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24-Feb-2006
Bootstrapping, by James Sogi

Use bootstrapping methods, following along with Chair and the Artful Simulator, to derive a mean and standard error statistic in the situation (4-year returns) where there was not enough existing data to get a robust result . The re-sampled data with replacement using the sampled variance of yield converges to the true mean and variance as n approaches infinity under the central limit theory. The distributions of the multiple samples is used as a simulated population to get the p and t scores for the sample. They converted the p and t scores to odds. Collett describes the process. "It is sometimes helpful to describe the chance that a binary response variable leads to a success in terms of the odds of that event. The odds of success is defined to be the ratio of the probability of a success to the probability of a failure. thus if p is the true success probability, the odds of a success is p/(1-p)."  The odds ratio is the ratio of the odds for two sets of binary data. The odds ratio can be set in a 2x2 contingency table and confidence levels can be computed.

R Cran has a library 'boot' to perform bootstrapping. Several of the other stat library functions have bootstrapping built in such as the chisq. test. The advent of modern statistical computing has changed statistical analysis and the theory by allowing robust and exact p scores in situations where it was not possible before.

R gives references explaining the bootstrap and its variations. Among them are:

  Booth, J.G., Hall, P. and Wood, A.T.A. (1993) Balanced importance
    resampling  for the bootstrap. _Annals of Statistics_, *21*,
    286-298.

    Davison, A.C. and Hinkley, D.V. (1997)  _Bootstrap Methods and
    Their Application_. Cambridge University Press.

    Davison, A.C., Hinkley, D.V. and Schechtman, E. (1986) Efficient
    bootstrap  simulation. _Biometrika_, *73*, 555-566.

    Efron, B. and Tibshirani, R. (1993) _An Introduction to the
    Bootstrap_. Chapman & Hall.

    Gleason, J.R. (1988) Algorithms for balanced bootstrap
    simulations. _ American Statistician_, *42*, 263-266.

    Hall, P. (1989) Antithetic resampling for the bootstrap.
    _Biometrika_, *73*, 713-724.

    Hinkley, D.V. (1988) Bootstrap methods (with Discussion).
    _Journal of the  Royal Statistical Society, B_, *50*, 312-337,
    355-370.

    Hinkley, D.V. and Shi, S. (1989) Importance sampling and the
    nested bootstrap. _Biometrika_, *76*, 435-446.

    Johns M.V. (1988) Importance sampling for bootstrap confidence
    intervals. _Journal of the American Statistical Association_,
    *83*, 709-714.

    Noreen, E.W. (1989) _Computer Intensive Methods for Testing
    Hypotheses_.  John Wiley & Sons.