Let T n \u00ba The variance of \u00ba is V n E T n E T n 2 Let T n \u00ba It has variance V n

# Let t n º the variance of º is v n e t n e t n 2

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Let T n = ^ º: The variance of ^ º is V n = E( T n ± E T n ) 2 : Let T ± n = ^ º ± : It has variance V ± n = E( T ± n ± E T ± n ) 2 : The simulation estimate is ^ V ± n = 1 B B X b =1 ´ ^ º ± b ± ^ º ± µ 2 : A bootstrap standard error for ^ º is the square root of the bootstrap estimate of variance, s ± ( ^ º ) = q ^ V ± n : While this standard error may be calculated and reported, it is not clear if it is useful. The primary use of asymptotic standard errors is to construct asymptotic con±dence intervals, which are based on the asymptotic normal approximation to the t-ratio. However, the use of the bootstrap presumes that such asymptotic approximations might be poor, in which case the normal approxi- mation is suspected. It appears superior to calculate bootstrap con±dence intervals, and we turn to this next. 6.5 Percentile Intervals For a distribution function G n ( u; F ) ; let q n ( ·; F ) denote its quantile function. This is the function which solves G n ( q n ( ·; F ) ; F ) = ·: [When G n ( u; F ) is discrete, q n ( ·; F ) may be non-unique, but we will ignore such complications.] Let q n ( · ) denote the quantile function of the true sampling distribution, and q ± n ( · ) = q n ( ·; F n ) denote the quantile function of the bootstrap distribution. Note that this function will change depending on the underlying statistic T n whose distribution is G n : Let T n = ^ º; an estimate of a parameter of interest. In (1 ± · )% of samples, ^ º lies in the region [ q n ( ·= 2) ; q n (1 ± ·= 2)] : This motivates a con±dence interval proposed by Efron: C 1 = [ q ± n ( ·= 2) ; q ± n (1 ± ·= 2)] : This is often called the percentile con°dence interval . Computationally, the quantile q ± n ( · ) is estimated by ^ q ± n ( · ) ; the · ³th sample quantile of the simulated statistics f T ± n 1 ; :::; T ± nB g ; as discussed in the section on Monte Carlo simulation. The (1 ± · )% Efron percentile interval is then [^ q ± n ( ·= 2) ; ^ q ± n (1 ± ·= 2)] : 86
The interval C 1 is a popular bootstrap con±dence interval often used in empirical practice. This is because it is easy to compute, simple to motivate, was popularized by Efron early in the history of the bootstrap, and also has the feature that it is translation invariant. That is, if we de±ne ¸ = f ( º ) as the parameter of interest for a monotonically increasing function f; then percentile method applied to this problem will produce the con±dence interval [ f ( q ± n ( ·= 2)) ; f ( q ± n (1 ± ·= 2))] ; which is a naturally good property. However, as we show now, C 1 is in a deep sense very poorly motivated. It will be useful if we introduce an alternative de±nition C 1 . Let T n ( º ) = ^ º ± º and let q n ( · ) be the quantile function of its distribution. (These are the original quantiles, with º subtracted.) Then C 1 can alternatively be written as C 1 = [ ^ º + q ± n ( ·= 2) ; ^ º + q ± n (1 ± ·= 2)] : This is a bootstrap estimate of the ´idealµcon±dence interval C 0 1 = [ ^ º + q n ( ·= 2) ; ^ º + q n (1 ± ·= 2)] : The latter has coverage probability P ° º 0 2 C 0 1 ± = P ´ ^ º + q n ( ·= 2) º 0 ^ º + q n (1 ± ·= 2) µ = P ´ ± q n (1 ± ·= 2) ^ º ± º 0 ¶ ± q n ( ·= 2) µ = G n ( ± q n ( ·= 2) ; F 0 ) ± G n ( ± q n (1 ± ·= 2) ; F 0 )

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