```Date: Jan 18, 2013 5:23 AM
Author: Torsten
Subject: Re: Maximum Likelihood Estimation and Confidence Intervals

"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*(((t-mu)/sigma).^2);> > So, after some calculations, the negative log-likelihood function becomes:> > function loglik = gauss_mle_bin(w,t,y,n)> % gauss_mle The log-likelihood 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*(((t-mu)/sigma).^2);> loglik = (-1) * (sum(log(a*exp(h)).*x) + sum(log(1-a*exp(h)).*(n-x)));> > 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 (variance-covariance 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> Finlandhelp nlparcihelp nlpredciBest wishesTorsten.
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