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Torsten
Posts:
1,515
Registered:
11/8/10


Re: Maximum Likelihood Estimation and Confidence Intervals
Posted:
Jan 18, 2013 5:23 AM


"Topi Kaaresoja" wrote in message <kdb1a9$75m$1@newscl01ah.mathworks.com>... > Hi, > > I have conducted an experiment where users needed to judge where two events were simultaneous or not. The time (milliseconds) between the two events were varied like this: > > t=[0, 10, 20, 30, 50, 70, 100, 150, 300]; > > The question was repeated n=96 times for each value of t. The proportion of "YES" answers is in a response vector > > y=[0.875 0.89583 0.89583 0.86458 0.69792 0.63542 0.34375 0.1875 0]; > x=n*y; > > I wanted to apply a Gaussian model to that data using maximum likelihood estimation (MLE) similarly than in an MLE Tutorial (http://people.physics.anu.edu.au/~tas110/Teaching/Lectures/L3/Material/Myung03.pdf). > > So the PDF of one x is binomial distribution, and my model is > a*exp(.5*(((tmu)/sigma).^2); > > So, after some calculations, the negative loglikelihood function becomes: > > function loglik = gauss_mle_bin(w,t,y,n) > % gauss_mle The loglikelihood function of the gaussian model > % based on binary distribution on separate responses (Myung MLE tutorial) > % t = vector or m latencies > % y = vector of m proportion of ?simultaneous? responses > % Estimates: w(1) = mu, w(2) = sigma, w(3) = a > x=y*n; > mu = w(1); > sigma = w(2); > a = w(3); > h = .5*(((tmu)/sigma).^2); > loglik = (1) * (sum(log(a*exp(h)).*x) + sum(log(1a*exp(h)).*(nx))); > > I get nice estimates for mu, sigma and a with fminsearch or fminunc. > 2.4895 77.6230 0.8956 > > My problem is: How to get confidence intervals for these parameters? Function mle does them automatically, but I have no idea how to pass my data to mle, because my data is not equally spaced. > > I know that I can get Hessian matrix from fminunc and by computing an inverse (variancecovariance matrix) and taking square root I will get approximations of standard errors for the parameters. From them I can calculate the 95% confidence intervals as follows: > > hessian = hessian returned by fminunc > varcov = inv(hessian); > stderrs = sqrt(diag(varcov)); > c= stderrs*sqrt(chi2inv(0.95,1)); > > I would like to double check if these are correct with mle function (or by other means) if possible. > > Best, > > Topi Kaaresoja > Nokia Research Center > Finland
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Best wishes Torsten.



