--- main/statistics/inst/jackknife.m.orig 2011-12-24 08:27:40.000000000 -0800
+++ main/statistics/inst/jackknife.m 2011-12-24 11:34:17.000000000 -0800
@@ -15,24 +15,27 @@
## .
## -*- texinfo -*-
-## @deftypefn{Function File} { [ @var{t}, @var{v} ] } = jackknife ( @var{E}, @var{x}, ... )
-## Compute jackknife estimates of a parameter taking one or more given samples as parameters,
-## as well as its variance.
+## @deftypefn{Function File} { @var{jackstat} } = jackknife ( @var{E}, @var{x}, ... )
+## Compute jackknife estimates of a parameter taking one or more given samples as parameters.
## In particular, @var{E} is the estimator to be jackknifed as a function name, handle,
## or inline function, and @var{x} is the sample for which the estimate is to be taken.
-## @var{t} will be the estimate for the parameter, and @var{v} the estimate of its variance.
+## The @var{i}-th entry of @var{jackstat} will contain the value of the estimator
+## on the sample @var{x} with its @var{i}-th row omitted.
+## @code{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:
## If only one sample is used, then the estimator need not be concerned with cell arrays,
## for example jackknifing the standard deviation of a sample can be performed with
-## @code{ [ @var{t}, @var{v} ] = jackknife(@@std, rand (100, 1))}.
+## @code{ @var{jackstat} = jackknife(@@std, rand (100, 1))}.
## If, however, more than one sample is to be used, the samples must all be of equal size,
## and the estimator must address them as elements of a cell-array,
## in which they are aggregated in their order of appearance:
-## @code{ [ @var{t}, @var{v} ] = jackknife(@@(x) std(x@{1@})/var(x@{2@}), rand (100, 1), randn (100, 1)}.
+## @code{ @var{jackstat} = jackknife(@@(x) std(x@{1@})/var(x@{2@}), rand (100, 1), randn (100, 1)}.
##
## If all goes well, a theoretical value @var{P} for the parameter is already known,
-## and @var{n} is the sample size, then
+## @var{n} is the sample size,
+## @code{ @var{t} = @var{n} * @var{E}(@var{x}) - (@var{n} - 1) * mean(@var{jackstat}) }, and
+## @code{ @var{v} = sumsq(@var{n} * @var{E}(@var{x}) - (@var{n} - 1) * @var{jackstat} - @var{t}) / (@var{n} * (@var{n} - 1)) }, then
## @code{ (@var{t}-@var{P})/sqrt(@var{v}) } should follow a t-distribution with @var{n}-1 degrees of freedom.
##
## Jackknifing is a well known method to reduce bias; further details can be found in:
@@ -46,7 +49,7 @@
## Author: Alexander Klein
## Created: 2011-11-25
-function [ theta_tilde, var_theta_tilde ] = jackknife ( anEstimator, varargin )
+function jackstat = jackknife ( anEstimator, varargin )
## Convert function name to handle if necessary, or throw
@@ -72,20 +75,13 @@
aSample = varargin { 1 };
g = length ( aSample );
- indices = 1 : g;
- theta_hat = anEstimator ( aSample );
-
- theta_hat_minus_i = zeros ( 1, g );
+ jackstat = zeros ( 1, g );
for k = 1 : g
- theta_hat_minus_i ( k ) = anEstimator ( aSample ( [ 1 : k - 1, k + 1 : g ] ) );
+ jackstat ( k ) = anEstimator ( aSample ( [ 1 : k - 1, k + 1 : g ] ) );
end
- theta_tilde = g * theta_hat - ( g - 1 ) * mean ( theta_hat_minus_i );
-
- var_theta_tilde = sumsq ( ( g * theta_hat - ( g - 1 ) * theta_hat_minus_i ) - theta_tilde ) / ( g * ( g - 1 ) );
-
## More complicated input requires more work, however.
else
@@ -98,21 +94,13 @@
g = g ( 1 );
- theta_hat = anEstimator ( varargin );
-
- theta_hat_minus_i = zeros ( 1, g );
-
- flags = 1 - eye ( 1, g );
+ jackstat = zeros ( 1, g );
for k = 1 : g
- theta_hat_minus_i ( k ) = anEstimator ( cellfun ( @(x) x( [ 1 : k - 1, k + 1 : g ] ), varargin, "UniformOutput", false ) );
+ jackstat ( k ) = anEstimator ( cellfun ( @(x) x( [ 1 : k - 1, k + 1 : g ] ), varargin, "UniformOutput", false ) );
end
- theta_tilde = g * theta_hat - ( g - 1 ) * mean ( theta_hat_minus_i );
-
- var_theta_tilde = sumsq ( ( g * theta_hat - ( g - 1 ) * theta_hat_minus_i ) - theta_tilde ) / ( g * ( g - 1 ) );
-
end
endfunction
@@ -120,14 +108,15 @@
%!test
%! ##Example from Quenouille, Table 1
%! d=[0.18 4.00 1.04 0.85 2.14 1.01 3.01 2.33 1.57 2.19];
-%! t = jackknife ( @(x) 1/mean(x), d );
-%! assert ( t, 0.5240, 1e-5 );
+%! jackstat = jackknife ( @(x) 1/mean(x), d );
+%! assert ( 10 / mean(d) - 9 * mean(jackstat), 0.5240, 1e-5 );
%!demo
%! for k = 1:1000
%! x=rand(10,1);
%! s(k)=std(x);
-%! j(k)=jackknife(@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))
@@ -135,7 +124,9 @@
%! for k = 1:1000
%! x=randn(1,50);
%! y=rand(1,50);
-%! [j(k),v(k)]=jackknife(@(x) std(x{1})/std(x{2}),y,x);
+%! 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.;")