help mata _u2jackpseud-------------------------------------------------------------------------------

Title

_u2jackpseud() -- Jackknife pseudovalue functions used by somersd

Syntax

void_u2jackpseud(phiidot[,phiii,fweight])

void_v2jackpseud(phiidot[,phiii,fweight])where

phiidot:numeric matrixphiii:numeric matrixfweight:numeric colvector

DescriptionThese functions are used by the

somersdpackage 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 filesomersd.pdf, which is distributed with thesomersdpackage. 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 matrixphiidot, with one column for each of a set of degree-2 Hoffding U statistics. On entry, theith row of each column ofphiidotcontains theith kernel total of the corresponding degree-2 Hoeffding U statistic. This kernel total might be denoted asphi_i.in the notation of (19) to (24) of the filesomersd.pdf, which is distributed with thesomersdpackage. On exit, theith row of each column of the matrixphiidotcontains theith jackknife pseudovalue of the same degree-2 Hoeffding U statistic. This pseudovalue might be denoted aspsi_iin the notation of (19) to (24) ofsomersd.pdf. The input matrixphiiicontains, in theith row of each column, the degree-2 kernel function of theith sampling unit with itself, which might be denotedphi_iiin the notation ofsomersd.pdf. The input column vectorfweightcontains frequency weights, implying that theith rows ofphiidotandphiiirepresent a number of sampling units stored in theith row offweight. Bothphiiiandfweightare unchanged on exit. The matrixphiiimay 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 withphiidot. The column vectorfweightmay have one row and is then input into the calculations as if the row were duplicated as many times as necessary for conformability withphiidot. Ifphiiiis absent, then it is set to a scalar with value 0. Iffweightis absent, then it is set to a scalar with value 1._u2jackpseud()still works ifphiidot,phiii, andfweightare views onto the dataset in memory.

_v2jackpseud(phiidot,phiii,fweight)inputs and modifies a matrixphiidot, using the additional input matrixphiiiand the additional input weight vectorfweight. The function_v2jackpseud()is similar to the function_u2jackpseud(), except that each column ofphiidotcontains 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.

RemarksThe use of the jackknife is discussed in Miller (1974). The application of the jackknife specifically to U statistics is discussed in Arvesen (1969). The

somersdpackage 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 filesomersd.pdf, which is distributed with thesomersdpackage.

Conformability

_u2jackpseud(phiidot,phiii,fweight),_v2jackpseud(phiidot,phiii,fweight):phiidot:N x Kphiii:N x KorN x1 or 1x Kor 1x1fweight:N x1 or 1x1

Diagnostics

_u2jackpseud()and_v2jackpseud()carry out no checks for missing values. Therefore, an entry in the matrixphiidoton output will be missing if any entry in the input matrices affecting its value is missing.

Source code_u2jackpseud.mata, _v2jackpseud.mata

AuthorRoger Newson, Imperial College London, UK. Email: r.newson@imperial.ac.uk

ReferencesArvesen, J. N. 1969. Jackknifing U-statistics.

Annals of MathematicalStatistics40: 2076-2100.Miller, R. G. 1974. The jackknife--a review.

Biometrika61: 1-15.Serfling, R. 1980.

Approximation Theorems of Mathematical Statistics. New York: Wiley.

Also seeManual:

[M-0] introOnline:

mata,somersd(if installed)