Multivariate approximation in total variation

Transcription

1 Multivariate approximation in total variation A. D. Barbour, M. J. Luczak and A. Xia Department of Mathematics and Statistics The University of Melbourne, VIC May, 2015 [Slide 1]

2 Background Information Two corner-stone distributions Continuous: normal and the central limit theorem, in the Kolmogorov distance Discrete: Poisson and the law of small numbers, in the total variation [Slide 2]

3 Poisson approximation The Stein Chen method (Chen, 1975) Barbour and Hall (1984) { } n min λ, 1 p 2 i d T V (L(W ), Pn(λ)) 1 e λ λ i=1 n p 2 i. i=1 The message: Poisson has only one parameter, so to improve on the quality of approximation, more parameters should be introduced. [Slide 3]

4 Improvements Translated Poisson (Röllin 2005), perturbation (Barbour and X. 1999, Barbour, Čekanavičius and X. 2007), polynomial birth-death (Brown and X. 2001), zero biasing with birth-death process (Goldstein and X. 2006), discretised normal (Roy 2003, Chen and Leong (2010), and Fang (2014)). [Slide 4]

5 Multivariate approximation Normal: Gotze 1991, Goldstein and Rinott 1996, Rinott and Rotar (1996),... Discrete version? Multivariate Poisson (Barbour 1988, Roos 1999) How to introduce more parameters? Multivariate Poisson with negative correlation? Transform a multivariate Poisson? Discretise multivariate normal? Stein s identity? How to extract Stein s constants? [Slide 5]

6 One dimension Stein s equation for Poisson(λ): λg(i + 1) ig(i) = f(i) Pn(λ)(f). Birth-death process interpretation (Barbour 1988): by taking g(i) = h(i) h(i 1), Ah(i) := λ(h(i+1) h(i))+i(h(i 1) h(i)) = f(i) Pn(λ)(f). A is the generator of the birth-death process with birth rate λ and each individual has lifetime of exp(1), independent of others. [Slide 6]

7 The solution of the Stein equation is h(i) = 0 [Ef(Z i (t)) Pn(λ)(f)]dt, where Z i is a birth-death process with generator A and initial value Z i (0) = i. Stein s constants are from estimates of h := sup i h(i + 1) h(i) and 2 h := sup i h(i + 1) h(i). [Slide 7]

8 Discrete CLT: Goldstein and X Ah(i) := α i (h(i + 1) h(i)) + β i (h(i 1) h(i)), where α i = σ 2, i κ, σ 2 + µ i i κ 1, β i = σ 2 + i µ i κ, σ 2 i κ 1. A is the generator of the bilateral birth-death processes with birth rates {α i : i Z} and death rates {β i : i Z}. Similarly, h can be represented in terms of Z i (t), t 0, the Markov process with generator A and initial value i. If µ σ 2 is an integer and κ = µ σ 2, the states < κ are transient: translated Poisson. [Slide 8]

9 Fang 2014 Works on σ 2 f (s) (s µ)f(s) = h(s) Eh(Z µ,σ 2), where h is constant on [i 0.5, i + 0.5) for i Z, so can extract Stein s factors for discretised normal from those of normal approximation. [Slide 9]

10 Multivariate d-dimensional Ornstein-Uhlenbeck operator Ah(w) = n 2 Tr(σ2 D 2 h(w)) + Dh T (w)a(w nc), w R d on twice differentiable functions h : R d R. A is a matrix whose eigenvalues all have negative real parts. σ 2 is a positive definite symmetric matrix. The equilibrium distribution is N d (nc, nσ) with Σ = 0 e At σ 2 e AT t dt. AΣ + ΣA T + σ 2 = 0. [Slide 10]

11 Discrete version Consider DN d (nc, nσ) as discretised version of N d (nc, nσ): for (i 1,..., i d ) Z d, DN d (nc, nσ){(i 1,..., i d )} = N d (nc, nσ)([i 1 0.5, i ) [i d 0.5, i d + 0.5)). Fix c R d. For h : Z d R, define the generator A n h(w) = n 2 Tr ( σ 2 2 h(w) ) + h T (w)a(w nc), w Z d. A, σ 2 : as above. j h(w) = h(w + e (j) ) h(w), 2 jk h(w) = j ( k h)(w), 1 j, k d. [Slide 11]

12 Markov population process X n has transition rates ng J (n 1 w) from w to w + J, w Z d, J J, where J is a finite subset of Z d. g J is twice continuously differentiable. Define F (ξ) = J J JgJ (ξ), we assume dξ dt = F (ξ) has an equilibrium point c so F (c) = 0 and A = DF (c). This Markov population process has the equilibrium distribution Π n. [Slide 12]

13 Modification of the Markov population process X n has small chance to drift away from its centre nc but if it happens,... DN d (nc, nσ) essentially distributes in a region of radius nδ around nc: { I n,δ = ξ Z d : } (ξ nc) T Σ 1 (ξ nc) nδ. We can focus on approximating any arbitrary Z d -valued random element W in I n,δ first and then add in the errors outside this region We modify X n to keep it in I n,δ [Slide 13]

14 Modification (2) Modify X n to Xn δ with transition rates from w to w + J: ng J (n 1 w), if w nc, w + J nc ngδ J (n 1 w) = I n,δ, 0, otherwise. If Xn δ starts in I n,δ, then it stays in I n,δ. We assume conditions to ensure it is irreducible so it has the unique equilibrium distribution Π δ n [Slide 14]

15 Π δ n approximation Generator A δ n(w) = n J J g J δ (n 1 w){h(w + J) h(w)}, w Z d Stein s equation: A δ nh B (w) = 1 B (w) Π δ n(b), B I n,δ. The solution h B can be written as h B (w) = 0 (P(X δ n(t) B X δ n(0) = w) Π δ n(b))dt. [Slide 15]

16 Estimates: h B (w) c 0 ln n, h B (w) c 1 n 1/2 ln n; 2 h B (w) c 2 n 1 ln n for w in a nδ/4-neighbourhood of nc and n sufficiently large. [Slide 16]

17 For any Z d valued random element W, B I n,δ, we have P(W B) Π δ n(b) = E{(1 B (W ) Π δ n(b))1 W In,δ } Π n (B)P(W I n,δ ) = E{A δ nh B (W )1 W In,δ } Π n (B)P(W I n,δ ), hence d T V (L(W ), Π δ n) sup E{A δ nh B (W )1 W In,δ } + P(W I n,δ ). B I n,δ }{{} Bienaymé-Chebyshev ineq. [Slide 17]

18 Multivariate in total variation Assume E W nc 2 V n, d T V (L(W ), L(W + e (j) )) ε 1, 1 j d, and E{A n h(w )}1 [ W nc c1 n] c 2 h ε }{{} 20 + c 3 n 1/2 h }{{} c ln n c ln n ε 21 + c 4 n 2 h ε }{{} 22, c ln n where h is the solution for A δ n, h, h and 2 h are estimates around an ηn neighbourhood of nc. [Slide 18]

19 Then d T V (L(W ), DN d (nc, nσ)) O ln n n} 1/2 {{} +ε 1 + ε 20 + ε 21 + ε 22. diff of DN d and Π n [Slide 19]