(16)This subchapter is divided into the following topics:
Confidence Bounds on Mean Life
The mean life for the Eyring model is given by Eqn. ( 1) 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:
(16)
(17)
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 A and B are estimated from
the local 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. (16) and (17).
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. (16) and (17).
Bounds on the Parameters
From the asymptotically normal property of the maximum likelihood estimators
and since 
is a positive parameter, ln
() can then be treated as normally distributed.
After performing this transformation, the bounds on the parameters are
estimated from:
Also:
and:
The variances and covariances of β, A
and B are estimated from
the Fisher Matrix (evaluated at , , )
as follows:
Confidence Bounds on Reliability
The reliability function for the Eyring-Weibull (ML estimate) is given by:
or:
Setting:
or:
The reliability function now becomes:
The next step is to find the upper and lower bounds on 
:
(18)
(19)
where:
or:
The upper and lower bounds on reliability are:
where uU and uL are estimated using Eqns (18) and (19).
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:
or:
where = ln
. The upper and lower bounds on are then estimated from:
(20)
(21)
where:
or:
The upper and lower bounds on time are then found by:
where uU and uL are estimated using Eqns. (20) and (21).
Bounds on the Parameters
The lower and upper bounds on A and B are estimated from:
(upper bound)
(lower bound)
and
(upper bound)
(lower bound)
Since the standard deviation, 
is a
positive parameter, ln ()
is treated as normally distributed and the bounds 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 given by:
Let 
(t,
V; A, B,
) = 
,
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 yields:
(upper bound)
(lower bound)
See Also:
Eyring Relationship
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