Abstract
This paper introduces the “piggyback bootstrap.” Like the weighted bootstrap, this bootstrap procedure can be used to generate random draws that approximate the joint sampling distribution of the parametric and nonparametric maximum likelihood estimators in various semiparametric models, but the dimension of the maximization problem for each bootstrapped likelihood is smaller. This reduction results in significant computational savings in comparison to the weighted bootstrap. The procedure can be stated quite simply. First obtain a valid random draw for the parametric component of the model. Then take the draw for the nonparametric component to be the maximizer of the weighted bootstrap likelihood with the parametric component fixed at the parametric draw. We prove the procedure is valid for a class of semiparametric models that includes frailty regression models airsing in survival analysis and biased sampling models that have application to vaccine efficacy trials. Bootstrap confidence sets from the piggyback, and weighted bootstraps are compared for biased sampling data from simulated vaccine efficacy trials.
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References
Abraham, R., Marsden, J. E. and Ratiu T. (1988).Manifolds, Tensor Analysis, and Applications, Springer-Verlag, New York.
Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993).Efficient and Adaptive Estimation for Semiparametric Models, Johns Hopkins University Press, Baltimore, Maryland.
Bilias, Y., Gu, M. and Ying, Z. (1997). Towards a general asymptotic theory for Cox model with staggered entry,Annals of Statistics,25, 662–682.
Dabrowska, D. M. and Doksum, K. A. (1988). Estimation and testing in a two-sample generalized odds-rate model,Journal of the American Statistical Association,83, 744–749.
Dixon, J. R. (2003). The piggyback bootstrap for functional inference in semiparametric models, Ph.D. Dissertation, Department of Statistics, University of, Wisconsin-Madison.
Dixon, J. R., Kosorok, M. R. and Lee, B. L. (2004). Technical report to accompany ‘Functional inference in semiparametric models using the piggyback bootstrap’, Technical Report, Department of Statistics, Florida State University.
Gilbert, P. B. (1996). Sieve analysis: Statistical methods for assessing differential vaccine protection against HIV types, Ph.D. Dissertation, Department of Biostatistics, University of Washington.
Gilbert, P. B. (2000). Large sample theory of maximum likelihood estimates in semiparametric biased sampling models.Annals of Statistics,28, 151–194.
Gilbert, P. B., Self, S. G. and Ashby, M. A. (1998). Statistical methods for assessing differential vaccine protection against human immunodeficiency virus types,Biometrics,54, 799–814.
Kim, Y. and Lee, J. (2003). Bayesian bootstrap for proportional hazards models,Annals of Statistics,31, 1905–1922.
Kosorok, M. R., Lee, B. L. and Fine, J. P. (2004). Robust inference for proportional hazards univariate frailty regression models,Annals of Statistics,32, 1448–1491.
Kress, R. (1989).Linear Integral Equations, Springer-Verlag, Berlin.
Lee, B. L. (2000). Efficient semiparametric estimation using Markov chain Monte Carlo, Ph.D. Dissertation, Department of Statistics, University of Wisconsin-Madison.
Lin, D. Y., Fleming, T. R. and Wei, L. J. (1994). Confidence bands for survival curves under the proportional hazards model,Biometrika,81, 73–81.
Murphy, S. A. (1994). Consistency in a proportional hazards model incorporating a random effect,Annals of Statistics,22, 712–731.
Murphy, S. A. and van der Vaart, A. W. (2000). On profile likelihood,Journal of the American Statistical Association,95, 449–465.
Newton, M. A. and Raftery, A. E. (1994). Approximate Bayesian inference with the weighted likelihood bootstrap,Journal of the Royal Statistical Society, Series B,56, 3–26.
Parner, E. (1998). Asymptotic theory for the correlated gamma-frailty model,Annals of Statistics,26, 183–214.
Praestgaard, J. and Wellner, J. A. (1993). Exchangeably weighted bootstraps of the general empirical process,Annals of Probability,21, 2053–2086.
Press, W., Flannery, B., Teukolsky, S. and Vetterling, W. (1994).Numerical Recipes in Pascal: The Art of Scientific Computing, Cambridge University Press, New York.
Rubin, D. B. (1981). The Bayesian bootstrap,Annals of Statistics,9, 130–134.
Scharfstein, D. O., Tsiatis, A. A. and Gilbert, P. B. (1998). Semiparametric efficient estimation in the generalized odds-rate class of regression models for right-censored time-to-event data,Lifetime Data Analysis,4, 355–391.
Shen, X. (2002). Asymptotic normality of semiparametric and nonparametric posterior distributions,Journal of the American Statistical Association,97, 222–235.
Tsodikov, A. (2003). Semiparametric models: A generalized self-consistency approach,Journal of the Royal Statistical Society, Series B,65, 759–774.
van der Vaart, A. W. (2000).Asymptotic Statistics, Cambridge University Press, New York.
van der Vaart, A. W. and Wellner, J. A. (1996).Weak Convergence and Empirical Processes: With Applications to Statistics, Springer, New York.
Vaida, F. and Xu, R. (2000). Proportional hazards model with random effects,Statistics in Medicine,19, 3309–3324.
Vardi, Y. (1985). Empirical distributions in selection bias models,Annals of Statistics,13, 178–203.
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Dixon, J.R., Kosorok, M.R. & Lee, B.L. Functional inference in semiparametric models using the piggyback bootstrap. Ann Inst Stat Math 57, 255–277 (2005). https://doi.org/10.1007/BF02507025
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DOI: https://doi.org/10.1007/BF02507025