Non Uniform Bounds on Geometric Approximation Via Stein s Method and w-functions

Communications in Statistics Theory and Methods, 40: 45 58, 20 Copyright Taylor & Francis Group, LLC ISSN: 036-0926 print/532-45x online DOI: 0.080/036092090337778 Non Uniform Bounds on Geometric Approximation Via Stein s Method and w-functions K. TEERAPABOLARN Department of Mathematics, Faculty of Science, Burapha University, Chonburi, Thailand. Introduction In this article, we use Stein s method and w-functions to give uniform and non uniform bounds in the geometric approximation of a non negative integer-valued random variable. We give some applications of the results of this approximation concerning the beta-geometric, Pólya, and Poisson distributions. Keywords Geometric approximation; Non uniform bounds; Stein s method; w- functions. Mathematics Subject Classification 60F05; 62E20. The context of geometric approximation via Stein s method has yielded useful results in applications. The first work of geometric approximation, for the problem of finding the first sum of a random positive integer sequence with given divider, was presented by Barbour and Grübel (995). Pekoz (996) gave two uniform bounds for measuring the error in the geometric approximation of a random variable counting the number of failures before the first success in a sequence of dependent Bernoulli trials. He applied the results to Markov hitting time and sequence pattern applications. Brown and Phillips (999) considered this approximation in connection with a sum of indicator random variables, and they gave a uniform bound on the rate of convergence of the Pólya distribution. Phillips and Weinberg (2000) gave a non uniform bound for approximating the distribution of a sum of indicator random variables by improving the bound in Brown and Phillips (999), and Teerapabolarn (2008) gave a better uniform bound on the rate of convergence of the Pólya distribution by a different manner in the recent article. However, all bounds as mentioned above are given as total variation distance bounds. In this article we use Stein s method and w-functions to give uniform and non uniform bounds in the geometric approximation of a non negative integer-valued random variable Received December 24, 2008; Accepted September 30, 2009 Address correspondence to Teerapabolarn, Department of Mathematics, Faculty of Science, Burapha University, Chonburi 203, Thailand; E-mail: kanint@buu.ac.th 45

46 Teerapabolarn for the total variation distance between two distribitions and the difference of two distribution functions. Let X be a non negative integer-valued random variable with probability function px = PX = x > 0 for every x in the support of X, x. Let LX denote the distribution of X and x 0 = x 0 k=0 pk, for x 0 0, denote the distribution function of X at x 0 and and 2 (0 < 2 < ) denote the mean and variance of X, respectively. It is well known that the distribution of X can be approximated by some discrete distributions if their parameters are satisfied under certain conditions. Let Gep denote the geometric distribution with parameter p = q 0 and p x 0 = x 0 k=0 pqk, x 0 0, denote the geometric distribution function with parameter p at x 0. If we expect LX to be closer to Gep than other distributions, then it is reasonable to estimate LX by Gep. Correspondingly, it is reasonable to estimate x 0 by p x 0 as well. For approximating LX by Gep and x 0 by p x 0, each upper bound for the total variation distance between LX and Gep and the difference of x 0 and p x 0 is a criterion for measuring the accuracy of the approximation. It should be noted that the supremum over all steps x 0 0 of the difference of x 0 and p x 0 is less than or equal to the total variation distance between LX and Gep, that is, sup x 0 p x 0 sup LXA GepA (.) x 0 0 where LXA = x A px and GepA = x A pq x, and then all upper bounds of the right-hand side of (.) are also upper bounds of the left-hand side of (.). The work of geometric approximation for a sum of m random indicators, X = m j= X j, was started by Brown and Phillips (999). They used Stein s method to give a uniform bound for the total variation distance between LX and Gep as the following result: sup LXA GepA 2 p m j= EX j EX + Z X j (.2) where EX = q, Z is a geometric random variable with parameter p and p independent of X and, for each j, Xj is a random variable with distribution as X X j conditional on X j =. Afterwards, Phillips and Weinberg (2000) improved the uniform bound (.2) to be the better result sup LXA GepA 2 p m X + Z X EX j E j Xj + j= (.3) The bounds (.2) and (.3) can be used to measure the error between the distribution of a sum of random indicators X and the geometric distribution only. It may not be easily applied to the case that X is a non negative integer-valued random variable. In this study, we derive a uniform bound for the error on sup LXA GepA and non uniform bounds for x 0 p x 0, where X is a non negative integer-valued random variable. The tools for giving our main results consist of the so-called w-functions and Stein s equation for the geometric distribution, which are the same tools as in Teerapabolarn (2008) and similar to the tools of the Poisson approximation in Majsnerowska (998). These are in Sec. 2 and we give

Non Uniform Bounds on Geometric Approximation 47 some applications of the results concerning the beta-geometric, Pólya, and Poisson distributions in the last section. 2. The Main Results We will prove our main results by using the same methodology as in Teerapabolarn (2008), which consists of Stein s method and w-functions. For w-functions, Cacoullos and Papathanasiou (989), defined a function w associated with the non negative integer-valued random variable X in the relation wkpk = 2 k ipi k x (2.) and Majsnerowska (998) expressed the relation (2.) as the form i=0 w0 = wk = { } + 2 wk pk k k x\0 (2.2) 2 2 pk where pk > 0 for every k x. The next relation is an important property for obtaining our main results, which was stated by Cacoullos and Papathanasiou (989). If a non negative integer-valued random variable X is defined as in Sec., then CovX gx = 2 EwXgX (2.3) for any function g 0 for which EwXgX <, where gx = gx + gx. By taking gx = x, EwX = is obtained. For Stein s method, we start it by using Stein s equation in Brown and Phillips (999). Stein s equation for the geometric distribution with parameter p = q 0 is, for given h, of the form hx G p h = q + xgx + xgx (2.4) where G p h = l=0 hlpql and g and h are bounded real-valued functions defined on 0. For A 0, let h A 0 be defined by h A x = { if x A 0 if x A (2.5) Following Brown and Phillips (999) and writing C x = 0x, the solution g A of (2.4) can be written as g A x = xpq G ph x A Cx G p h A G p h Cx if x 0 if x = 0 (2.6)

48 Teerapabolarn Note that, for obtaining a bounded solution g A of (2.4), it can be constructed by choosing g A 0 arbitrarily, here we choose g A 0 = 0. For A = C x0, the solution g = g Cx0 of (2.4) xpq G ph x Cx0 G p h Cx if x 0 <x gx = xpq G ph x Cx G p h Cx0 if x 0 x 0 if x = 0 (2.7) is an immediate consequence of (2.6). Observe that gx > 0 for every x x 0. Let gx = gx + gx. Then, by (2.7) and for x, we have that G p h Cx0 [ pq x x + q G p h Cx ] x G p h Cx gx = G p h Cx0 [ pq x x + q G ph Cx ] x G ph Cx x 0 ] q k [ xx + k=0 = [ q k x + q x+ k=x 0 + x j=0 ] pq j x pq j xq x j=0 if x 0 <x if x 0 x if x 0 <x if x 0 x (2.8) For bounding (2.7) and (2.8) and proving the main results, we first need the following lemmas. Lemma 2.. For n k, if0 <q ( n n+k) k, then nq (2.9) n n + kqn+k Proof. It is easy to see that = qk nq n nq n+k. n+kq n+k Lemma 2.2. We have the following.. For A 0 and x 0, sup g A x A if x = 0 x if x>0 (2.0) and sup g A x (2.) A p

Non Uniform Bounds on Geometric Approximation 49 2. For x>0, 3. For x 0 > 0 and q /2, = x gx xp if x 0 = 0 if x 0 > 0 (2.2) gx (2.3) x 0 + p 4. For x>0, 5. For x 0 > 0 and q 2/3, = if x xx + 0 = 0 gx if x x + 0 > 0 (2.4) gx (2.5) x 0 + Proof.. For x = 0, (2.0) follows from Lemma 5 in Brown and Phillips (999) and, for x>0, it follows from Lemma of Phillips and Weinberg (2000), and (2.) is also directly obtained from Brown and Phillips (999, p. 4). 2. We shall show (2.2) holds. Following (2.7), for x 0 = 0, we get gx = xq x pq k k=x = (2.6) x for 0 <x 0 <x, gx = xpq x x 0 k=0 = qx 0+ xp xp pq k j=x pq j pq j x j=x (2.7) and, for x 0 x, we have gx = x pq k xpq x k=0 j=x 0 + pq j

50 Teerapabolarn = qx xp j=x 0 + pq j x (2.8) xp Hence, by (2.6), (2.7), and (2.8), (2.2) holds. 3. For obtaining (2.3), we shall show that gx. x 0 +p From (2.7), for 0 <x 0 <x, we have and, for x 0 x, gx xq x p = j=x 0 + gx pq j x 0 + q x 0+ p x 0 + p j=x 0 + pq j (by Lemma 2.) (2.9) (2.20) x 0 + p Therefore, by (2.9) and (2.20), (2.3) is obtained. It should be note that 0 <q /2 ( x k where k = x x+k) 0 + x. 4. From (2.8), it is obvious for x 0 = 0 that For 0 <x 0 <x, we have gx = (2.2) xx + x0 k=0 gx = qk xx + x + (2.22) and, for x 0 x, gx = q k x+ x pq j x pq j+ k=x 0 x + x + j=0 j=0 xp q q x p xx + = x x k=0 qk+ (2.23) xx + x + Thus, by (2.2), (2.22), and (2.23), we have (2.4).

Non Uniform Bounds on Geometric Approximation 5 5. For 0 <x 0 <x, it follows from (2.22) that gx x 0 + (2.24) and, for x 0 x, gx = this yields (2.5). = = k=x 0 + k=x 0 + k=x 0 + q k x+ x + x q k x+ xp q q x xx + x pq j pq j+ x j=0 j=0 pq k x+ x x [ ] k=0 qk+ pq k xx + k=x 0 x + q + x+ [ ] pq k (by Lemma 2.) x 0 + q x 0+ k=x 0 + = (2.25) x 0 + Theorem 2.. Let X be a non negative integer-valued random variable defined as mentioned above and having corresponding w-function wx. Then the following inequalities hold: sup LXA GepA k + q p 2 wkpk k + q p p0q (2.26) k x\0 and, if q =, then p sup LXA GepA k x\0 k + q p 2 wkpk (2.27) k Proof. Putting h = h A in Eq. (2.4), and applying the proof of Theorem 2. in Teerapabolarn (2008), we have: LXA GepA = qeg A X + + qexg A X pcovx g A X + Eg A X where g A is defined as in (2.6). = qeg A X + qexg A X + q peg A X pcovx g A X

52 Teerapabolarn Since EwX = and using Lemma 2.2 (), we have EwXg A X <. Thus, by (2.3), LXA GepA =qeg A X + qexg A X + q peg A X p 2 EwXg A X =E + Xq p 2 wxg A X + q peg A X E + Xq p 2 wxg A X +q peg A X = k + q p 2 wkg A kpk k x + k x q pg A kpk (2.28) Therefore, by using Lemma 2.2 () and g A 0 = 0, the theorem is obtained. In case of A = 0 x 0 and g A is defined as in (2.7), by using the same argument detailed as in the proof of Theorem 2. together with Lemma 2.2 (2 5), we also have the following theorems. Theorem 2.2. For the difference of x 0 and p x 0, we have the following.. For x 0 = 0, 0 p 0 and, if q =, then p k x\0 { q k 2 wkp kk + pk +q p p0 + 0 p 0 k x\0 k x\0 } pk k (2.29) q k 2 wkp kk + pk (2.30) 2. For x 0 > 0, x 0 p x 0 k x q 2 wkp k + pk + q p k x\0 pk k (2.3) and, if q =, then p x 0 p x 0 k x k x q 2 wkp k + pk (2.32) Theorem 2.3. For x 0 0, if0 <q /2, then x 0 p x 0 k + q 2 wkp pk + q x 0 + p p0 x 0 + (2.33)

Non Uniform Bounds on Geometric Approximation 53 and, if 0 <q 2/3 and q =, then p Remark 2.. x 0 p x 0 x 0 + k x k + q 2 wkp pk (2.34). It is seen that the bound in Theorem 2.2 (2) is a bound for the supremum over all steps x 0 0 of the difference of x 0 and p x 0, that is, sup x 0 p x 0 x 0 0 and, if q =, then p k x sup x 0 p x 0 x 0 0 q 2 wkp k + pk + q p k x k x\0 pk k (2.35) q 2 wkp k + pk (2.36) 2. If the mean q/p of the geometric random variable equals the mean of the non negative integer-valued random variable X, then we get the better results for this approximation as in (2.30), (2.32), and (2.34). The following corollary is a consequence of Theorem 2.3. Corollary 2.. If k + q/p 2 wk>/<0 for every k x, then x 0 p x 0 + q p2 p +q p p0 (2.37) x 0 + p where 0 <q /2 and, for 0 <q 2/3 and q p =, x 0 p x 0 2 + 2 p (2.38) x 0 + 3. Applications We use the results in the Theorems 2. and 2.2 and Corollary 2., (2.27), (2.30), (2.32) and (2.38), to illustrate some applications of the geometric approximation concerning the beta-geometric, Pólya, and Poisson distributions. 3.. Application to the Beta-Geometric Distribution In the case that the probability of success parameter p of a geometric distribution has a beta distribution with shape parameters >0 and >0, the resulting distribution is referred to as the beta-geometric distribution with parameters and. For a standard geometric distribution, p is usually assumed to be fixed for successive trials, but the value of p changes for each trial for the beta-geometric

54 Teerapabolarn distribution. Let X be the beta-geometric random variable with probability function given by pk = + k + + + k + k = 0 and the mean and variance of X are = and 2 = +, respectively, where 2 2 >2. Using the relation (2.2), the w-function associated with the beta-geometric random variable X is wk = k++k, k = 0. The following theorem is an 2 application of the results in the Theorems 2. and 2.2 and Corollary 3., respectively. Theorem 3.. Let BG denote the beta-geometric distribution with parameters and and G x 0 denote the beta-geometric distribution function at x 0 0. If p =, then, for >2, we have the following. 2. + 2 + sup BG A GepA (3.) + + 3. For 2, Remark 3.. + + G x 0 p x 0 + G x 0 p x 0 if x 0 = 0 if x 0 > 0 (3.2) 2 (3.3) 2 x 0 +. If is large and is small, then each result of (3.), (3.2), and (3.3) gives a good geometric approximation. 2. In the case where = c a and = a, the beta-geometric distribution is the Waring distribution that was developed by Irwin (963), see also Johnson et al. (2005). The probability function is given by pk = c aa + k!c! ca!c + k! k = 0 In the case that a =, the distribution is the Yule distribution (Johnson et al., 2005) and the probability function is written as pk = c k!c! cc + k! k = 0 Also, we can apply all results of geometric approximation in the theorems and corollary for these distributions.

Non Uniform Bounds on Geometric Approximation 55 3.2. Application to the Pólya Distribution Let us consider the random assignment of m balls into d compartments such that all partitions have equal probability. Let X be the number of balls in the first compartment, then the distribution of X is the special case r = of the Pólya distribution in Phillips and Weinberg (2000, p. 30) and the probability function of X is as follows: ( d+m k 2 ) m k pk = ) k = 0m ( d+m m Following Phillips and Weinberg (2000), the mean and variance of X are = m d and 2 = md+md, respectively. As d m such that the mean m tends to d 2 d+ d a constant c, the Pólya distribution with parameters d and m converges to the geometric distribution with parameter / + c. For this case, bounds on the rate of convergence of the Pólya distribution can be obtained as the following. From (2.2), we then have the w-function associated with X is wk = k+m k for k = 0m. 2 d Therefore, by applying (2.27), (2.30), (2.32), and (2.38), the following theorem is obtained. Theorem 3.2. Let Y m d denote the Pólya distribution with parameters m and d and Y md x 0 denote the Pólya distribution function at x 0 0. Ifp = d, then d+m we have the following.. 2. m2d + m sup Y m da GepA (3.4) dd + m d + m m d + m d + m Y md x 0 p x 0 m dd + m if x 0 = 0 if x 0 > 0 (3.5) 3. For m 2d, Y md x 0 p x 0 2m (3.6) dd + x 0 + Remark 3.2.. It is noted that if the mean m is fixed and d is large, then each result of (3.4), d (3.5), and (3.6) yields a good approximation. 2. For this application, Brown and Phillips (999) used Stein s method to give a bound in the form of 2md + 2m sup Y m da GepA (3.7) d + d 2

56 Teerapabolarn and Phillips and Weinberg (2000) used the same method to improve the bound (3.7) as 2md + 2m sup Y m da GepA (3.8) dd + d + m In a recent article, Teerapabolarn (2008) gave a bound for this approximation by using Stein s method and the w-function associated with the Pólya random variable X as follows: 2m sup Y m da GepA (3.9) dd + By comparing the bounds in (3.4) and (3.7) (3.9), the bound (3.9) is better than the bounds (3.7) and (3.8) and the bound (3.4) is better than the bound (3.9). Thus, for the total variation distance between the Pólya and geometric distributions, our bound, (3.4), is better than the bounds of Brown and Phillips (999), Phillips and Weinberg (2000) and Teerapabolarn (2008), respectively. 3. Following (3.5), we have m sup Y md x 0 p x 0 (3.0) x 0 0 dd + m 4. In the case where m = N M and d = M +, the Pólya distribution is the negative hypergeometric distribution that has the probability function as follows: pk = ( N k N M k ( N N M ) ) k = 0N M Therefore, we can also apply all results in Theorem 3.2 to the negative hypergeometric distribution. 3.3. Application to the Poisson Distribution It is well known that many discrete distribution can be approximated by Poisson distribution. For this application, we need to approximate the Poisson distribution function with mean by the geometric distribution function with parameter 0 < p <, which is an application of the geometric approximation. Let X be the Poisson random variable with mean and probability function pk = e k k! k = 0 Its mean and variance are and the w-function associated with the X is wk = for all k 0. Let denote the Poisson distribution with mean and x 0 denote the Poisson distribution function at x 0 0. By applying (2.27), (2.30), (2.32), and (2.38) to this case, we then have the results of the geometric approximation to the Poisson distribution as the following.

Non Uniform Bounds on Geometric Approximation 57 Theorem 3.3. If p =, then we have the following. +. sup A GepA e (3.) + 2. 3. For 2, x 0 p x 0 x 0 p x 0 e + + e + + if x 0 = 0 if x 0 > 0 (3.2) 2 (3.3) + x 0 + Remark 3.3.. It is easy to see that the results in (3.), (3.2), and (3.3) yield good approximations provided that is small. 2. It is well known that the Poisson distribution can be applied to deal with many problems in probability convergence and approximation. So, in the case where the Poisson mean is small, we can apply these results, via the Poisson distribution, to some Poisson approximation problems. For example, in the somatic cell hybrid model, random graphs, the birthday problem and the classical occupancy problem. 3. Let us consider other bounds for the total variation distance in Vervaat (969), Romanowska (977), Gerber (984), Pfeifer (987), Barbour et al. (992), and Roos (2003) as follows: sup A GepA Vervaat 969 (3.4) sup A GepA Romanowska 977 (3.5) 2 sup A GepA + Gerber 984 (3.6) sup A GepA 2 (Pfeifer, 987) (3.7) sup A GepA e (Barbour et al., 992) (3.8) { } 3 sup A GepA min 4e 2 (Roos, 2003) (3.9) where = q/p and the bound (3.8) is derived from Theorem.C (ii) of Barbour et al. (992, p. 2). It should be noted that all bounds in (3.4) (3.9) are useful results in applications when. In this case, e < 2, it can be observed that the bound in (3.) is better than all bounds in (3.4) (3.8) and the second 2

58 Teerapabolarn bound in (3.9). By comparing the bound (3.) and Roos s bound, the first bound in (3.9), it follows the bound (3.) is better than Roos s bound when 0565. Therefore, if 0565 or p 0639, then the bound (3.) is better than all bounds as mentioned above. 4. It follows (3.) that sup x 0 p x 0 e + (3.20) x 0 0 + It is seen that all bounds in (3.4) (3.9), via the inequality (.), are also bounds on the error of sup x0 0 x 0 p x 0. For this case, it can be seen that the bound (3.20) is better than the bounds (3.4) (3.9) because e + < 2 and + 2+ e = e 2 > 2 and 3/4e = 3+ > +. Therefore, for 2 2 /2+ 2e 2 2 the difference of Poisson and geometric distribution functions, our bound (3.20) is better than all bounds in (3.4) (3.9) for all. Acknowledgment The author would like to thank the referees for their useful comments and suggestions. References Barbour, A. D., Holst, L., Janson, S. (992). Poisson Approximation. Oxford Studies in Probability 2, Oxford: Clarendon Press. Barbour, A. D., Grübel, R. (995). The first divisible sum. J. Theor. Probab. 8:39 47. Brown, T. C., Phillips, M. J. (999). Negative binomial approximation with Stein s method. Meth. Comp. Appl. Probab. :407 42. Cacoullos, T., Papathanasiou, V. (989). Characterization of distributions by variance bounds. Statist. Probab. Lett. 7:35 356. Gerber, H. U. (984). Error bounds for the compound Poisson approximation. Insur. Math. Econom. 3:9 94. Irwin, J. O. (963). The place of mathematics in medical and biological statistics. J. Roy. Statist. Soc. Ser. A 26: 44. Johnson, N. L., Kotz, S., Kemp, A. W. (2005). Univariate Discrete Distributions. 3rd ed. New York: Wiley. Majsnerowska, M. (998). A note on Poisson approximation by w-functions. Appl. Math. 25:387 392. Pekoz, E. (996). Stein s method for geometric approximation. J. Appl. Probab. 33:707 73. Pfeifer, D. (987). On the distance between mixed Poisson and Poisson distributions. Statist. Decis. 5:367 379. Phillips, M. J., Weinberg, G. V. (2000). Non-uniform bounds for geometric approximation. Statist. Probab. Lett. 49:305 3. Romanowska, M. (977). A note on the upper bound for the distance in total variation between the binomial and the Poisson distributions. Statist. Neerlandica 3:27 30. Roos, B. (2003). Improvements in the Poisson approximation of mixed Poisson distributions. J. Statist. Plann. Infer. 3:467 483. Teerapabolarn, K. (2008). On the geometric approximation to the Polya distribution. Int. Math. Forum 3:359 362. Vervaat, W. (969). Upper bound for distance in total variation between the binomial or negative binomial and the Poisson distribution. Statist. Neerlandica 23:79 86.