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:
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:
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 lnβ and lnη are treated as being normally distributed as well.
The lower and upper bounds on the parameters are estimated from :
where Kα is defined by:
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:
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:
By transforming t = lnT and converting ρ1 = lnη, , the above equation becomes the Weibull reliability function:
set u = β(lnT - lnη). The reliability function now becomes:
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 :
The upper and lower bounds on reliability are:
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 lnT. (Note that this is an approximation since it eliminates the third parameter and assumes that Var() = 0.)
For the one-parameter, Var() = 0, thus:
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 T0 from the adjusted plotted line, then these bounds should be obtained for a T0 - 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]:
where u = lnT.
The upper and lower bounds on u are estimated from:
The upper and lower bounds are then found by:
As covered in the Confidence Bounds chapter, the likelihood confidence bounds are calculated by finding values for θ1 and θ2 that satisfy:
This equation can be rewritten as:
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 = 2δ - 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:
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:
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:
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.
The Weibull Distribution
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