JACKKNIFE procedure

Produces Jackknife estimates and standard errors (R.W. Payne).


Options

PRINT = string
Controls printed output (estimates, vcovariance); default esti

DATA = variates, factors or texts
Data vectors from which the statistics are to be calculated

ANCILLARY = any type
Other relevant information needed to calculate the statistics

VCOVARIANCE = symmetric matrix
Saves the variance-covariance matrix for the statistics


Parameters

LABEL = texts
Texts, each containing a single line, to label the statistics

ESTIMATE = scalars
Saves the Jackknife estimate for each statistic

SE = scalars
Saves Jackknife estimates of the standard errors

PSEUDOVALUES = variates
Saves the Jackknife pseudo-values

ACCELERATION = scalars
Saves the acceleration parameter for bias-corrected and accelerated bootstrap confidence intervals


Description

The Jackknife provides a way of decreasing bias and obtaining standard errors in situations where the standard methods might be expected to be inappropriate. The basic form of the Jackknife method works by calculating the statistic (or statistics) of interest omitting each data value in turn. Thus, if there are n data values, n "partial estimates" T-1 ... T-n are obtained (where T-j is the estimate omitting value j). These are combined with the estimate T obtained from all the data, to produce n pseudo-values:

Pj = n × T - (n - 1) × T-j : j = 1 ... n

The Jackknife estimate of the statistic is given by the mean of the pseudo-values, and the standard error by the standard error of the mean of the pseudo-values.

   The Jackknife can be shown to eliminate the term proportional to 1/n from a bias of the form

T = t + a/n + O(1/n2)

where t is the true value of the estimate and O(1/n2) is a term of order one divided by the square of the number of observations (Quenouille 1956). However, it is not appropriate in all situations. In particular the statistic needs to be "smooth" (small changes in the data set should cause only small changes in the statistic); it will not work for example with medians or order statistics. Further details and advice are given by Miller (1974), Bissell & Ferguson (1975), Hinkley (1983) and Efron & Tibshirani (1993).

   The data for JACKKNIFE are provided as a list of vectors (variates, factors or texts) using the DATA option. From this, new vectors are formed omitting each unit of the original vectors in turn, and a subsidiary procedure RESAMPLE is called to calculate the statistics. Other relevant information can be provided for passing to RESAMPLE, in any type of data structure, using the ANCILLARY option. To use JACKKNIFE, you need to provide a version of RESAMPLE to calculate the particular statistics that you require. The default RESAMPLE procedure, which accompanies JACKKNIFE in the library, merely prints details of the syntax (also described in the Methods Section).

   A label should be provided for each statistic, using the LABEL parameter; by default, there is assumed to be a single statistic labelled simply as Statistic. The estimates, their standard errors and variates of corresponding pseudo-values for each statistic can be saved by the ESTIMATE, SE and PSEUDOVALUES parameters, respectively. Also, if there is more than one statistic, a variance-covariance matrix can be saved for the estimates using the VCOVARIANCE option.

   Printed output is controlled by the PRINT option, with settings estimates for the estimates and their standard errors, and vcovariance for the variance-covariance matrix; by default PRINT=estimates.

   The jackknife is also required for the calculation of bias-corrected and accelerated confidence limits for bootstrap statistics (as given by the BOOTSTRAP procedure). The necessary acceleration quantities can be saved using the ACCELERATION parameter. For details see Efron & Tibshirani, 1993, Section 14.3.


Options: PRINT, DATA, ANCILLARY, VCOVARIANCE.

Parameters: LABEL, ESTIMATE, SE, PSEUDOVALUES, ACCELERATION.


Method

The original papers describing the Jackknife technique are by Quenouille (1949, 1956) and by Tukey (1958). Good expository accounts are provided by Hinkley (1983) or Bissell & Ferguson (1975).

   JACKKNIFE needs a subsidiary procedure RESAMPLE to calculate the statistics of interest. RESAMPLE has an option, DATA, which is used to supply the data vectors (variates, factors or texts) from which the statistics are to be calculated. (On the first occasion that RESAMPLE is called, these will be the original vectors as supplied to JACKNIFE, in order to calculate the estimate T; subsequently, they will be new vectors containing all except one of the units.) Other relevant information can can be supplied through the ANCILLARY option, which corresponds to the ANCILLARY option of JACKKNIFE itself. RESAMPLE can be called by the BOOTSTRAP procedure, and it then also has an AUXILIARY option, but this is not relevant to JACKKNIFE.

   There are two parameters: STATISTICS supplies a list of scalars to store the estimates of each statistic, and EXIT a list of scalars which should be set to zero or one according to whether or not each statistic could be estimated successfully with the supplied data vectors. If the value of EXIT is not calculated in RESAMPLE, JACKKNIFE assumes that the calculations succeeded. This example shows a version of RESAMPLE which calculates the correlation between two variates.

PROCEDURE [PARAMETER=pointer] 'RESAMPLE'

OPTION 'DATA', " (I: variates, factors or texts) data

                         vectors from which to calculate the

                         statistics; no default"\

          'ANCILLARY'; " (I: any type of structure) other

                         relevant information needed to

                         calculate the statistics "\

          MODE=p; TYPE=!t(variate,factor,text),*;\

          SET=yes,no; LIST=yes; DECLARED=yes; PRESENT=yes

PARAMETER 'STATISTIC', " (O: scalars) to save the calculated

                         statistics "\

          'EXIT'; " (O: scalars) to save an exit code

                         to indicate failure (EXIT[i]=1) or

                         success (EXIT[i]=0) when calculating

                         each STATISTIC[i]"\

          MODE=p; TYPE='scalar'; SET=yes

 

CALCULATE STATISTIC[1] = CORRELATION(DATA[1]; DATA[2])

& EXIT[1] = STATISTIC[1]==C('missing')

 

ENDPROCEDURE


Action with RESTRICT

If any of the data vectors is restricted, JACKKNIFE will use only the units that are not restricted for any of the vectors.


References

Bissell, A.F. & Ferguson, R.A. (1975). The jackknife - toy, tool or two-edged weapon. The Statistician, 24, 79-100.

Efron, B. & Tibshirani, R.J. (1993). An Introduction to the Bootstrap. Chapman & Hall, London.

Hinkley, D. (1983). Jackknife methods. In: Encyclopedia of Statistics, Volume 4 (ed: S. Kotz, N.L. Johnson & C.B. Read). Wiley, New York.

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

Quenouille, M.H. (1949). Approximate tests of correlation in time series. Journal of the Royal Statistical Society, Series B, 11, 18-44.

Quenouille, M.H. (1956). Notes on bias in estimation. Biometrika, 61, 353-360.