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1 Hug ad Giag SprigerPlus 20165:79 DOI /s y RESEARCH O bouds i Poisso approximatio for distributios of idepedet egative biomial distributed radom variables Tra Loc Hug * ad Le Truog Giag Ope Access *Correspodece: tlhugv@gmailcom; tlhug@ufmeduv Uiversity of Fiace ad Maretig, 306 Nguye Trog Tuye Street, Ta Bih District, Ho Chi Mih City, Vietam Abstract Usig the Stei Che method some upper bouds i Poisso approximatio for distributios of row-wise triagular arrays of idepedet egative-biomial distributed radom variables are established i this ote Keywords: Stei Che method, Poisso approximatio, Le Cam s iequality, Negative-biomial variable Mathematics Subject Classificatio: 60F05, 60G50, 41A36 Bacgroud Let X,1, X,2, ; = 1, 2, be a row-wise triagular array of idepedet egativebiomial distributed radom variables with probabilities P X = = Cr p r + 1 p, 1 where p 0, 1; r = 1, 2, ; i = 1, 2, ; = 0, 1, It is worth poitig out that if all r,1 = r,2 = =1; = 1, 2,, the we have the sequece of idepedet geometric distributed radom variables with success probabilities p,1, p,2, ; = 1, 2, Write W = X ad λ = EW = r 1 p p We will deote by Z λ the Poisso radom variable with positive mea λ The mai aim of this paper is to establish some upper bouds i Poisso approximatio for =1 PW = PZ λ = for the sequece X,1, X,2, ; = 1, 2, by the well-ow Stei Che method It has log bee ow that the remarable Le Cam s iequality i Poisso approximatio for the row-wise triagular array of idepedet Beroulli distributed radom variables Y,1, Y,2, ; = 1, 2, with probabilities PY = 1 = p = 1 PY = 0, i = 1, 2, is defied as follows: PS = PZ β = 2 p 2, =1 2 where S = Y ad β = ES = p [see Le Cam 1960, Neammaee 2003 for more details] Moreover, a shape iequality has bee established as follows: 2016 Hug ad Giag This article is distributed uder the terms of the Creative Commos Attributio 40 Iteratioal Licese which permits urestricted use, distributio, ad reproductio i ay medium, provided you give appropriate credit to the origial authors ad the source, provide a li to the Creative Commos licese, ad idicate if chages were made

2 Hug ad Giag SprigerPlus 20165:79 Page 2 of 12 =1 PS = PZ β = 21 e β β p 2 3 [We refer the reader to Barbour et al 1992 ad Che 1975] As far as we ow the Stei Che method is the well-ow method have bee used i Poisso approximatio problems ad it ca be applied to a wide class of discrete radom variables as geometric distributed radom variables ad egative-biomial distributed radom variables I recet years, usig the Stei Che method, may results related to Poisso approximatio for various discrete radom variables are established i Teerapabolar ad Wogasem 2007, Teerapabolar 2009, 2013 These results are icluded here for the sae of completeess Let Z 1, Z 2, be a sequece of idepedet geometric distributed radom variables with probabilities PZ i = = 1 p i p i, = 0, 1, 2, ; i = 1, 2, The, for A Z + := 0, 1, 2, }, sup A PV A γ e γ A γ 1 e γ } mi,1 1 p i 2 pi, p i 4 ad for A Z +, w 0 Z + w 0 γ e γ PV w 0 =0 γ e γ 1 mi p i w 0 + 1,1 } 1 p i 2 p i, 5 where V = Z i, γ = EV = 1 p i pi [see Teerapabolar ad Wogasem 2007, for more details] It should be oted that i case whe the mea γ = EV will be replaced by a parameter γ = 1 p i, aother results will be established as follows: γ e γ } 1 mi p i w 0 + 1,1 w 0 1 p i 2 γ e γ PV w 0 =0 0, 6 ad w 0 PV w 0 w0 =0 =0 e γ γ e γ γ 1 w 0 e γ γ w 0 +1 w 0 +1! =0 e γ γ for A Z + [results of this ature may be foud i Teerapabolar 2013] It is easy to chec that whe the values r,1 = r,2 = =1; = 1, 2, the desired sequece X, 1 will become the sequece Z 1, Z 2, Therefore, it maes sese to cosider the results i 4, 5, 6, ad 7 for egative-biomial radom variables with probabilities i term of 1 It should be oted that i recet years the same problem was tacled i Upadhye ad Vellaisamy 2014 ad Vellaisamy ad Upadhye 2009 by usig Kerstas method 1964 ad the method of expoets [see Upadhye ad Vellaisamy 2013, 2014 ad Vellaisamy } 1 mi p i w 0 + 1,1 1 p i 2, 7

3 Hug ad Giag SprigerPlus 20165:79 Page 3 of 12 ad Upadhye 2009, for more details] The compoud egative biomial ad compoud Poisso approximatios to the geeralized Poisso biomial distributio are studied ad applicatios are also discussed [see Upadhye ad Vellaisamy 2013, 2014, for more details] Specifically, usig Kerstas method 1964 ad the method of expoets, Vellaisamy ad Upadhye 2009 have established the bouds i Poisso approximatio as followig iequality: α j qj 2 d TV S, Z λ j=1 p j 1 mi 1, }, 2λe where λ = α i q i = αq, for X 1, X 2,, X are idepedet egative biomial distributed radom variables with parameters α j ad q j, j = 1, 2,, ad Z λ is a Poisso radom variable with mea λ It is worth poitig out that compariso of bouds i egative biomial approximatio ad Poisso approximatio is showig that a egative biomial approximatio is better tha Poisso approximatio i the case X j, j = 1, 2, are idepedet egative biomial radom variables [see Theorem 22 ad Theorem 24 i Vellaisamy ad Upadhye 2009] Besides, Poisso approximatio is also cosidered for a wide class of discrete radom variables via operator method ad method of probability distace [see Hug ad Thao 2013 ad Hug ad Giag 2014, for more details] The mai purpose of this paper is to use the Stei Che method for providig the bouds of Le Cam-type iequality 2 ad 3 i Poisso approximatio for row-wise arrays of idepedet egative-biomial distributed radom variables The results obtaied i this paper are extesios ad geeralizatios of some results i Teerapabolar ad Wogasem 2007, Teerapabolar 2009, 2013 Prelimiaries Durig the last several decades the Stei Che method has rise to become oe of the most importat tools available for studyig i Poisso approximatio problems The Stei Che method has bee dealt with i detail i may articles [the reader is referred to Stei 1972, Che 1975, Che ad Rölli 2013, Barbour et al 1992 ad Barbour ad Che 2004 for fuller developmet] The Stei Che method ca be summarized as follows: Let us deote by F X A the probability distributio fuctio of a discrete radom variable X A ad we will deoted by P α A = α A e α the Poisso distributio fuctio, defied o the set A Z + The best ow method for estimatig = sup x F X A P α A is basig o the followig argumets [see Che 1975 for more details]: Assume that hu is a real-valued bouded fuctio ad P α h = e α =0 h α Cosider the fuctio f which is a solutio of the differetial equatio α f x + 1 xf x = hx P α h

4 Hug ad Giag SprigerPlus 20165:79 Page 4 of 12 Settig hx = h A x = Puttig x = X ad taig the expectatio of both sides of the above differetial equatio, we have Thus, the problem of estimatig ca be reduced to that of estimatig the differece of the expectatios Eα f X + 1 EXf X Before startig the mai results i the ext sectio we first recall the followig remarable lemmas: Lemma 1 Barbour et al 1992 Let Vf A w = f A w + 1 f A w The, for A Z + ad Z + \0}, Lemma 2 Teerapabolar ad Wogasem 2007 Let w 0 Z + ad Z + \0}, we have Lemma 3 1, if x A, 0, if x / A F X A P α A = E[α f X + 1 Xf X] sup VfA w mi α 1 e α, 1 } w sup Vf Cw0 w γ e γ 1 1 mi w 0 + 1, 1 } w Teerapabolar 2009 Let w 0 Z + ad Z + \0, 1} The, we have } 0 < sup f w γ e γ 1 1 mi w, 1 w Lemma 4 Teerapabolar 2013 For w 0 Z + ad Z + \0, 1}, let e γ γ w 0 p γ w 0 = ad P γ w 0 = w 0 γ e γ The the followig iequality is true w 0! =0 sup f Cw0 w P w γ w 0 1 P γ w 0 p γ w mi w 0 + 1, 1 } Results Throughout the forthcomig, uless otherwise specified, we shall deote by X,1, X,2, ; = 1, 2, a row-wise triagular array of idepedet egative-biomial distributed radom variables with probabilities P X = = Cr p r + 1 p,

5 Hug ad Giag SprigerPlus 20165:79 Page 5 of 12 where p 0, 1; r = 1, 2, ; i = 1, 2, ; = 0, 1, Let W = X ad set λ = EW = r 1 p p The, for r 1, 2, } we have the followig theorems: Theorem 1 For A Z +, Proof Let f ad h are bouded real-valued fuctios defied o Z + For w = 0, 1, we have the Stei s equatio for Poisso distributio with a mea λ where P λ h = e λ hλ =0 For A Z +, let us deote by h A : Z + R ad by f A w the fuctios defied by 1, if w A, h A w = 0, if w / A ad where C w = 0, 1,, w} Give f = f A ad h = h A, We have the followig Stei s equatio: where PW A λ e λ A mi 1 e λ sup A λ λ f w + 1 wf w = hw P λ h, f A w = } r 1 p p,1 pr 1 p p w w 1!λ e λ [ ] P λ ha Cw Pλ h A P λ hcw, if w 1, 0, if w = 0, λ f w + 1 wf w = h A w P λ h A, P λ h A = e λ Therefore, the Stei s equatio ca be writte as follows: h A w A =0 h A λ = e λ λ A e λ λ = λ f w + 1 wf w Taig expectatios of both sides of above equatio, we have λ e λ PW A A = E[λ f W + 1 W f W ] It follows that

6 Hug ad Giag SprigerPlus 20165:79 Page 6 of 12 PW A A λ e λ =E[λ f W + 1 W f W ] E[r p 1f W + 1 X f W ] 8 Let W i = W X The, for each i, we get E[r p 1f W + 1 X f W ] = E[r p 1f W i + X + 1 X f W i + X ] = E[E[r p 1f W i + X + 1 X f W i + X X ]] = E[r p 1f W i + X + 1 X f W i + X X = 0]p r + E[r p 1f W i + X + 1 X f W i + X X = 1]r p r 1 p + E[ r p 1f W i + X X f W i + X X = ]C r + pr 1 p = E[r p 1p r f W i + 1] + E[r 2 2p r 1 p f W i + 2 r p r 1 p f W i + 1] + E[Cr + r 1 p +1 p r f W i C r + pr 1 p f W i + ] 2p r = r 1 p E[f W i + 1]+E[r 2 1 p 2 p r f W i + 2] + E[Cr + r 1 p +1 p r f W i C r + pr 1 p f W i + ] = r 1 p 2p r E[f W i + 1] + 2 r + 2 r 1 p p r f W i + Cr + pr 1 p f W i + ] E[C = r 1 p 2p r E[f W i + 1] + 2 E[C r + 2 r 1 p p r f W i + r + 1 C r + 2 pr 1 p f W i + ]

7 Hug ad Giag SprigerPlus 20165:79 Page 7 of 12 2p r = r 1 p E[f W i + 1] + [ r + 1 E C r + 2 r p r 1 p f W i + 2 r r + 1 C r + 2 pr f i 1 p Wi + 2 r + 1 r 1 C r + 2 r ] p r 1 p E[f W i + ] 2p r = r 1 p E[f W i + 1] + r + 1 C r p +1 p r E[f W i + ] 2 r + 1 Cr r + r 1 p +1 p r E[f W i + + 1] 2 r 1 p 2p r E[f W i + 2] 2p r = r 1 2p r p E[f W i + 1] r 1 p E[f W i + 2] + Cr + 1 p +1 p r E[f W i + ] 2 2 C r + 1 p +1 p r E[f W i + + 1] = r 1 p 2p r E[f W i + 1 f W i + 2] + 2 = 1 C r + 1 p +1 p r E[f W i + f W i + + 1] C r + 1 p +1 p r E[f W i + f W i + + 1] By usig Lemma 1, we have E[r p 1f W + 1 X f W ] 1 C r + 1 p +1 p r E f W i + f W i Cr + 1 p +1 p r sup Vf w 1 w mi λ 1 e λ p r Cr + 1 p +1, 1 p r Cr + 1 p +1 1 = mi λ 1 e λ p r 1 p Cr + 1 p, 1 p r } 1 p p r 1 = mi λ 1 e λ p r r 1 p r 1 p p, pi r 1 p 1 p } = mi λ } 1 e λ r 1 p p,1 pr 1 p p 9

8 Hug ad Giag SprigerPlus 20165:79 Page 8 of 12 To combie 8 ad 9, we have PW A λ e λ A mi 1 e λ sup A λ The proof is complete } r 1 p p,1 pr 1 p p Remar 1 It is easily see that the 4 is a special case of the Theorem 1 with r = 1; = 1, 2, ; i = 1, 2, Theorem 2 Let W ad λ be defied as i Theorem 1 The, for w 0 N, PW w 0 λ e λ w 0 λ e λ r 1 p 1 mi p w 0 + 1,1 pr } 1 p p Proof For C w =0,, w} ad w 0 N, let h w0 : Z + R, f Cw0 w 0 be defied by h Cw0 w = 1 if w w0, 0 if w > w 0 w 1!λ w e λ [ ] P λ hcw0 Pλ 1 hcw if w 0 < w, f Cw0 w = w 1!λ w e λ ] [P λ hcw Pλ 1 h Cw0 if w 0 w, 0 if w = 0 Give f = f Cw0 ad h = h Cw0 We have the Stei s equatio h Cw0 w w 0 e λ λ = λ f w + 1 wf w Taig expectatios of both sides ad arguig similarly to the proof of Theorem 1 we prove that 10 PW w 0 e λ E[r p 1f W + 1 X f W ] w 0 λ Accordig to the Theorem 1, we have E[r p 1f W + 1 X f W ] = 1 C r + 1 p +1 p r E[f W i + f W i + + 1] 11

9 Hug ad Giag SprigerPlus 20165:79 Page 9 of 12 Hece, by 10, 11 ad Lemma 2, we have PW w 0 λ e λ w 0 E[r p 1f W + 1 X f W ] Cr + 1 p +1 p r sup Vf w 1 w λ e λ p r 1 mi C r w p +1, 1 p r Cr + 1 p +1 1 λ e λ p r 1 p 1 mi C r w p, 1 p r } 1 p p r 1 r = λ e λ p 1 p r 1 p 1 mi w 0 + 1p r, 1 p p r +1 1 p } } = λ e λ r 1 p 1 mi p w 0 + 1,1 pr 1 p p Thus PW w 0 λ e λ w 0 λ e λ r 1 p 1 mi This fiishes the proof p w 0 + 1,1 pr } 1 p p Remar 2 It is easy to chec that the 5 is a special case of Theorem 2 with r = 1; = 1, 2, ; i = 1, 2, Theorem 3 have λ λ e 1 Let W = X i ad λ = r q with q = 1 p The, we mi α i, β } i α i w With α i = 1 p r r q p r, β i = r p r w 0 PW w 0 =0 1 r q p r λ e λ 0, Proof Arguig as i theorem 3, we have the Stei s equatio w 0 h w0 w =0 e λ λ = λ f w + 1 wf w

10 Hug ad Giag SprigerPlus 20165:79 Page 10 of 12 Taig expectatios of both sides, we get w 0 λ e λ PW w 0 =0 = E[ λ f W + 1 W f W = E[r q f W + 1 X f W ] ] 12 Let W i = W X The, for each i, we deduce E[r q f W + 1 X f W ] = E[E[r q f W i + X + 1 X f W i + X X ]] = E [ r q p r f W i + 1 ] + E[r 2 q2 pr f W i + 2 r q p r f W i + X ] + 2 E[r Cr + q+1 p r f W i Cr + q pr f W i + ] = E[r C r + 2 q pr f W i + Cr + q pr f W i + ] 2 = r E[ r + 1 C r + q pr f W i + Cr + q pr f W i + ] 2 = 1 r + 1 C r + q pr f W i + 1 r + 1 C r + q pr By usig Lemma 3, the we have sup w f w 1 r + 1 C r + q pr λ λ e 1 p r mi 1 w Moreover, we have sup w 1 r + 1 C r + q f w 1 r + 1 C r + q, } p r 1 r + 1 C r + q 1 pr r q p r ad p r 1 r + 1 C r + q r p r 1 r q p r 1 p r r q p r 15

11 Hug ad Giag SprigerPlus 20165:79 Page 11 of 12 Hece, by 12, 13, 14 ad 15, we ca assert that λ λ e 1 mi α i, β } i α i w w 0 PW w 0 =0 λ e λ 0 The proof is complete Remar 3 Whe r = 1, we have α i = 1 p q p = 1 p 1 p = q 2, β i = p 1 q p = 1 p 1 p 1 p 2 1 p p = p p 1 + β i α i = q 2 p 1 = q2 p p It is clear that the 6 is a special case of Theorem 3 with r = 1; = 1, 2, ; i = 1, 2, Theorem 4 Let W = X ad λ = r q i with q = 1 p The, for w 0 N we have where PW w 0 P λ w 0 P λ w 0 1 P λ w 0 p λ w α i = 1 p r r q p r ad β i = r p r mi α i, β } i α i, w r q p r = q p, p Proof Accordig to Theorem 3 we obtai the followig iequality PW w 0 Pλ w 0 with α i = 1 p r r q p r, β i = r p r Hece, the theorem is proved 1 r + 1 C r + q pr By usig Lemma 4, the we have PW w 0 Pλ w 0 Pλ w 0 1 Pλ w 0 p λ w Pλ w 0 1 Pλ w 0 mi p λ w r q p r sup w f Cw0 w 1 r + 1 C r + q pr 1 mi w 0 + 1, 1 } α i, β } i α i, w 0 + 1

12 Hug ad Giag SprigerPlus 20165:79 Page 12 of 12 Remar 4 I the same way as i Remars 3, we otice that 7 is a special case of Theorem 4 with r = 1; = 1, 2, ; i = 1, 2, Coclusios We coclude this paper with the followig commets The received results i this paper are extesios ad geeralizatios of results i Teerapabolar ad Wogasem 2007, Teerapabolar 2009, 2013 The results would be more iterestig ad valuable if the discussed egative-biomial radom variables i this paper are depedet We shall tae this up i the ext study Authors cotributios All authors cotributed equally ad sigificatly to this wor All authors drafted the mauscript Both authors read ad approved the fial mauscript Acowledgemets The authors would lie to express their gratitude to the referees for valuable commets ad suggestios The research was supported by the Vietam s Natioal Foudatio for Sciece ad Techology Developmet NAFOSTED, Vietam uder Grat Competig iterests The authors declare that they have o competig iterests Received: 22 Jue 2015 Accepted: 12 Jauary 2016 Refereces Barbour AD, Holst L, Jaso S 1992 Poisso approximatio Claredo Press, Oxford Barbour AD, Che LHY 2004 A itroductio to Stei s method, Lecture Notes Series, Istitute for Mathematical Scieces Natioal Uiversity of Sigapore, vol 4 Che LHY 1975 Poisso approximatio for depedet trials A Probab 3: Che LHY, Rölli A 2013 Approximatig depedet rare evets Beroulli 194: Hug TL, Thao VT 2013 Bouds for the Approximatio of Poisso-biomial distributio by Poisso distributio J Iequal Appl 2013:30 Hug TL, Giag LT 2014 O bouds i Poisso approximatio for iteger-valued idepedet radom variables J Iequal Appl 2014:291 Kersta J 1964 Verallgemeierug eies Satzes vo Prochorow ud Le Cam Z Wahrsch Verw Gebiete 2: Le Cam L 1960 A approximatio theorem for the Poisso biomial distributio Pacific J Math 10: Neammaee K 2003 A ouiform boud for the approximatio of Poisso biomial by Poisso distributio IJMMS 48: Stei CM 1972 A boud for the error i ormal approximatio to the distributio of a sum of depedet radom variables I: Proceedigs of sixth Bereley symposium mathematical statistics ad probability, vol 3, pp Teerapabolar K, Wogasem P 2007 Poisso approximatio for idepedet geometric radom variables It Math Forum 2: Teerapabolar K 2009 A ote o Poisso approximatio for idepedet geometric radom variables It Math Forum 4: Teerapabolar K 2013 A ew boud o Poisso approximatio for idepedet geometric variables It J Pure Appl Math 844: Upadhye NS, Vellaisamy P 2013 Improved bouds for approximatios to compoud distributios Stat Probab Lett 832: Upadhye NS, Vellaisamy P 2014 Compoud Poisso approximatio to covolutios of compoud egative biomial variables Methodol Comput Appl Probab 164: Vellaisamy P, Upadhye NS 2009 Compoud egative biomial approximatios for sums of radom variables Probab Math Stat 292: