(12)This subchapter is made up of the following topics:
Confidence Bounds on the Mean Life
From the inverse power law relationship the mean life for the exponential distribution is given by Eqn. ( 30) by setting m = L(V). The upper (mU) and lower (mL) bounds on the mean life (ML estimate of the mean life) are estimated by:
(12)
(13)
where Kα is defined by:
If 
is the confidence level,
then α = 
for
the two-sided bounds and α
= 1- for the one-sided bounds.
The variance of 
is given
by:
or:
The variances and covariance of K
and n are estimated from the
Fisher Matrix (evaluated at 
,
) as follows:
Confidence Bounds on Reliability
The bounds on reliability at a given time, T, are estimated by:
where mU and mL are estimated using Eqns. ( 12) and ( 13).
Confidence Bounds on Time
The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:
The corresponding confidence bounds are estimated from:
where mU and mL are estimated using Eqns. ( 12) and ( 13).
Bounds on the Parameters
Using the same approach as previously discussed (
and positive parameters):
and:
The variances and covariances of β, K
and n are estimated from
the local Fisher Matrix (evaluated at ,
, ) as
follows:
Confidence Bounds on Reliability
The reliability function (ML estimate) for the IPL-Weibull model is given by:
or:
Setting:
or:
(14)
The reliability function now becomes:
The next step is to find the upper and lower bounds on 
:
(15)
(16)
where:
or:
The upper and lower bounds on reliability are:
where uU and uL are estimated using Eqns. (15) and (16).
Confidence Bounds on Time
The bounds on time for a given reliability (ML estimate of time) are estimated by first solving the reliability function with respect to time:
or:
where = ln
. The upper and lower bounds on u are estimated from:
(17)
(18)
where:
or:
The upper and lower bounds on time are then found by:
where uU and uL are estimated using Eqns. (17) and (18).
Bounds on the Parameters
Since the standard deviation, 
and
are positive parameters, then ln ()
and ln ()
are treated as normally distributed, and the bounds are estimated from:
(upper bound)
(lower bound)
and:
(upper bound)
(lower bound)
The lower and upper bounds on n, are estimated from:
(upper bound)
(lower bound)
The variances and covariances of A, B and 
are estimated from the local Fisher Matrix (evaluated at 
,
, ), as
follows:
where:
Bounds on Reliability
The reliability of the lognormal distribution is:
Let 
(t,
V; K, n, 
)
= 
, then = 
.
For t = 
,
= 
and for t = 
,
= .
The above equation then becomes:
The bounds on z are estimated from:
where:
or:
The upper and lower bounds on reliability are:
(upper bound)
(lower bound)
Confidence Bounds on Time
The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:
where:
and:
The next step is to calculate the variance of (V; , ,
):
or:
The upper and lower bounds are then found by:
Solving for TU and TL we get:
(upper bound)
(lower bound)
See Also:
Inverse Power Law Relationship
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