Eyring Confidence Bounds

This subchapter is divided into the following topics:

Approximate Confidence Bounds for the Eyring Exponential
Approximate Confidence Bounds for the Eyring Weibull
Approximate Confidence Bounds for the Eyring Lognormal

Approximate Confidence Bounds for the Eyring Exponential

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).

Approximate Confidence Bounds for the Eyring Weibull

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).

Approximate Confidence Bounds for the Eyring Lognormal

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)