_u2jackpseud() -- Jackknife pseudovalue functions used by somersd
void _u2jackpseud(phiidot [, phiii, fweight])
void _v2jackpseud(phiidot [, phiii, fweight])
phiidot: numeric matrix phiii: numeric matrix fweight: numeric colvector
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.
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 infinitesimal 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.
_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
_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.
Roger Newson, Imperial College London, UK. Email: firstname.lastname@example.org
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.
Manual: [M-0] intro
Online: mata, somersd (if installed)