the number of failures before the nth success in n sequences of independent. Bernoulli trials, where, in the sequence ith, success occurs on each trial with a.
International Journal of Pure and Applied Mathematics Volume 76 No. 5 2012, 727-732 ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu
AP ijpam.eu
A POINTWISE APPROXIMATION FOR INDEPENDENT GEOMETRIC RANDOM VARIABLES K. Teerapabolarn Department of Mathematics Faculty of Science Burapha University Chonburi, 20131, THAILAND
Abstract: The Stein-Chen method is used to determine a non-uniform bound on pointwise approximation of the distribution of a sum of n independent P geometric random variables by the Poisson distribution with mean λ = ni=1 qi , in terms of the point metric of two such distributions. AMS Subject Classification: 60F05, 60G50 Key Words: geometric random variable, point metric, Poisson distribution, non-uniform bound, Stein-Chen method
1. Introduction Let Y1 , ..., Yn be n independent geometric random variables Pnwith, for each i ∈ k {1, ..., n}, P (Yi = k) = pi qi , k = 0, 1, ..., and let X = i=1 Yi . Then X is the number of failures before the nth success in n sequences of independent Bernoulli trials, where, in the sequence ith , success occurs on each trial with a probability of pi and failure occurs on each trial with a probability of qi = 1−pi . If pi ’s are identical to p, then X has the Pascal distribution with parameters n and p. It is well known that if all qi are small, then the distribution of X can be approximated by a Poisson distribution. In this case, for A ⊆ N ∪ {0}, Teerapabolarn and Wongkasem [5] used the Stein-Chen method to obtain a Received:
December 27, 2011
c 2012 Academic Publications, Ltd.
url: www.acadpubl.eu
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uniform bound −1 2 n k e−λ X λ λ (1 − e−λ ) q X min ,1 i P (X ∈ A) − ≤ k! pi pi
(1.1)
i=1
k∈A
for the difference Pn qiof the distribution of X and the Poisson distribution with mean λ = i=1 pi , and for A = {0, ..., x0 }, x0 ∈ N ∪ {0}, they also gave a non-uniform bound for the difference of the distribution function of X and the Poisson distribution function with the same mean as follows: 2 x0 n X X 1 qi λk e−λ −1 λ min , 1 . (1.2) ≤ λ (e − 1) P (X ≤ x ) − 0 k! pi (x0 + 1) pi i=1 k=0 Pn For the Poisson mean λ = i=1 qi , Vellaisamy and Upadhye [6] used Kerstan’s method to obtain a uniform bound, in the same form as in (1.1), as follows: 2 n X λk e−λ X 0.42888 q √ min (1.3) ,1 i , P (X ∈ A) − ≤ k! pi λ i=1
k∈A
and Teerapabolarn [4] used the Stein-Chen method to obtain a better result of (1.2) in the following: x0 n X X 1 λk e−λ −1 λ min , 1 qi2 , (1.4) ≤ λ (e − 1) P (X ≤ x0 ) − k! pi (x0 + 1) i=1
k=0
where x0 ∈ N ∪ {0}. Consider the results in (1.1) and (1.3), when A = {x0 } for x0 ∈ N ∪ {0}, the results become −1 2 n x0 −λ X λ (1 − e−λ ) q P (X = x0 ) − λ e ≤ min ,1 i (1.5) x0 ! pi pi i=1
for λ =
Pn
qi i=1 pi ,
and for λ =
Pn
i=1 qi ,
X 2 n x0 −λ 0.42888 q P (X = x0 ) − λ e ≤ √ min ,1 i , x0 ! pi λ
(1.6)
i=1
which are uniform bounds, with respect to x0 , for the point metric of two such distributions. In this situation, a non-uniform bound with respect to x0 is more appropriate for measuring the accuracy of the approximation. In this study, we use the Stein-Chen method to determine a non-uniform bound for the pointPmetric of the distribution of X and the Poisson distribution with mean λ = ni=1 qi .
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2. Method Stein [3] introduced a powerful and general method for bounding the error in the normal approximation. This method was first developed and applied in the setting of Poisson approximation by Chen [2], which is refer to as the SteinChen method. Stein’s equation for Poisson distribution with mean λ > 0 is, for given h, of the form h(x) − ℘λ (h) = λf (x + 1) − xf (x),
(2.1)
P λl where ℘λ (h) = e−λ ∞ l=0 h(l) l! and f and h are bounded real valued functions defined on N ∪ {0}. For A ⊆ N ∪ {0}, let function hA : N ∪ {0} → R be defined by ( 1 if x ∈ A, hA (x) = 0 if x ∈ / A. For A = {x0 } where x0 ∈ N ∪ {0}, let Cx = {0, ..., x} and hx0 = h{x0 } , then following Barbour et al. [1], the solution fx0 = f{x0 } of (2.1) is of the form −x λ if x ≤ x0 , −(x − 1)!λ e [℘λ (hx0 )℘λ (hCx−1 )] −x λ fx0 (x) = (x − 1)!λ e [℘λ (hx0 )℘λ (1 − hCx−1 )] if x > x0 , 0 if x = 0.
(2.2)
The following lemma gives a non-uniform bound for fx0 .
Lemma 2.1. Let x0 , x ∈ N and k ∈ N \ {1}, then the following inequality holds: ( min nλ−1 (1 − e−λ ), 12 , k1 o if x0 = 1, (2.3) sup |fx0 (x)| ≤ min λ−1 (1 − e−λ ), x10 , k1 if x0 ≥ 2. x≥k
Proof. For k ≤ x ≤ x0 , it follows from Barbour et al. [1] that fx0 is negative and decreasing in x ≤ x0 , thus we obtain 0 ≤ −fx0 (x) ≤ −fx0 (x0 )
= (x0 − 1)!λ−x0 eλ [℘λ (hx0 )℘λ (hCx0 −1 )]
x0 −1 i 1 −λ X λ = e x0 i! i=0
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= λ−1 e−λ
x0 X λi i i! x0 i=1
1 −1 −λ , ≤ min λ (1 − e ), x0 o n this yields sup |fx0 (x)|≤ min λ−1 (1 − e−λ ), x10 , k1 . x≥k
For k ≤ x and x0 < x,
0 ≤ fx0 (x) = (x − 1)!λ−x eλ [℘λ (hx0 )℘λ (1 − hCx−1 )] ∞ (x − 1)! −λ X λi+x0 −x = e x0 ! i! i=x x0 λx0 +1 λx0 +2 1 −λ λ + + + ··· = e x x0 ! x0 !(x + 1) x0 !(x + 1)(x + 2) 1 1 , ≤ min λ−1 (1 − e−λ ), , x0 k n o 1 1 −1 −λ we have sup |fx0 (x)|≤ min λ (1 − e ), x0 , k . x≥k
Hence, from two cases, the inequality (2.3) holds.
3. Result We use the Stein-Chen method to determine a result in the Poisson approximation to the distribution of X, in terms of the point metric of two such distributions together with its non-uniform bound, in the following theorem. Theorem 3.1. For x0 ∈ N, let λ =
Pn
i=1 qi ,
then we have
2 n X qi −1 −λ 1 −λ min λ (1 − e ), , pi P (X = 1) − λe ≤ 2 pi
(3.1)
i=1
and for x0 ≥ 2, X 2 n x0 1 qi λ −1 −λ −λ ≤ P (X = x0 ) − e min λ (1 − e ), , p , i x0 ! x0 pi i=1
where P (X = 0) =
n Y i=1
pi .
(3.2)
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Proof. Substituting h by hx0 and x by X and taking expectation in (2.1), it yields −λ λx0 e P (X = x0 ) − = |E[λf (X + 1) − Xf (X)]| x0 ! n X |E[qi f (X + 1) − Yi f (X)]| , (3.3) ≤ i=1
where f is defined as in (2.2). For i ∈ {1, ..., n}, let Xi = X − Yi , then by using the proof detailed as in Teerapabolarn [4], it follows that X k |E[qi f (X + 1) − Yi f (X)]| = (1 − k)pi qi E[f (Xi + k)] k≥2 X ≤ (1 − k)pi qik E|f (Xi + k)| k≥2
X ≤ (k − 1)pi qik sup |f (x)|, k≥2
x≥k
and from (3.3), we obtain X n X −λ x0 P (X = x0 ) − e λ ≤ (k − 1)pi qik sup |f (x)|. x0 ! x≥k i=1 k≥2
Hence, by Lemma 2.1, the theorem is proved. Immediately from Theorem 3.1, it is easily obtained a uniform bound of this approximation as follows. Corollary 3.1. We have 2 n x0 X qi −1 −λ 1 P (X = x0 ) − e−λ λ ≤ min λ (1 − e ), , pi x0 ! 2 pi
(3.4)
i=1
for every x0 ∈ N.
Corollary 3.2. For x0 ∈ N, if pi = p for every i ∈ {1, ..., n} and λ = nq, then q −λ −λ λ (3.5) P (X = 1) − λe ≤ min 1 − e , , λp 2 p
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and for x0 ≥ 2, x0 P (X = x0 ) − e−λ λ ≤ min 1 − e−λ , λ , λp q , x0 ! x0 p
(3.6)
where P (X = 0) = pn .
Remark. By simple comparison between the bound in Corollary 3.1 and the bounds in (1.5) and (1.6), it can be seen that the bound in Corollary 3.1 is sharper than the bound in (1.5) and the second bound in (1.6), and it is also sharper than the first bound in (1.6) when λ < 0.72 or λ > 5.39.
References [1] A.D. Barbour, L. Holst, S. Janson, Poisson Approximation, Oxford Studies in Probability 2, Clarendon Press, Oxford (1992). [2] L.H.Y. Chen, Poisson approximation for dependent trials, Ann. Probab., 3 (1975), 534-545. [3] C.M. Stein, A bound for the error in normal approximation to the distribution of a sum of dependent random variables, Proc. Sixth Berkeley Sympos. Math. Statist. Probab., 3 (1972), 583-602. [4] K. Teerapabolarn, A note on Poisson approximation for independent geometric random variables, Int. Math. Forum, 4 (2009), 531-535. [5] K. Teerapabolarn, P. Wongkasem, Poisson approximation for independent geometric random variables, Int. Math. Forum, 2 (2007), 3211-3218. [6] P. Vellaisamy, N.S. Upadhye, Compound negative binomial approximations for sums of random variables, Probab. Math. Statist., 29 (2009), 205-226.
