.
Then, in general, the expected value of G()
can be found by:This section presents an overview of the theory on obtaining approximate confidence bounds on suspended (multiply censored) data. The methodology used is the so-called Fisher Matrix Bounds, described in Nelson [ 30] and Lloyd & Lipow [ 25].
Approximate Estimates of the Mean and Variance of a Function
Single Parameter Case
For simplicity, consider a one-parameter distribution represented by
a general function G, which is a function
of one parameter estimator, say (G).
Then, in general, the expected value of G()
can be found by:
(7)
where G(
)
is some function of , such as the reliability
function, and is the population
moment or parameter such that E() = as n 
. The term O (
)
is a function of n, the sample size,
and tends to zero, as fast as
as n
. For example, in the case
of = 
and G(x) = 
,
then E(G()) = 
+ O(
)
where O () = 
, thus as n
, E(G())
= (μ and
σ are
the mean and standard deviation, respectively). Using the same one parameter
distribution, the variance of the function G()
can then be estimated by:
(8)
Two Parameter Case
Repeating the previous method for the case of a two parameter distribution,
it is generally true that for a function G, which is a function
of two parameter estimators, say G(
,
), that,
(9)
and:
(10)
Note that the derivatives of Eqn. (10) are evaluated at
= 
and
= , where E() 
and E() 
.
Variance and Covariance Determination of the Parameters
The determination of the variance and covariance of the parameters is accomplished via the use of the Fisher information matrix. For a two-parameter distribution, and using maximum likelihood estimates, the log likelihood function for censored data (without the constant coefficient) is given by:
Then the Fisher information matrix is given by:
where = 
and = 
.
So for a sample of N units where R units have failed, S have been suspended and P have failed within a time interval, and N = R + M + P, one could obtain the sample local information matrix by:
(11)
By substituting in the values of the estimated parameters, in this case
and ,
and inverting the matrix, one can then obtain the local estimate of the
covariance matrix or:
(12)
Then the variance of a function (Var(G)) can be estimated using Eqn. ( 10). Values for the variance and covariance of the parameters are obtained from Eqn. (12).
Once they are obtained, the approximate confidence bounds on the function are given as:
(13)
Approximate Confidence Intervals on the Parameters
In general, MLE estimates of the parameters are asymptotically normal,
thus if
is the MLE estimator for , in
the case of a single parameter distribution, estimated from a sample of
n
units, and if:
then:
(14)
for large n. If one now wishes
to place confidence bounds on ,
at some confidence level 
, bounded
by the two end points C1 and C2,
and where:
then from Eqn. (14):
where Kα is defined by:
Now by simplifying Eqn. (15), one can obtain the approximate confidence
bounds on the parameter at
a confidence level 
or:
If must be positive, then
ln is treated
as normally distributed. The two-sided approximate confidence bounds on
the parameter , at confidence
level , then become:
(two-sided upper) (16)
(two-sided lower) (17)
The one-sided approximate confidence bounds on the parameter ,
at confidence level can be found from:
(one-sided upper)
(one-sided lower)
The same procedure can be repeated for the case of a two or more parameter distribution. Lloyd and Lipow [ 24] elaborate on this procedure.
Percentile Confidence Bounds (Type 1 in ALTA)
Percentile confidence bounds are confidence bounds around time. For example, when using the 1-parameter exponential distribution, the corresponding time for a given exponential percentile (i.e. y-ordinate or unreliability, Q = 1 - R) is determined by solving the unreliability function for the time, T, or:
(18)
Percentile bounds (Type 1) return the confidence bounds by determining
the confidence intervals around 
and
substituting into Eqn. (18). The bounds on
were determined using Eqns. ( 16)
and ( 17), with its variance obtained
from Eqn. ( 12).
Reliability Confidence Bounds (Type 2 in ALTA)
Type 2 bounds in ALTA are confidence bounds around reliability. For example, when using the 1-parameter exponential distribution, the reliability function is:
(19)
Reliability bounds (Type 2) return the confidence bounds by determining
the confidence intervals around and
substituting into Eqn. (19). The bounds on
were determined using Eqns. ( 16)
and ( 17), with its variance obtained
from Eqn. ( 12).
See Also:
Appendix A: Brief Statistical Background
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