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August 10th, 2012, 01:19 AM   #1
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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<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)*(N-1))+1) Value a(floor(m2(1)*(N-1))+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 bound-Value, Value-Right bound if you increase the bias....

P.S. Note I do NOT divide by 0!
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