BULLETIN of the Malaysian Mathematical Sciences Society http://mathusmmy/bulletin Bull Malays Math Sci Soc (2) 34(2) (2011), 331 336 On the Poisson Appoximation to the Negative Hypegeometic Distibution K Teeapabolan Depatment of Mathematics, Faculty of Science, Buapha Univesity, Chonbui 20131, Thailand kanint@buuacth Abstact In this pape, the Stein-Chen method and the w-function associated with the negative hypegeometic andom vaiable ae used to give a esult of the Poisson appoximation to the negative hypegeometic distibution in tems of the total vaiation distance Some numeical examples ae pesented to illustate the esult obtained 2010 Mathematics Subject Classification: 60F05 Keywods and phases: Negative hypegeometic distibution, Poisson appoximation, Stein-Chen method, w-function 1 Intoduction Suppose a set o a population of items of size S + R consists of S items of special type and R items of non-special type, espectively Items ae dawn at andom, one at a time, without eplacement fom this population until the numbe of non-special items eaches a fixed numbe Let X be the numbe of special items in the sample Then X has a negative hypegeometic distibution, sometimes called invese hypegeometic distibution, with paametes R, S and, denoted by N H(R, S, ) Its pobability function can be expessed as ) (11) p X (k) ( +k 1 )( R +S k k S k ( R+S S ), k 0, 1,, S, whee R, S N and {1,, R} The mean and vaiance of X ae µ E(X) S and σ 2 Va(X) S(R+S+1)(R +1) () 2 (R+2), espectively This distibution was used by Kaigh and Lachenbuch [7] in esampling fo nonpaametic quantile estimation Its othe applications can be found in [5, 6, 9, 13, 14], etc Note that this distibution is a finite sample analogy to the negative binomial distibution, which aises in a scheme of sampling with eplacement Moeove, if Communicated by Anton Abdulbasah Kamil Received: Novemebe 12, 2008; Revised: Januay 19, 2010
332 K Teeapabolan S R, S in such a way tends to a constant θ, then the negative hypegeometic distibution conveges to the negative binomial with paametes and θ 1+θ Similaly, this distibution may convege to the binomial o Poisson o nomal distibution if the conditions on thei paametes ae appopiate Let us conside the pobability function in (11), it can be seen that, fo k {0, 1,, S}, (12) ( ) S ( + k 1)! p X (k) k ( 1)! ( S k (R + S k)! (R )! ) ( + k 1)( + k 2) R! (R + S)! (R + S k)(r + S k 1) (R + 1) (R + S)(R + S 1) (R + 1) ( ) S + k 1 + k 2 k R + 1 R + 1 R + 1 R +S k R +S k 1 R+S R+S 1 R +1 ( ) ( ) k ( ) S k ( S R + 1 k R + 1 R + 1 ( ) ( ) 1 + S k 1 R +1 1 + S k 2 R +1 1 ( ) ( ), 1 + S 1 1 + S 2 1 1 + k 1 ) ( 1 + k 2 ) 1 and if R, in such a way p 1 q, then p X(k) ( S k) p k q S k, k 0, 1,, S, which is the binomial distibution with paametes S and p, denoted by B(S, p) It is well known that if p is small and Sp λ as S, then B(S, p) can be appoximated by Po(λ), whee Po(λ) is the Poisson distibution with mean λ Theefoe, in view of (12), N H(R, S, ) can also be appoximated by Po(λ), unde appopiate conditions on thei paametes It should be noted that if is not small and S is sufficiently lage, then N H(R, S, ) can also be appoximated by the nomal distibution with mean S S(R +1) and vaiance () In this case, a 2 bound on the nomal appoximation can be deived by using the same method in Boonta and Neammanee [2] In this pape, the negative hypegeometic distibution has been appoximated by Poisson distibution, and the accuacy of the appoximation is measued in tems of an uppe bound fo the total vaiation distance The total vaiation distance between N H(R, S, ) and Po(λ) is defined by (13) d T V (N H(R, S, ), Po(λ)) sup A sup A N H(R, S, ){A} Po(λ){A} p X (k) e λ λ k k!, k A k A whee A is a subset of N {0} and p X (k) is defined in (11) The tools fo giving an uppe bound fo the total vaiation distance ae the Stein-Chen method and the w-function associated with the negative hypegeometic
On the Poisson Appoximation to the Negative Hypegeometic Distibution 333 andom vaiable, which ae intoduced and applied in Section 2 In Section 3, some numeical examples ae pesented to illustate the obtained esult 2 Main esult We will pove ou main esult using the Stein-Chen method togethe with the w- function associated with the negative hypegeometic andom vaiable 21 The w-function The w-functions wee studied and used by many authos, among othes by Cacoullos and Papathanasiou [3], Papathanasiou and Utev [10], and Majsneowska [8] In the note by the latte, the following ecuence elation can be found: (21) w(k) p X(k 1) w(k 1) + µ k p X (k) σ 2 0, k S(x) \ {0}, whee w(0) µ σ, S(x) is suppot of X, p 2 X (k) > 0 fo all k S(x) and µ and σ 2 (0, ) ae mean and vaiance of X, espectively Using the elation (21), we give the fom of the w-function associated with the negative hypegeometic andom vaiable in the following lemma Lemma 21 Let w(x) be the w-function associated with the negative hypegeometic andom vaiable X, then (22) w(k) whee σ 2 S(R+S+1)(R +1) () 2 (R+2) ( + k)(s k) (R + 1)σ 2, k 0, 1,, S, Poof Following (21), the ecuence elation of w-function associated with the andom vaiable X can be witten as S k(r + S k + 1) (23) w(k) + w(k 1) (R + 1)σ2 ( + k 1)(S k + 1) k, k 1,, S, σ2 S whee w(0) ()σ In the next step, we shall show that (22) holds fo evey 2 k {1,, S} Fom (23), w(1) S (R + 1)σ 2 + w(0)r + S 1 ( + 1)(S 1) S σ2 (R + 1)σ 2 Assuming that w(i 1) (+i 1)(S i+1) ()σ, fo 1 < i 1 < S, we have 2 w(i) S i(r + S i + 1) + w(i 1) (R + 1)σ2 ( + i 1)(S i + 1) i ( + i)(s i) σ2 (R + 1)σ 2 Theefoe, by mathematical induction, (22) holds fo evey k {1,, S} The next elation stated by Cacoullos and Papathanasiou [3] is cucial fo obtaining ou main esult If a non-negative intege-valued andom vaiable X has p X (k) > 0 fo evey k S(x) and 0 < σ 2 Va(X) <, then (24) Cov(X, g(x)) σ 2 E[w(X) g(x)],
334 K Teeapabolan fo any function g : N {0} R fo which E w(x) g(x) <, whee g(x) g(x + 1) g(x) 22 The Stein-Chen method The classical Stein s method was fist intoduced by Stein [11] in 1972 It is a stating tool fo appoximating the distibution of andom elements His oiginal wok was applied to cental limit theoem fo sums of andom vaiables The vesion appopiate fo the Poisson case was fist developed by Chen [4] in 1975 It is efeed to as the Stein-Chen method The Stein-Chen equation fo the Poisson distibution with paamete λ > 0 is, fo given h, of the fom (25) h(x) P λ (h) λg(x + 1) xg(x), whee P λ (h) l0 h(l) e λ λ l l! and g and h ae bounded eal-valued functions defined on N {0} Fo A N {0}, let h A : N {0} R be defined by { 1 if x A, (26) h A (x) 0 if x / A Babou et al showed in [1] that the solution g g A of (25) can be expessed in the fom { (x 1)!λ x e λ [P λ (h A Cx 1 ) P λ (h A )P λ (h Cx 1 )] if x 1, (27) g(x) 0 if x 0, whee C x {0,, x}, and that fo any subset A of N {0} and k N, the following bound is valid (28) sup A,k g(k) sup g(k + 1) g(k) λ 1 (1 e λ ) A,k Putting h h A and taking expectation in (25), we get (29) N H(R, S, ){A} Po(λ){A} E[λg(X + 1) Xg(X)], whee g is defined in (27) In the following theoem using the Stein-Chen method, we pesent an uppe bound fo the total vaiation distance between the negative hypegeometic and Poisson distibutions Theoem 21 Let X be the negative hypegeometic andom vaiable, λ and (S 1) Then, fo A N {0}, (210) d T V (N H(R, S, ), Po(λ)) ( 1 e λ) (R + 1)( + 1) S(R + 1) Poof Fom (29) and (24), it follows that N H(R, S, ){A} Po(λ){A} λe[g(x + 1)] E[Xg(X)] λe[g(x + 1)] Cov(X, g(x)) µe[g(x)] λe[ g(x)] Cov(X, g(x)) λe[ g(x)] σ 2 E[w(X) g(x)] S
On the Poisson Appoximation to the Negative Hypegeometic Distibution 335 Thus, by (28), E [λ σ 2 w(x)] g(x) sup g(x) E λ σ 2 w(x) x 1 (211) d T V (N H(R, S, ), Po(λ)) λ 1 (1 e λ )E λ σ 2 w(x) In view of Lemma 21, we have, fo k {0, 1,, S}, which implies that λ σ 2 w(k) S ( + k)(s k) R + 1 R + 1 ( S + k)k 0, R + 1 E λ σ 2 w(x) λ σ 2 E[w(X)] λ σ 2 λ (R + 1)( + 1) S(R + 1) Substituting this esult into (211), finishes the poof of the theoem Coollay 21 Fo S 1, we have ( ) 1 e λ S(S 1) d T V (N H(R, S, ), Po(λ)) Remak 21 It is obseved fom Theoem 21 that d T V (N H(R, S, ), Po(λ)) ( 1 e λ) (R + 1)( + 1) S(R + 1) < R, that is, if R peviously is small, then (210) yields a good Poisson appoximation, as mentioned 3 Numeical examples The following numeical examples ae given to illustate how well the Poisson distibution appoximates the negative hypegeometic distibution and to see how tight is the uppe bound fo the total vaiation distance between two distibutions given in Theoem 21 Table 1 povides numeical examples of the total vaiation distance between negative hypegeometic and Poisson distibutions and its uppe bound fo given R 100, 300, 600, S 5, 10, 15, 20, 25, 30, 35, 40, 45 and 15, 30, 45 The numeical examples in the table suggest that the Poisson appoximation to the negative hypegeometic distibution is quite efficient povided that R is small, that is, the estimate of the total vaiation distance between two distibutions is close to the tue value of the distance povided that is small and close to S and R is lage
336 K Teeapabolan Table 1 Numeical examples of the total vaiation distance between negative hypegeometic and Poisson distibutions and its uppe bound R S λ d T V (N H, Po) uppe bound 100 5 15 074257 003107 006034 100 10 15 148515 001809 005676 100 15 15 222772 000727 002823 300 20 30 199336 000993 003716 300 25 30 249169 000674 002579 300 30 30 299003 000298 001255 600 35 45 262063 000547 002098 600 40 45 299501 000347 001419 600 45 45 336938 000170 000701 Acknowledgement The autho would like to thank the efeees fo thei constuctive suggestions that helped impove the pesentation of the pape Refeences [1] A D Babou, L Holst and S Janson, Poisson Appoximation, Oxfod Studies in Pobability, 2, Oxfod Univ Pess, New Yok, 1992 [2] S Boonta and K Neammanee, Bounds on andom infinite un model, Bull Malays Math Sci Soc (2) 30 (2007), no 2, 121 128 [3] T Cacoullos and V Papathanasiou, Chaacteizations of distibutions by vaiance bounds, Statist Pobab Lett 7 (1989), no 5, 351 356 [4] L H Y Chen, Poisson appoximation fo dependent tials, Ann Pobability 3 (1975), no 3, 534 545 [5] J Cuzick, Event-based analysis times fo andomised clinical tials, Biometika 88 (2001), no 1, 245 253 [6] P G Gaz, On the systematic and system-based study of gapheme fequencies: A e-analysis of Geman lette fequencies, Glottometics, 15 (2007), 82 91 [7] W D Kaigh and P A Lachenbuch, A genealized quantile estimato, Comm Statist A Theoy Methods 11 (1982), no 19, 2217 2238 [8] M Majsneowska, A note on Poisson appoximation by w-functions, Appl Math (Wasaw) 25 (1998), no 3, 387 392 [9] T Obilade, A weighted likelihood atio of two elated negative hypegeomeic distibutions, Acta Math Appl Sin Engl Se 20 (2004), no 4, 647 654 [10] V Papathanasiou and S A Utev, Intego-diffeential inequalities and the Poisson appoximation, Sibeian Adv Math 5 (1995), no 1, 120 132 [11] C Stein, A bound fo the eo in the nomal appoximation to the distibution of a sum of dependent andom vaiables, in Poceedings of the Sixth Bekeley Symposium on Mathematical Statistics and Pobability (Univ Califonia, Bekeley, Calif, 1970/1971), Vol II: Pobability theoy, 583 602, Univ Califonia Pess, Bekeley, CA [12] C M Stein, Appoximate Computation of Expectations, IMS, Haywad Califonia, 1986 [13] X-G Wang, Phase popeties of hypegeometic states and negative hypegeometic states, J Opt B Quantum Semiclass Opt 2 (2000), no 1, 29 32 [14] J Weso lowski and M Ahsanullah, Lineaity of egession fo non-adjacent weak ecods, Statist Sinica 11 (2001), no 1, 39 52