Page 257 - Computational Statistics Handbook with MATLAB

Chapter 7: Data Partitioning 245

)
3. Calculate the pseudo-value T i using Equation 7.12.
4. Repeat steps 2 and 3 for the remaining data points, yielding n values
)
of T i .
5. Determine the jackknife estimate of the standard error of T using
Equation 7.14.

Example 7.6
We now repeat Example 7.4 using the jackknife pseudo-value approach and
compare estimates of the standard error of the correlation coefficient for these
data. The following MATLAB code implements the pseudo-value procedure.
% Loads up a matrix.
load law
lsat = law(:,1);
gpa = law(:,2);
% Get the statistic from the original sample
tmp = corrcoef(gpa,lsat);
T = tmp(1,2);
% Set up memory for jackknife replicates
n = length(gpa);
reps = zeros(1,n);
for i = 1:n
% store as temporary vector
gpat = gpa;
lsatt = lsat;
% leave i-th point out
gpat(i) = [];
lsatt(i) = [];
% get correlation coefficient
tmp = corrcoef(gpat,lsatt);
% In this example, is off-diagonal element.
% Get the jackknife pseudo-value for the i-th point.
reps(i) = n*T-(n-1)*tmp(1,2);
end
JT = mean(reps);
sehatpv = sqrt(1/(n*(n-1))*sum((reps - JT).^2));
ˆ ˆ
We obtain an estimated standard error of SE JackP ρ() = 0.14 , which is the
same result we had before.

Efron and Tibshirani [1993] describe a situation where the jackknife proce-
dure does not work and suggest that the bootstrap be used instead. These are
applications where the statistic is not smooth. An example of this type of sta-
tistic is the median. Here smoothness refers to statistics where small changes

© 2002 by Chapman & Hall/CRC