By Bradley Efron, Robert J. Tibshirani (auth.)

ISBN-10: 0412042312

ISBN-13: 9780412042317

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**Extra resources for An Introduction to the Bootstrap**

**Example text**

6) directly from the data (see Chapters 12-14 on bootstrap confidence intervals). 6), which is correct for large values of n, can sometimes be quite inaccurate for the sample size actually available. Keeping this in mind, it is still true that the standard error of an estimate usually gives a good idea of its accuracy. A simple example shows the limitations of the central limit theorem approximation. Suppose that F is a distribution that puts 1 In some books, the term "standard error" is used to denote an estimated standard deviation, that is, an estimate of t:rF based on the data.

Use a basic result of probability theory to show that 0 0 0 0 0 0 s se(x) = ,jii," (b) t Suppose instead that x 1 , x 2 , · · · , Xn is selected by random sampling without replacement (so we must have PROBLEMS 29 n::::; N), show that [N- n] ~ se(x) = _§___ y'n N-1 (c) We see that sampling without replacement gives a smaller standard error for x. Proportionally how much smaller will it be in the case of the law school data? r;b{A} = #{x; E A}jn. 9). (b) The standard error of an empirical probability is [Prob{A} · (1- Prob{A} )/nJl1 2.

5), s "'Bi(n,p). 30) satisfies n · :Pr";b{A} "'Bi(n, Prob{A} ). , Bi( n, p), then E( x) = np. 34) as 3. L,CJ 2). 8 Suppose that y and z are independent random variables, with variances a~ and a;. ) (b) Suppose F _, (x1,x2, · · · ,xn) where the probability distribution F has expectation p and variance a 2 • Show that x has expectation p and variance a 2 jn. 8) to show that a;= np(1- p) for x "'Bi(n,p). 0. Why do we know that LSAT and GPA are not independent? 11 In the discussion of random sampling, j 1 , )2, · · · , Jn were taken to be independent integers having a uniform distribution on the numbers 1,2, ..