jackknife
Compute jackknife estimates of a parameter taking one or more given samples as parameters.
In particular, E is the estimator to be jackknifed as a function name, handle, or inline function, and x is the sample for which the estimate is to be taken. The i-th entry of jackstat will contain the value of the estimator on the sample x with its i-th row omitted.
jackstat (i) = E(x(1 : i - 1, i + 1 : length(x))) |
Depending on the number of samples to be used, the estimator must have the appropriate form:
jackstat = jackknife (@std, rand (100, 1)).
jackstat = jackknife (@(x) std(x{1})/var(x{2}),
rand (100, 1), randn (100, 1))
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If all goes well, a theoretical value P for the parameter is already known, n is the sample size,
t = n * E(x) - (n - 1) *
mean(jackstat)
and
v = sumsq(n * E(x) - (n - 1) *
jackstat - t) / (n * (n - 1))
then
(t-P)/sqrt(v) should follow a t-distribution with
n-1 degrees of freedom.
Jackknifing is a well known method to reduce bias. Further details can be found in:
Source Code: jackknife
for k = 1:1000
x=rand(10,1);
s(k)=std(x);
jackstat=jackknife(@std,x);
j(k)=10*std(x) - 9*mean(jackstat);
end
figure();hist([s',j'], 0:sqrt(1/12)/10:2*sqrt(1/12))
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for k = 1:1000
x=randn(1,50);
y=rand(1,50);
jackstat=jackknife(@(x) std(x{1})/std(x{2}),y,x);
j(k)=50*std(y)/std(x) - 49*mean(jackstat);
v(k)=sumsq((50*std(y)/std(x) - 49*jackstat) - j(k)) / (50 * 49);
end
t=(j-sqrt(1/12))./sqrt(v);
figure();plot(sort(tcdf(t,49)),"-;Almost linear mapping indicates good fit with t-distribution.;")
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