Communications in Statistics—Theory and Methods, 40: 145–158, 2011 Copyright © Taylor & Francis Group, LLC ISSN: 0361-0926 print/1532-415X online DOI: 10.1080/03610920903377781
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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 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; wfunctions. Mathematics Subject Classification 60F05; 62E20.
1. Introduction 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 (1995). Pekoz (1996) 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 (1999) 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 (1999), 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 20131, Thailand; E-mail:
[email protected]
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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 X x0 denote the distribution of X and x0 = k=0 pk, for x0 ≥ 0, denote the distribution function of X at x0 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 x0 p = 1 − q ∈ 0 1 and p x0 = k=0 pq k , x0 ≥ 0, denote the geometric distribution function with parameter p at x0 . If we expect X to be closer to Gep than other distributions, then it is reasonable to estimate X by Gep. Correspondingly, it is reasonable to estimate x0 by p x0 as well. For approximating X by Gep and x0 by p x0 , each upper bound for the total variation distance between X and Gep and the difference of x0 and p x0 is a criterion for measuring the accuracy of the approximation. It should be noted that the supremum over all steps x0 ≥ 0 of the difference of x0 and p x0 is less than or equal to the total variation distance between X and Gep, that is, sup x0 − p x0 ≤ sup XA − GepA x0 ≥0
(1.1)
A⊆∪0
where XA = x∈A px and GepA = x∈A pq x , and then all upper bounds of the right-hand side of (1.1) are also upper bounds of the left-hand side of (1.1). The work of geometric approximation for a sum of m random indicators, X = m j=1 Xj , was started by Brown and Phillips (1999). They used Stein’s method to give a uniform bound for the total variation distance between X and Gep as the following result: sup XA − GepA ≤ 2 − p A⊆∪0
m
EXj EX + Z − Xj∗
(1.2)
j=1
where EX = pq , Z is a geometric random variable with parameter p and independent of X and, for each j, Xj∗ is a random variable with distribution as X − Xj conditional on Xj = 1. Afterwards, Phillips and Weinberg (2000) improved the uniform bound (1.2) to be the better result sup XA − GepA ≤ 2 − p A⊆∪0
X + Z − Xj∗ EXj E Xj∗ + 1 j=1
m
(1.3)
The bounds (1.2) and (1.3) 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 supA⊆∪0 XA − GepA and non uniform bounds for x0 − p x0 , 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 (1998). These are in Sec. 2 and we give
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some applications of the results concerning the beta-geometric, Pólya, and Poisson distributions in the last section.
2. The Main Results
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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 (1989), defined a function w associated with the non negative integer-valued random variable X in the relation wkpk =
k 1 − ipi k ∈ x 2 i=0
(2.1)
and Majsnerowska (1998) expressed the relation (2.1) as the form 2 wk − 1pk − 1 1 w0 = 2 wk = 2 + − k k ∈ x\0 pk
(2.2)
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 (1989). If a non negative integer-valued random variable X is defined as in Sec. 1, then CovX gX = 2 E wX gX
(2.3)
for any function g ∪ 0 → for which EwX gX < , where gx = gx + 1 − gx. By taking gx = x, E wX = 1 is obtained. For Stein’s method, we start it by using Stein’s equation in Brown and Phillips (1999). Stein’s equation for the geometric distribution with parameter p = 1 − q ∈ 0 1 is, for given h, of the form hx − Gp h = q1 + xgx + 1 − xgx
(2.4)
l where Gp h = l=0 hlpq and g and h are bounded real-valued functions defined on ∪ 0. For A ⊆ ∪ 0, let hA ∪ 0 → be defined by hA x =
1 if x ∈ A 0 if x A
(2.5)
Following Brown and Phillips (1999) and writing Cx = 0 x, the solution gA of (2.4) can be written as 1 G h p A∩Cx−1 − Gp hA Gp hCx−1 if x ≥ 1 gA x = xpq x 0 if x = 0
(2.6)
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Note that, for obtaining a bounded solution gA of (2.4), it can be constructed by choosing gA 0 arbitrarily, here we choose gA 0 = 0. For A = Cx0 , the solution g = gCx of (2.4) 0
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1 G h G 1 − hCx−1 if x0 < x xpq x p Cx0 p 1 gx = G h G 1 − hCx if x0 ≥ x 0 xpq x p Cx−1 p 0 if x = 0
(2.7)
is an immediate consequence of (2.6). Observe that gx > 0 for every x x0 ∈ . Let
gx = gx + 1 − gx. Then, by (2.7) and for x ≥ 1, we have that Gp hCx 1 1 0 G 1 − hCx − Gp 1 − hCx−1 if x0 < x pq x x + 1q p x
gx = Gp 1 − hCx0 1 1 G G h − h if x0 ≥ x pq x x + 1q p Cx x p Cx−1 x 0 1 q k − if x0 < x k=0 xx + 1
= (2.8) x j 1 1 x−1 k j if x0 ≥ x q pq − x pq x + 1q x+1 j=0 xq j=0 k=x0 +1 For bounding (2.7) and (2.8) and proving the main results, we first need the following lemmas. Lemma 2.1. For n k ∈ , if 0 < q ≤
n n+k
k1
, then
1 1 ≤ nq n n + kq n+k Proof. It is easy to see that
1 nq n
=
qk nq n+k
≤
(2.9)
1 . n+kq n+k
Lemma 2.2. We have the following. 1. For A ⊆ ∪ 0 and x ≥ 0,
sup gA x ≤ A
1 if x = 0 1 if x > 0 x
(2.10)
and sup gA x ≤ A
1 p
(2.11)
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2. For x > 0, 1 = if x0 = 0 x gx 1 ≤ if x0 > 0 xp
(2.12)
3. For x0 > 0 and q ≤ 1/2,
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gx ≤
1 x0 + 1p
(2.13)
4. For x > 0, =
1 if x0 = 0 xx + 1 gx ≤ 1 if x0 > 0 x+1
(2.14)
5. For x0 > 0 and q ≤ 2/3, gx ≤
1 x0 + 1
(2.15)
Proof. 1. For x = 0, (2.10) follows from Lemma 5 in Brown and Phillips (1999) and, for x > 0, it follows from Lemma 1 of Phillips and Weinberg (2000), and (2.11) is also directly obtained from Brown and Phillips (1999, p. 411) . 2. We shall show (2.12) holds. Following (2.7), for x0 = 0, we get gx = =
1 pq k xq x k=x
1 x
(2.16)
for 0 < x0 < x, gx =
x0 1 pq k pq j x xpq k=0 j=x
=
1 − q x0 +1 pq j−x xp j=x
≤
1 xp
and, for x0 ≥ x, we have gx =
k 1 x−1 pq pq j xpq x k=0 j=x0 +1
(2.17)
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1 − qx pq j−x xp j=x0 +1
≤
1 xp
(2.18)
Hence, by (2.16), (2.17), and (2.18), (2.12) holds.
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3. For obtaining (2.13), we shall show that gx ≤ From (2.7), for 0 < x0 < x, we have gx ≤
1 . x0 +1p
1 x0 + 1p
(2.19)
and, for x0 ≥ x, gx ≤
1 pq j xq x p j=x0 +1
≤
1 pq j (by Lemma 2.1) x +1 x0 + 1q 0 p j=x0 +1
=
1 x0 + 1p
(2.20)
Therefore, by (2.19) and (2.20), (2.13) is obtained. It should be note that 0 < q ≤ x k1 1/2 ≤ x+k where k = x0 + 1 − x. 4.
From (2.8), it is obvious for x0 = 0 that gx =
For 0 < x0 < x, we have
(2.21)
x0
gx = ≤ and, for x0 ≥ x,
1 xx + 1
qk xx + 1 k=0
1 x+1
1 x x−1 1 gx = q k−x+1 pq j − pq j+1 x + 1 j=0 x j=0 k=x0 +1 1 xp − q1 − q x ≤ p xx + 1 k+1 x − x−1 1 k=0 q = ≤ xx + 1 x+1
(2.22)
Thus, by (2.21), (2.22), and (2.23), we have (2.14).
(2.23)
Non Uniform Bounds on Geometric Approximation 5.
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For 0 < x0 < x, it follows from (2.22) that gx ≤
1 x0 + 1
(2.24)
and, for x0 ≥ x,
1 x j+1 1 x−1 j gx = q pq − pq x + 1 j=0 x j=0 k=x0 +1 xp − q1 − q x = q k−x+1 xx + 1 k=x0 +1 k+1 1 x − x−1 k=0 q pq k−x+1 pq k = ≤ xx + 1 x + 1q x+1 k=x0 +1 k=x0 +1 1 k ≤ (by Lemma 2.1) pq x0 + 1q x0 +1 k=x0 +1
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=
k−x+1
1 x0 + 1
(2.25)
this yields (2.15). Theorem 2.1. 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 XA − GepA ≤ A⊆∪0
and, if
q p
k + 1q − p2 wkpk k k∈ x\0 q + − 1 − p0q p
(2.26)
= , then
sup XA − GepA ≤ A⊆∪0
k + 1q − p2 wkpk k k∈ x\0
(2.27)
Proof. Putting h = hA in Eq. (2.4), and applying the proof of Theorem 2.1 in Teerapabolarn (2008), we have: XA − GepA = qE gA X + 1 + qE X gA X − pCovX gA X + E gA X = qE gA X + qE X gA X + q − pE gA X − pCovX gA X where gA is defined as in (2.6).
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Since E wX = 1 and using Lemma 2.2 (1), we have EwX gA X < . Thus, by (2.3), XA − GepA = qE gA X + qE X gA X + q − pE gA X −p2 E wX gA X = E 1 + Xq − p2 wX gA X + q − pE gA X ≤ E1 + Xq − p2 wX gA X + q − pEgA X = k + 1q − p2 wk gA kpk Downloaded By: [Teerapabolarn, Kanint] At: 12:41 9 February 2011
k∈ x
+
q − pgA kpk
(2.28)
k∈ x
Therefore, by using Lemma 2.2 (1) and gA 0 = 0, the theorem is obtained. In case of A = 0 1 x0 and gA is defined as in (2.7), by using the same argument detailed as in the proof of Theorem 2.1 together with Lemma 2.2 (2–5), we also have the following theorems. Theorem 2.2. For the difference of x0 and p x0 , we have the following. 1. For x0 = 0, 2 q pk wkp − 0 − p 0 ≤ k kk + 1 pk + q − p p0 + k k∈ x\0 k∈ x\0 (2.29)
and, if
q p
= , then q 2 wkp 0 − p 0 ≤ k − kk + 1 pk k∈ x\0
(2.30)
2. For x0 > 0, q pk 2 wkp x0 − p x0 ≤ q − k + 1 pk + p − k k∈ x k∈ x\0 and, if
q p
(2.31)
= , then 2 wkp x0 − p x0 ≤ q − k + 1 pk k∈ x
(2.32)
Theorem 2.3. For x0 ∈ ∪ 0, if 0 < q ≤ 1/2, then q 1 2 k + 1q − wkp pk + − 1 − p0 x0 − p x0 ≤ x +1 x0 + 1 k∈ x p 0 (2.33)
Non Uniform Bounds on Geometric Approximation and, if 0 < q ≤ 2/3 and
q p
153
= , then
x0 − p x0 ≤
1 k + 1q − 2 wkp pk x0 + 1 k∈ x
(2.34)
Remark 2.1.
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1. It is seen that the bound in Theorem 2.2 (2) is a bound for the supremum over all steps x0 ≥ 0 of the difference of x0 and p x0 , that is, q pk 2 wkp q− pk + − (2.35) sup x0 − p x0 ≤ k+1 p k x0 ≥0 k∈ x k∈ x\0 and, if
q p
= , then 2 wkp sup x0 − p x0 ≤ q − k + 1 pk x0 ≥0 k∈ x
(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 2.1. If k + 1q/p − 2 wk > / < 0 for every k ∈ x, then x0 − p x0 ≤
+ 1q − p2 p + q − p1 − p0 x0 + 1p
where 0 < q ≤ 1/2 and, for 0 < q ≤ 2/3 and x0 − p x0 ≤
q p
(2.37)
= ,
2 + − 2 p x0 + 1
(2.38)
3. Applications We use the results in the Theorems 2.1 and 2.2 and Corollary 2.1, (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. 3.1. 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
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distribution. Let X be the beta-geometric random variable with probability function given by pk =
+ k + k = 0 1 + + k + 1
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+−1 and the mean and variance of X are = −1 and 2 = −2 −1 2 , respectively, where
> 2. Using the relation (2.2), the w-function associated with the beta-geometric , k = 0 1 . The following theorem is an random variable X is wk = k+1+k −12 application of the results in the Theorems 2.1 and 2.2 and Corollary 3.1, respectively.
Theorem 3.1. Let G denote the beta-geometric distribution with parameters and and G x0 denote the beta-geometric distribution function at x0 ∈ ∪ 0. If
−1 , then, for > 2, we have the following p = +−1 1. sup G A − GepA ≤ A⊆∪0
2.
2 + − 1 − 1 + − 1 +
(3.1)
if x0 = 0 + − 1 + G x0 − p x0 ≤ if x0 > 0 − 1 + − 1
(3.2)
3. For ≤ 2 − 1, G x0 − p x0 ≤
2 − 2 − 1x0 + 1
(3.3)
Remark 3.1. 1. If is large and is small, then each result of (3.1), (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 (1963), see also Johnson et al. (2005). The probability function is given by pk =
c − aa + k − 1!c! k = 0 1 ca − 1!c + k!
In the case that a = 1, the distribution is the Yule distribution (Johnson et al., 2005) and the probability function is written as pk =
c − 1k!c! k = 0 1 cc + k!
Also, we can apply all results of geometric approximation in the theorems and corollary for these distributions.
Non Uniform Bounds on Geometric Approximation
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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 = 1 of the Pólya distribution in Phillips and Weinberg (2000, p. 310) and the probability function of X is as follows: d+m−k−2 m−k k = 0 m pk = d+m−1
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m
Following Phillips and Weinberg (2000), the mean and variance of X are = md and 2 = md+md−1 , respectively. As d m → such that the mean md tends to d2 d+1 a constant c, the Pólya distribution with parameters d and m converges to the geometric distribution with parameter 1/1 + 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+1m−k for k = 0 m. 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 d and Y md x0 denote the Pólya distribution function at x0 ∈ ∪ 0. If p = d+m , then we have the following. 1. sup Y m dA − GepA ≤ A⊆∪0
m2d + m − 1 dd + m − 1d + m
(3.4)
2.
Y md x0 − p x0 ≤
m d + m − 1d + m if x0 = 0
m dd + m
(3.5)
if x0 > 0
3. For m ≤ 2d, Y md x0 − p x0 ≤
2m dd + 1x0 + 1
(3.6)
Remark 3.2. 1. It is noted that if the mean md is fixed and d is large, then each result of (3.4), (3.5), and (3.6) yields a good approximation. 2. For this application, Brown and Phillips (1999) used Stein’s method to give a bound in the form of sup Y m dA − GepA ≤ A⊆∪0
2md + 2m d + 1d2
(3.7)
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and Phillips and Weinberg (2000) used the same method to improve the bound (3.7) as sup Y m dA − GepA ≤ A⊆∪0
2md + 2m dd + 1d + m − 1
(3.8)
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:
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sup Y m dA − GepA ≤ A⊆∪0
2m dd + 1
(3.9)
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 (1999), Phillips and Weinberg (2000) and Teerapabolarn (2008), respectively. 3. Following (3.5), we have sup Y md x0 − p x0 ≤ x0 ≥0
m dd + m
(3.10)
4. In the case where m = N − M and d = M + 1, the Pólya distribution is the negative hypergeometric distribution that has the probability function as follows: pk =
N −k−1 N −M−k N N −M
k = 0 N − 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 < 1, which is an application of the geometric approximation. Let X be the Poisson random variable with mean and probability function pk =
e− k k = 0 1 k!
Its mean and variance are and the w-function associated with the X is wk = 1 for all k ≥ 0. Let denote the Poisson distribution with mean and x0 denote the Poisson distribution function at x0 ≥ 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 Theorem 3.3. If p =
1 , 1+
157
then we have the following.
1. sup A − GepA ≤ A⊆∪0
1 − e− 1+
(3.11)
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2. 1 − e− 1 + if x0 = 0 1+ x0 − p x0 ≤ e− + − 1 if x0 > 0 1+
(3.12)
3. For ≤ 2, x0 − p x0 ≤
2 + 1x0 + 1
(3.13)
Remark 3.3. 1. It is easy to see that the results in (3.11), (3.12), and (3.13) 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 (1969), Romanowska (1977), Gerber (1984), Pfeifer (1987), Barbour et al. (1992), and Roos (2003) as follows: sup A − GepA ≤ Vervaat 1969
(3.14)
A⊆∪0
sup A − GepA ≤ √ Romanowska 1977 A⊆∪0 2 sup A − GepA ≤ A⊆∪0
2 Gerber 1984 1+
sup A − GepA ≤ 2 (Pfeifer, 1987)
(3.15) (3.16) (3.17)
A⊆∪0
sup A − GepA ≤ 1 − e− (Barbour et al., 1992) A⊆∪0
3 2 sup A − GepA ≤ min 4e A⊆∪0
(3.18)
(Roos, 2003)
(3.19)
where = q/p and the bound (3.18) is derived from Theorem 1.C (ii) of Barbour et al. (1992, p. 12). It should be noted that all bounds in (3.14)–(3.19) are useful results in applications when ≤ 1. In this case, 1 − e− < √12 , it can be observed that the bound in (3.11) is better than all bounds in (3.14)–(3.18) and the second
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bound in (3.19). By comparing the bound (3.11) and Roos’s bound, the first bound in (3.19), it follows the bound (3.11) is better than Roos’s bound when ≤ 0565. Therefore, if ≤ 0565 or p ≥ 0639, then the bound (3.11) is better than all bounds as mentioned above. 4.
It follows (3.11) that sup x0 − p x0 ≤
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x0 ≥0
e− + − 1 1+
(3.20)
It is seen that all bounds in (3.14)–(3.19), via the inequality (1.1), are also bounds on the error of supx0 ≥0 x0 − p x0 . For this case, it can be seen that the − +−1 2 bound (3.20) is better than the bounds (3.14)–(3.19) because e 1+ < 21+ and 2 31+ 3/4e 1 1 − −1 − 2 1 − e = 1 − e > 2 and 2 / 21+ = 2e > 2 + 2 ≥ 1. Therefore, for the difference of Poisson and geometric distribution functions, our bound (3.20) is better than all bounds in (3.14)–(3.19) for all ≤ 1.
Acknowledgment The author would like to thank the referees for their useful comments and suggestions.
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