Arrhenius Confidence Bounds

This subchapter is made up of the following topics:

Approximate Confidence Bounds for the Arrhenius-Exponential
Approximate Confidence Bounds for the Arrhenius Weibull
Approximate Confidence Bounds for the Arrhenius-Lognormal

Approximate Confidence Bounds for the Arrhenius Exponential

There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher Matrix Bounds.

Confidence Bounds on the Mean Life

The Arrhenius-exponential distribution is given by Eqn. (1) by setting m = L(V) as shown in Eqn. (6). The upper mU and lower mL bounds on the mean life are then estimated by:

(19)

(20)

where Kα is defined by:

(21)

If is the confidence level (i.e. 95% = 0.95), then α = for the two-sided bounds, and α = 1 - for the one-sided bounds. The variance of is given by:

or:

(22)

The variances and covariance of B and C are estimated from the local Fisher Matrix (evaluated at , ) as follows:

(23)

Confidence Bounds on Reliability

The bounds on reliability for any given time, T, are estimated by:

where mU and mL are estimated using Eqns. (19) and (20).

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 then estimated from:

where mU and mL are estimated using Eqns. (19) and (20).

Approximate Confidence Bounds for the Arrhenius Weibull

Bounds on the Parameters

From the asymptotically normal property of the maximum likelihood estimators, and since , and are positive parameters, ln () and ln () can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

Also:

and:

The variances and covariances of β, B and C are estimated from the local Fisher Matrix (evaluated at , , , as follows:

Confidence Bounds on Reliability

The reliability function for the Arrhenius-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 :

(24)

(25)

where:

or:

The upper and lower bounds on reliability are:

where uU and uL are estimated from Eqns. (24) and (25).

Confidence Bounds on Time

The bounds on 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 u are estimated from:

(26)

(27)

where:

or:

The upper and lower bounds on time can then found by:

where uU and uL are estimated using Eqns. (26) and (27).

Approximate Confidence Bounds for the Arrhenius-Lognormal

Bounds on the Parameters

The lower and upper bounds on B are estimated from:

Since the standard deviation, and the parameter C are positive parameters, ln() and ln(C) are treated as normally distributed. The bounds are estimated from:

and:

The variances and covariances of B, C and are estimated from the local Fisher Matrix (evaluated at , , , as follows:

Bounds on Reliability

The reliability of the lognormal distribution is:

Let (t, V; B, C, ) = , 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:

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: