BOOTSTRAP CONFIDENCE INTERVALS 193 0.5 1.0 1.520 2.5 3.0 *+ FIG.2.Bootstrap distributions for the maximum eigenvalue of the covariance matrix,cd4 data:(left)2,000 parametric bootstrap replications assuming a bivariate normal distribution;(right)2,000 nonparametric bootstrap replications,discussed in Section 6.The solid lines indicate the limits of the BCa 0.90 central confidence intervals,compared to the standard intervals (dashed lines). If a and zo are zero,then OBc [a]=G-1(a),the thatξ=专+W for all values ofg,with W always 100ath percentile of the bootstrap replications.In having the same distribution.This is a translation this case the 0.90 BCa interval is the interval be- problem so we know how to set confidence limits tween the 5th and 95th percentiles of the bootstrap a]for replications.If in addition G is perfectly normal, then Ogc,[a]=0+z(),the standard interval end- (2.6) a=专-W1-, point.In general,(2.3)makes three distinct correc- where wd-a)is the 100(1-a)th percentile of W tions to the standard intervals,improving their The BCa interval(2.3)is exactly equivalent to the coverage accuracy from first to second order. translation interval(2.6),and in this sense it is cor- The c.d.f.G is markedly long-tailed to the rect as well as accurate. right,on the normal-theory side of Figure 2. The bias-correction constant zo is easy to inter- Also a and zo are both estimated to be positive, pret in (2.5)since (a,)=(0.105,0.226),further shifting 0nc [a]to the right of OsTAN[a]=+()The 0.90 BC (2.7) Prob{中<中}=(zo): interval for 6 is Then Prob<}=(zo)because of monotonicity. (2.4)(G-1(0.157),G-1(0.995)=(1.10,3.18) The BC algorithm,in its simplest form,estimates zo by compared to the standard interval(0.80,2.55). The following argument motivates the BCa def- 0=-1 #{*(b)<创 (2.8) inition (2.3),as well as the parameters a and zo. B Suppose that there exists a monotone increasing -1 of the proportion of the bootstrap replications transformation中=m(f)such that中=m(0is less than 6.Of the 2,000 normal-theory bootstrap normally distributed for every choice of 0,but pos- replications shown in the left panel of Fig- sibly with a bias and a nonconstant variance, ure 2,1179 were less than 6=1.68.This gave (2.5)中~N(中-z00b,o6),06=1+a中 2o=-(0.593)=0.226,a positive bias correction since is biased downward relative to 6.An often Then(2.3)gives exactly accurate and correct confi- more accurate method of estimating zo is described dence limits for 0 having observed 6. in Section 4. The argument in Section 3 of Efron(1987)shows The acceleration a in(2.5)measures how quickly that in situation (2.5)there is another monotone the standard error is changing on the normalized transformation,.say专=M(a)and专=M(a),such scale.The value a =0.105 in (2.4),obtained fromBOOTSTRAP CONFIDENCE INTERVALS 193 Fig. 2. Bootstrap distributions for the maximum eigenvalue of the covariance matrix, cd4 data: (left) 2,000 parametric bootstrap replications assuming a bivariate normal distribution; (right) 2,000 nonparametric bootstrap replications, discussed in Section 6. The solid lines indicate the limits of the BCa 0:90 central confidence intervals, compared to the standard intervals (dashed lines). If a and z0 are zero, then θˆBCa ’α = Gˆ −1 α‘, the 100αth percentile of the bootstrap replications. In this case the 0.90 BCa interval is the interval be￾tween the 5th and 95th percentiles of the bootstrap replications. If in addition Gˆ is perfectly normal, then θˆBCa ’α = θˆ + z α‘σˆ , the standard interval end￾point. In general, (2.3) makes three distinct correc￾tions to the standard intervals, improving their coverage accuracy from first to second order. The c.d.f. Gˆ is markedly long-tailed to the right, on the normal-theory side of Figure 2. Also a and z0 are both estimated to be positive, aˆ; zˆ0 ‘ = 0:105; 0:226‘, further shifting θˆBCa ’α to the right of θˆ STAN’α = θˆ + z α‘σˆ . The 0.90 BCa interval for θ is 2:4‘ Gˆ −1 0:157‘; Gˆ −1 0:995‘‘ = 1:10; 3:18‘; compared to the standard interval (0.80, 2.55). The following argument motivates the BCa def￾inition (2.3), as well as the parameters a and z0 . Suppose that there exists a monotone increasing transformation φ = mθ‘ such that φˆ = mθˆ‘ is normally distributed for every choice of θ, but pos￾sibly with a bias and a nonconstant variance, 2:5‘ φˆ ∼ Nφ − z0σφ; σ 2 φ ‘; σφ = 1 + aφ: Then (2.3) gives exactly accurate and correct confi- dence limits for θ having observed θˆ. The argument in Section 3 of Efron (1987) shows that in situation (2.5) there is another monotone transformation, say ξ = Mθ‘ and ξˆ = Mθˆ‘, such that ξˆ = ξ + W for all values of ξ, with W always having the same distribution. This is a translation problem so we know how to set confidence limits ξˆ’α for ξ, 2:6‘ ξˆ’α = ξ − W1−α‘ ; where W1−α‘ is the 1001 − α‘th percentile of W. The BCa interval (2.3) is exactly equivalent to the translation interval (2.6), and in this sense it is cor￾rect as well as accurate. The bias-correction constant z0 is easy to inter￾pret in (2.5) since 2:7‘ Probφˆ < φ• = 8z0 ‘: Then Probθˆ < θ• = 8z0 ‘ because of monotonicity. The BCa algorithm, in its simplest form, estimates z0 by 2:8‘ zˆ0 = 8 −1  #θˆ ∗ b‘ < θˆ• B  ; 8−1 of the proportion of the bootstrap replications less than θˆ. Of the 2,000 normal-theory bootstrap replications θˆ ∗ shown in the left panel of Fig￾ure 2, 1179 were less than θˆ = 1:68. This gave zˆ0 = 8−1 0:593‘ = 0:226, a positive bias correction since θˆ ∗ is biased downward relative to θˆ. An often more accurate method of estimating z0 is described in Section 4. The acceleration a in (2.5) measures how quickly the standard error is changing on the normalized scale. The value aˆ = 0:105 in (2.4), obtained from