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How to solve for Weibull distribution parameters using Excel's Solver?

  1. Nov 2, 2012 #1
    1. The problem statement, all variables and given/known data

    Asked to find Weibull distribution parameters, [itex]\theta[/itex] and [itex]\tau[/itex], using Excel's Solver feature. I know that the 25th percentile is 1500 and the 75th percentile is 250000.

    2. Relevant equations

    Weibull CDF = 1-exp(-(x/[itex]\theta[/itex])^[itex]\tau[/itex])

    3. The attempt at a solution

    I have solved the solution by paper to yeild: [itex]\theta[/itex]=86384.36181, [itex]\tau[/itex]=0.307375855.

    In Excel's Solver:
    Objective cell is the formula: =EXP(-(1500/C2)^C3)-0.75
    Optimize value of: 0
    By Changing Cells: C2 and C3, which are my [itex]\theta[/itex] and [itex]\tau[/itex] values respectively.
    Constraints: cell with formula: EXP(-(250000/C2)^C3)-0.25 constrained to be 0.

    The solver is not able to find the solution. What am I doing wrong?
     
  2. jcsd
  3. Nov 2, 2012 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    The EXCEL Solver can sometimes choke on highly nonlinear problems, because it tries to solve equations using quasi-Newton methods but with derivatives replaced by finite differences; or it may try to perform a sequence of successive quadratic fits, etc. So, your problem has two issues: (i) difficult nonlinearity; and (ii) bad scaling. One of the EXCEL Solver options is an "automatic scaling" setting, but I would not trust it---in fact, it seemed to not work when I tried it; instead, I would scale manually (using percentiles 1.5 and 250 instead of 1500 and 250000). Also, rather than solving the equations, I would suggest minimizing the sum of squared deviations.

    Even so, I could not get it to work until I re-cast the problem: for
    [tex] W(x) = 1 - \exp(-(x/a)^b),[/tex] (using a and b instead of θ and τ) we can solve W(x) = p as x = x(p), where
    [tex] x(p) = a\, \exp\left( \frac{\ln(-\ln(1-p))}{b}\right).[/tex]
    Now minimizing the sum of squares (x(.25) - 1.5)^2 + (x(.75) - 250)^2 seems to work well: it gives a = 86.14852 and b = 0.306587316 (close to what you want, re-scaled). By re-starting the Solver from this point and re-setting some of the parameters, better accuracy can perhaps be obtained; however, there is a fundamental limit to the attainable precision, due to the Solver's use of finite-differences in place of analytical derivatives.

    RGV
     
  4. Nov 4, 2012 #3
    Thank you for replying Ray, I appreciate it.

    Here's how I ended up solving the problem. The Weibull distribution has a VaR(p) formula, which is the "Value at Risk"; essentially it directly translates into the X value of the distribution that corresponds to a percentile for the distribution.

    The formula for the Weibull is: VaR(p) = [itex]\theta[/itex](-LN(1-p))1/[itex]\tau[/itex], where p is the percentile.

    Then I entered this formula into my Solver as the objective using the same methods I prescribed earlier and voila! It worked.

    I hope this helps anyone else that has a similar problem.
     
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