(12)This subchapter is made up of the following topics:
Confidence Bounds on the Mean Life
The mean life for the T-H exponential distribution 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:
(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 A,
b
and 
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. (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, or:
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, A,
b and are
estimated from the local Fisher Matrix (evaluated at ,
, , ) as follows:
where:
Confidence Bounds on Reliability
The reliability function (ML estimate) for the T-H Weibull model is given by:
or:
Setting:
or:
The reliability function now becomes:
The next step is to find the upper and lower bounds on u:
(14)
(15)
where:
or:
The upper and lower bounds on reliability are:
Where uU and uL are estimated using Eqns. (14) and (15).
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 as follows:
or:
Where 
= ln
.
The upper and lower bounds on u are estimated from:
(16)
(17)
where:
or:
The upper and lower bounds on time are then found by:
where uU and uL are estimated using Eqns. (16) and (17).
Bounds on the Parameters
Since the standard deviation, 
and are
positive parameters, 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
and b, are estimated from:
(upper bound)
(lower bound)
and:
(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, U; 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, U; , , , )
as follows:
or:
The upper and lower bounds are then found by:
Solving for TU and TL yields:
(upper bound)
(lower bound)
See Also:
Temperature-Humidity Relationship
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