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Title

_u2jackpseud() -- Jackknife pseudovalue functions used by somersd

Syntax

void
_u2jackpseud(phiidot [,
phiii,
fweight])

void
_v2jackpseud(phiidot [,
phiii,
fweight])

where

phiidot: numeric matrix
phiii: numeric matrix
fweight: numeric colvector

Description

These functions are used by the somersd package to calculate jackknife
pseudovalues for Hoeffding U statistics and von Mises V statistics
of degree 2, on the basis of kernel totals provided as input by the user.
Applications of these functions are discussed in the file somersd.pdf,
which is distributed with the somersd package.
The theory of Hoeffding U statistics, von Mises V statistics, and
their kernel functions is presented in chapter 5 of Serfling (1980).

_u2jackpseud(phiidot, phiii, fweight)
inputs and modifies a matrix phiidot, with one column for each of a set
of degree-2 Hoffding U statistics. On entry, the ith row of each
column of phiidot contains the ith kernel total of the corresponding
degree-2 Hoeffding U statistic. This kernel total might be denoted as
phi_i. in the notation of (19) to (24) of the file
somersd.pdf, which is distributed with the somersd package. On
exit, the ith row of each column of the matrix phiidot contains the
ith jackknife pseudovalue of the same degree-2 Hoeffding U
statistic. This pseudovalue might be denoted as psi_i in the
notation of (19) to (24) of somersd.pdf. The input matrix
phiii contains, in the ith row of each column, the degree-2 kernel
function of the ith sampling unit with itself, which might be denoted
phi_ii in the notation of somersd.pdf. The input column vector
fweight contains frequency weights, implying that the ith rows of
phiidot and phiii represent a number of sampling units stored in the
ith row of fweight. Both phiii and fweight are unchanged
on exit. The matrix phiii may have one row and/or one column and is then
input into the calculation as if the row and/or column were duplicated as many
times as necessary for conformability with phiidot. The column vector
fweight may have one row and is then input into the calculations as if
the row were duplicated as many times as necessary for conformability with
phiidot. If phiii is absent, then it is set to a scalar with value
0. If fweight is absent, then it is set to a scalar with value 1.
_u2jackpseud() still works if phiidot, phiii, and
fweight are views onto the dataset in memory.

_v2jackpseud(phiidot, phiii, fweight)
inputs and modifies a matrix phiidot, using the additional input matrix
phiii and the additional input weight vector fweight. The function
_v2jackpseud() is similar to the function _u2jackpseud(), except
that each column of phiidot contains on input the kernel totals and
contains on output the jackknife pseudovalues of a degree-2 von Mises V
statistic rather than a degree-2 Hoeffding U statistic.

Remarks

The use of the jackknife is discussed in Miller (1974). The application of
the jackknife specifically to U statistics is discussed in Arvesen
(1969). The somersd package uses the delta-jackknife; that
is, it uses the jackknife to define standard errors for means, U
statistics or V statistics and then uses Taylor polynomials to define
standard errors for ratios of these means, U statistics, V
statistics, or transformations of these ratios. The formulas used are given
in detail in the file somersd.pdf, which is distributed with the
somersd package.

Conformability

_u2jackpseud(phiidot, phiii, fweight), _v2jackpseud(phiidot, phiii, fweight):

phiidot: N x K
phiii: N x K or N x 1 or 1 x K or 1 x 1
fweight: N x 1 or 1 x 1

Diagnostics

_u2jackpseud() and _v2jackpseud() carry out no checks for missing
values. Therefore, an entry in the matrix phiidot on output will be
missing if any entry in the input matrices affecting its value is missing.

Source code

_u2jackpseud.mata,
_v2jackpseud.mata

Author

Roger Newson, Imperial College London, UK.

Email: r.newson@imperial.ac.uk

References

Arvesen, J. N. 1969.
Jackknifing U-statistics.
Annals of Mathematical Statistics 40: 2076-2100.

Miller, R. G. 1974.
The jackknife--a review.
Biometrika 61: 1-15.

Serfling, R. 1980.
Approximation Theorems of Mathematical Statistics.
New York: Wiley.

Also see

Manual: [M-0] intro

Online: mata,

somersd (if installed)