On bounds in multivariate Poisson approximation

Transcription

1 Workshop on New Directions in Stein s Method IMS (NUS, Singapore), May 18-29, 2015

2 Acknowledgments Thanks Institute for Mathematical Sciences (IMS, NUS), for the invitation, hospitality and accommodation support. Professor Chen Louis H. for helpful discussions on Stein-Chen method, Poisson approximation and related topics.

3 Talk Title Note: This talk is based on two our papers published in years 2013 and T. L. Hung and V. T. Thao, (2013), Bounds for the Approximation of Poisson-binomial distribution by Poisson distribution, Journal of Inequalities and Applications, 2013:30. 2 T. L. Hung and L. T. Giang, (2104), On bounds in Poisson approximation for integer-valued independent random variables, Journal of Inequalities and Applications, 2014:291.

4 Outline 1 A few words about Poisson approximation. 2 Motivation (Multivariate Poisson approximation). 3 Mathematical tools (A linear oprator, A probability distance). 4 Results (Le Cam type bounds for sums and random sums of d-dimensional independent Bernoulli distributed random vectors) 5 Remarks 6 References

5 Bernoulli s formula Let X 1, X 2,... be a sequence of independent Bernoulli random variables with success probability p, such that P (X j = 1) = p = 1 P (X j = 0), p (0, 1) Set S n = X X n the number of successes. Then Formula P (S n = k) = n! k!(n k)! pk (1 p) n k ; n 1, k = 1, 2,..., n. (1)

6 Approximation formula Suppose that the number n is large enough, and the success probability p is small, such that n p = λ. Then, Formula P (S n = k) P (Z λ = k); k = 0, 1, 2,...,. (2) where λ = E(S n ) = np = E(Z λ ) and P (Z λ = k) = e λ λ k k!. We denote Z λ P oi(λ) the Poisson random variable with mean λ.

7 Poisson limit theorem Suppose that n, p 0 such that n p λ. Then, Formula S n d Zλ, as n (3) where the notation d denotes the convergence in distribution. Note:The Poisson limit theorem suggests that the distribution of a sum of independent Bernoulli random variables with small success probabilities (p) can be approximated by the Poisson distribution with the same mean (np), if the success probabilities (p) are small and the number (n) of random variables is large.

8 Le Cam s bound, (1960) Let X 1, X 2,... be a sequence of independent Bernoulli distributed random variables such that P (X j = 1) = p j = 1 P (X j = 0) = 1 p j ; p j (0, 1). Let S n = X X n and λ n = E(S n ) = n p j. Then, Le Cam s result k=0 P (S n = k) P (Z λn = k 2 n p 2 j. (4)

9 Chen s bound, (1975) Chen s result P (S n = k) P (Z λn = k 2 ( 1 e λn) k=0 λ n n p 2 j (5)

10 Mathematical tools Operator theoretic method (Le Cam, (1960)) Stein-Chen method: 1 the local approach (Chen (1975), Arratia, Goldstein and Gordon (1989, 1990) 2 the size-bias coupling approach (Barbour((1982), Barbour, Holst and Janson (1992), Chatterjee, Diaconis and Meckes (2005))

11 Total variation distance A measure of the accuracy of the approximation is the total variation distance. For two distributions P and Q over Z + = {0, 1, 2,...}, the total variation distance between them is defined by d T V (P, Q) = sup A Z + P (A) Q(A) which is also equal to 1 2 sup hdp h =1 hdq = 1 P {i} Q{i}. 2 i Z +

12 Well-known results For the binomial distribution Bin n (p), Prohorov ((1953)) proved that [ ] 1 1 d T V (Bin n (p), P oi(np)) p + O(inf(1, ) + p 2πe n (6) Using the method of convolution operators, Le Cam ((1960)) obtained the error bounds n d T V (L(S n ), P oi(λ n )) 2 p 2 j (7) and, if max 1 j n p j 1 4, d T V (L(S n ), P oi(λ n )) 8 λ n n p 2 j (8)

13 Motivation A. Renyi, Probability Theory, ((1970)) Poisson approximation Theorem: Let X n1, X n2,... be a sequence of independent random variables which assume only the values 0 and 1, with P (X nj = 1) = p nj = 1 P (X nj = 0). Put λ n = n p nj and suppose that and Then, X n X nn d Zλ as n. lim λ n = λ (9) n lim max p nj = 0. (10) n 1 j n

14 Linear Operator Let K denote the set of all real-valued bounded functions f(x), defined on the nonnegative integers Z + = {0, 1, 2,...}. Put f = sup f(x). Let there be associated with every probability distribution P = {p 0, p 1,...} an operator defined by for every f K. A P f = f(x + r)p r, (11) r=0

15 Properties Operator A P maps the set K into itself. A P is a linear contraction operator If P and Q are any two distributions defined on the nonnegative integers, then A P A Q = A R, where R is the convolution of the distributions P and Q. The sufficient condition for proof of Poisson approximation Theorem is lim n A P Sn A PZλ = 0 (12) for f K, S n = n P oi(λ). X nj, λ n = E(S n ), λ = lim n λ n, Z λ

16 Le Cam s type bounds in Poisson approximation via operator introduced by A. Renyi Hung T. L. and Thao V. T., Journal of Inequalities and Applications, ((2013)) Theorem 1. Let X n1, X n2,... be a row-wise triangular array of independent, Bernoulli random variables with success probabilities P (X nj = 1) = 1 P (X nj = 0) = p nj, p nj (0, 1), j = 1, 2,..., n; n = 1, 2,... Let us write S n = n X nj, λ n = E(S n ). We denote by Z λn the Poisson random variable with the parameter λ n. Then, for all real-valued bounded functions f K, we have A PSn f A PZλn n 2 f p 2 nj. (13)

17 Le Cam s type bounds in Poisson approximation for Random sums Theorem 2. Let X n1, X n2,... be a row-wise triangular array of independent, Bernoulli random variables. Let {N n, n 1} be a sequence of non-negative integer-valued random variables independent of all X nj. Let us write S Nn = Nn X nj, λ Nn = E(S Nn ). Then, for all real-valued bounded functions f K, we have N n A PSNn f A PZλNn 2 f E p 2 N nj. (14)

18 Le Cam s type bounds in Poisson approximation for sum of independent integer-valued random variables Hung T. L. and Giang L. T., Journal of Inequalities and Applications, ((2014)) Theorem 3. Let X n1, X n2,... be a row-wise triangular array of independent, integer-valued random variables with success probabilities P (X nj = 1) = p nj, P (X nj = 0) = 1 p nj q nj, p nj, q nj (0, 1), p nj + q nj (0, 1), j = 1, 2,..., n; n = 1, 2,... Then, for all real-valued bounded functions f K, we have A PSn f A PZλn n 2 f (p 2 nj + q nj ). (15)

19 Le Cam s type bounds in Poisson approximation for random sum of independent integer-valued random variables Theorem 4. Let X n1, X n2,... be a row-wise triangular array of independent, integer-valued random variables with success probabilities P (X nj = 1) = p nj, P (X nj = 0) = 1 p nj q nj, p nj, q nj (0, 1), p nj + q nj (0, 1), j = 1, 2,..., n; n = 1, 2,... Let {N n, n 1} be a sequence of non-negative integer-valued random variables independent of all X nj. Then, for all real-valued bounded functions f K, we have N n A PSNn f A PZλNn 2 f E (p 2 N nj + q Nnj). (16)

20 Multivariate Poisson approximation Consider independent Bernoulli random d-vectors, X 1, X 2,... with P (X j = e (i) ) = p j,i, P (X j = 0) = 1 p j, 1 i d, 1 j n, where e (i) denotes the ith coordinate vector in R d and p j = p j,i. 1 i d Let S n = n X j, λ n = n p j.

21 Multivariate Poisson approximation McDonald (1980): for the independent Bernoulli d- random vectors d T V (P Sn, P Zλn ) := sup P (S n A) P (Z λn A) A Z+ d ( n d ) 2 (17) p j,i i=1 for S n = X 1 + X X n, and λ n = E(S n ) = (λ 1,..., λ d ), Z λn P oi(λ n ),

22 Multivariate Poisson approximation Using the Stein-Chen method, Barbour (1988) showed that d T V (P Sn, P Z ) n { ( c λn min λ n n p 2 j,i µ i=1 i ) ( d ) 2 }, p j,i i=1 (18) where c λn = log+ (2λ n ), µ i = λ 1 n p j,i, Z = (Z 1,..., Z d ), and Z 1,..., Z d are independent Poisson random variables with means λµ 1,..., λµ d. This bound is sharper than the bound in McDonald (1980).

23 Multivariate Poisson approximation Using the multivariate adaption of Kerstan s generating function method, Roos (1999) proved that d T V (P Sn, P Z ) 8.8 n { ( 1 min p j 2, λ n d i=1 p 2 j,i µ i ) } (19) which improved over (18)in removing c λn from the error bound although the absolute constant is increased to 8.8

24 Operator for d-dimensional random vector Definition Let X be a d-dinensional random vector with probability distribution P X. A linear operator A PX : K K such that (A PX f) (x) := f(x + m)p (X = m) (20) m Z d + where f K, the class of all real-valued bounded functions f on the set of all non-negative integers Z+ d and x = (x 1,..., x d ), m = (m 1,..., m d ) Z+. d The norm of function f is f = sup f(x) x Z+ d

25 Properties 1 Operator A PX is a contraction operator A PX f f 2 For the convolution of two distributions: A PX P Y f = A PX (A PY f) = A PY (A PX f) 3 Let A PXn f A PX f = o(1) as n. Then X n d X as n.

26 Inequalities 1 Let X 1,..., X n and Y 1,..., Y n be two sequences of independent d-random vectors in each group (and they are independent). Then, for f K, A n f A n f P Xj P Yj n A PXj f A PYj f 2 Let N n, n 1 be a sequence of non-negative integer random variables, independent from all X j and Y j. Then A Nn f A Nn P Xj P Yj f P (N n = k) k=1 k A PYj f A PYj f

27 Inequalities for iid. d-random vectors 1 Let X 1,..., X n and Y 1,..., Y n be two sequences of iid. d-random vectors in each group (and they are independent). Then, for f K, A n f A n f n A PX1 f A PY1 f P Xj P Yj 2 Let {N n, n 1} be a sequence of non-negative integer random variables, independent from all iid. X j and Y j. Then A Nn f A Nn P Xj P Yj f E(N n ) A PY1 f A PY1 f

28 Notations Denote by P Z d by the distribution of Poisson d-random vector λn Zλ d n, where λ n = (λ 1,..., λ d ) and P (Z λn = m) = where m = (m 1,..., λ d ) Z d +. d ( e λ j λ m j j m j! ),

29 Notations Denote P Sn by the distribution of sum S d n = X d Xd n, where X d 1,..., Xd n are independent Bernoulii d-random vectors with success probabilities P (X d k = e j) = p j,k [0, 1], k = 1, 2,..., n; j = 1, 2,... d. and d P (Xk d = (0,..., 0)) = 1 p j,k [0, 1] Here e j R d denotes the vector with entry 1 at position j and entry 0 otherwise.

30 Le Cam type bound in multivariate Poisson approximation Theorem 5. For the independent Bernoulli distributed d-dimensional random vectors. For f K, A PSn f A PZλn 2 ( n d ) 2 p j,i (21) i=1

31 Le Cam type bound in multivariate Poisson approximation for random sum Theorem 6. For the independent Bernoulli distributed d-dimensional random vectors. For f K, For {N n, n 1} the sequence of non-negative integer-value random variables independent from all X 1, X 2,..., Z λ1, Z λ2,... A PSNn f A PZλNn ( N n d ) 2 2E p j,i (22) i=1

32 Concluding Remarks 1 The operator method is elementary and elegant, but efficient in use. 2 This operator method can be applicable to a wide class of discrete d-dimensional random vectors as geometric, hypergeometric, negative binomial, etc. 3 This operator theoretic method can be applicaable to compound Poisson approximation and Poisson process approximation problems. 4 Based on this operator we can define a probability distance having a role in stydy of Poisson approximation problems.

33 Probability distance based on operator A PX A probability distance for two probability distributions P X and P Y is defined by Definition for f K. d(p X, P Y, f) := A PX f A PY = sup x Z d + Ef(X + x) Ef(Y + x) (23)

34 Properties 1 Distance d(p X, P Y, f) is a probability distance, i.e. d(p X, P Y, f) 0. d(p X, P Y, f) = 0, if P (X = Y ) = 1. d(p X, P Y, f) = d(p Y, P X, f) d(p X, P Z, f) d(p X, P Y, f) + d(p Y, P Z, f). 2 Convergence d(p Xn, P X, f) 0 as n implies convergence in distribution for X n d X, as n.

35 Properties 1 Suppose that X 1, X 2,... X n ; Y 1, Y 2,... Y n are d-dimensional independent random variables (in each group). Then, for every f K, ( ) n d P n X, j P n Y ; f d(p j Xj, P Yj ; f). (24) 2 Moreover, let {N n, n 1} be a sequence of positive integer-valued random variables that independent of X 1, X 2,..., X n and Y 1, Y 2,..., Y n. Then, for every f K, ( ) d P Nn X, P Nn j Y ; f j P (N n = k) k=1 k d(p Xj, P Yj ; f). (25)

36 Comparision with total variation distance Use the simple inequality (A. D. Barbour), for every f K, d(p X, P Y, f) 2 f d T V (P X, P Y ) Thus, we can apply the probability distance d(p X, P Y, f) to some problems related to Poisson approximation.

37 References Barbour A. D., L. Holst and S. Janson, (1992) Poisson Approximation, Clarendon Press-Oxford. Chen L. H. Y. and Röllin A., (2013), Approximating dependent rare events, Bernoulli19(4), Hung T. L., and Thao V. T.,(2013), Bounds for the Approximation of Poisson-binomial distribution by Poisson distribution, Journal of Inequalities and Applications. Hung T. L., and Giang L. T., (2014), On bounds in Poisson approximation for integer-valued independent random variables, Journal of Inequalities and Applications.

38 References Renyi A.(1970), Probability Theorem, A. Kiado, Budapest. Roos B.,(1998), Metric multivariate Poisson approximation of the generalized multinomial distribution, Teor. Veroyatnost. i Primenen. 43, Roos B.,(1999) On the rate of multivariate Poisson convergence, Journal of Multivariate Analysis, V. 69, Teerapabolarn K., (2013), A new bound on Poisson approximation for independent geometric variables, International Journal of Pure and Applied Mathematics, Vol. 84, No. 4,

39 Thanks Thanks for your attention!