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August 10th, 2012, 02:19 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 NonParametric 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 nonlinear function). (These best fitted parameters fit quite nicely!) For the bootstrap method I use NonParametric resampling (I.E. randomly resample 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 dataset 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 = mu2*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<Value)/N; %K = normcdf(Value,mu,sigma); %"exact solution",I mean no numerical approximation z0= norminv(K,0,1); za1 = norminv(0.025,0,1); % 97.5 percentile for upper bound za2 = norminv(0.975,0,1); % 2.5 percentile for lower bound, I.E. 95% CI m1 = normcdf(2*z0+za1,0,1); m2 = normcdf(2*z0+za2,0,1); CIinterval = [a(floor(m1(1)*(N1))+1) Value a(floor(m2(1)*(N1))+1)] %this might induce a (very very weak) bias, but makes sure that it will not crash/try to find a to big/low index for any mu & sigma/Value %CIinterval = [norminv(m1,mu,sigma) Value norminv(m2,mu,sigma)] %"again exact solution" Doing this will yield a CI of [2.3471 < 0 < 0.0469] or if you use no "numerical approximation" [3.9843 < 0< 0.0644] As you can see 0 does NOT lie within the interval, this get worse if the bias is bigger.... Also note the growing disparity between the distance: Left boundValue, ValueRight bound if you increase the bias.... P.S. Note I do NOT divide by 0! 

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