In this section, we present the methods used in the application to estimate the different types of confidence bounds for the Weibull distribution. The complete derivations were presented in detail (for a general function) in the Confidence Bounds chapter.

This section includes the following subsections:

Fisher Matrix Confidence Bounds for the Weibull Distribution

Likelihood Ratio Confidence Bounds for the Weibull Distribution

One of the methods used by the application in estimating the different types of confidence bounds for Weibull data, the Fisher matrix method, is presented in this section. The complete derivations were presented in detail (for a general function) in the Confidence Bounds chapter.

This subsection includes the following subsubsections:

Fisher Matrix Confidence Bounds on the Parameters for the Weibull Distribution

Fisher Matrix Confidence Bounds on Reliability for the Weibull Distribution

Fisher Matrix Confidence Bounds on Time for the Weibull Distribution

One of the properties of maximum likelihood estimators is that they
are asymptotically normal, meaning that for large samples they are normally
distributed. Additionally, since both the shape parameter estimate, , and the scale parameter
estimate, , must
be positive, thus ln*β*
and ln*η*
are treated as being normally distributed as well.

The lower and upper bounds on the parameters are estimated from [30]:

(19)

(20)

and:

(21)

(22)

where *K**α*
is defined by:

(23)

If *δ* is the
confidence level, then for the two-sided bounds and
*α* = 1 - *δ*
for the one-sided bounds.

The variances and covariances of and are estimated from the inverse local Fisher matrix, as follows:

(24)

Note that the variance and covariance of the parameters are obtained from the inverse Fisher information matrix as described in this section. The local Fisher information matrix is obtained from the second partials of the likelihood function, by substituting the solved parameter estimates into the particular functions. This method is based on maximum likelihood theory and is derived from the fact that the parameter estimates were computed using maximum likelihood estimation methods. When one uses least squares or regression analysis for the parameter estimates, this methodology is theoretically then not applicable. However, if one assumes that the variance and covariance of the parameters will be similar regardless of the underlying solution method, then the above methodology can also be used in regression analysis. (Note: One also assumes similar properties for both estimators.)

The Fisher matrix is one of the methodologies that Weibull++ uses for both MLE and regression analysis.
Specifically, Weibull++ uses the likelihood function and computes the
local Fisher information matrix based on the estimates of the parameters
and the current data. This gives consistent confidence bounds regardless
of the underlying method of solution, *i.e*.
MLE or regression. In addition, Weibull++ checks this assumption and proceeds
with it if it considers it to be acceptable. In some instances, Weibull++
will prompt you with an "Unable to Compute Confidence Bounds"
message when using regression analysis. This is an indication that these
assumptions were violated.

The bounds on reliability can easily be derived by first looking at the general extreme value distribution (EVD). Its reliability function is given by:

(25)

By transforming *t*
= ln*T*
and converting *ρ*1
= ln*η*,
, the above equation becomes
the Weibull reliability function:

With:

(26)

set *u* = *β*(ln*T* - ln*η*). The reliability function now
becomes:

(27)

The next step is to find the upper and lower bounds on *u*.
Using the equations derived in the Confidence
Bounds chapter, the bounds on *u*
are then estimated from [30]:

(28)

(29)

where:

(30)

or:

(31)

The upper and lower bounds on reliability are:

(32)

(33)

Weibull++ makes the following assumptions/substitutions in Eqn. (26) to Eqn. (31) when using the three-parameter or one-parameter forms:

For the three-parameter case, substitute

*t*= ln(*T*- ) (and by definition*γ*<*T*), instead of ln*T*. (Note that this is an approximation since it eliminates the third parameter and assumes that*Var*() = 0.)-
For the one-parameter,

*Var*() = 0, thus:(34)

Also note that the time axis (x-axis) in the three-parameter Weibull
plot in Weibull++ 7 is not *T*
but *T* - *γ*.
This means that one must be cautious when obtaining confidence bounds
from the plot. If one desires to estimate the confidence bounds on reliability
for a given time *T*0
from the adjusted plotted line, then these bounds should be obtained for
a *T*0
- *γ* entry on
the time axis.

The bounds around the time estimate or reliable life estimate, for a given Weibull percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows [24, 30]:

or:

(35)

where *u*
= ln*T.*

The upper and lower bounds on *u*
are estimated from:

(36)

(37)

where:

or:

(38)

The upper and lower bounds are then found by:

(39)

(40)

As covered in the Confidence Bounds
chapter, the likelihood confidence bounds are calculated by finding
values for *θ*1
and *θ*2
that satisfy:

(41)

This equation can be rewritten as:

(42)

For complete data, the likelihood function for the Weibull distribution is given by:

For a given value of *α*, values
for *β* and *η*
can be found which represent the maximum and minimum values that satisfy
Eqn. (16).
These represent the confidence bounds for the parameters at a confidence
level *δ*, where *α*
= *δ* for two-sided
bounds and *α* = 2*δ*
- 1 for one-sided.

Similarly, the bounds on time and reliability can be found by substituting
the Weibull reliability equation into the likelihood function so that
it is in terms of *β* and time
or reliability, as discussed in the Confidence
Bounds chapter. The likelihood ratio equation used to solve for bounds
on time (Type 1) is:

The likelihood ratio equation used to solve for bounds on reliability (Type 2) is:

Bayesian Bounds use non-informative prior distributions for both parameters.
From the Confidence Bounds chapter,
we know that if the prior distribution of *η*
and *β* are independent,
the posterior joint distribution of *η*
and *β* can be
written as:

The marginal distribution of *η*
is:

where:

is the non-informative prior
of *β*.

is the non-informative prior
of *η*.

Using these non-informative prior distributions, *f*(*η**|Data)*
can be rewritten as:

The one-sided upper bounds of *η*
is:

The one-sided lower bounds of *η*
is:

The two-sided bounds of *η*
is:

Same method is used to obtain the bounds of *β*.

From the Confidence Bounds chapter, we know that:

From the posterior distribution of *η*,
we have:

(43)

Eqn. (43) is solved numerically for *TU*.
The same method can be applied to calculate one sided lower bounds and
two-sided bounds on time.

From the posterior distribution of *η*,
we have:

(44)

Eqn. (44) is solved numerically for *RU*.
The same method can be used to calculate the one sided lower bounds and
two-sided bounds on reliability.

See Also:

The Weibull Distribution

Go
to weibull.com

Go to ReliaSoft.com

©1996-2006. ReliaSoft Corporation. ALL RIGHTS RESERVED.