Coupling Binomial with normal

Transcription

1 Appendix A Coupling Binomial with normal A.1 Cutpoints At the heart of the construction in Chapter 7 lies the quantile coupling of a random variable distributed Bin(m, 1 / 2 ) with a random variable Y distributed N(m/2, m/4). The coupling relies heavily on very accurate approximations for the cutpoints = β < β 1 < < β n < β m+1 = for which P{X k} = P{Y > β k } for k =, 1,..., m. It is more convenient to work with the the tails of the standard normal Φ(z) = P{N(, 1) > z}, and the standardized cutpoint z k = 2(β k m/2)/ m, thereby replacing P{Y > β k } by Φ(z k ). The coupling is then defined via a Z distributed N(, 1) by Y = m 2 + m 4 Z X = γ m (Z) := m k= k{z J k} where J k := (z k, z k+1 ] γ m (z) γ m (z) k ( ] k ( ] m = 2L m = 2L-1 ( ]( ] ( ] z L-1 z L z L+1 z k z k+1 J k z ( ]( ] ( ] z L-1 z L z L+1 z k z k+1 Symmetry considerations show that z m k+1 = z k, so that it suffices to consider only half the range for k. ore precisely, when m = 2L is even we have only to consider k L + 1 = (m + 2)/2. When m = 2L 1 is odd we have only to consider k L + 1 = (m + 3)/2. The usual normal approximation with continuity correction suggests that β k k 1/2 or z k (2k 1 m)/ m. For the purposes of Chapter 7, we need only a sharp lower bound, J k z <1> β k k 1 2 cm 1/2 for m 1 and (m + 1)/2 k m. version: 18 July 21 printed: 18 July 21 Beijing lectures c David Pollard

2 2 Coupling Binomial with normal for some universal constant c. What follows is based on the method used by Carter and Pollard (24), who derived a slightly sharper form of the Tusnády coupling inequality (Csörgő and Révész 1981, Section 4.4). See also Pollard (21, Appendix D) for a slightly different version of the argument. A.2 Proof of inequality <1> The argument involves two steps. First obtain an upper bound for the tail probability Φ(z k ) = P{Y > β k } using Laplace s method with the exact representation of the Binomial tail as a beta integral. Then invert the inequality to obtain a lower bound for z k. A.2.1 Binomial tail probability Start from the well known (Feller 1971, Section 1.7) relationship between the tails of the Binomial and beta distributions. <2> P{X k} = m! (k 1)!(m k)! 1/2 t k 1 (1 t) m k dt. The representation takes a more convenient form with the reparametrizations = m 1, K = k 1, N = m k, and the change of variable s = 1 2t: P{X k} = m! 1 K!N! 2 1 ( ) 1 s K ( ) 1 + s N ds. 2 2 <3> For the purposes of a Laplace approximation, write the integrand as exp ( h(s)) where h(s) = 1 + ɛ ( ) 1 s log + 1 ɛ ( ) 1 + s log = h() log(1 s2 ) + ɛ ( ) 1 s 2 log 1 + s = h() 1 2 (s2 + s4 2 + s6 s ) ɛ(s s ) if s < 1. 5 The function h is maximized at s = ɛ, where ɛ := (2K )/, and <4> h( ɛ) h() = 1 2 ɛ2 + ɛ 4 G(ɛ) where G(ɛ) := r= ɛ 2r (2r + 3)(2r + 4).

3 A.2 Proof of inequality <1> 3 <5> <6> The function G is strictly increasing on [, 1] with 1 12 = G() G(ɛ) G(1) = 1 + log Stirling s formula (Feller 1968, Section II.9), n! = 2πn exp (log n n + λ n ) with 1 12n + 1 λ n 1 12n, and the representations K/ = (1 + ɛ)/2 and N/ = (1 ɛ)/2 simplifies the ratio m!/k!n! to m 2πKN exp ( log K log K N log N + Λ ) where Λ = Λ m,k := λ λ K λ N = m ( ( ) ( ) ) 4 K N 2π(1 ɛ 2 ) exp K log N log + Λ = 2 m ( 2π exp h( ɛ) + Λ 1 ) 2 log(1 ɛ2 ) In summary, to determine the standardized cutpoint z k we need to solve the equation Φ(z k ) = e 2π 1 exp ( h(s) (h( ɛ) ) ds where := log(1 + 1 ) 1 2 log(1 ɛ2 ) + Λ m,k. To get a lower bound for z k we need an upper bound for the right-hand side of <6>. Expansions <3> and <4> imply h(s) h( ɛ) 1 2 s2 ɛs 1 2 ɛ2 ɛ 4 G(ɛ) which gives Φ(z k ) exp( ɛ 4 G(ɛ)) 2π = exp( ɛ 4 G(ɛ)) Φ(ɛ ) ( exp 12 ) (s + ɛ)2 ds It is convenient to reexpress the last inequality, using the increasing function Ψ(x) := log Φ(x), as <7> Ψ(z k ) Ψ(ɛ ) + ɛ 4 G(ɛ) log(1 ɛ2 ) log(1 + 1 ) Λ m,k.

4 4 Coupling Binomial with normal A.2.2 Inversion of the tail bound The following Lemma, whose proof is given in Section 3, will covert inequality <7> to a lower bound for z k. <8> Lemma. Define Then ρ(x) := d dx Ψ(x) = φ(x)/ Φ(x) and r(x) := ρ(x) x. (i) ρ is an increasing, nonnegative function with ρ( ) = and ρ() = 2/ 2π (ii) r is a decreasing nonnegative, function with r( ) = and r() = ρ() and r(x) 2/x for x 2. oreover, for all x R and δ, the increments of Ψ satisfy the inequalities (iii) δρ(x) Ψ(x + δ) Ψ(x) δρ(x + δ), (iv) xδ δ2 Ψ(x + δ) Ψ(x) ρ(x)δ δ2. With ɛ = (2K )/, inequality <1> is equivalent to z k ɛ 2c for ɛ 1 if 1. A large enough choice of c would take care of all m small enough. Thus it more than suffices to find a constant C and an for which <9> z k ɛ C/ for all. It is easiest to split the argument into three parts, for three ranges of ɛ. In what follows, C, C 1, C 2,... will be various universal constants. Case: ɛ C 1 /. For this range, inequality <7> implies Ψ(z k ) Ψ(ɛ ) C 2 / where the constant C 2 depends on C 1. Invoke Lemma 8(iii) with δ = C 3 / and x = ɛ δ to get Ψ(ɛ ) Ψ(ɛ δ) + ρ( δ)δ

5 A.3 Tails of the normal distributions 5 If C 3 and are large enough, the δρ( δ) term is larger than C 2 /, so that Ψ(z k ) Ψ(ɛ δ), implying z k ɛ C 3 /. Case: C 1 / ɛ ɛ < 1. For this range, if C 1 is large enough (depending on ɛ ) the ɛ 4 G(ɛ) term is much bigger than the other remainder terms in <7>, implying Ψ(z k ) Ψ(ɛ ) ɛ4 Choose x = ɛ and δ = C 4 ɛ 3 in Lemma 8(iv) to get Ψ(x + δ) Ψ(x) + δρ(x) δ2 Ψ(x) + 2C 4 ɛ C2 4ɛ 4 Choose C 4 so that 2C C2 4 < 1/13 to conclude that z k ɛ (1 + C 4 ɛ 2 ). Case: ɛ ɛ 1. If ɛ is close enough to 1 then ɛ (1+C 4 ɛ 2 ) 1 ɛ. The desired lower bound for z k is trivially true. A.3 Tails of the normal distributions Needs editing <1> The classical tail bounds for the normal distribution (cf. Feller (1968), Section VII.1 and Problem 7.1) show that Φ(x) behaves roughly like the density φ(x): ( 1 x 1 ) x 3 φ(x) < Φ(x) < 1 x φ(x) Φ(x) < 1 2 exp ( x 2 /2 ) for x > The first upper bound is good for large x, the second for x. Lemma 8 interpolates smoothly between the different cases in <1>. Recall that Ψ(x) := log Φ(x) and ρ(x) = d Ψ(x) = φ(x)/ Φ(x). dx To a first approximation, the positive function ρ(x) increases like x. By inequality <1>, the error of approximation, r(x) := ρ(x) x, is positive for x > and, for x > 1, r(x) < x x 2 = O(1/x) as x. 1

6 6 Coupling Binomial with normal In fact, as shown by the proof of the next lemma, ρ( ) is increasing and r( ) is decreasing and positive, on the whole real line. <11> Lemma. The function ρ( ) is increasing and the function r( ) is decreasing, with r( ) = ρ( ) = and r() = ρ() = 2/ 2π For all x R and δ, the increments of the function Ψ(x) := log Φ(x) satisfy the inequalities (i) δρ(x) Ψ(x + δ) Ψ(x) δρ(x + δ), (ii) δr(x + δ) Ψ(x + δ) Ψ(x) 1 2 (x + δ) x2 δr(x), (iii) xδ δ2 Ψ(x + δ) Ψ(x) ρ(x)δ δ2. Proof Let Z be N(, 1) distributed. Define (x) = Pe x Z, a decreasing function of x with log (x) strictly convex. Notice that 1/ρ(x) = 2π exp ( x 2 /2 ) φ(z+x) dz = exp ( xz z 2 /2 ) π dz = 2 (x). Thus log (x) log π/2 = log ρ(x) = Ψ(x) x 2 /2 log 2π is a concave, increasing function of x with derivative ρ(x) x = r(x). It follows that r( ) is a decreasing function, because r (x) = d2 log (x) < dx2 Inequality (i) follows from the equality by convexity of log (x). Ψ(x + δ) Ψ(x) = δψ (y ) = δρ(y ) for some x < y < x + δ, together with the fact that ρ( ) is an increasing function. Similarly, the fact that d ( Ψ(y) 1 dy 2 y2) = ρ(y) y = r(y) which is a decreasing function gives inequality (ii). Inequality (iii) follows from (ii) because δr(x + δ) and xδ + r(x)δ = ρ(x)δ. A.4 Notes Carter and Pollard (24) KT method (with refinements as in the exposition by Csörgő and Révész (1981), Section 4.4)

7 References for Appendix A 7 References Carter, A. and D. Pollard (24). Tusnády s inequality revisited. Annals of Statistics 32 (6), Csörgő,. and P. Révész (1981). Strong Approximations in Probability and Statistics. New York: Academic Press. Feller, W. (1968). An Introduction to Probability Theory and Its Applications (third ed.), Volume 1. New York: Wiley. Feller, W. (1971). An Introduction to Probability Theory and Its Applications (second ed.), Volume 2. New York: Wiley. Pollard, D. (21). A User s Guide to easure Theoretic Probability. Cambridge University Press.