2. BOOTSTRAP METHODS 9 Moreover,from occupancy theory for urn models vn(n-1Nn-(1-1/m))aN(0,e-1(1-2e-1)=N(0,.09720887.…)月 see e.g.Johnson and Kotz(1977),page 317,3.with r =0.]By using some other vector of exchangeable weights W rather than Mn~Multn(n,(1/n,...,1/n)),we might be able to avoid some of this discreteness caused by multinomial weights. Since the resulting measure should be a probability measure,it seems reasonable to require that the components of W should sum to n.Since the multinomial random vector with cell probabilities all equal to 1/n is exchangeable,it seems reasonable to require that the vector W have an exchangeable distribution:i.e.W=(W(1),...,W(n))4W for all permutations of {1,..,n}.Then PW Wni6X:(w) i=1 is called the exchangeably weighted bootstrap empirical measure corresponding to the weight vector W.Here are several examples. Example 2.6 (Dirichlet weights).Suppose that Yi,Y2,...are i.i.d.exponential(1)random vari- ables,and set nYi Wni三 yi+…+Yn i=1,.,n. The resulting random vector W/n has a Dirichlet(1,...,1)distribution;i.e.n-WD where the Di's are the spacings of a random sample of n-1 Uniform(0,1)random variables Example 2.7 (More general continuous weights).Other weights W of the same for as in example 1.6 are obtained by replacing the exponential distribution of the Y's by some other distribution on R+.It will turn out that the limit theory can be established for any of these weights as long as the Yi's satisfy YiL2.1;i.e.P(Y>t)dt <oo. Example 2.8 (Jackknife weights).Suppose that w=(wn,1,...,wn,n)is a vector of constants which sum ton:n=n.Let Wbearandom permutation of the coordinates of w:if Ris uni- formly distributed overΠ≡{all permutations of{l,.,n}},then W≡Ew≡(wn,1,.,wn,rn): If we take w =(n/(n-d))1n-d (n/(n-d)(1,...1,0,...0)where In-d is the vector with all 1's in the first n-d coordinates and 0's in the remaining d coordinates,then these weights Wn.i corre- spond to the delete-d jackknife.It turns out that these weights yield behavior like that of Efron's nonparametric bootstrap (with multinomial weights)only if d=dn satisfies n-dna >0. Other weights W based on various urn schemes are also possible;see Praestgaard and Wellner (1993)for some of these.2. BOOTSTRAP METHODS 9 [Moreover, from occupancy theory for urn models √ n(n −1Nn − (1 − 1/n) n ) →d N(0, e−1 (1 − 2e −1 )) = N(0, .09720887 . . .); see e.g. Johnson and Kotz (1977), page 317, 3. with r = 0.] By using some other vector of exchangeable weights W rather than Mn ∼ Multn(n,(1/n, . . . , 1/n)), we might be able to avoid some of this discreteness caused by multinomial weights. Since the resulting measure should be a probability measure, it seems reasonable to require that the components of W should sum to n. Since the multinomial random vector with cell probabilities all equal to 1/n is exchangeable, it seems reasonable to require that the vector W have an exchangeable distribution: i.e. πW ≡ (Wπ(1), . . . , Wπ(n) ) d= W for all permutations π of {1, . . . , n}. Then P W n ≡ 1 n Xn i=1 WniδXi(ω) is called the exchangeably weighted bootstrap empirical measure corresponding to the weight vector W. Here are several examples. Example 2.6 (Dirichlet weights). Suppose that Y1, Y2, . . . are i.i.d. exponential(1) random vari￾ables, and set Wni ≡ nYi Y1 + · · · + Yn , i = 1, . . . , n. The resulting random vector W/n has a Dirichlet(1, . . . , 1) distribution; i.e. n −1W d= D where the Di ’s are the spacings of a random sample of n − 1 Uniform(0, 1) random variables. Example 2.7 (More general continuous weights). Other weights W of the same for as in example 1.6 are obtained by replacing the exponential distribution of the Y ’s by some other distribution on R +. It will turn out that the limit theory can be established for any of these weights as long as the Yi ’s satisfy Yi ∈ L2,1; i.e. R ∞ 0 p P(|Y | > t)dt < ∞. Example 2.8 (Jackknife weights). Suppose that w = (wn,1, . . . , wn,n) is a vector of constants which sum to n: Pn i=1 wn,i = n. Let W be a random permutation of the coordinates of w: if R is uni￾formly distributed over Π ≡ {all permutations of {1, . . . , n}}, then W ≡ R w ≡ (wn,R1 , . . . , wn,Rn ). If we take w = (n/(n − d))1n−d = (n/(n − d)(1, . . . 1, 0, . . . 0) where 1n−d is the vector with all 1’s in the first n−d coordinates and 0’s in the remaining d coordinates, then these weights Wn,i corre￾spond to the delete -d jackknife. It turns out that these weights yield behavior like that of Efron’s nonparametric bootstrap (with multinomial weights) only if d = dn satisfies n −1dn → α > 0. Other weights W based on various urn schemes are also possible; see Praestgaard and Wellner (1993) for some of these