This is just an Excerpt from a larger document, click here to view the entire document.Putting the Problem in Perspective
Assume that we need to estimate the reliability of a device, R(T), for a Mission Time T, based on some life data (X1, ..., Xn). First, consider that the distribution of the life of a device (times to failure) is Weibull (Figure 1) and then that it is Exponential (Figure 2), having the same mean = 10. Figures 1 and 2 were obtained from 5000 data points from each of these two distributions. The Weibull, in addition, has shape parameter β = 1.23 and scale parameter α = 11.
The descriptive statistics for these 5000 data points are shown in Table 1. Notice how the two means are 10. The two distributions differ mainly in that Weibull clusters about the mean and is therefore, less variable than the Exponential (contrast the StDev values).
Table 1. Descriptive Statistics for the Data Sets
Variable
N
Mean
Median
StDev
Min
Max
Q1
Q3
W(11,1.23)
5000
10.106
7.936
8.338
0.010
77.834
3.875
14.010
Expon(10)
5000
9.996
6.868
10.174
0.001
92.951
2.736
13.933
There are some practical connotations of belonging to one of these two distributions. The Weibull distribution with shape parameter larger than unity (β > 1) characterizes a life that deteriorates with time, i.e., device lives whose failure rate increases with time (reliability decay). On the other hand, when the shape parameter is unity (β = 1), Weibull becomes an Exponential distribution. Hence, the device failure rate is constant and there is no reliability growth or decay. Finally, if the shape parameter is smaller than unity (β < 1), there is reliability growth because the failure rate of the device decreases with time.
Thus, a point estimator based on the life data is obtained by calculating such reliability according to some "formula." However, reliability is defined as the probability that a device life X outlasts the device mission time T (formally, R(T) = P{X > T}). As a result, the assumption of a specific statistical distribution for the device life determines which "formula" we use, as well as which parameters it includes.
For example, assume the data are distributed as a Weibull, with shape parameter β and scale parameter α. Then, the "formula" of the Weibull reliability point estimator is:
R(T) = P{X > T} = Exp{-(T/α)β}
However, if the data are assumed Exponential, with mean , the Exponential reliability estimator becomes:
R(T) = P{X > T} = Exp{-T/θ}
Because the two distributions are different the two reliability estimations will differ (they have different formulas and parameters) except when the shape parameter β = 1 and the Weibull distribution becomes an Exponential.
For example, if the required Mission Time is T = 3 and the parameters are known and equal to α = 11, β = 1.23 and θ = 10, the two respective reliabilities are as follows:
If the true distribution of lives were Weibull (11,1.23):
R(T) = Exp{-(T/)} = Exp{-(3/11)1.23} = 0.81
If the true distribution of lives were Exponential (10):
R(T) = Exp{-T/} = Exp(-3/10) = 0.74
The difference between the two reliabilities is close to 10%! Thus, it is very importance to assess (via the sample data) whether or not that our distribution assumption is correct.
Finally, the problem becomes yet more complex when the distribution parameters are unknown. For then we also need to estimate these parameters from the samples and the uncertainty increases even more.
