<- file stat 97boot.html ->

bootstrap t test

=======================Rick McFarland, 17 Apr 1997==========ssc,sse,ssm From: Rick McFarland <hrm3c@virginia.edu> Subject: Re: Need clariciation on bootstrap for Ho test Message-ID: <3356786D.167EB0E7@virginia.edu> dandrade@nwlink.com wrote: > > Can someone provide a clear and concise explanation of how to use a > bootstrap for a 2 sided (or 1 sided) hypothesis test? > > I think the steps are: > > 1. form new sample by sampling with replacement from original population > 2. obtain estimator, SE, t-stat for the new pseudo-sample data > 3. repeat steps 1 and 2 say, m times (what is a good choice for m?) > 4. order the m t-stats and calculate the lower and upper alpha/2 > percentiles of t1,....,tm > 5. reject Ho at level alpha if the original t-stat falls outside the > bootstrapped t-interval > > In particular, if m=100 and alpha = 5%, at step 4 I don't understand if > I am to simply choose the 6th and 94th bootstrap t-statistics or do > something more complicated using the sampling distribution of the > bootstrap t-stats?????? > > > David Andrade There have been many good papers written about confidence intervals (which can be constued for p-values) calculation using Bootstrap: One that comes imediately to mind: Hall, Peter. Ann Stat 1986 V14,No4 p1431-1462 Again in 1988 Ann Stat V16 No3 You have many options. From a practical standpoint, I use what most people call the percentile-method which amounts to a non-parametric confidence interval and requires about 2000 bootstrap samples (what you outlied). A t-method based on the t-statistic is also popular but not theoretically sound since you are not taking independent simple random samples. I hope this is of service. -Rick

bootstrap, simplest example

=======================Tim Hesterberg, 23 Apr 1997==========sse Message-ID: <m0wK8V0-000RW3C@quake.statsci.com> From: Tim Hesterberg <timh@statsci.com> Subject: Re: EDSTAT-L digest 1418 dandrade@nwlink.com (David Andrade) wrote: > Can someone provide a clear and concise explanation of how to use a > bootstrap for a 2 sided (or 1 sided) hypothesis test? > > I think the steps are: > > 1. form new sample by sampling with replacement from original population > 2. obtain estimator, SE, t-stat for the new pseudo-sample data > 3. repeat steps 1 and 2 say, m times (what is a good choice for m?) > 4. order the m t-stats and calculate the lower and upper alpha/2 > percentiles of t1,....,tm > 5. reject Ho at level alpha if the original t-stat falls outside the > bootstrapped t-interval > > In particular, if m=100 and alpha = 5%, at step 4 I don't understand if > I am to simply choose the 6th and 94th bootstrap t-statistics or do > something more complicated using the sampling distribution of the > bootstrap t-stats?????? A good introduction to the bootstrap is: @book{efro93, author ={Efron, B. and Tibshirani, R. J.}, year =1993, title ={An Introduction to the Bootstrap}, publisher ={Chapman and Hall} } They include one chapter specifically on hypothesis testing. Most of the book, and other bootstrap literature, deals with confidence intervals rather than hypothesis tests, and most often people use CI's for hypothesis testing or to obtain p-values. The procedure you outline above corresponds to the "bootstap t-interval", which has good asymptotic properties (errors in coverage probability are O(1/n)) but often has problems in practice when used with nonlinear statistics. Rick FcFarland wrote: >You have many options. From a practical standpoint, I use >what most people call the percentile-method which amounts >to a non-parametric confidence interval and requires about 2000 >bootstrap samples (what you outlied). A t-method based on the >t-statistic is also popular but not theoretically sound since >you are not taking independent simple random samples. >From the description above I believe the bootstrap-t is theoretically sound here. I'm assuming that the original data were obtained using a simple random sample. If obtained by some other procedure such as stratified sampling both the bootstrap-t and percentile intervals would be incorrect, unless the bootstrap sampling method were modified to match the original sampling method. The percentile method is less accurate, with coverage errors O(1/sqrt(n)). There are other procedures which have the same error rate as the bootstrap-t, but tend to be better in practice, such as the bootstrap-bca interval, which involves doing "something more complicated" with the sampling distribution of the bootstrap estimates (not t-values). See the Efron and Tibshirani book for a description. There are ways to get accurate answers with fewer than 2000 replications, though they involve extra programming. See the following or the references contained therein: @Article{hest95b, author = {Tim C. Hesterberg}, title = {Tail-Specific Linear Approximations for Efficient Bootstrap Simulations}, journal = {Journal of Computational and Graphical Statistics}, year = 1995, volume = 4, pages = {113--133} } @Article{hest96a, author = {Tim C. Hesterberg}, title = {Control Variates and Importance Sampling for Efficient Bootstrap Simulations}, journal = {Statistics and Computing}, year = 1996, volume = 6, pages = {147--157} }

Bootstrap references (1993)

=======================Rob Malouf, 03 Jun 1993==========sms Subject: Re: Bootstrapping Message-ID: <1993Jun3.065430.11434@leland.Stanford.EDU> In article <11237@ncrwat.Waterloo.NCR.COM> ishay@53iss5.Waterloo.NCR.COM (Ishay Friedman) writes: >There is a curve fitting technique called "bootstrapping" which I >intend to use for some research in finance ( Option pricing, more >specifically). I cannot find any information on it. I would >greatly appreciate receiving any leads. Here are some references: @BOOK{Statistics, AUTHOR = "Donald A. Berry and Bernard W. Lindgren", TITLE = "Statistcs: Theory and Methods", YEAR = 1990, PUBLISHER = "Brooks/Cole Publishing Co", ADDRESS = "Pacific Grove, CA"} @ARTICLE{SciAm, AUTHOR = "Persi Diaconis and Bradley Efron", TITLE = "Computer-intensive methods in statistics", JOURNAL = "Scientific American", YEAR = 1983, VOLUME = 248, PAGES = "116--130"} @ARTICLE{Algorithm, AUTHOR = "John R. Gleason", TITLE = "Algorithms for balanced bootstrap simulations", JOURNAL = "The American Statistician", YEAR = 1988, VOLUME = 42, PAGES = "263--266"} @ARTICLE{Balance, AUTHOR = "A.C. Davison, D.V. Hinkley and E. Schechtman", TITLE = "Efficient bootstrap simulation", JOURNAL = "Biometrika", YEAR = 1986, VOLUME = 73, PAGES = "555--566"} @ARTICLE{Wu, AUTHOR = "C.F.J. Wu", TITLE = "Jackknife, bootstrap, and other resampling methods in regression analysis", JOURNAL = "Annals of Statistics", YEAR = 1986, VOLUME = 14, PAGES = "1261--1295"} @ARTICLE{Algae, AUTHOR = "E.P. Smith and R.B. Genter and J. Cairns", TITLE = "Confidence intervals for similarity between algal communities", JOURNAL = "Hydrobiologica", YEAR = 1986, VOLUME = 139, PAGES = "237--245"} @ARTICLE{Efron88, AUTHOR = "B. Efron", TITLE = "Bootstrap confidence intervals: good or bad?", JOURNAL = "Psychological Bulletin", YEAR = 1988, VOLUME = 104, PAGES = "293--296"} @ARTICLE{Water, AUTHOR = "Antonio G.-Valdecasas and Angel Baltan\'as", TITLE = "Jackknife and bootstrap estimation of biological index of water quality", JOURNAL = "Water Research", YEAR = 1990, VOLUME = 24, PAGES = "1279--1283"} @ARTICLE{Audit, AUTHOR = "K. Muralidhar and G.A. Ames and R. Sarathy", TITLE = "Bootstrap confidence intervals for estimating audit value from skewed populations and small samples", JOURNAL = "Simulation", YEAR = 1991, VOLUME = 56, PAGES = "119--127"} @ARTICLE{Love, AUTHOR = "Gloria M. Borrello and Bruce Thompson", TITLE = "A replication bootstrap analysis of the structure underlying perceptions of stereotypic love", JOURNAL = "The Journal of General Psychology", YEAR = 1989, VOLUME = 116, PAGES = "317--327"} @ARTICLE{Error, AUTHOR = "Michael J. Strube", TITLE = "Bootstrap type {I} error rates for the correlation coefficient: an analysis of alternate procedures", JOURNAL = "Psychological Bulletin", YEAR = 1988, VOLUME = 104, PAGES = "290--292"} @BOOK{Efron82, AUTHOR = "Bradley Efron", TITLE = "The jackknife, the bootstrap, and other resampling plans", YEAR = 1982, PUBLISHER = "Society for Industrial and Applied Mathematics", ADDRESS = "Philadelphia"} @ARTICLE{Hall, AUTHOR = "Peter Hall", TITLE = "On the number of bootstrap simulations required to construct a confidence interval", JOURNAL = "Annals of Statistics", VOLUME = 14, YEAR =1986, PAGES = "1453--1462"} @ARTICLE{Efron86, AUTHOR = "B. Efron and R. Tibshirani", TITLE = "Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy", JOURNAL = "Statistical Science", YEAR = 1986, VOLUME = 1, PAGES = "54-77"} @ARTICLE{Rasmussen, AUTHOR = "Jeffrey Lee Rasmussen", TITLE = "``Bootstrap confidence intervals: good or bad'': comments on {Efron} (1988) and {Strube} (1988) and further evaluation", JOURNAL = "Phsychological Bulletin", YEAR = 1988, VOLUME = 104, PAGES = "297--299"} @ARTICLE{Feinstein, AUTHOR = "J. Feinstein", TITLE = "Discussion of {Wu} (1986)", JOURNAL = "Annals of Statistics", YEAR = 1986, VOLUME = 14} * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

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