INTRODUCTION
In Accelerated Life Tests (ALT), products are exposed to higher stress
conditions (e.g., higher voltage, pressure, or temperature) to produce
failures earlier than at typical conditions.
Acceleration in lab tests is justified because products are expected
to survive a considerable length of time (e.g., months, years, decades)
under typical operating conditions. A model is then fit to the data collected
and the results are used to estimate quantities of interest, e.g., quantiles
and hazard rates at use conditions through extrapolation. The first important
step in accelerated life testing is to determine a test plan given constraints
such as maximum test duration and test unit availability. In general,
this involves specifying at what levels to test (e.g. temperature settings)
and how many units to test at each level.
The inference procedures (or models) are classified into three types:
statistics based models. Physicsstatistics based models and physicsexperimental
based models (Pham, 2003). Furthermore, the statisticsbased models are
classified into two categories parametric and nonparametric models. Parametric
models that are based on AFT assumption, assume that the failure time
data follow a distribution such as exponential or Weibull. It is also
assumed that the failure times at different stress levels are linearly
related to each other. Moreover, the failure time distribution at stress
level s_{1} is expected to be the same at different stress levels
s_{2},s_{3},... as well as under normal operating conditions.
In other words, the shape parameters of the distributions are the same
for all stress levels (including normal conditions), but the scale parameters
may be different. Thus, the relationship between the operating conditions
and stress conditions is:
where, λ(t,z) is the hazard function at time t and stress vector
Z, λ_{0}(t) is the baseline hazard rate function; and β
is the coefficient of stress covariate Z (Sarhan, 2007).
In the nonparametric models there is no assumption of the failure time
distribution, i.e. no predetermined failure time distribution is required.
Cox (1972) proportional hazards (PH) model is the most widely used among
the nonparametric models. It is expressed as:
where, z = (z_{1},z_{2},Y,z_{p})` is a column
vector of the covariates (or applied stresses), β = (β_{1}β_{2}...β_{p})is
a column vector of the unknown coefficients and λ_{0}(t)
is the baseline hazard rate function.
The PH model implies that λ(t; z_{1}) is directly proportional
to λ(t; z_{2}). This is the socalled PH model`s hazard rate
proportionality assumption.
If the proportionality assumption is violated and there are one or more
covariates that totally occur on q levels, a simple extension of the PH
model is stratification (Kalbfleisch and Prentice, 2002), as given below:
A partial likelihood function can be obtained for each of the q strata
and β is estimated by maximizing the multiplication of the partial
likelihood of the q strata. The baseline hazard rate λ_{0j}
(t), estimated as before, is completely unrelated among the strata. This
model is most useful when the covariate is categorical and of no direct
interest.
Another extension of the PH model includes the use of timedependent
covariates. Kalbfleisch and Prentice (2002) classified the timedependent
covariates as internal and external. An internal covariate is the output
of a stochastic process generated by the unit under study and can be observed
as long as the unit survives and is uncensored. An external covariate
has a fixed value or is defined in advance for each unit under study.
Many other extensions exist in the literature (Pham, 2003). However,
one of the most generalized extensions is the extended linear hazard regression
model proposed by Elsayed et al. (2006). Both accelerated failure
time model and PH model are indeed special cases of the generalized model,
whose hazard function is:
where, λ(t; z) is the hazard rate at time t and stress vector z;
λ_{0}($) i s the baseline hazard function; and β_{0},
β_{1}, α_{0}, α_{1} are constants.
This model has been validated through extensive experimentation and simulation
testing (Pham, 2003).
All the above models are based on failurerates proportionality and failuretimes
proportionality. Oakes and Dasu (1990) originally proposed the concept
of the proportional mean residual life (PMRL) by analogy with Cox (1972)
PH model. The concept of MRL is based on conditional expectations and
has been of much interest in actuarial science, survival studies and reliability
theory. In the last two decades, however, reliability engineers, statisticians
and others have shown increasing interest in the MRL and derived many
useful results.
Two distributions with reliability functions R_{0} and R are
said to have proportional MRL functions if, for some θ > 0, e(t)
= θe_{0}(t), ∀ t where, e(t) is the mean residual
life under accelerated conditions, θ is a constant and e_{0}(t)
is the mean residual life under normal conditions.
In reliability studies of repair and replacement strategies, the MRL
function may be more relevant than the hazard function or failure time
distribution function, although they are mathematically equivalent, that
is by knowing one, the other can be determined:
However, for modeling purposes, MRL and the hazard function play slightly
different roles. The former summarizes the entire residual life distribution
of time t and the latter only the risk of immediate failure at time t.
It is possible for the MRL function to exist, but for the failure rate
function not to exist (e.g., the standard Cantor ternary function), although
sometimes it is possible for the failure rate function to exist but the
MRL function not to exist. For example,
When, the main concern is the risk of immediate failure, the proportional
hazards model has proved to be extremely useful. However, it may not be
the most appropriate model when one is concerned with the remaining lifetime
of an individual at time t. In this case, when the ALT model is developed
based on the PMRL, it will provide a useful alternative to the standard
models for ALT, which includes the accelerated failure model and the proportional
hazards.
Let X ≥ 0 be a continuous random variable with reliability function
R(x) and finite expectation μ. The MRL is defined to be:
Gupta and Kirmani (1998) presented details of the implications of the
PMRL and Zhao and Elsayed (2005a, b) modeled an accelerated life test
based on mean residual life.
Chen et al. (2005) presented a semiparametric estimation of the
Proportional Mean Residual Life model in the presence of censoring. In
this study, an approach to ALT planning when product failure is caused
by two or more failure modes is presented.
The problem of multiple failure modes encompasses the study of any failure
process in which there is more than one distinct cause or type of failure
(Balasooriya and Low, 2004; Zhang and Elsayed, 2006; Kundu and Sarhan,
2006). This is very important and should be considered in reliability
studies. The problem may be described as follows:
For a given unit, let T_{i} be a random variable with cumulative
distribution function F_{i}(t), i ={1,2,..., k}, where k is the
number of failure modes or types. It is assumed that the T_{i}
`s are not observable and only the time of failure, T ≥ 0, T = min(T_{1},T_{2}
... , T_{k}) and the cause of the failure, j, among a finite set
of possible causes, say j _ {1,2, ... , k}, which may be censored may
be observed. A regression vector z may also be available to record characteristics
of the under study subject. Some components of z may be time dependent,
that is z = z(t). For the accelerated life testing problem with multiple
failure modes, a set of covariates z can also be observed to reflect the
characteristics of the tested components, for example, the environmental
conditions that the components experienced in the test.
However, existing reliability methods for multiple failure mode problems deal
only with products operating at normal conditions subject to hard (catastrophic)
failures, which imply abrupt and complete cessation of the product`s function.
NONPARAMETRIC APPROACHES WITH MULTIPLE FAILURE MODES PROBLEM
The analysis of failure data with applied stresses at accelerated conditions
often involves complex and not wellknown shapes of time to failure distributions.
To avoid making additional assumptions that would be difficult to test,
nonparametric regression models appear to be more attractive than the
parametric ones which assume a specific distribution. Further contributions
on the subject are given in Kalbfleisch and Prentice (2002) and Park (2005).
Consider inference on the relationship between causespecific hazard
functions and regression vectors or function z. For example, proportional
hazards modeling in which the causespecific hazard function at time t
depends on z only in terms of the concurrent value z(t) is:
Both the shape functions λ_{0j} and regression coefficients
β_{j} have been permitted to vary arbitrarily over the m
failure types.
Let t_{ji} <... < t_{ikj} denote the time
of k_{j} failures of type j, j = 1, Y, m and let Z_{ji}
be the regression function for the individual that fails at t_{ji}.
The method of partial likelihood then gives:
Estimation and comparison of the β_{j}`s can be conducted
by applying standard asymptotic likelihood techniques individually to
the m factors. The functions R_{j}(t; z) can be estimated at specified
Z upon inserting the maximum likelihood estimators from the above Equation.
The corresponding estimators of the cumulative incidence can be obtained
simply by inserting the appropriate estimators for R and λ_{i}
functions.
The causespecific hazard functions could similarly be modeled using
an accelerated failure time model:
It would be necessary to restrict the covariate to be fixed or a step
function in order to preserve the multiplicative relationship between
covariates and failure time.
PMRL MODEL AND STATISTICAL INFERENCE FOR MULTIPLE FAILURE MODES
Two distributions with reliability functions R_{0} and R and
with mean residual lives at time x of e(x) and e_{0}(x), respectively,
are said to have proportional MRL functions, if they are related as follows:
therefore:
Where:
μ_{0} = e_{0}(0)
It is assumed that there are more than one independent failure modes.
That is, time to failure is viewed as the minimum of p latent failure
times. Here only two failure modes are considered. The procedure is similar
for the cases with p failure modes.
Consider N components subjected to accelerated life testing with an applied
stress z. The test is conducted until all components fail, i.e., no censoring.
A component may fail in two failure modes: failure in mode 1 and failure
in mode 2.
The collected data from the test units with two failure modes have three
features:
• 
Failure mode δ_{1}, δ_{2} 
• 
Failure times T_{1}, T_{2} respectively 
U = min (T_{1}, T_{2}) 
(14) 
(U,δ) are observable.
T_{1} and T_{2} are independent
then
and
but
From the assumptions, the likelihood function can be represented as:
and
where, lnL is a function of λ_{1}(•), λ_{2}(•).
We can maximize lnL to estimate λ_{1}(•), λ_{2}(•).
Suppose λ_{1}(•), λ_{2}(•) can be
represented by the PMRL model separately:
and
Where:
Substitute the specific form of e_{10}, e_{20} into the
log likelihood function and consider censoring, then we have:
Let`s only consider the scenario that there is no censoring, thus the
log likelihood function is as follows:
SIMULATION RESULTS
MATLAB was used for computer simulations to demonstrate the use of the
proportional mean residual life model in modeling the failure times obtained
from accelerated life testing with multiple failure modes in order to
show its applicability in the reliability field.
The exponential function was used as the baseline function of the PMRL
model in which there are 6 unknown parameters, a_{1}, a_{2},
b_{1}, b_{2}, β_{1}, β_{2} and
the Baseline exponential Function is:
substituting (25) in the PMRL model, we obtain:
In order to perform the simulation, four groups of weibull distribution
data with two failure modes and two stress levels were generated.
Table 1: 
Estimated parameters of the PMRL model 


Fig. 1: 
Generated data 
Then
failure times for two stress levels were extracted as shown in Fig.
1. The MATLAB optimization toolbox was used for maximizing lnL. Maximizing
lnL is equivalent to minimizing (lnL). The selection of the initial value
is very important. For the exponential baseline mean residual life case,
the generated data of failure modes 1 and 2 was fitted with the Weibull
model separately, then let the initial value of the parameter a_{1}
and a_{2} in our model,
a_{i} = ln( MTTF_{i}) 
(27) 
β_{1 }, β_{2} equal to zero in present model.
After generating the failure time data, the PH, the KaplanMeier and
the PMRL models were fitted to the data. The estimated parameters are
shown in Table 1 based on which this model can be used
to estimate the reliability at a specific stress level. Figure
2 shows the reliability estimation at stress level 2 for the PMRL
model, PH model and the true model.
Figure 3 shows the probability distribution function
(pdf) of the true model and the PMRL model.
In Fig. 4, the PMRL model and the PH models are compared
with the KapalnMeier model and Fig. 5 shows all models
at stress level 1.

Fig. 2: 
Reliability estimation at stress level 2 for the PMRL
Model, the PH Model and the true Model 

Fig. 3: 
Pdf of the true and PMRL model 

Fig. 4: 
Reliability estimation at stress level 2 for the PMRL
model, the PH model and the K and M model 
Figure 5 shows that the PMRL model has better results
in modeling the data obtained from accelerated test with multiple failure
modes.
Table 2 shows the sum of squared error (SSE) between
PMRL estimates, true data and results obtained from K and M model at stress
level 1 and Table 3 shows the same statistical measures
at stress level 2. Based on the results shown in Table 2
and 3, the PMRL model gives very promising results,
where the sum of squared error between PH estimates and the observed data
is 0.7141 and the same statistical measure for PMRL model is only 0.5037
at stress level 1 and the sum of squared errors at stress level 2 for
the PH model and PMRL are 0.3036 and 0.1925.

Fig. 5: 
Reliability estimation at stress level 1 for the PMRL
model, the PH model, the true model and the K and M model 
Table 2: 
SSE for models at stress 1 

Table 3: 
SSE for models at stress 2 

Table 4: 
Goodness of fit for model 

Therefore, the PMRL model
is a useful alternative model. In some cases, it is even better than the
PH model.
MODEL VERIFICATION
In order to compare the proposed model with the PH model, 120 data sets
distributed as Weibull were generated each of which included four groups
of Weibull distribution data with different stress levels. After generating
the failure data, the PH model and the PMRL model were fitted to the data
and the sum of squared error between model estimates and the observed
data was calculated, as shown in Table 4.
Based on this Table 4 are 77 data sets in which the
PMRL model provides a better estimate than the PH model. These results
show that the PMRL model provides a useful alternative to the PH model
and gives very promising results.
Similar with the PH model, the PMRL model implies that the ratio of the
MRL functions for any two units associated with different vectors of covariates,
z_{1} and z_{2}, is constant with respect to time. This
means that e(t; z_{1}) is directly proportional to e(t; z_{2}).
The PMRL model is a valid model to analyze ALT data only when the data
satisfy its proportional mean residual life assumption. Therefore, it
is very important to check the validity of the PMRL model and the assumption
of the covariates` multiplicative effect before applying it to the failure
data. Weibull distribution was used in this research to generate data
to satisfy this assumption.
CONCLUSIONS
In this study, a new accelerated life testing based on Proportional Mean
Residual Life model in the presence of more than one failure mode is presented
and its applicability in reliability is shown. The model utilizes the
data at accelerated conditions to estimate the reliability measures at
normal operating conditions.
Moreover, the use of the PMRL that is modeled is shown in modeling ALT`s
with multiple failure modes, where exponential function is used to present
the baseline mean residual function. The average square error between
the PMRL model based on this baseline function and the true data is very
small. This means that the model can provide accurate reliability estimates
for multiple failure mode problems and is a useful alternative to the
accelerated failure time (AFT) and the proportional hazards (PH) models.
In some cases, this model is even better than the PH model.