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    %% Jackknife Resampling
%

% Copyright 2015 The MathWorks, Inc.


%%
% Similar to the bootstrap is the jackknife, which uses resampling to
% estimate the bias of a sample statistic. Sometimes it is also used to
% estimate standard error of the sample statistic. The jackknife is
% implemented by the Statistics and Machine Learning Toolbox(TM) function
% |jackknife|.

%%
% The jackknife resamples systematically, rather than at random as the
% bootstrap does. For a sample with |n| points, the jackknife computes
% sample statistics on |n| separate samples of size |n|-1. Each sample is
% the original data with a single observation omitted.

%%
% In the bootstrap example, you measured the uncertainty in estimating the
% correlation coefficient. You can use the jackknife to estimate the bias,
% which is the tendency of the sample correlation to over-estimate or
% under-estimate the true, unknown correlation. First compute the sample
% correlation on the data.
load lawdata
rhohat = corr(lsat,gpa)

%%
% Next compute the correlations for jackknife samples, and compute their
% mean.
rng default;  % For reproducibility
jackrho = jackknife(@corr,lsat,gpa);
meanrho = mean(jackrho)

%%
% Now compute an estimate of the bias.
n = length(lsat);
biasrho = (n-1) * (meanrho-rhohat)

%%
% The sample correlation probably underestimates the true correlation by
% about this amount.