estimates are based on large sample normal
theory. However, when there are only few failures, the large sample normal theory
is not very accurate. Thus, the ML estimates could be very different from the true
values. The LRB method is based on the --squared distribution assumption. For
example, Vander Wiel and Meeker [18] investigated the accuracy of the likelihood
ratio (LR)-based confidence bounds and asymptotic s-normal-based confidence
bounds using censored Wiebull regression data from constant-stress ALT. It is
noted that the LR bounds perform better than the ML-based bounds when the
sample size is small. In this subsection we will use the LRB method to obtain the
confidence bounds of the model parameters when the sample size is small.
Here, we will apply the LRB method to derive the confidence bounds for a
vector of unknown parameters � � � �( , , ) when the sample size is small. We treat
the LR as a function on � defined by
LR
L
L
( )
( , , )
( � , � , � )
,�
� � �
� � �
�
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 89
Estimating the Generalized Exponential Distribution Parameters ...
where L( , , )� � � is the likelihood function with three parameters �, �, and �,
and � ,� � ,� and �� are their estimated values.
Because the log likelihood ratio statistic is X 2 distributed, that is,
� .2 2log ( ) ,,LR X k� � with k degrees of freedom (the number of quantities jointly
estimated), then, the confidence bounds over which LR e
X k( ) , /
� ��
2 2
is the
100 1( )%�� LRB for �.
There is no closed-form solution available. Therefore, it is very difficult (there
are more than two parameters) to discuss LR bounds except for simulation study
which has been done by others (e.g., McSorley et al. [19]). The simulation results
of the LR bounds of our model parameters are presented in Tables 1 and 2. For the
LR bounds of the model parameters when the sample size is small, the simulation
results are calculated based on the --squared distribution and are reported in these
tables. These results support the theoretical findings.
3. Simulation Studies. In this section, a simulation study is conducted to
illustrate the theoretical results given in this paper and to investigate the performance
of the MLEs of the model parameters via their mean square error (MSE). Moreover,
the performance of the various approximate intervals presented in this paper is
studied. The simulation algorithm or procedure can be described as follows.
Step 1. 10,000 random samples of sizes 30, 50, 75, and 100 are generated
from the GE distribution. Two sets of the true parameter values �, �, and � are
considered to be 1.3, 0.5, 0.7 and 2.2, 1.4, 1.1, respectively.
Step 2. Considering the allocation parameter
to be
� 0.5.
Step 3. For the two sets of parameters and for each sample size, the two
parameters of the GE distribution and the acceleration factor are estimated under
CSPALT with type-II censored data using the maximum likelihood approach.
90 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
A. A. Ismail
T a b l e 1
Simulation Results of the LR Bounds of the Model Parameters Using Small Samples
(� � 1.3, � � 0.5, and � � 0.7)
n Parameter Lower bound
95%
Upper bound
95%
Lower bound
99%
Upper bound
99%
10 �
�
�
0.88330
0.33932
0.36953
2.363765
0.673620
0.948625
0.794970
0.305388
0.332577
2.659236
0.757823
1.067203
15 �
�
�
0.81750
0.37065
0.24325
2.199545
0.650370
0.743185
0.735750
0.333585
0.218925
2.474488
0.731666
0.836083
20 �
�
�
0.93918
0.42480
0.42942
1.870935
0.632850
0.813705
0.845262
0.382320
0.386478
2.104802
0.711956
0.915418
25 �
�
�
0.99072
0.45559
0.48816
1.641350
0.623440
0.802060
0.891648
0.410031
0.439344
1.846519
0.701370
0.902318
Step 4. The Newton–Raphson method is used to solve the nonlinear likelihood
equations for �, �, and � numerically.
Step 5. The MSE of the MLEs of model parameters is computed for different
sized samples and different sets of parameter values.
Step 6. The asymptotic variances of the estimators of model parameters are
estimated for different sized samples and different sets of parameter values.
Step 7. The approximate confidence bounds with confidence level �� 0.95
and 0.99 are obtained for the three parameters of the model.
By conducting the above steps using a computer program written in the Pascal
language, the simulation results are reported in Tables 3 and 4. As shown from the
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 91
Estimating the Generalized Exponential Distribution Parameters ...
T a b l e 2
Simulation Results of the LR Bounds of the Model Parameters Using Small Samples
(� � 2.2, � � 1.4, and � � 1.1)
n Parameter Lower bound
95%
Upper bound
95%
Lower bound
99%
Upper bound
99%
10 �
�
�
1.44683
0.70889
0.16919
5.771750
2.826875
1.813125
1.229806
0.602557
0.143812
6.060338
2.968219
1.903781
15 �
�
�
1.51844
0.75075
0.35182
4.416750
2.357875
1.674500
1.290674
0.638138
0.299047
4.637588
2.475769
1.758225
20 �
�
�
1.67937
0.92281
0.71729
3.528875
1.810375
1.558125
1.127465
0.784389
0.609697
3.705319
1.900894
1.636031
25 �
�
�
1.72627
0.86569
0.84035
3.322625
1.615125
1.549625
1.267330
0.735837
0.714298
3.488756
1.695881
1.627106
T a b l e 3
Average Values of the MLEs, MSE, Variance, IW95 , and IW99
(� � 1.3, � � 0.5, and � � 0.7)
n Parameter MLE MSE Variance IW95 IW99
30 �
�
�
1.193727
0.757161
0.528947
0.326016
0.231624
0.104616
0.132593
0.004154
0.010251
1.427402
0.252650
0.396889
1.878927
0.332570
0.522436
50 �
�
�
1.233921
0.664767
0.590717
0.175032
0.141264
0.070704
0.132593
0.004154
0.010251
1.266506
0.210406
0.276019
1.667135
0.276963
0.363331
75 �
�
�
1.272472
0.568719
0.664158
0.114624
0.081504
0.035064
0.008442
0.001742
0.002948
0.360171
0.163610
0.212838
0.474103
0.215364
0.280165
100 �
�
�
1.318393
0.501048
0.682955
0.053496
0.030888
0.062024
0.001005
0.000670
0.000871
0.124271
0.101467
0.115690
0.163581
0.133563
0.152286
numerical results, the maximum likelihood estimators have good statistical
properties. The estimates of the parameters approach the true values as the sample
size increases. Also, the estimated asymptotic variances of the estimators decrease
as the sample size increases. Moreover, the estimated approximate confidence
intervals for the three parameters are to be narrower when the sample size is
getting to be larger.
Conclusions. In this paper the problem of estimating the generalized
exponential distribution parameters and the acceleration factor in the case of
constant-stress partially accelerated life tests was considered under type-II censoring.
The maximum likelihood method was used to estimate the model parameters using
the Newton–Raphson method. The performance of the estimators was investigated
numerically for different parameter values and different sample sizes. Moreover,
the approximate confidence bounds of the model parameters were obtained at 95
and 99% levels of confidence.
It is concluded that the numerical results support the theoretical findings. That
is, the maximum likelihood estimators are consistent and their asymptotic variances
decrease as the sample size increases. Moreover, the estimated approximate
confidence intervals for the three parameters are to be smaller when the sample
size is getting to be larger. In addition, the confidence intervals obtained at
confidence level �� 0.95 are narrower than those at �� 0.99. Another method,
namely, the LRB method was used to obtain the confidence bounds of the model
parameters when the sample size is small. The simulation results of this method
support the theoretical findings. That is, good estimations were obtained using this
method when the sample size is small. As a future work, this study can be extended
to deal with the problem of estimation using progressively type-II censored data
assuming the same distribution under CSPALT.
Acknowledgements. This project was supported by King Saud University,
Deanship of Scientific Research, College of Science Research Center.
92 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
A. A. Ismail
T a b l e 4
Average Values of the MLEs, MSE, Variance, IW95 , and IW99
(� � 2.2, � � 1.4, and � � 1.1)
n Parameter MLE MSE Variance IW95 IW99
30 �
�
�
3.412658
0.919338
0.797132
0.304116
0.103726
0.171306
0.106191
0.089262
0.033372
1.277409
1.171168
0.716106
1.681487
1.541640
0.942629
50 �
�
�
2.905014
1.083096
0.985296
0.178002
0.044950
0.082088
0.061803
0.048843
0.009639
0.974520
0.866338
0.384859
1.282787
1.140383
0.506601
75 �
�
�
2.303450
1.321138
1.035468
0.113274
0.015686
0.037882
0.021789
0.026487
0.002835
0.578635
0.637973
0.208719
0.761673
0.839781
0.274743
100 �
�
�
2.225172
1.394834
1.067906
0.067518
0.001744
0.009688
0.001134
0.000972
0.001458
0.132006
0.122214
0.149680
0.173763
0.160873
0.197028
Ð å ç þ ì å
Ïðèñêîðåí³ òà ÷àñòêîâî ïðèñêîðåí³ ðåñóðñí³ âèïðîáóâàííÿ ÷àñòî ïðîâîäÿòüñÿ
äëÿ çàáåçïå÷åííÿ íàä³éíîñò³ ñó÷àñíî¿ òåõí³êè. Ìåòîþ òàêèõ âèïðîáóâàíü º çà
êîðîòêèé ïðîì³æîê ÷àñó òà ç íàéìåíøèìè çàòðàòàìè îòðèìàòè ³íôîðìàö³þ
ïðî ðåñóðñ âèðîá³â ³ ìàòåð³àë³â. Âèïðîáóâàííÿ ìîæíà ïðîâîäèòè çà ïîñò³é-
íèõ, ñõ³ä÷àñòèõ, ïðîãðåñóþ÷èõ ³ öèêë³÷íèõ íàâàíòàæåíü, à òàêîæ ïðè íàâàí-
òàæåíí³ âèïàäêîâèì íàïðóæåííÿì. Ðîçãëÿäàºòüñÿ çàäà÷à îö³íêè ïàðàìåòð³â
çàãàëüíîãî åêñïîíåíö³àëüíîãî ðîçïîä³ëó òà êîåô³ö³ºíòà ïðèñêîðåííÿ â óìî-
âàõ ÷àñòêîâî ïðèñêîðåíîãî ðåñóðñíîãî âèïðîáóâàííÿ ç ïîñò³éíîþ íàïðóãîþ.
²ç âèêîðèñòàííÿì ñåðåäíüîêâàäðàòè÷íî¿ ïîõèáêè ïðîâåäåíî ÷èñåëüíå äîñë³ä-
æåííÿ åôåêòèâíîñò³ îö³íêè ìåòîäîì ìàêñèìàëüíî¿ ïðàâäîïîä³áíîñò³ äëÿ ð³ç-
íèõ ðîçì³ð³â çðàçêà òà çíà÷åíü ïàðàìåòðà. Äëÿ ïàðàìåòð³â ìîäåë³ ïîáóäîâàíî
ïðèáëèçí³ äîâ³ð÷³ ãðàíèö³. Ùîá îòðèìàòè äîâ³ð÷³ ãðàíèö³ äëÿ ïàðàìåòðà
ìîäåë³ ó âèïàäêó çðàçêà ìàëåíüêîãî ðîçì³ðó, âèêîðèñòîâóâàëè ìåòîä â³ä-
íîøåííÿ ïðàâäîïîä³áíîñò³. ßê ïðèêëàä ïðîâåäåíî äîñë³äæåííÿ çà äîïîìîãîþ
ìîäåëþâàííÿ. Ïîêàçàíî, ùî ðåçóëüòàòè ìîäåëþâàííÿ çá³ãàþòüñÿ ç äàíèìè
ðîçðàõóíê³â.
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Received 30. 05. 2013
94 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
A. A. Ismail
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| id | nasplib_isofts_kiev_ua-123456789-112673 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T16:18:37Z |
| publishDate | 2013 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Ismail, A.A. 2017-01-25T20:20:22Z 2017-01-25T20:20:22Z 2013 Estimating the Generalized Exponential Distribution Parameters and the Acceleration Factor under Constant-Stress Partially Accelerated Life Testing with Type-II Censoring / A.A. Ismail // Проблемы прочности. — 2013. — № 6. — С. 82-94. — Бібліогр.: 19 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/112673 539.4 Accelerated life testing (ALT) and partially accelerated life testing (PALT) are frequently used in modern reliability engineering. ALT and PALT are run to obtain information on the life of the products and materials in a shorter time and at lower cost. The experimental units are subject to stress conditions that are more severe than those encountered in normal use condition to induce early failures. ALT or PALT can be carried out using constant, step, progressive, cyclic and random stress loadings. This paper considers the problem of estimating the generalized exponential (GE) distribution parameters and the acceleration factor under constant-stress PALT model. The main objective is to derive the maximum likelihood estimators (MLEs) of the parameters of the GE distribution and the acceleration factor when the data are type-II censored from constant-stress PALT. Also, the performance of the MLEs is investigated numerically for different sample sizes and different parameter values using the mean square error. In addition, the approximate confidence intervals of the model parameters are constructed. Moreover, the likelihood ratio bounds (LRB) method is used to obtain confidence bounds of the model parameters when the sample size is small. For illustration, a simulation study is conducted. It is observed that the simulation results support the theoretical findings. Ускоренные и частично ускоренные ресурсные испытания часто проводятся для обеспечения надежности современной техники. Цель таких испытаний – за короткий срок и с меньшими затратами получить информацию о ресурсе изделий и материалов. Испытания можно проводить при постоянных, ступенчатых, прогрессирующих и циклических нагрузках, а также при нагрузке случайным напряжением. Рассматривается задача оценки параметров обобщенного экспоненциального распределения и коэффициента ускорения в условиях частично ускоренного ресурсного испытания с постоянным напряжением. С использованием среднеквадратической погрешности проведено численное исследование эффективности оценки методом максимального правдоподобия для разных размеров образцов и значений параметра. Для параметров модели построены приблизительные доверительные границы. Чтобы получить доверительные границы для параметров модели в случае образца маленького размера, использовали метод отношения правдоподобия. В качестве примера проведено исследование с помощью моделирования. Показано, что результаты моделирования согласуются с данными расчетов. Прискорені та частково прискорені ресурсні випробування часто проводяться для забезпечення надійності сучасної техніки. Метою таких випробувань є за короткий проміжок часу та з найменшими затратами отримати інформацію про ресурс виробів і матеріалів. Випробування можна проводити за постійних, східчастих, прогресуючих і циклічних навантажень, а також при навантаженні випадковим напруженням. Розглядається задача оцінки параметрів загального експоненціального розподілу та коефіцієнта прискорення в умовах частково прискореного ресурсного випробування з постійною напругою. Із використанням середньоквадратичної похибки проведено чисельне дослідження ефективності оцінки методом максимальної правдоподібності для різних розмірів зразка та значень параметра. Для параметрів моделі побудовано приблизні довірчі границі. Щоб отримати довірчі границі для параметра моделі у випадку зразка маленького розміру, використовували метод відношення правдоподібності. Як приклад проведено дослідження за допомогою моделювання. Показано, що результати моделювання збігаються з даними розрахунків. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Estimating the Generalized Exponential Distribution Parameters and the Acceleration Factor under Constant-Stress Partially Accelerated Life Testing with Type-II Censoring Оценка параметров обобщенного экспоненциального распределения и коэффициента ускорения в условиях частично ускоренного ресурсного испытания с постоянным напряжением при цензурировании типа II Article published earlier |
| spellingShingle | Estimating the Generalized Exponential Distribution Parameters and the Acceleration Factor under Constant-Stress Partially Accelerated Life Testing with Type-II Censoring Ismail, A.A. Научно-технический раздел |
| title | Estimating the Generalized Exponential Distribution Parameters and the Acceleration Factor under Constant-Stress Partially Accelerated Life Testing with Type-II Censoring |
| title_alt | Оценка параметров обобщенного экспоненциального распределения и коэффициента ускорения в условиях частично ускоренного ресурсного испытания с постоянным напряжением при цензурировании типа II |
| title_full | Estimating the Generalized Exponential Distribution Parameters and the Acceleration Factor under Constant-Stress Partially Accelerated Life Testing with Type-II Censoring |
| title_fullStr | Estimating the Generalized Exponential Distribution Parameters and the Acceleration Factor under Constant-Stress Partially Accelerated Life Testing with Type-II Censoring |
| title_full_unstemmed | Estimating the Generalized Exponential Distribution Parameters and the Acceleration Factor under Constant-Stress Partially Accelerated Life Testing with Type-II Censoring |
| title_short | Estimating the Generalized Exponential Distribution Parameters and the Acceleration Factor under Constant-Stress Partially Accelerated Life Testing with Type-II Censoring |
| title_sort | estimating the generalized exponential distribution parameters and the acceleration factor under constant-stress partially accelerated life testing with type-ii censoring |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112673 |
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