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Reliability Basics | |||

Until the release of Version 7, Weibull++ has dealt exclusively with what is commonly
referred to as
This type of analysis is particularly useful if there is a lack of current test data and when there is a strong prior understanding about the parameter of the assumed life model and a distribution can be used to model the parameter. By incorporating prior information about parameter(s), a posterior distribution for the parameter(s) can be obtained and inferences on the model parameters and their functions can be made. This article describes the Bayesian concepts, their applications in life data analysis and the Bayesian-Weibull distribution in Weibull++.
*θ*is a vector of the parameters of the chosen distribution- ς is the range of
*θ* *L(Data|θ)*is the likelihood function based on the chosen distribution and data*φ(θ)*is the prior distribution for each of the parameters
The
integral in Eqn. (1) is often referred to as the It can be seen from Eqn. (1) that there is
a significant difference between classical and Bayesian statistics. First,
the idea of prior information does not exist in classical statistics. All
inferences in classical statistics are based on the sample data. On the
other hand, in the Bayesian framework, prior information constitutes the
basis of the theory. Another difference is in the overall approach of making
inferences and in their interpretation. For example, in Bayesian analysis
the parameters of the distribution to be ''fitted'' are the random
variables. In reality, there is no distribution fitted to the data in the
Bayesian case. For instance, consider the case where a data set is obtained
from a reliability test. Based on prior experience on a similar product, the
analyst believes that the shape parameter of the Weibull distribution has a
value between
and
β,
and wants to utilize this information. This can be achieved by using the
Bayes theorem. At this point, the analyst is automatically forcing the
Weibull distribution as a model for the data with a shape parameter between
_{2}β_{1}
and
βIn this example, the range of
values for the shape parameter is the prior distribution, which in this case
is Uniform. By applying Eqn. (1), the posterior distribution of the shape
parameter will be obtained. So the analyst will end up with a distribution
for the parameter rather than an estimate of the parameter as in classical
statistics._{2}.
The posterior distribution of failure time
where:
is the This model considers prior knowledge on the beta parameter of the Weibull distribution when it is chosen to be fitted on a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the one-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes'' model, unlike the one-parameter Weibull that is commonly referred to as such) offers an alternative to the one-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Initial studies performed by the ReliaSoft R&D group show some very promising results using this model. In this model,
On the other hand, informative priors have a stronger influence on the posterior distribution. The influence of the prior distribution on the posterior is related to the sample size of the data and the form of the prior. Generally speaking, large sample sizes are required to modify strong priors, where weak priors are overwhelmed by even relatively small sample sizes. Informative priors are typically obtained from past data.
The
expected reliability at time where: The expected failure rate at time or: where: is the two-parameter Weibull failure rate function.
Confidence bounds estimations will be discussed in future issues of the | |||

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