My Math Forum Unbiased bootstrap confidence interval for fitted parameters

 August 10th, 2012, 01:19 AM #1 Newbie   Joined: Aug 2012 Posts: 1 Thanks: 0 Unbiased bootstrap confidence interval for fitted parameters Hello, I was wondering if anyone could explain me the odd behavior (in my eyes) of the confidence interval for the unbiased Non-Parametric bootstrap (see http://projecteuclid.org/DPubS/Reposito ... 1177013815 table 6, the bias corrected bootstrap). If you are bored of text, just read what is marked in red. The problem I'm solving: I am trying to find the confidence interval for the two (best) fitted parameters (of a highly non-linear function). (These best fitted parameters fit quite nicely!) For the bootstrap method I use Non-Parametric re-sampling (I.E. randomly re-sample the residues that were obtained by the best fit as the # of residues are too low to fit a certain error density curve through them), add these 'new errors' to the original data-set and compute the best fit (about 3000 times). From this I get a nicely normal looking distribution when all parameters are put in histograms BUT! there is a big bias for one of the two parameters (mean of bootstrap parameters vs original best fit), a 'big' bias means that ~90% of the bootstrap values are larger than the parameter found by the original fit. So naturally I would like to correct for this bias and thus use bias correction. [color=#FF0000]Problem(?): Upon application of the bias correction, the best fitted parameter falls outside the CI, and very very unsymmetrical![/color] My question: What is the cause of this? - Is it a case of bad implementation? (Likely) - Should this be a known flaw for the bias correction (unlikely?) - Is the CI correct but my brain capacity is to low to understand it? (Maybe) -Could this be caused by some unknown (at least to me) condition that says that for my case the bootstrap is not applicable? (Unlikely) - Other? Repeatability?, Try for your self! I found that it is easy to simulate the problem as followed (at least in Matlab or similar programs): 1: Assume you have a normal distribution that represents the found parameters of your bootstrap (with a certain mu,sigma). 2: Assume your found parameter lies on the 10th or 90th percentile of this distribution (the further away, the more clear the problem) 3: Try to find the confidence interval for this parameter using the distribution of 1! In Matlab it looks as followed: N = 100000; %the so called number of bootstrap runs mu = 2; %the bootstrap distribution mean sigma = 1; Value = mu-2*sigma; %value from which the CI should be created, now 2 sigma away, can make it 3 or even 4 to make the point more clear. a = sort(normrnd(mu,sigma,N,1)); %my off center distribution from whom I would like to determine the 95% CI if it were centered around 0 % sorting is just done to make nice plots, should have no effect on calculations K = sum(a

 Tags bootstrap, confidence, fitted, interval, parameters, unbiased

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