STAT4290 April 15th 2011 Notes

# STAT4290 April 15th 2011 Notes - The bootstrap is a general...

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The bootstrap is a general methodology. For any estimator of a parameter, call in θ , = - = ( *- ) Varθ 1nn 1i 1nn θi θ ^2 The resamples can also be used to get an empirical distribution for theta-hat -- this lead also to relatively simple hypothesis tests and confidence intervals. Problem: the total number of possible resample, n^n, gets prohibitively lard vary quickly. In practice – taking a sample of the resamples in good enough. Thus, for a sample (X1, X2…Xn) and an estimator theta-hat, select B resamples, or bootstrap samples, and calculate e.g. Nonparametric Bootstrap! = - = ( *- ) Varθ 1B 1i 1nn θi θ ^2 Parametric bootstrap – we have a sample (X1, X2…Xn) assumed to come from a distribution F(x| theta) wuih parameter the theta. We estimate theta with thetahat. Draw X1-star, X2-star from F(x| theta) for each of the B bootstrap samples. How big should B be? Usually, B=500 or 1000 is sufficient. Goodness to Fit Test

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We’ve now interested in the unknown probability distribution of the random variable being observe. Prior to this, we’ve mostly focused on specific features of the distribution – mean,
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