A Non-uniform Bound on Poisson Approximation in Beta Negative Binomial Distribution

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1 International Mathematical Forum, Vol. 14, 2019, no. 2, HIKARI Ltd, A Non-uniform Bound on Poisson Approximation in Beta Negative Binomial Distribution Chanokgan Sahatsathatsana Department of Science and Mathematics Kalasin University, Thailand This article is distributed under the Creative Commons BY-NC-ND Attribution License. Copyright c 2019 Hikari Ltd. Abstract In this artical, we use the Stein-Chen identity and the w- function associated with Beta negative binomial distribution to determine a formula of non-uniform upper bound on the Poisson approximation to beta negative binomial distribution in terms of the total variation distance. We give some applications of the result of this approximation concerning the beta-geometric and Po lya distributions. Mathematics Subject Classification: 60F05 Keywords: Beta negative binomial distribution, Poisson Approximation, Stein-Chen identity, w-function 1 Introduction Regarding theories of probability and statistics, the beta negative binomial distribution (represents a discrete statistical distribution defined at integer values 1, 2,..., n) refers to the negative binomial distribution which its probability of success parameter p follows a beta distribution in terms of shape parameters α and β. On contrary, if the probability of success parameter p of a negative binomial distribution (with parameters n and p) has a beta distribution in terms of shape parameters α and β, the resulting distribution is, therefore, referred to the beta-negative binomial distribution with parameters α, β and n, denoted by BN (α,β,r). The p is usually assumed to be fixed for successive trials for a standard negative binomial distribution, but the value of p changes in each trial according

2 58 Chanokgan Sahatsathatsana to the beta-negative binomial distribution. The beta negative binomial distribution is sometimes referred to the inverse Markov-Plya distribution, or the beta-pascal distribution, and or the generalized Waring distribution. For a standard negative binomial distribution, p is usually assumed to be fixed for successive trials, but the value of p changes for each trial for the beta-negative binomial distribution. The beta negative binomial distribution is sometimes referred to as the inverse Markov-Po lya distribution, the beta- Pascal distribution, and the generalized Waring distribution. Let X be the beta negative binomial variable, see also Johnson et al [4]. For k {0, 1, 2,...,, its probability distribution is defined by p(k). (1.1) In particular case r 1, the distribution of X is beta geometric distribution with parameters α and β, where α, β are positive real numbers, r be the number of failures until the experiment is stopped (integer but can be extended to real), Γ be the gamma function, B be the beta function and mean and variance are and (α r 1)(α β 1) (α 2)(α 1) 2, respectively. α 1 It is generally accepted that the negative binomial distribution with parameters r and p can be approximated by the Poisson distribution with the mean λ, which is denoted by P (λ), under some conditions concerning parameters r, p and λ. As mentioned above, the beta-negative binomial distribution is obtained from a negative binomial distribution. Consequently, we expect that the beta-negative binomial distribution can be approximated by the Poisson distribution as well. The purpose of this paper is to approximate the distribution of beta negative binomial distribution by using the w-function associated with beta negative binomial distribution with Stein-Chen identity which is introduced in Section 2 and we give some applications of the result concerning the betageometric and Po lya distributions in the last section. The following theorem is our main result. Theorem 1.1. Let X be the beta negative binomial random variable, for A N {0, α > 2, then we have d T V (L (X) {A, BN (α,β,r) {A) λ { (λ r β 1) (α 2) (1.2) where d T V (L (X) {A, BN (α,β,r) {A) sup A L (X) {A BN (α,β,r) {A

3 A non-uniform bound on Poisson approximation 59 2 The w-function and Poisson Approximation via Stein-Chen Method We will prove our main result by using the w-function associated with beta negative binomial random variable X and the Stein-Chen identity. 2.1 The w-functions. The w-functions were studied and used by many authors, among others by Cacoullos and Papathanasiou [2], Papathanasiou and Utev [7] and Majsnerowska [5]. In the note by the latter, the following recurrence relation can be found where k S(x)\{0, w(0) w(k) p x(k 1) w(k 1) µ k 0, (2.1) p x (k) σ 2 µ σ 2, S(x) is support of X, p x(k) > 0 for all k S(x), µ and σ 2 (0, ) are mean and variance of X, respectively. The next relation was stated by Cacoullos and Papathanasiou [2], if a function f satisfies E w(x) f(x) <, E (X µ)f(x) <, then Cov(x, f(x)) σ 2 E[w(X) f(x)]. (2.2) where f(x) f(x 1) f(x) and E[w(x)] 1 Using the relation (2.1), we give the form of the w-function associated with the beta negative binomial random variable X in the following lemma. Lemma 2.1. Let w(x) be the w-function associated with beta negative binomial random variable X and p x (k) > 0, then we have w(k) (r k)(β k) (α 1)σ 2, k {0, 1, 2... (2.3) where σ 2 (α r 1)(α β 1) (α 2)(α 1) 2 and α > 2 Proof. Of Lemma 2.1. The recurrence relation of w-function associated with the random variable X can be written as w(k) w(k 1) w(k 1) Γ(rk 1)B(αr,βk 1) (k 1)!Γ(r)B(α,β) Γ(rk)B(αr,βk) k!γ(r)b(α,β) (α 1)σ 2 k σ 2 kγ(r k 1)B(α r, β k 1) (α 1)σ 2 k σ 2 (2.4)

4 60 Chanokgan Sahatsathatsana where w(0) (α 1)σ. 2 In the next step, we shall show that (2.3) holds for every k {1, 2,...,. From (2.4), we have w(1) (α 1)σ w(0) 2 Γ(r)B(αr,β) (0)!Γ(r)B(α,β) Γ(r1)B(αr,β1) 1!Γ(r)B(α,β) 1 σ 2 (α 1)σ Γ(r)B(α r, β) 2 (α 1)σ 2 Γ(r 1)B(α r, β 1) 1 σ 2 α r β 1 σ 2 r β 1 (r 1)(β 1) Assuming that w(i 1) (r i 1)(β i 1), then we have (α r β i 1) w(i) i ri βi i2 (r i)(β i). i σ 2 Therefore, by mathematical induction, (2.3) holds for every. For k {0, 1, 2, Stein-Chen Method. For the Stein-Chen identity, Stein [9] introduced a new powerful technique for the obtaining the rate of convergence to standard normal distribution. Chen [3] adapted and applied idea of normal case to the Poisson setting. The Stein- Chen identity or the Stein identity for Poisson distribution with a parameter λ which, given h, is defined by λf(x 1) xf(x) h(x) BN (α,β,r) (h), (2.5) and g, f are bounded real- where BN (α,β,r) (h) k0 valued function defined on N {0. h(k) Γ(rk)B(αr,βk) k!γ(r)b(α,β)

5 A non-uniform bound on Poisson approximation 61 For A N {0, let h A : N {0 R be defined by { 1 ; x A, h A (x) 0 ; x A. The solution f f A of (2.5) can be written as [BN (α,β,r) (h A Cx 1 ) BN (α,β,r) (h A )BN (α,β,r) (h Cx 1 )] ; x 1, f A (x) x Γ(rk)B(αr,βk) x!γ(r)b(α,β) 0 ; x 0 (2.6) where C x {0, 1,..., x. and it follows from Brown and Phillips [1] that f A f A c (2.7) where A c is the complement of A, from which it also yields f A f A c (2.8) where f A (x) f A (x 1) f A (x). For x 0 N {0 and A x 0, we let h x0 h{x 0, then the solution f x0 f{x 0 of (2.6), we get BN (α,β,r) (h x0 )BN (α,β,r) (h Cx 1 )] ; x x 0, xbn (α,β,r) (h x ) f x0 (x) BN (α,β,r) (h x0 )BN (α,β,r) (1 h Cx 1 )] (2.9) ; x > x 0, xbn (α,β,r) (h x ) 0 ; x 0 From Malingam and Teerapabolarn [8], we have that { < 0 ; x x0, f A (x) f A (x 1) f A (x) (2.10) > 0 ; x x 0 The following lemmas give new bounds for f x and f A, A N {0, which are needed to improve the desired result. Lemma 2.2. For x N, then we have x 1 k0 < x x k0 (2.11)

6 62 Chanokgan Sahatsathatsana Proof. We assume that P (k) : k 1 k0 < x k k0 (2.12) In the next step, we shall show that (2.11) holds for k N. From (2.12), we have Γ(r)B(α r, β) P (1) : Γ(r)B(α, β) Γ(r)B(α r, β) Γ(r 1)B(α r, β 1) < Γ(r)B(α, β) Γ(r)B(α, β) x r, β) Γ(r 1)B(α r, β 1) < [Γ(r)B(α ] Γ(r)B(α, β) Γ(r)B(α, β) (2.13) Assuming that P (k) : k 1 k0 < x k k0. (2.14) Then we have P (k 1) : k k0 Γ(r)B(α r, β) Γ(r)B(α, β) x < { < { x { x x k k0 k k0 Γ(r k 1)B(α r, β k 1) (k 1)!Γ(r)B(α, β) (k)!γ(r)b(α, β) (k)!γ(r)b(α, β) Γ(r k 1)B(α r, β k 1) (k 1)!Γ(r)B(α, β) k1 k0 Hence, by mathematical induction, (2.11) holds for every. For k N (2.15)

7 A non-uniform bound on Poisson approximation 63 Lemma 2.3. Let x N, then we have following: f x (x) Proof. We shall show that f x X Lemma 2.2. it yields f x (x) BN (α,β,r)(h x )BN (α,β,r) (1 h Cx )] (x 1)BN (α,β,r) (h x1 ) kx1 { kx1 x 1 x k0 { kx1 (2.16), by using (2.9) and BN (α,β,r)(h x )BN (α,β,r) (h Cx 1 )] xbn (α,β,r) (h x ) 1 x x 1 k0 x k0 (2.17) Lemma 2.4. Let A N {0 and x N, then we have following: sup f x (x) A Proof. From (2.8), (2.10) and lemma 2.3, we have f x(x) k A f x (x) f A (x) f x c(x) f x (x) (2.18) (2.19)

8 64 Chanokgan Sahatsathatsana 3 Proof of Main Result Proof. Of Theorem 1.1. In view of lemma 2.1, for k {0, 1, 2,...,, then we have λ σ 2 w(k) (r k)(β k) σ2 (α 1) (α 1)σ 2 k(r β k) (α 1) 0 (3.1) From lemma 2.4, (2.5) and (4.6), when h h A and f f A, then we have d T V (L (X) {A, BN (α,β,r) {A) E λf(x 1) Xf(X) Therefore, by λ (α r 1)(α β 1) (α 2)(α 1) 2 and (3.2), this implies (1.2). α 1, σ2 λe[f(x 1)] E[Xf(X)] λe[f(x 1)] Cov(X, f(x)) µe[f(x)] λe[f(x 1)] Cov(X, f(x)) λe[f(x)] λe[ f x (X)] Cov(X, f(x)) λe[ f x (X)] E[σ 2 w(x) f x (X)] E [λ σ 2 w(x)] f x (X) sup f x (X) E [λ σ 2 w(x)] A E[λ σ2 w(x)] E[σ2 w(x) λ] (σ2 λ) (3.2) 4 Applications 4.1 An application on 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 α > 0 and β > 0, the

9 A non-uniform bound on Poisson approximation 65 resulting distribution is referred to as the beta-geometric distribution with parameters α > 0 and β > 0. 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 distribution. Let X be the beta-geometric random variable. For k {0, 1, 2,...,,with probability function given by p(k) αγ(β k)γ(α β) Γ(β)Γ(α β k 1). (4.1) the mean and variance of X are µ β α 1 and αβ(α β 1) σ2 (α 2)(α 1), respectively, where α > 2. Using the relation (2.1), the w-function associated with 2 the beta-geometric random variable X is w(k) (k1)(βk). In view of lemma (α 1)α 2 2.1, for k {0, 1, 2,...,, then we have λ σ 2 β (k 1)(β k) w(k) σ2 (α 1) (α 1)σ 2 k(1 β k) (α 1) 0 (4.2) The following theorem is an application of the results in (3.2). Theorem 4.1. Let BG (α,β) denote the beta-geometric distribution with parameters α and β, for A N {0, α > 2, then we have d T V (BG (α,β) {A, BN (α,β,r) {A) λ { (λ β 2) (α 2) (4.3) Remark: If β a and α c a then the beta-geometric distribution is the so-called generalized Waring distribution [4], for k {0, 1, 2,...,,with probability function given by p(k) (c a)(a k 1)!c! c(a 1)!(c k)! (4.4) We can apply all results of beta-geometric approximation for these distribution. 4.2 An application on the Po lya distribution The Po lya or Po lya-eggenberger distribution is typically and simply introduced by an urn model. Suppose that a single urn contain r red and Nr black balls. Draw a ball at random, note the color, and return it into the urn together with c additional balls of the same color. Repeat this way for m draws.

10 66 Chanokgan Sahatsathatsana 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 Po lya distribution in Phillips and Weinberg [6], for k {0, 1, 2,..., m,with probability function given by ) p(k) ( dm k 2 m k ( dm 1 m ), (4.5) the mean and variance of X are µ m d and m(d m)(d 1) σ2, respectively. As d, m such that the mean m d tends to a constant c, the Po d 2 (d 1) lya distribution with parameters d and m converges to the geometric distribution 1 with parameter. For this case, bounds on the rate of convergence of the (1c) Po lya distribution can be obtained as the following. Using the relation (2.1), the w-function associated with X is w(k) (k1)(m k). In view of lemma 2.1, σ 2 d for k {0, 1, 2,..., m, then we have λ σ 2 w(k) m (k 1)(m k) σ2 d σ 2 d k( m k 1) d 0 when k m 1 (4.6) The following theorem is an application of the results in (3.2). Theorem 4.2. Let PY (m,d) denote the Po lya distribution with parameters m and d, for A N {0, then we have d T V (PY (m,d) {A, BN (α,β,r) {A) λ { ( 2 m λ) (d 1) (4.7) Acknowledgments. The author would, sincerely, like to thank the anonymous referees for their valuable suggestions and constructive criticism which helped for improving the presentation of the paper.

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