Bootstrap - theoretical problems

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1 Date: January 23th 2006 Bootstrap - theoretical problems This is a new version of the problems. There is an added subproblem in problem 4, problem 6 is completely rewritten, the assumptions in problem 7 have been sharpened and subproblem 7.2 has been rewritten to conform with problem 6. Finally there is a new problem 8. The new problems are not added to increase the workload, they are merely added as a help. If you can solve the old problems without solving the new problems on the way, then fine by me. 4. Marcinkiewicz-Zygmund SLLN Let X, X 2,... be independent and identically distributed variables with values in R k, with E X i 2 <. Assume E X i = 0, and let V X i = Σ. OPGAVE 4.. Show that n X i 2 Tr Σ a.e. for n OPGAVE 4.2. Show that for any c > 0 it holds that Hint: Borel-Cantellis lemma. OPGAVE 4.3. Show that P ( X n > c infinitely often ) = 0 max,...,n X i 0 a.e. for n. OPGAVE 4.4. Combine the pieces to obtain the following version of the Marcinkiewicz- Zygmund SLLN: n 3/2 X i 3 0 a.e. for n. (This is trivial if E X i 3 <, but the catch is that we have only assumed the existence of second moments.)

2 2 OPGAVE 4.5. Extend the Marcinkiewicz-Zygmund SLLN to the following situation: n 3/2 where X n = n n X i. X i X n 3 0 a.e. for n, 5. CLT for a simple triangular array Let (X nm ) be a triangular array of stochastic variables with values in R k. We assume that the variables in each row of the array are independent and identically distributed. That is, X n, X n2,..., X nn are iid for each n. But two variables in different rows may well be dependent, and they may well have different distributions. Assume that E X nm 3 < for all n N and all m =,..., n. Let Σ n = V X nm. OPGAVE 5.. Show that if E X nm = 0 for all n and m, and if lim Σ n = Σ, n lim n n 3/2 E X nm 3 = 0, then D X nm N(0, Σ) m= for n 6. Multivariate bootstrap CLT Let X, X 2,... be a sequence of independent and identically distributed stochastic variables with values in R k. Assume that E X n 2 < and that EX n = 0. Let Σ = V X n. We use the notation X = (X, X 2,...) to refer to the entire sequence. Let ˆµ n = n X i.

3 3 Let (Xnm ) be a bootstrap array corresponding to X. That is, conditionally on the entire sequence X the variables Xn, X n2,..., X nn in the n th row of the array are independent and identically distributed with distribution ˆν n = n ε Xi OPGAVE 6.. Show that conditionally on X it holds that ( X ) D nm ˆµ n N(0, Σ) a.e. m= EXPLANATION: The statement is probably hard to digest... If we condition on X = x for some specific R k -sequence x, then it may or may not happen that ( X ) D nm ˆµ n N(0, Σ) m= Whether it holds or not, depends on x. The claim is the set A of x s for which the weak convergence does take place, is so large that P(X A) =. 7. A general bootstrap consistency theorem Let Y, Y 2,... be a sequence of independent and identically distributed variables with values in a space (Y, K). The distribution ν of the individual variables is assumed completely unknown, except for a moment conditions to be made precise in (7.) below. We write Y = (Y, Y 2,...) when we refer to the entire sequence of variables. Let h : Y R k be a measurable mapping, and let g : R k R be a C -map. We consider the parameter function ( ) τ(ν) = g h(y) dν(y) We consider only measures ν satisfying h(y) 2 dν(y) < (7.)

4 4 We apply the plug-in estimator ˆτ n = g n h(y i ) = τ(ˆν n) and consider the combinant ˆτ n τ(ν). OPGAVE 7.. Show that ˆτ n τ(ν) as N (0, n ) σ(ν)2 when ν is the true probability measure. Here the variance is given by the formula Σ(ν) = V ν h(y i ), ( σ(ν) 2 = g T ( h dν) Σ(ν) g ) h dν Let (Ynm ) be a bootstrap array corresponding to Y. That is, conditionally on the entire sequence Y, the variables Yn, Y n2,..., Y nn in the n th row of the array are independent and identically distributed with distribution ˆν n = n ε Yi Consider the bootstrap analogue of the combinant in question, τ(ˆν n ) τ(ˆν n), where ˆν n is the empirical distribution based on Y n, Y n2,..., Y nn. OPGAVE 7.2. Show that conditionally on Y it holds that ˆτ n τ(ˆν n ) as N (0, n ) σ(ν)2 a.e. Hint: use problem 6 and the version of the delta method provided in problem 8. Let F n be the distribution function corresponding to the combinant ˆτ n τ(ν) and let F n be the (stochastic) distribution function corresponding to the bootstrap analogue τ(ˆν n) τ(ˆν n ). OPGAVE 7.3. Show that sup F n (x) Fn(x) 0 for n a.e. x R

5 5 8. A uniform delta method Let Z, Z, Z 2,... be stochastic variables with values in R, and let g : R R be a C -map. Let a, a 2,... be a sequence of real numbers, converging to a limit a for n. Assume that (Zn a n ) D Z for n. and consider the variable W n = g(z n ) g(a n ) g (a) (Z n a n ) Note that the derivative is calculated in the limit point a. OPGAVE 8.. Show that Z n P a for n. OPGAVE 8.2. Pick for ɛ > 0 an r > 0 such that Show that for any δ > 0 it holds that for n large enough. g (x) g (a) < ɛ when x a < r P ( n Wn ) ( > δ P n (Zn a n ) δ ) > + P ( Z n a r) ɛ OPGAVE 8.3. Show that W n P 0 and conclude that ( g(zn ) g(a n ) ) D g (a) Z for n OPGAVE 8.4. Generalize this result to hold if Z, Z 2,... have values in R k, if a, a 2,... is a convergent sequence in R k, and if g is a C -map R k R.