Calculation of parameter confidence intervals

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Calculation of parameter confidence intervals

Charles Ll
Dear all,

I asked the following question in the Julia-Opt mailing list, but was advised to ask for some help here.

I am trying to estimate the confidence intervals of parameters that have been estimated using the JuMP optimisation framework and the Ipopt solver.

For now, I've been using parametric and non-parametric bootstrapping, in the first case resampling my dataset with a given PDF for the y values or in the second case just resampling my dataset with replacement.

However, for one of my problems, it takes too much time: I have approximately 50 different large datasets to fit with the same procedure, such that bootstrapping starts to consume too much time, and hence, becomes non-realistic to perform.

I would like to have another "quicker" but still robust way to get the confidence interval for my parameters, and I was wondering if anybody has an idea for implementing such function...

From the LsqFit.jl package (, I was thinking about writing something like:

    r = y - y_calc # the residuals

    dof = length(r) - length(parameters) # the degrees of freedom

    mse = sumabs2(r)/dof    # mean square error is: standard sum square error / degrees of freedom

    Q,R = qr(J) # compute the covariance matrix from the QR decomposition

    Rinv = inv(R)

    covar = Rinv*Rinv'*mse

    std_error = sqrt(diag(covar)) # standard errors

If I continue following the example of the LsqFit package:
dist = TDist(fit.dof)
std_error *= quantile(dist, alpha)

Does such code seem good? Could I think about other alternatives, such as Log-likelihood or F-test? Any advise to implement them? I never implemented those tests yet...
Thanks in advance!

Charles LL.

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