help mata _u2jackpseud
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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 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.


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)