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• 1. PART I Approximation, Bounds, and Inequalities &#xA9;2001 CRC Press LLC
• 2. 1 Nonuniform Bounds in Probability Approximations Using Stein&#x2019;s Method Louis H. Y. Chen National University of Singapore, Republic of Singapore ABSTRACT Most of the work on Stein&#x2019;s method deals with uniform error bounds. In this paper, we discuss non-uniform error bounds using Stein&#x2019;s method in Poisson, binomial, and normal approximations. Keywords and phrases Stein&#x2019;s method, non-uniform bounds, proba- bility approximations, Poisson approximation, binomial approximation, normal approximation, concentration inequality approach, binary expan- sion of a random integer 1.1 Introduction In 1972 Stein introduced a method of normal approximation which does not depend on Fourier analysis but involves solving a di&#xFB00;erential equa- tion. Although his method was for normal approximation, his ideas are applicable to other probability approximations. The method also works better than the Fourier analytic method for dependent random variables, particularly if the dependence is local or of a combinatorial nature. Since the publication of this seminal work of Stein, numerous papers have been written and Stein&#x2019;s ideas applied in many di&#xFB00;erent con- texts of probability approximation. Most notable of these works are in normal approximation, Poisson approximation, Poisson process approxi- mation, compound Poisson approximation and binomial approximation. An account of Stein&#x2019;s method and a brief history of its developments can be found in Chen (1998). In this paper we discuss another aspect of the application of Stein&#x2019;s method, not in terms of the approximating distribution but in terms of the nature of the error bound. Most of the papers on Stein&#x2019;s method deal with uniform error bounds. We show that Stein&#x2019;s method can also &#xA9;2001 CRC Press LLC
• 3. be applied to obtain non-uniform error bounds and of the best possible order. Roughly speaking, a uniform bound is one on a metric between two distributions. Whereas a non-uniform bound on the discrepancy between two distributions, L(W) and L(Z), is one on |Eh(W)&#x2212;Eh(Z)|, which depends on h for every h in a separating class. We will consider non-uniform bounds in three di&#xFB00;erent contexts, Poisson, binomial, and normal. In the exposition below, we will focus more on ideas than on technical details. 1.2 Poisson Approximation Poisson approximation using Stein&#x2019;s method was &#xFB01;rst investigated by Chen (1975a). Since then many developments have taken place and Poisson approximation has been applied to such diverse &#xFB01;elds as ran- dom graphs, molecular biology, computer science, probabilistic number theory, extreme value theory, spatial statistics, and reliability theory, where many problems can be phrased in terms of dependent events. See for example Arratia, Goldstein, and Gordon (1990), Barbour, Holst, and Janson (1992) and Chen (1993). All these results of Poisson approxima- tion concern error bounds on the total variation distance between the distribution of a sum of dependent indicator random variables and a Poisson distribution. These bounds are therefore uniform bounds. The possibility of nonuniform bounds in Poisson approximation using Stein&#x2019;s method was &#xFB01;rst mentioned in Chen (1975b). For independent indicator random variables, nonuniform bounds were &#xFB01;rst obtained for small and moderate &#x3BB; by Chen and Choi (1992) and for unrestricted &#x3BB; with improved results by Barbour, Chen, and Choi (1995). To explain the ideas behind obtaining nonuniform bounds, we &#xFB01;rst illustrate how a uniform bound is obtained in the context of independent indicator random variables. Let X1, . . . , Xn be independent indicator random variables with P(Xi = 1) = 1 &#x2212; P(Xi = 0) = pi, i = 1, ..., n. De&#xFB01;ne W = n i=1 Xi, W(i) = W &#x2212; Xi, &#x3BB; = n i=1 pi and Z to be a Poisson random variable with mean &#x3BB;. Let fh be the solution (which is unique except at 0) of the Stein equation &#x3BB;f(w + 1) &#x2212; wf(w) = h(w) &#x2212; Eh(Z) where h is a bounded real-valued function de&#xFB01;ned on Z+ = {0, 1, 2, . . .}. Then we have &#xA9;2001 CRC Press LLC
• 4. Eh(W) &#x2212; Eh(Z) = E {h(W) &#x2212; Eh(Z)} = E {&#x3BB;fh(W + 1) &#x2212; Wfh(W)} = n i=1 p2 i E fh(W(i) + 1) (1.2.1) where f(w) = f(w + 1) &#x2212; f(w). A result of Barbour and Eagleson (1983) states that fh &#x221E; &#x2264; 2(1 &#x2227; &#x3BB;&#x2212;1 ) h &#x221E;. Applying this result, we obtain dT V (L(W), L(Z)) = sup A |P(W &#x2208; A) &#x2212; P(Z &#x2208; A)| = (1/2) sup |h|=1 |Eh(W) &#x2212; Eh(Z)| &#x2264; (1 &#x2227; &#x3BB;&#x2212;1 ) n i=1 p2 i (1.2.2) where dT V denotes the total variation distance. It is known that the absolute constant 1 is best possible and the factor (1 &#x2227; &#x3BB;&#x2212;1 ) has the correct order for both small and large values of &#x3BB;. The signi&#xFB01;cance of the factor (1 &#x2227; &#x3BB;&#x2212;1 ) is explained in Chapter 1 of Barbour, Holst, and Janson (1992). To obtain a nonuniform bound, we let Ai(r) = P(W(i) = r) P(Z = r) . Then (1.2.1) can be rewritten as Eh(W) &#x2212; Eh(Z) = n i=1 p2 i EAi(Z) fh(Z + 1) (1.2.3) where h is no longer assumed to be bounded. Let C&#x2217; = sup1&#x2264;i&#x2264;n supr&#x2265;0 Ai(r). Then |Eh(W) &#x2212; Eh(Z)| &#x2264; C&#x2217; n i=1 p2 i E| fh(Z + 1)|. What remains to be done is to calculate or bound C&#x2217; and E| fh(Z+1)|. In Barbour, Chen and Choi (1995), it is shown that for max1&#x2264;i&#x2264;n pi &#x2264; 1/2, C&#x2217; &#x2264; 4e13/12 &#x221A; &#x3C0; and the following theorem was proved. &#xA9;2001 CRC Press LLC
• 5. THEOREM 1.2.1 [Theorem 3.1 in Barbour, Chen, and Choi (1995)] Let h be a real-valued function de&#xFB01;ned on Z+ such that EZ2 |h(Z)| &lt; &#x221E;. We have |Eh(W) &#x2212; Eh(Z)| &#x2264; C&#x2217; n i=1 p2 i [4(1 &#x2227; &#x3BB;&#x2212;1 )E|h(Z + 1)| + E|h(Z + 2)| &#x2212;2E|h(Z + 1)| + E|h(Z)|]/2. (1.2.4) If |h| = 1, then we have dT V (L(W), L(Z)) = (1/2) sup |h|=1 |Eh(W) &#x2212; Eh(Z)| &#x2264; C&#x2217; (1 &#x2227; &#x3BB;&#x2212;1 ) n i=1 p2 i where the upper bound has the same order as that of (2.2), but it has a larger absolute constant. However, the bound in (2.4) allows a very wide choice of possible functions h, and therefore contains more information than the total variation distance bound in (2.2). By iterating (2.1), we obtain Eh(W) &#x2212; Eh(Z) = n i=1 p2 i E fh(Z + 1) + second order terms = &#x2212; 1 2 n i=1 p2 i E 2 h(Z) + second order terms where E&#x2206;fh(Z + 1) = &#x2212;(1/2)E&#x2206;2 h(Z) (see, for example, Chen and Choi (1992), p.1871). In Barbour, Chen, and Choi (1995), a more re&#xFB01;ned result (Theorem 3.2) was obtained by bounding the second order error terms in the same way the &#xFB01;rst order error terms were bounded. From this theorem, a large deviation result (Theorem 4.2) was proved which produces the following corollary. COROLLARY 1.2.2 Let z = &#x3BB; + &#x3BE; &#x221A; &#x3BB;. Suppose max1&#x2264;i&#x2264;n pi &#x2192; 0 and &#x3BE; = o [&#x3BB;/ n i=1 p2 i ]1/2 as n &#x2192; &#x221E;. Then, as n, z and &#x3BE; &#x2192; &#x221E;, P(W &#x2265; z) P(Z &#x2265; z) &#x2212; 1 &#x223C; &#x2212; &#x3BE;2 2&#x3BB; n i=1 p2 i . &#xA9;2001 CRC Press LLC
• 6. The following asymptotic result was also deduced. THEOREM 1.2.3 Let N be a standard normal random variable. Let h be a nonnegative function de&#xFB01;ned on R which is continuous almost everywhere and not identically zero. Suppose that (Z&#x2212;&#x3BB;&#x221A; &#x3BB; )4 h(Z&#x2212;&#x3BB;&#x221A; &#x3BB; ) : &#x3BB; &#x2265; 1 is uniformly in- tegrable. Then as &#x3BB; &#x2192; &#x221E; such that max1&#x2264;i&#x2264;n pi &#x2192; 0, &#x221E; r=0 h( r &#x2212; &#x3BB; &#x221A; &#x3BB; )|P(W = r) &#x2212; P(Z = r)| &#x223C; 1 2&#x3BB; ( n i=1 p2 i )E|N2 &#x2212; 1|h(N). By letting h &#x2261; 1, E|N2 &#x2212; 1|h(N) = E|N2 &#x2212; 1| = 2 2/(&#x3C0;e), and Theorem 2.3 yields a result of Barbour and Hall (1984a, p. 477) and Theorem 1.2 of Deheuvels and Pfeifer (1986). Nonuniform bounds in compound Poisson approximation on a group for small and moderate &#x3BB; were &#xFB01;rst obtained by Chen (1975b) and later generalized and re&#xFB01;ned by Chen and Roos (1995). In these papers, the techniques were inspired by Stein&#x2019;s method. The &#xFB01;rst paper on com- pound Poisson approximation using Stein&#x2019;s method directly was by Bar- bour, Chen, and Loh (1992). 1.3 Binomial Approximation: Binary Expansion of a Random Integer In his monograph, Stein (1986) considered the following problem. Let n be a natural number and let X denote a random variable uniformly distributed over the set {0, 1, , n &#x2212; 1}. Let W denote the number of ones in the binary expansion of X and let Z be a binomial random variable with parameters (k, 1/2), where k is the unique integer such that 2k&#x2212;1 &lt; n &#x2264; 2k . If n = 2k , then W has the same distribution as Z, otherwise it is a sum of dependent indicator random variables. By using the solution of the Stein equation (k &#x2212; x)f(x) &#x2212; xf(x &#x2212; 1) = h(x) &#x2212; Eh(Z) (1.3.1) where h = I{r} and r = 0, 1, . . . , k, Stein (1986) proved that sup 0&#x2264;r&#x2264;k |P(W = r) &#x2212; P(Z = r)| &#x2264; 4/k. Diaconis (1977), jointly with Stein, proved a normal approximation re- sult for W with an error bound of order 1/ &#x221A; k. A combination of this &#xA9;2001 CRC Press LLC
• 7. result with the normal approximation to the binomial distribution shows that sup0&#x2264;r&#x2264;k |P(W &#x2264; r) &#x2212; P(Z &#x2264; r)| is of the order of 1/ &#x221A; k. Loh (1992) obtained a bound on the solution of a multivariate version of (3.1) using the probabilistic approach of Barbour (1988). Using this result of Loh and arguments in Stein (1986), we can obtain a bound of order 1/ &#x221A; k on the total variation distance between L(W) and L(Z). In an unpublished work of Chen and Soon (1994) which was based on the Ph.D. dissertation of the latter, the method of obtaining nonuni- form bounds in Poisson approximation was applied to the approximation of L(W) by L(Z). Apart from proving other results, this work shows that the total variation distance between L(W) and L(Z) is, in many instances, of much small order than 1/ &#x221A; k. Let X = k i=1 Xi2k&#x2212;i for the binary expansion of X and W = k i=1 Xi. In Stein (1986, pp. 44&#x2013;45), it is shown that Eh(W) &#x2212; Eh(Z) = EQfh(W) (1.3.2) where Q = |{j : Xj = 0 or X +2k&#x2212;j &#x2265; n}| and fh is the solution of (3.1) with h being a real-valued function de&#xFB01;ned on {0, 1, . . . , k}. De&#xFB01;ne &#x3C8;(r) = E[Q|W = r] and A(r) = P(W = r) P(Z = r) . Then (3.2) can be written as Eh(W) &#x2212; Eh(Z) = E&#x3C8;(Z)A(Z)fh(Z). (1.3.3) Let lk be the number of consecutive 1s, starting from the beginning in the binary expansion of n &#x2212; 1. The relationship between n &#x2212; 1 and lk is given by n &#x2212; 1 = lk i=1 2k&#x2212;i + m where 0 &#x2264; m &lt; 2k&#x2212;lk&#x2212;1 . It is shown in Chen and Soon (1994) that for 0 &#x2264; r &#x2264; k &#x2212; 1, lk/k &#x2264; A(r) &#x2264; 2. By obtaining upper and lower bounds on the right hand side of (3.3), the following theorem was proved. THEOREM 1.3.1 Assume that 2k&#x2212;1 &lt; n &lt; 2k . (i) If limk&#x2192;&#x221E; lk&#x221A; k = &#x221E;, then dT V (L(W), L(Z)) 2&#x2212;lk . (ii) If lim supk&#x2192;&#x221E; lk&#x221A; k &lt; &#x221E;, then dT V (L(W), L(Z)) 2&#x2212;lk lk &#x221A; k &#xA9;2001 CRC Press LLC
• 8. where xk yk means that there exist positive constants a &lt; b such that a &#x2264; xk/yk &#x2264; b for su&#xFB03;ciently large k. From this theorem it follows that dT V (L(W), L(Z)) 1 &#x221A; k if and only if 0 &lt; lim inf k&#x2192;&#x221E; lk &#x2264; lim sup k&#x2192;&#x221E; lk &lt; &#x221E;. The following theorems were also proved. THEOREM 1.3.2 |Eh(W) &#x2212; Eh(Z)| &#x2264; 13 &#x221A; k E Z &#x2212; [k/2] &#x2212; 1 k/4 (|h(Z)| + |h(Z + 1)| + 2|Eh(Z)|) . THEOREM 1.3.3 Let a = [k/2] + bk where bk/ &#x221A; k &#x2192; &#x221E; and bk/k &#x2192; 0 as k &#x2192; &#x221E;. If lk = l for all su&#xFB03;ciently large k, then P(W &#x2265; a) P(Z &#x2265; a) &#x2212; 1 &#x223C; &#x2212;2&#x3C8; k 2 bk k as k &#x2192; &#x221E;, where l(1/2 &#x2212; (l &#x2212; 1)/[2(k &#x2212; l + 1)])l+1 &lt; &#x3C8;([k/2]) &#x2264; 3. Theorem 3.3 is in fact a corollary of a more general large deviation theorem. 1.4 Normal Approximation Let X1, . . . , Xn be independent random variables with EXi = 0, var(Xi) = &#x3C3;2 i , E|Xi|3 = &#x3B3;i &lt; &#x221E; and n i=1 &#x3C3;2 i = 1. Let F be the distribution function of n i=1 Xi and let &#x3A6; be the standard normal distribution func- tion. The Berry-Esseen Theorem states that sup &#x2212;&#x221E;&lt;x&lt;&#x221E; |F(x) &#x2212; &#x3A6;(x)| &#x2264; C n i=1 &#x3B3;i &#xA9;2001 CRC Press LLC
• 9. where C is an absolute constant. The smallest value of C, obtained so far by Van Beek (1972) (without using computers), is 0.7975. If X1, . . . , Xn are independent and identically distributed, then sup &#x2212;&#x221E;&lt;x&lt;&#x221E; |F(x) &#x2212; &#x3A6;(x)| &#x2264; Cn&#x3B3; where &#x3B3; = &#x3B3;i for i = 1, . . . , n. Nonuniform bounds were &#xFB01;rst obtained by Esseen (1945) who proved that for the i.i.d. case |F(x) &#x2212; &#x3A6;(x)| &#x2264; &#x3BB; log n &#x221A; n(1 + x2) and |F(x) &#x2212; &#x3A6;(x)| &#x2264; &#x3BB; log(2 + |x|) &#x221A; n(1 + x2) where &#x3BB; depends on n3/2 &#x3B3;. Nagaev (1965) improved the upper bounds to Cn&#x3B3;/(1+|x|3 ), also for the i.i.d. case. This was generalized by Bikelis (1966) who proved that, for independent and not necessarily identically distributed random variables, |F(x) &#x2212; &#x3A6;(x)| &#x2264; C n i=1 &#x3B3;i 1 + |x|3 where C is an absolute constant. Paditz (1977) calculated C to be 114.7 and Michel (1981) reduced it to 30.54 for the i.i.d. case. All the above proofs used the Fourier analytic method. Chen and Shao (2000) used Stein&#x2019;s method to prove the following more general result: |F(x)&#x2212;&#x3A6;(x)| &#x2264; C n i=1 EX2 i I(|Xi| &gt; 1 + |x|) (1 + |x|)2 + E|Xi|3 I(|Xi| &#x2264; 1 + |x|) (1 + |x|)3 where the existence of third moments is no longer assumed. Their proof is based on truncation and the concentration inequality approach. The concentration inequality approach was originally used by Stein for the i.i.d. case (see Ho and Chen (1978)). It was extended by Chen (1986) to dependent and non-identically distributed random variables with ar- bitrary index set. A proof of the Berry-Esseen Theorem for independent and non-identically distributed random variables using the concentration inequality approach is given in Section 2 of Chen (1998). The concentration inequality approach is not the only approach for obtaining Berry-Esseen bounds using Stein&#x2019;s method. Another approach based on inductive arguments has been used by Barbour and Hall (1984b), Bolthausen (1984) and Stroock (1993). We would like to mention in passing that Stein&#x2019;s method has also been applied to obtain bounds on the total variation distances between the standard normal distribution and distributions satisfying certain varia- tional inequalities. See Utev (1989) and Cacoullos, Papathanasiou, and Utev (1994). &#xA9;2001 CRC Press LLC
• 10. 1.5 Conclusion We would like to conclude by saying that there is much more to be done in the direction of nonuniform bounds, particularly for dependent ran- dom variables both in Poisson approximation and normal approximation. The large deviation results referred to in the above sections are actually those of moderate deviation. A related question therefore is how Stein&#x2019;s method can be applied to obtain results which cover both moderate and really large deviations. Acknowledgement This work is partially supported by grant RP3982719 at the National University of Singapore. I would like to thank K. P. Choi and Qi-Man Shao for their help in preparing the manuscript and for their helpful comments. References 1. Arratia, R., Goldstein, L., and Gordon, L. (1990). Poisson approxi- mation and the Chen-Stein method. Statistical Science 5, 403&#x2013;434. 2. Barbour, A. D. (1988). Stein&#x2019;s method and Poisson process con- vergence. Journal of Applied Probability 25 (A), 175&#x2013;184. 3. Barbour, A. D., Chen, L. H. Y., and Choi, K. P. (1995). Poisson approximation for unbounded functions, I: independent summands. Statistica Sinica 5, 749&#x2013;766. 4. Barbour, A. D., Chen, L. H. Y., and Loh, W. L. (1992). Com- pound Poisson approximation for nonnegative random variables via Stein&#x2019;s method. Annals of Probability 20, 1843&#x2013;1866. 5. Barbour, A. D. and Eagleson, G. (1983). Poisson approximation for some statistics based on exchangeable trials. Advances in Applied Probability 15, 585&#x2013;600. 6. Barbour, A. D. and Hall, P. (1984a). On the rate of Poisson conver- gence. Mathematical Proceedings of the Cambridge Philosophical Society 95, 473&#x2013;480. 7. Barbour, A. D. and Hall, P. (1984b). Stein&#x2019;s method and the Berry- Esseen theorem. The Australian Journal Statistics 26, 8&#x2013;15. 8. Barbour, A. D., Holst, L., and Janson, S. (1992). Poisson Ap- proximation. Oxford Studies in Probability 2, Clarendon Press, Oxford. 9. Bikelis, A. (1966). Estimates of the remainder in the central limit &#xA9;2001 CRC Press LLC
• 11. theorem. Litovsk. Mat. Sb. 6(3), 323&#x2013;346 (in Russian). 10. Bolthausen, E. (1984). An estimate of the remainder in a combina- torial central limit theorem. Zeitschrift Wahrscheinlichkeitstheorie und Verwandte Gebiete 66, 379&#x2013;386. 11. Cacoullos, T., Papathanasiou, V. and Utev, S. A. (1994). Varia- tional inequalities with examples and an application to the central limit theorem. Annals of Probability 22, 1607&#x2013;1618. 12. Chen, L. H. Y. (1975a). Poisson approximation for dependent tri- als. Annals of Probability 3, 534&#x2013;545. 13. Chen, L. H. Y. (1975b). An approximation theorem for convolu- tions of probability measures. Annals of Probability 3, 992&#x2013;999. 14. Chen, L. H. Y. (1986). The rate of convergence in a central limit theorem for dependent random variables with arbitrary index set. IMA Preprint Series #243, University of Minnesota. 15. Chen, L. H. Y. (1993). Extending the Poisson approximation. Sci- ence 262, 379&#x2013;380. 16. Chen, L. H. Y. (1998). Stein&#x2019;s method: some perspectives with applications. Probability Towards 2000 (Eds., L. Accardi and C. Heyde), pp. 97&#x2013;122. Lecture Notes in Statistics No. 128. Springer Verlag. 17. Chen, L. H. Y. and Choi, K. P. (1992). Some asymptotic and large deviation results in Poisson approximation. Annals of Probability 20, 1867&#x2013;1876. 18. Chen, L. H. Y. and Roos, M. (1995). Compound Poisson approx- imation for unbounded functions on a group, with application to large deviations. Probability Theory and Related Fields 103, 515&#x2013; 528. 19. Chen, L. H. Y. and Shao, Q. M. (2000). A non-uniform Berry- Esseen bound via Stein&#x2019;s method. Preprint. 20. Chen, L. H. Y. and Soon, S. Y. T. (1994). On the number of ones in the binary expansion of a random integer. Unpublished manuscript. 21. Deheuvels, P. and Pfeifer, D. (1986). A semigroup approach to Poisson approximation. Annals of Probability 14, 663&#x2013;676. 22. Diaconis, P. (1977). The distribution of leading digits and uniform distribution mod 1. Annals of Probability 5, 72&#x2013;81. 23. Esseen, C.-G. (1945). Fourier analysis of distribution functions: a mathematical study of the Laplace-Gaussian law. Acta Mathemat- ica 77 1&#x2013;125. &#xA9;2001 CRC Press LLC
• 12. 24. Ho, S. T. and Chen, L. H. Y. (1978). An Lp bound for the remain- der in a combinatorial central limit theorem. Annals of Probability 6, 231&#x2013;249. 25. Loh, W. L. (1992). Stein&#x2019;s method and multinomial approximation. Annals of Applied Probability 2, 536&#x2013;554. 26. Michel, R. (1981). On the constant in the non-uniform version of the Berry-Esseen Theorem. Zeitschrift Wahrscheinlichkeitstheorie und Verwandte Gebiete 55, 109&#x2013;117. 27. Nagaev, S. V. (1965). Some limit theorems for large deviations. Theory of Probability and its Applications 10, 214&#x2013;235. 28. Paditz, L. (1977). &#xA8;Uber die Ann&#xA8;aherung der Verteilungsfunktionen von Summen unabh&#xA8;angiger Zufallsgr&#xA8;oben gegen unberrenzt teil- bare Verteilungsfunktionen unter besonderer berchtung der Verteilungsfunktion der standarddisierten Normalverteilung. Dis- sertation, A.TU Dresden. 29. Soon, S. Y. T. (1993). Some Problems in Binomial and Compound Poisson Approximations. Ph.D. dissertation, National University of Singapore. 30. Stein, C. (1972). A bound for the error in the normal approx- imation to the distribution of a sum of dependent random vari- ables. Proceedings of the Sixth Berkeley Symposium on Mathe- matics, Statistics and Probability 2, 583&#x2013;602, University California Press. Berkeley, California. 31. Stein, C. (1986). Approximation Computation of Expectations. Lecture Notes 7, Institute of Mathematics and Statistics, Hayward, California. 32. Stroock, D. W. (1993). Probability Theory: An Analytic View. Cambridge University Press, Cambridge, U.K. 33. Utev, S. A. (1989). Probability problems connected with a certain integrodi&#xFB00;erential inequality. Siberian Mathematics Journal 30, 490&#x2013;493. 34. Van Beek, P. (1972). An approximation of Fourier methods to the problem of sharpening the Berry-Esseen inequality. Zeitschrift Wahrscheinlichkeitstheorie und Verwandte Gebiete 23, 187&#x2013;196. &#xA9;2001 CRC Press LLC