A uniform bound on negative binomial approximation with w ... - Hikari

Applied Mathematical Sciences, Vol. 9, 2015, no. 57, 2831 - 2841 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.52160

A Uniform Bound on Negative Binomial Approximation with w-Functions K. Jaioun and K. Teerapabolarn* Department of Mathematics, Faculty of Science Burapha University, Chonburi 20131, Thailand * Corresponding author Copyright © 2015 K. Jaioun and K. Teerapabolarn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract This paper uses Stein’s method and w-functions to determine a uniform bound for the Kolmogorov distance between the cumulative distribution function of a non-negative integer-valued random variable X and the negative binomial cumulative distribution function with parameters r > 0 and p = 1 − q ∈ (0,1) , where

rq p

= E ( X ) and E ( X ) is the mean of X. Two examples are provided to

illustrate applications of the result obtained. Mathematics Subject Classification: 62E17, 60F05 Keywords: Cumulative distribution function, Kolmogorov distance, Negative binomial approximation, Stein’s method, w-function

1 Introduction The topics related to the negative binomial approximation have yielded useful results in applied probability and statistics. The first study of negative binomial approximation has concerned an approximation of the distribution of a sum of dependent Bernoulli random variables, which was presented by Brown and Phillips [1]. They used Stein’s method to give a uniform bound on the negative binomial approximation to the distribution of a sum of dependent Bernoulli random variables, and they also applied the result to approximate the Pólya distribution. For independent geometric summands, Vellaisamy and Upadhye [9]

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K. Jaioun and K. Teerapabolarn

used Kerstan’s method to give a uniform bound for approximating the distribution of a sum of independent geometric random variables by a negative binomial distribution. However, the results in [1] and [9] are inappropriate for a nonnegative integer-valued random variable. Later, Teerapabolarn and Boondirek [8] adapted and applied Stein’s method and w-functions to approximate the distribution of a non-negative integer-valued random variable by an appropriate negative binomial distribution as follows. Let NB be the negative random variable with parameter r > 0 and p = 1 − q ∈ ( 0,1) that has probabilities pNB (k ) = and has mean E ( NB) =

Γ(r + k ) r k p q , k ∈ ` ∪ {0}, k !Γ ( r ) rq p

and variance Var ( NB ) =

(1.1) rq p2

. For r = 1 , the negative

random variable is the geometric random variable with parameter p , denoted by G. Let X be a non-negative integer-valued random variable with probability mass function p X ( x) > 0 for every x in the support of X , denoted by S ( X ), have mean μ = E ( X ) and variance σ 2 = Var ( X ) . For A ⊆ ` ∪ {0} , a uniform bound for the total variation distance between the distributions of X and NB, presented in Teerapabolarn and Boondirek [8], is of the form dTV ( X , NB ) ≤

p(1 − p r ) (r + X )q rq E − w( X ) + −μ , rq p p

(1.2) x

dTV ( X , NB) = sup P( X ∈ A) − P( NB ∈ A)

where

and

w( x) =

A

x ∈ S ( X ), is the w-function associated with X. In addition, if

∑ ( μ −k ) p X ( k )

k =0

σ 2 pX ( x)

,

rq = μ then the p

result in (1.2) becomes dTV

p(1 − p r ) (r + X )q E − w( X ) . ( X , NB ) ≤ rq p

(1.3)

For A = {x0 } , x0 ∈ S ( X ) , Malingam and Teerapabolarn [4] and Teerapabolarn [7] used the same tools in [8] to give a non-uniform bound for the point metric between the distributions of X and NB as follows: ⎫ (1 − p r ) p ⎧ (r + X )q rq | p X (0) − pNB (0) |≤ − σ 2 w( X ) + (1 − p X (0)) − μ ⎬ (1.4) ⎨Ε rq p p ⎩ ⎭ and if

rq p

= μ , then [7] improved the bound (1.4) to be the form

A uniform bound on negative binomial approximation with w–functions

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rq − (1 − p r ) p Ε (r + X )q − pσ 2 w( X ) . | p X (0) − pNB (0) |≤ 2 r (r + 1)q

(1.5)

When x0 ∈ S ( X ) \{0} , for r ∈ (0,1) , [7] showed that (r + X )q ⎪⎧ 1 1 − p r ⎪⎫ − σ 2 w( X ) | p X ( x0 ) − pNB ( x0 ) | ≤ min ⎨ , ⎬ pΕ p ⎩⎪ x0 rq ⎭⎪ + (1 − p X (0))

and if

rq p

⎫ rq −μ ⎬ p ⎭

(1.6)

= μ , then

⎧⎪ 1 1 − p r ⎫⎪ (r + X )q − σ 2 w( X ) | p X ( x0 ) − pNB ( x0 ) |≤ min ⎨ , ⎬ pΕ p ⎪⎩ x0 rq ⎪⎭

(1.7)

For r ≥ 1 , [4] showed that 1 − p r ⎪⎫ ⎧ (r + X )q ⎪⎧ 1 − σ 2 w( X ) | p X ( x0 ) − pNB ( x0 ) |≤ min ⎨ , ⎬ p ⎨Ε ( 1) x r x q p + − 0 ⎩⎪ 0 ⎭⎪ ⎩ + (1 − p X (0))

and if

rq p

⎫ rq −μ ⎬ p ⎭

(1.8)

= μ , then

⎧⎪ 1 1 − p r ⎫⎪ (r + X )q | p X ( x0 ) − pNB ( x0 ) | ≤ min ⎨ , − σ 2 w( X ) . (1.9) ⎬ pΕ p ⎩⎪ x0 (r + x0 − 1)q ⎭⎪ Consider the bounds in (1.2)−(1.9), it is seen that if the mean E ( NB) = rq p

= μ , then the bounds in (1.3), (1.5), (1.7) and (1.9) are sharper than those

reported in (1,2), (1.4), (1.6) and (1.8), respectively. For x0 ∈ S ( X ), if A = {0,..., x0 }, then the result in (1.3) can be expressed in the following form d K ( X , NB ) ≤

p(1 − p r ) (r + X )q E − w( X ) , rq p

(1.10)

where d K ( X , NB ) = sup P ( X ≤ x0 ) − P( NB ≤ x0 ) is the Kolmogorov distance x0 ≥0

between the cumulative distribution functions of X and NB. In this paper, we are

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interested to improve the bound in (1.10) to be more appropriate for measuring the accuracy of this approximation when E ( NB) = rqp = μ . The tools for giving our main result are consist of Stein’s method and w-functions, which are utilized to provide the desired result as mentioned in Section 2. In Section 3, we use Stein’s method and w-functions to yield a new result of the approximation. In Section 4, we give two examples to illustrate applications of the result. Conclusion of this study is presented in the last section.

2 Method The methodology in this study consists of Stein’s method and w-functions. For w-functions, Cacoullos and Papathanasiou [2] defined a function w associated with the non-negative integer-valued random variable X in the relation w( x ) p X ( x ) =

1

x

∑ ( μ − i ) p X (i), x ∈ S ( X ) σ2

(2.1)

i =0

and Majsnerowska [3] expressed the relation (11) as the form

μ 1 w(0) = 2 , w( x ) = 2 σ σ

⎧⎪ σ 2 w( x − 1) p X ( x − 1) ⎫⎪ − x ⎬ , x ∈ S ( X ) {0}, ⎨μ + p X ( x) ⎪⎩ ⎪⎭

(2.2)

where w( x) ≥ 0 and p X ( x ) > 0 for every x ∈ S ( X ) . The next relation is an important property for obtaining the main result, which was started by [2]. If a non-negative integer-valued random variable X is defined as in Section 1, then E [ ( X − μ ) f ( X ) ] = σ 2 E[ w( X )Δf ( X )], for any function

f : ` ∪ {0} → \ for which

(2.3)

E w( X )Δf ( X ) < ∞ , where

Δf ( x ) = f ( x + 1) − f ( x) and E[w(X)]=1. For Stein’s method, Stein [5] introduced a powerful and general method for bounding the error in the normal approximation. This method was developed and applied in the setting of the negative binomial approximation by Brown and Phillips [1]. Stein’s equation for the negative binomial distribution with parameters r > 0 and p = 1 − q ∈ (0,1) is, for given h, of the form h ( x ) −ℵr , p ( h ) = (1 − p )( r + x ) f ( x + 1) − xf ( x ), ∞

where ℵr , p (h) = ∑ h(k ) k =0

Γ(r + k ) r k p q and f and h are bounded real-valued k !Γ ( r )

(2.4)

A uniform bound on negative binomial approximation with w–functions

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functions defined on ` ∪ {0} . For A ⊆ ` ∪ {0} , let hA : ` ∪ {0} → \ be defined by ⎧ 1 if x ∈ A hA ( x ) = ⎨ ⎩0 if x ∉ A.

(2.5)

Following [1] and [8] and writing Cx = {0,..., x} , the solution f A of (14) can be written as x !Γ ( r ) ⎧ ⎡ℵr , p (hA∩C ) −ℵr , p (hA )ℵr , p (hC ) ⎤ if x ≥ 1, ⎪ x −1 x −1 ⎦ f A ( x) = ⎨ Γ(r + x) xp r q x ⎣ ⎪ 0 if x = 0. ⎩

(2.6)

Note that, it follows from [1] that f A = − f Ac ,

(2.7)

where Ac is the complement of A. For x0 ∈ ` ∪ {0} and hx0 = h{ x0 } , by following (2.6) the solution f x0 = f{ x0 } is of the form x !Γ(r ) ⎧ ℵ (h )ℵr , p (hCx −1 ) if x ≤ x0 , ⎪− r x r , p x0 ( r x ) xp q Γ + ⎪ ⎪ x !Γ(r ) f x0 ( x) = ⎨ ℵ (h )ℵr , p (1 − hCx −1 ) if x > x0 , r x r , p x0 r x xp q ( ) Γ + ⎪ ⎪ 0 if x = 0. ⎪ ⎩

(2.8)

Moreover, it follows from [1] that ⎧< 0 if x0 > x, Δf x0 ( x) = f x0 ( x + 1) − f x0 ( x) ⎨ ⎩ > 0 if x0 = x.

(2.9)

For A = C x0 , x0 ∈ ` ∪ {0} , the solution fC x of (2.4) can be written as 0

x !Γ(r ) ⎧ )ℵr , p (1 − hC x ) if x ≤ x0 , ℵ (h ⎪ r x r , p C x −1 0 ( r x ) xp q Γ + ⎪ ⎪ x !Γ(r ) fC x ( x) = ⎨ ℵ (h )ℵr , p (1 − hC x −1 ) if x > x0 , r x r , p C x0 0 ⎪ Γ(r + x) xp q ⎪ 0 if x = 0. ⎪ ⎩

(2.10)

The following lemma gives a uniform bound for Δf x , which is also need for proving the desired result.

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K. Jaioun and K. Teerapabolarn

Lemma 2.1. Let x ∈ ` , then the following inequality holds: Δf x ( x ) ≤

1 − p r +1 . ( r + 1) q

(2.11)

Proof. [1] showed that Δf x to be a decreasing function in x ∈ ` . From which, it follows that Δf x ( x) − Δf x ( x + 1) > 0 for every x ∈ ` . Therefore, we obtain Δf x ( x ) ≤ Δf1 (1) = =

r r r ∞ 1 Γ(r + k ) p r q k r 1 − p − rp q + ( r + 1) p q + p = ∑ k !Γ ( r ) ( r + 1) q k = 2 ( r + 1) q

1 − p r +1 , ( r + 1) q

which implies that (2.11) holds. Lemma 2.2. For x0 ∈ ` ∪ {0} and x ∈ ` , then we have the following: 1 − p r +1 0 ( r + 1)q x0 , x 1 2. sup Δf C x ( x ) ≤ for r = 1 . 0 2 x0 , x

1. sup ΔfC x ( x ) ≤

(2.12) (2.13)

Proof. 1. With Lemma 2.1, (2.7) and (2.9), we have that 1 − p r +1 1 − p r +1 ≥ Δf x ( x) ≥ ∑ Δf k ( x) = ΔfCx ( x) ≥ Δf{ x}c ( x) = −Δf x ( x) ≥ − . 0 ( r + 1) q ( r + 1) q k∈C x 0

Hence, the inequality (2.12) is obtained. 2. It is directly obtained from [6].

3 Main result The following theorem presents a new uniform bound for the Kolmogorov distance between the cumulative distribution function of the non-negative integer-valued random variable X and an appropriate negative binomial cumulative distribution function with parameters r and p. Theorem 3.1. With the above definitions, let

rq p

= μ , then we have the following

inequality. d K ( X , NB) ≤

p(1 − p r +1 ) (r + X )q E − σ 2 w( X ) . (r + 1)q p

(3.1)

A uniform bound on negative binomial approximation with w–functions Proof. For x0 ∈ S ( X ) , replacing h by hCx

0

2837

and x by X and taking the

expectation in (2.4), yields P ( X ≤ x0 ) − P( NB ≤ x0 ) = E [ q(r + X ) f ( X + 1) − Xf ( X )] ,

(3.2)

where f = fCx is defined in (2.10). Let δ ( X ) = E [ q(r + X ) f ( X + 1) − Xf ( X )] 0

then we obtain

δ ( X ) = E [ rqf ( X + 1) + qXf ( X + 1) − Xf ( X )] = rqE [ f ( X +1)] + qE [ X Δf ( X )] − pE [ Xf ( X )]

= rqE [ f ( X + 1) ] + qE [ X Δf ( X ) ] − p { E[( X − μ ) f ( X )] + μ E [ f ( X ) ]}

= rqE [ Δf ( X )] + qE [ X Δf ( X )] + (rq − pμ)E[ f ( X )] − pE[( X − μ) f ( X )] = rqE [ Δf ( X )] + qE [ X Δf ( X )] − pE[( X − μ) f ( X )].

(3.3)

Since, E[ w( X )] = 1 and E w( X )Δf ( X ) = E ⎡⎣w( X ) Δf ( X ) ⎤⎦ < ∞ . Then by (2.3), (2.4) and (3.3), we have that d K ( X , NB ) = rqE [ Δf ( X ) ] + qE [ X Δf ( X ) ] − pσ 2 E[ w( X )Δf ( X )]

≤ E ⎡ (r + X )q − pσ 2 w( X ) Δf ( X ) ⎤ ⎣ ⎦

≤ sup Δf ( x) E (r + X )q − pσ 2 w( X ) x0 , x

≤

p(1 − p r +1 ) (r + X )q E − σ 2 w( X ) (r + 1)q p

(by (2.12)).

Hence, we have the inequality (3.1). Corollary 3.1. If

(r + X )q p

d K ( X , NB) ≤

− σ 2 w( X ) ≥ / < 0 for every x ∈ S ( X ) and

p(1 − p r +1 ) (r + μ )q −σ 2 . (r + 1)q p

rq p

= μ , then (3.4)

In the case of r = 1 , a new uniform bound on the geometric approximation with w-functions by using Lemma 2.2 (2) is as follows. Corollary 3.2. For geometric approximation, if

q p

= μ , then we have the

following: d K ( X , G) ≤

1 E (r + X )q − p 2 w( X ) 2

(3.5)

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and if

K. Jaioun and K. Teerapabolarn ( X +1) q p

− σ 2 w( X ) ≥ / < 0 for every x ∈ S ( X ) ,

d K ( X , G) ≤

1 (r + μ )q − pσ 2 . 2

(3.6)

Corollary 3.3. If p = 1 − q ∈ (0,1) and r > 0 , then

1 − p r +1 1 − p r < . (r + 1)q rq

(3.7)

Proof. Let λ = rq , then we have that eλ = 1 + λ + λ2! + λ3! +" . Therefore 2

3

(1 + λ )e − λ < 1 ⇔ e − λ + λ e− λ < 1 ⇔ e − λ < 1 − λ e − λ . Because e − q = 1 + ( − q ) +

( − q )2 2!

+

( − q )3 3!

+ " = 1 − (1 − p ) +

( − q )2 2!

(3.8) +

( − q )3 3!

+ ",

p < e − q ⇔ p r < e− rq = e− λ .

(3.9)

From (3.8) and (3.9), it follows that p r < 1 − λ e− λ ⇔ λ e−λ < 1 − p r ⇔ p r
0 and β > 0 , then the distribution is called the beta binomial distribution with parameters n , α and β . Let the beta binomial random variable X have probabilities

A uniform bound on negative binomial approximation with w–functions

2839

⎛ n ⎞ Γ(α + x)Γ( β + n − x)Γ(α + β ) , x = 0,1,..., n,   pX ( x ) = ⎜ ⎟ Γ(α + β + n)Γ( β )Γ(α ) ⎝ x⎠

 

and have mean μ = αn+αβ and variance σ 2 = in (2.2), we have w( x) = Theorem 3.1, then

(α + x )( n − x )

(α + β )σ 2

(α + x ) q p

nαβ (α + β + n )

(α + β )2 (α + β +1)

. Following the relation

. Putting p = αα++ββ+ n and r = α to the result in

− σ 2 w( x) = (αα++xβ) x ≥ 0 for all 0 ≤ x ≤ n . Applying the

Corollaries 3.1 and 3.2, the following corollary is as follows. Corollary 4.1. If r = α and p = αα++ββ+ n , then we have the following:

d K ( X , NB ) ≤

α (1 − pα +1 ) , (α + β + 1) p

and for r = 1 , d K ( X , G) ≤

n . ( β + 1)( β + 2)

The result gives a good approximation when β is large and n and α are small. Therefore, the negative binomial distribution with parameters α and α +β α + β +n

can be used as an approximation of the beta binomial distribution with

parameters n , α and β when β is large and n and α are both small. Example 4.2. An application to the beta negative binomial distribution It is well-known that the beta-negative binomial distribution is a negative binomial distribution whose probability of success parameter p follows a beta distribution with shape parameters α > 0 and β > 0 . Let X be the beta negative binomial random variable with probability mass function

p X ( x) =

Γ(r + α )Γ( x + β )Γ(r + x)Γ(α + β ) , x = 0,1,..., Γ(r + x + α + β )Γ(r )Γ( x + 1)Γ(α )Γ( β )

where α , β , r ∈ \ + . The mean and variance of X are μ = αr β−1

and

σ 2 = r β ( r +α −1)(α + β2 −1) , respectively. Following the relation in (2.2), we have (α − 2)(α −1)

w( x) = (r + x)q p

( r + x )( β + x )

(α −1)σ 2

. Setting p = αα+ β−1−1 to the result in Theorem 3.1, then

− σ 2 w( x) = − ( rα+−x1) x ≤ 0 for all x ≥ 0 . Using the Corollaries 3.1 and 3.2, the

following corollary is as follows.

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K. Jaioun and K. Teerapabolarn

Corollary 4.2. If p = αα+ β−1−1 , then we have the following:

d K ( X , NB ) ≤ and for r = 1 , d K ( X , G) ≤

r (1 − p r +1 ) , (α − 2) p

β . (α − 1)(α − 2)

The result gives a good approximation when α is large and r and β are small. Therefore, the negative binomial distribution with parameters r and α −1 can be used as an approximation of the beta negative binomial distribution α + β −1 with parameters α , β and r when α is large and r and β are both small.

5 Conclusion In this study, a uniform bound for the Kolmogorov distance between the cumulative distribution function of a non-negative integer-valued random variable and an appropriate negative binomial cumulative distribution function with parameters r and p was obtained by using Stein’s method and w-functions, where rq = μ . The main result gives a good negative binomial approximation when the p uniform bound is small. Furthermore, by theoretical comparison, the uniform bound in the present study is sharper than that presented in (1.10).

References [1] T. C. Brown, M. J. Phillips, Negative binomial approximation with Stein’s method, Methodology and Computing in Applied Probability, 1 (1999), 407-421. http://dx.doi.org/10.1023/a:1010094221471 [2] T. Cacoullos, V. Papathanasiou, Characterization of distributions by variance bounds, Statistics & Probability Letters, 7 (1989), 351-356. http://dx.doi.org/10.1016/0167-7152(89)90050-3 [3] M. Majsnerowska, A note on Poisson approximation by w-functions, Applicationes Mathematic, 25 (1998), 387-392. [4] P. Malingam, K. Teerapabolarn, A pointwise negative binomial approximation by w-functions, International journal of pure and applied mathematics, 69 (2011), 453-467. [5] C. M. Stein, A bound for the error in normal approximation to the distribution of a sum of dependent random variables, Proceedings of the Sixth Berkeley Sym-

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posium on Mathematical Statistics and Probability, 3 (1972), 583-602. [6] K. Teerapabolarn, An improved bound on geometric approximation by w-function, International journal of pure and applied mathematics, 84 (2013), 365-370. http://dx.doi.org/10.12732/ijpam.v84i4.6 [7] K. Teerapabolarn, An improvement of pointwise negative binomial approximation by w-functions, International journal of pure and applied mathematics, 98 (2015), 33-38. [8] K. Teerapabolarn, A. Boondirek, Negative Binomial Approximation with Stein's Method and Stein's Identity, International Mathematical Forum, 5 (2010), 2541-2551. [9] P. Vellaisamy, N.S. Upadhye, Compound negative binomial approximations for sums of random variables, Probability and Mathematical Statistics, 29 (2009), 205-226.

Received: March 8, 2015; Published: April 12, 2015