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All of Statistics: A Concise Course in Statistical Inference

All of Statistics: A Concise Course in Statistical Inference. Chapter 8 The Bootstrap Prepared by Gene Shiau. Terminology. : the distribution function of X : the empirical distribution function of X 1 , …, X n T n = g ( X 1 , …, X n ) is a function of the data, aka a statistic .

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All of Statistics: A Concise Course in Statistical Inference

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  1. All of Statistics:A Concise Course inStatistical Inference Chapter 8 The Bootstrap Prepared by Gene Shiau

  2. Terminology • : the distribution function of X • : the empirical distribution function of X1, …, Xn • Tn = g(X1, …, Xn) is a function of the data, aka a statistic. • VF(Tn) is the variance of Tn, usually dependant on F.

  3. BootstrapVariance Estimation Bootstrap • Draw X*1, …, X*n from (same as from X1, …, Xn with replacement) • Compute T*n = g(X*1, …, X*n) • Repeat previous steps B times to get T*n,1, …, T*n,B • Let the variance be

  4. BootstrapConfidence Intervals • The Normal Interval • Pivotal Intervals • Percentile Intervals

  5. Example: Nerve Data

  6. Bootstrap for the Skewness Given skewness Plug-in estimate of the skewness = 1.76 for (i in 1:B) { Draw IID data X* = (X*1, …, X*n) from Compute T*n,i = skew(X*) } vboot = variance(T*n,:) standard error = sqrt(vboot) Based on B = 1000, the standard error is 0.16

  7. Confidence Intervals

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