If the observation is an exact failure time, it contributes to the overall
negative loglikelihood by being transformed to a value of z and then z
is evaluated using the negative log of the gumbel pdf.
If the observation is a right censored removal time, it contributes
to the overall negative loglikelihood by being transformed to a value of
z and then z is evaluated using the negative log of the gumbel survival
function.
JMP's nonlinear fit platform will perform a complex search for optimal values of b0, b1, b2, and sigma that minimize the sum of all the observational summands of the negative loglikelihood of logdata.
To use JMP to find the maximum likelihood estimators (mle's) of this loss functions we will have to create a loss column. We will build a loss formula using the JMP calculator.
One can also see JMP's User's Guide p76-78 for assistance on column creation, and chapter 5 of the same manual for assistance on formula creation using the JMP calculator.
The finished loss function should look like the following:
Create a new column by clicking on "Cols" in the menu bar
Then click on "New Columns..."
Then name the column GumbelLoss, and change the data source to Formula:
The calculator window will appear, and now the gumbel loss function can be created.
First click on "Conditions" in the top middle column of choices, then click on "Assignments" in the top right column:
Next click on "If"
Point the mouse to the square box to the left of the "<<=" symbol
and click. Then click on the 
button to the right of the "Evaluate" button.
Now the parameters need to be identified, and given initial values.
Click on "Parameters" in the top middle column. Note: one will
need to scroll down the list.
Click on "New Parameter" in the top right column.
A parameter box will appear, name the first parameter, b0, and give
it an inital value of 84.45524. For all initial values use estimates
calculated from the Survival fit platform. See tutorial, "Weibull
Regression with voltage and temperature stresses" for estimates.
Click on OK, and you now have a usable parameter.
Repeat these steps to create parameters, b1, b2, and sigma. The initial value for b1 should be -20.0937. The initial value for b2 should be 0.3257. The initial value for sigma should be 2.3326.
Once all parameters are created the parameter list should look like:
Next lets create the variable z.
In the top middle column, click on "Variables" which is located right
below "Parameters". Then click on "New Variable"
A Variable window will appear. Name the new variable, z, and click
on OK.
Next create another variable mu just as you create the variable, z.
Now all parameters and variables are ready for placement in the loss function.
The finished loss function should look like the following:
Note: When you need a parameter, go to "Parameters" and click
on the needed parameter in the list.
Note: When you need a variable, go to "Variables" and click on
the needed variable in the list.
Note: When you create the temperature transformation as pictured
above,
Note: When you enter the z transformation,
- click on a parentheses
- click on division and place the cursor in the top then enter 11605
- place the cursor in the botton and click on parentheses
- click on Temperature and click on addition then enter 273.
Click here to return to tutorial2 "Using JMP's nonlinear fit platform to do weibull regression with voltage and temperature stresses."
- Enter the ln(voltage)-mu
- click on the substraction symbol to highlight the whole expression
- click on the division symbol
- click on the sigma parameter in the parameter list