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== Estimation of the Weibull Parameters ==
#REDIRECT [[The Weibull Distribution]]
 
The estimates of the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, either using least squares or maximum likelihood.
 
{{weibull parameters probability plotting}}
 
{{weibull parameters Rank Regression on Y}}
 
{{weibull parameters rank regression on x}}
 
=== Three-Parameter Weibull Regression ===
 
When the MR versus <span class="texhtml">''T''<sub>''j''</sub></span> points plotted on the Weibull probability paper do not fall on a satisfactory straight line and the points fall on a curve,(Note that other shapes, particularly shapes, might suggest the existence of more than one population. In these cases, the multiple population, mixed Weibull distribution, may be more appropriate. Chapter 10 presents the mixed Weibull distribution.) then a location parameter, <span class="texhtml">γ</span>, might exist which may straighten out these points. The goal in this case is to fit a curve, instead of a line, through the data points using nonlinear regression. The Gauss-Newton method can be used to solve for the parameters, <span class="texhtml">β</span>, <span class="texhtml">η</span> and <span class="texhtml">γ</span>, by performing a Taylor series expansion on <span class="texhtml">''F''(''T''<sub>''i''</sub>;β,η,γ)</span>. Then the nonlinear model is approximated with linear terms and ordinary least squares are employed to estimate the parameters. This procedure is iterated until a satisfactory solution is reached. Weibull++ 7 calculates the value of <span class="texhtml">γ</span> by utilizing an optimized Nelder-Mead algorithm, and adjusts the points by this value of <span class="texhtml">γ</span> such that they fall on a straight line, and then plots both the adjusted and the original unadjusted points. To draw a curve through the original unadjusted points, if so desired, select Weibull 3P Line Unadjusted for Gamma from the ''Show Plot Line'' submenu under the ''Plot Options'' menu.  The returned estimations of the parameters are the same when selecting RRX or RRY. To display the unadjusted data points and line along with the adjusted data points and line, select ''Show/Hide Items'' under the ''Plot Options ''menu and include the unadjusted data points and line as follows:
 
[[Image:showhideplotitems.png|thumb|center|300px]]
 
[[Image:showhideplotwindow.png|thumb|center|300px]]
 
The results and the associated graph for the previous example using the three-parameter Weibull case are shown next:
 
[[Image:3parameterweibullplot.png|thumb|center|400px| ]]
<br>
 
=== Maximum Likelihood Estimation ===
 
As outlined in Chapter 3, maximum likelihood estimation works by developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function.  This can be achieved by using iterative methods to determine the parameter estimate values that maximize the likelihood function, but this can be rather difficult and time-consuming, particularly when dealing with the three-parameter distribution.  Another method of finding the parameter estimates involves taking the partial derivatives of the likelihood function with respect to the parameters, setting the resulting equations equal to zero and solving simultaneously to determine the values of the parameter estimates. ( Note that MLE asymptotic properties do not hold when estimating <span class="texhtml">γ</span> using MLE [27].) The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the Weibull distribution are covered in Appendix C.
 
 
====Example 5====
Repeat Example 1 using maximum likelihood estimation.
 
 
=====Solution to Example 5=====
In this case, we have non-grouped data with no suspensions or intervals, i.e. complete data. The equations for the partial derivatives of the log-likelihood function are derived in Appendix C and given next:
::<math> \frac{\partial \Lambda }{\partial \beta }=\frac{6}{\beta } +\sum_{i=1}^{6}\ln \left( \frac{T_{i}}{\eta }\right) -\sum_{i=1}^{6}\left( \frac{T_{i}}{\eta }\right) ^{\beta }\ln \left( \frac{T_{i}}{\eta }\right) =0
</math>
 
:and:
 
::<math> \frac{\partial \Lambda }{\partial \eta }=\frac{-\beta }{\eta }\cdot 6+\frac{ \beta }{\eta }\sum\limits_{i=1}^{6}\left( \frac{T_{i}}{\eta }\right) ^{\beta }=0 </math>
 
Solving the above equations simultaneously we get:
 
::<math> \hat{\beta }=1.933,</math> <math>\hat{\eta }=73.526 </math>
 
<br>
The variance/covariance matrix is found to be,
 
::<math> \left[ \begin{array}{ccc} \hat{Var}\left( \hat{\beta }\right) =0.4211 & \hat{Cov}( \hat{\beta },\hat{\eta })=3.272  \\
 
\hat{Cov}(\hat{\beta },\hat{\eta })=3.272 & \hat{Var} \left( \hat{\eta }\right) =266.646 \end{array} \right] </math>
 
The results and the associated graph using Weibull++ (MLE) are shown next.
 
[[Image:weibullfolio16plot.png|thumb|center|400px| ]]
 
You can view the variance/covariance matrix directly by clicking the ''Quick Calculation Pad ''(QCP) icon
 
[[Image:qcpicon.gif|thumb|center|400px| ]]
 
 
[[Image:qcpfolio16.png|thumb|center|400px| ]]
 
 
<br> Note that the decimal accuracy displayed and used is based on your individual User Setup.

Latest revision as of 09:07, 3 August 2012

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