NONPARAMETRIC ESTIMATION IN RANDOM SUM MODELS

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1 STATISTICA, ao LXIX,., 009 NONPARAMETRIC ESTIMATION IN RANDOM SUM MODELS. INTRODUCTION The motvato for ths paper arses from the fact that the radom sum models are wdely used rsk theory, queueg systems, relablty theory, ecoomcs, commucatos, ad medce. I surace cotexts, the compoud dstrbuto of radom sum varables (see, e.g., Ca ad Wllmot (005), Charalambdes (005) ad Paer (006)) arses aturally as follows. Let X be the umber of persos volved the th accdet o a partcular day a certa cty ad let N be aother radom varable that represets the umber of accdets occurrg o that N day. The, the radom varable gve by SN = X, = S0 = X 0 =0, deotes the total umber of persos volved accdets a day, ad s called compoud radom varable. I queueg model, at a bus stato, assume that the umber of passegers o the th bus s X ad the umber of arrvg buses s the radom varable N ; the the umber of passegers arrvg buses durg a perod of tme s the radom sum S = N N X =. I commucatos, let us cosder the radom varable N as the umber of data packets trasmtted over a commucato lk oe mute such that each packet s successfully decoded wth probablty p, depedet of the decodg of ay other packet. Hece, the umber of successfully decoded packets oe mute spa s the radom sum S = N N X =, where X s f the th packet s decoded correctly ad 0 otherwse. That s, the compoud radom varable S N follows a bomal dstrbuto wth radom ad fxed parameters N ad p, respectvely. The dstrbuto of the radom varable S N, s kow as a compoud dstrbuto of the radom varables X ad N, where =,,..., N ; see, e.g., Chatfeld ad Thoebald (973), Ludberg (964), Medh (97), Peköz ad Ross (004), ad Sahoglu (99). Thus, the dstrbuto of a radom sum s based o two dstrbutos: the dstrbuto of N, whch we wll call the evet dstrbuto ad the dstrbuto of the radom varables X, X,, XN, whch we wll call the compoudg dstrbuto.

2 74 May authors (e.g. Ptts (994), Buchma ad Grübel (003, 004), ad Ca ad Wllmot (005)) have bee vestgated the dstrbutoal propertes of the radom sum S N cludg ts dstrbuto fucto ad the asymptotc behavour of the tal. Ptts (994) dscussed the oparametrc estmato of the dstrbuto fucto of S N based o topologcal cocepts ad assumed that the evet dstrbuto s kow ad the compoudg dstrbuto s ukow. Buchma ad Grübel (003, 004) cosdered the estmato problem of a probablty set of the compoudg dstrbuto of X, =,,..., N, ad the compoud dstrbuto of X ad N assumg that the evet dstrbuto follows Posso dstrbuto. Weba (007) vestgated predcto of the compoud mxed Posso process assumg that the evet dstrbuto s a mxed Posso process ad the compoudg dstrbuto s ukow. Meawhle, we here deal wth ukow evet ad compoudg dstrbutos. Also, our work s more geeral tha Ptts (994),ad Buchma ad Grübel (003, 004), sce t ca be appled to ay evet ad compoudg dstrbutos as log as they obey the assumptos that are secto 4.. Recetly, Bakouch ad Rstć (009), ad Rstć et al. (009) employed ths radom sum, S N, to model teger-valued tme seres that are used forecastg ad regresso aalyss of cout data. Also, they dscussed some propertes of S N for a kow evet dstrbuto ad a ukow compoudg dstrbuto, ad the obtaed estmators of the parameters of such dstrbutos after gvg a kow compoudg dstrbuto. I secto 4. of ths paper, we cosder a oparametrc estmato of the compoudg dstrbuto the compoud Posso model. The approach ths case s smlar to the estmato method of Buchma ad Grübel (003, 004) but our approach s of terest to ay problem cocerg the compoud Posso model. O the other had, Buchma ad Grübel (003, 004) are terested oly the compoud Posso model that s commoly used queueg theory for modelg the arrval dstrbuto of the umber of customers. Let Y,, Y deote depedet radom varables, each wth the dstrbuto of S N. Based o the formato provded by the radom varables Y,, Y, we wsh to estmate features of the dstrbutos of X ad N. A commo used approach to ths problem s to use parametrc models for the evet ad compoudg dstrbutos. Ths leads to a parametrc model for the dstrbuto of Y,, Y. Estmatg the parameters of ths model leads to estmators for the parameters of the dstrbutos of X ad N. The drawback of ths approach s that t requres strog assumptos about the dstrbutos of X ad N. However, sce we do ot observe a radom sample from both dstrbutos, these assumptos are dffcult to verfy emprcally. The purpose of ths paper s to cosder two methods of oparametrc estmato radom sum models; that s, we cosder methods that do ot requre full parametrc models for the compoudg dstrbuto ad the evet dstrbu-

3 Noparametrc estmato radom sum models 75 to. The frst cosders estmato of the meas of the dstrbutos of X ad N based o the secod-momet assumptos regardg compoudg ad evet dstrbutos. The secod problem s the oparametrc estmato of the compoudg dstrbuto based o a parametrc model for the evet dstrbuto. The outle of the paper s as follows. I secto some basc results regardg radom sum dstrbutos are preseted; although these results are well-kow, they are cetral to the proposed estmato procedures ad, hece, they are preseted for completeess. The approach based o secod-momet assumptos s preseted secto 3. I secto 4, oparametrc estmato of the compoudg dstrbuto s developed.. SOME PROPERTIES OF RANDOM SUMS Let S = N N X =, where X s depedet, detcally dstrbuted, oegatve, teger-valued radom varables ad let N be a o-egatve, tegervalued radom varable depedet of X, X,, XN. I ths secto, we brefly revew some statstcal propertes of S N (see, e.g., Klugma et al., 004). Let P N deote the probablty geeratg fucto of N ad let P X deote the probablty geeratg fucto of X. Therefore, the probablty geeratg fucto of S N s gve by P ()= t P [ P ()]. t (.) S N N X The fucto PS ( t ) s well defed for t. Smlarly, let M N N ad M X deote the momet geeratg fuctos of N ad X, respectvely. Hece, the momet geeratg fucto of S N s M ()= t P [ M ()]= t M [l M ()]. t (.) S N N X N X The fucto MS () t s well defed for t a for some a > 0. Hece, t follows N that E[ S ]= EN [ ] EX [ ], (.3) N N ES [ ]= ENEX [ ] [ ] + EN [ ( N )] EX [ ] EX [ ], (.4) Var[ S ]= E[ N ] Var[ X ] + Var[ N ]( E[ X ]). (.5) N

4 76 3. ESTIMATION OF THE DISTRIBUTION MEANS UNDER SECOND-MOMENT ASSUMPTIONS Let θ = EX ( ; θ ) ad let λ = EN ( ; λ ). Suppose we observe Y,, Y, depedet, detcally dstrbuted radom varables such that Y has the dstrbuto of SN = X + + XN. Note that ether N or X,, XN are drectly observed; the data oly cossts of realzatos of the radom varable S N ad are deoted by Y,, Y. Our goal s to estmate θ ad λ, the meas of the dstrbutos of X ad N, respectvely, based o ( Y,, Y ). Clearly, ths wll be mpossble wthout further assumptos regardg the dstrbutos of X ad N. Defe the fuctos V : Λ [0, ) ad W : Θ [0, ) by V( λ)= Var( N; λ) ad W ( θ)= Var( X ; θ ) ad assume that V ad W are kow. Usg equatos (.3) ad (.5) we ca show that Y has mea θλ ad varace θ V( λ) + λw( θ). We propose to estmate ( θ, λ ) usg the frst two momets of the dstrbuto of Y. Expressg the parameters θ ad λ terms of the frst two momets of the dstrbuto of Y proceeds ths approach. The assumpto that V ad W are kow s meat to be a weaker verso of the assumpto of parametrc models for the dstrbutos. Ths method mght be used whe oe s wllg to assume that V ( λ )= λ, as the Posso dstrbuto, but s ot wllg to assume that the Posso dstrbuto actually holds. Thus, our approach ths secto s smlar to the quas-lkelhood method descrbed, e.g., McCullagh ad Nelder (989, chapter 9). Sce the frst momet of Y s θλ, t s ecessary that the secod momet of Y s ot a fucto of θλ ; hece, we requre the followg assumpto: Assumpto ad For ( θ, λ) Θ Λ, Let Y = Y = θ V( λ) + λw( θ) s ot a fucto of θλ.

5 Noparametrc estmato radom sum models 77 S = ( Y Y). = Defe the estmator ( θ, λ ) as the soluto to θλ= Y, θ V( λ) + λw( θ)= S, (3.) provded that such soluto exsts. Determato of the asymptotc dstrbuto of ( θ, λ ) requres the followg regularty codtos: Assumpto The parameter space Θ Λ s a ope subset of R. Uder assumpto, the true parameter value ( θ, λ ) s a teror pot of the parameter space. Assumpto 3 Defe a fucto H : Θ Λ A by H( θ, λ)=( θλ, θ V( λ) + λw( θ)). The A s a ope subset of R ad H( θ, λ ) s a oe-to-oe fucto wth a cotuously dfferetable verse. Uder assumpto 3, the estmators θ ad λ ca be wrtte as fuctos of Y ad S. Ths fact, together wth the asymptotc dstrbuto of ( YS, ) ca be used to determe the asymptotc dstrbuto of ( θ, λ ). Assumpto 4 4 For all ( θ, λ) Θ Λ, EY ( ; θ, λ)<. Assumpto 4 s used to apply the strog law of large umbers ad the cetral lmt theorem to sums of the form Y = ad Y =. For a o-egatve defte matrx L, let N( 0, L) deote a radom varable wth a bvarate ormal dstrbuto wth mea vector 0 ad covarace matrx L ; let µ 3 ( θλ, ) ad µ 4 ( θλ, ) deote the thrd ad fourth cetral momets, respectvely, of Y.

6 78 Theorem 3. Assume that assumptos -4 hold. The, wth probablty approachg as, the estmator ( θ, λ ) exsts ad where θ θ N( 0, Σ( θ, λ)) as λ λ Var( Y; θ, λ) µ 3( θ, λ) Σ( θ, λ)= M( θ, λ) M(, ) θ λ µ 3( θ, λ ) µ 4( θ, λ) Var( Y; θ, λ) ad (, )= λ θ M θ λ. ' ' θv( λ) + λw ( θ) V ( λ) θ + W( θ) Proof: Sce here we are cocered wth the lmtg behavour of the estmators θ, λ, we wrte Y ad S for Y ad S, respectvely, to emphasze ther depedece o. Frst cosder cosstecy of ( θ, λ ). Uder assumpto 4, Y E( Y ; θ, λ)= θλw. p. as, where w.p. meas wth probablty, ad S Var( Y ; θ, λ)= θv( λ) + λw( θ) w. p. as so that ( Y, S ) coverges to H( θ, λ ) wth probablty. It follows that, wth probablty, ( Y, S ) les A for suffcetly large. By assumpto 3 there exsts a cotuous fucto h= H (.,.), the verse of H (.,.), such that, whe ( Y, S ) A, the ( θ, λ )= hy (, S ). Thus, wth probablty, ( θ, λ )= hy (, S ) for suffcetly large ad ( Y, S ) coverges to H( θ, λ ). It follows that ( θ, λ ) s a cosstet estmator of ( θ, λ ). Now cosder the asymptotc dstrbuto of ( θ, λ ). Uder assumpto 4, Y (, ) H θλ N( 0, B( θλ, )) as S T

7 Noparametrc estmato radom sum models 79 where Var( Y ; θ, λ) µ 3( θ, λ) B( θ, λ)= ; µ 3( θ, λ ) µ 4( θ, λ) Var( Y; θ, λ) see, e.g., example 3.3 of Sever (005). Sce, for suffcetly large, ( θ, λ )= hy (, S ), t follows from the δ -method (see, e.g., Sever, 005, secto 3.), that where θ θ N ( 0, D( θ, λ λ λ )) D( θ, λ)= h ( H( θ, λ)) B( θ, λ) h ( H( θ, λ)) T ad h deotes the matrx of partal dervatves of h. Note that, sce h= H (.,.), hhθ ( (, λ))=( θ, λ ); t follows that h ( H( θ, λ))= H ( θ, λ), the matrx verse of H (.,.), where H deotes the matrx of dervatves of H( θ, λ ) wth respect to ( θ, λ ). It follows mmedately from the expresso for H( θ, λ ) that λ θ H ( θ, λ)=. θv( λ) + λw ( θ) V ( λ) θ + W( θ) The result follows. ad Let 3 µ 3 = ( Y Y), = µ 4 = ( Y Y) = = S µ 3 B 4. µ 3 µ 4 S 4

8 80 Defe T Σ = M( θ, λ) BM( θ, λ). Theorem 3. Assume that assumptos -4 hold. The Σ Σ( θ, λ) as. Proof: Uder assumpto 4, as, S Var( Y ; θ, λ ), µ µ ( θ, λ ), ad µ µ ( θ, λ) It follows that B B( θ, λ) as. Sce M( θ, λ ) s a cotuous fucto of ( θ, λ ), ad ( θ, λ ) s a cosstet estmator of ( θ, λ ), t follows that M( θ, λ) M( θ, λ) as. The result follows. 4. NONPARAMETRIC ESTIMATION OF THE COMPOUNDING DISTRIBUTION 4. Geeral approach Let P deote the probablty geeratg fucto of Y, let R deote the probablty geeratg fucto of the evet dstrbuto, ad let Q deote the probablty geeratg fucto of the compoudg dstrbuto. Assume that we have a parametrc model wth parameter λ for the evet dstrbuto; hece, we wrte R( ; λ) for R as Let R( Q( t); λ )= P( t), t. Pt ( )= t = Y deote the emprcal probablty geeratg fucto based o Y,, Y. The Pt ( ) s a cosstet estmator of Pt ( ) for t. Hece, a estmator of the compoudg dstrbuto ca be obtaed by settg R( Q( t); λ ) equal to Pt ( ), where λ s a approprate estmator of λ.

9 Noparametrc estmato radom sum models 8 I order to carry out ths approach, there are several ssues whch must be cosdered. Frst, we must esure that λ ad Q are detfed ths model. Secod, a estmator of λ must be costructed. Fally, we must be able to use the detty R( Q( t); λ )= P ( t) (4.) order to obta a estmator of the compoudg dstrbuto. Frst cosder detfcato. The followg assumpto s suffcet for Q ad λ to be detfed. Assumpto 5 Rt ( ; λ)= Rt ( ; λ ) f ad oly f t = t.. R(0; λ )= R(0; λ ) f ad oly f λ= λ.. Lemma 4. Assume that assumpto 5 s satsfed. The R( Q( t); λ)= R( Q( t); λ ) for t f ad oly f Q= Q ad λ= λ. Proof: Clearly, f Q= Q ad λ = λ, the RQ ( ( t); λ)= RQ ( ( t); λ ) for t. Hece, assume that R( Q( t); λ)= R( Q( t); λ ) for t. Note that Q(0)= Q (0)= 0. Hece, settg t = 0, t follows that R(0; λ)= R(0; λ ) ; by part () of assumpto 5, t follows that λ= λ. Fx t. By part () of assumpto 5, RQ ( ( t); λ)= RQ ( ( t); λ ) mples that Q()= t Q() t. Sce t s arbtrary, Q= Q, provg the result. Now cosder estmato of λ. It follows from the detfcato result that a estmator of λ ca be obtaed by solvg R(0; λ )= P (0) (4.) for λ. However, the estmator gve (4.) wll oly be useful for those dstrbutos for whch P (0) s ot close to 0. A alteratve approach s to base a estmator of a secod-momet assumpto, as dscussed secto 3. Fally, we must use the detty (4.) to obta a estmator of the compoudg dstrbuto. The detals of ths wll deped o the parametrc model used for the evet dstrbuto. I the remader of ths secto we cosder two choces for ths dstrbuto. I secto 4., the evet dstrbuto s take to be a

10 8 Posso dstrbuto; ths case was also cosdered by Buchma ad Grübel (003), who developed a smlar method of estmato. I secto 4.3, the evet dstrbuto s take to be a geometrc dstrbuto. 4. Estmato uder a compoud Posso model The compoud Posso dstrbuto has a maor mportace the class of compoud dstrbutos. Ths s because of usg t the probablty theory ad ts applcatos bology, rsk theory, meteorology, health scece, etc. Uder the assumpto that N has a Posso dstrbuto wth mea λ, t follows from the dscusso secto 4. that Pt ( )= exp{ λ[ Qt ( ) ]}, t. Suppose that the parameter λ s kow. Hece, we ca estmate Q by Q where exp{ λ[ Q ( t) ]}= P ( t) (4.3) ad let p = k I{ Y = k} = so that P ca also be wrtte as k Pt ()= t p. k=0 k Here I { = } = f Y = k ad 0 otherwse. Y k For =,,, let q = Pr( X = ). Our goal s to estmate q, q, based o p 0, p, or, equvaletly, based o Pt ( ). To do ths, we make the followg assumpto: Assumpto 6 There exsts a postve teger M such that Pr( X > M )= 0 ad, hece, q = q = =0. M+ M+ Uder assumpto 6, the umber of ukow parameters s fte, whch smplfes certa techcal argumets. Note that M ca be take to be very large so that assumpto 6 s early always satsfed practce.

11 Noparametrc estmato radom sum models 83 We ca estmate q, q, by fdg Q to solve (4.3) ad the dfferetatg order to obta the estmators q, q, : q = Q (0), q = Q (0)/!, q = Q (0)/3! 3 ad so o. Defe Q to be the soluto to exp{ Qt ( ) }= Pt ( ), t. (4.4) The Qt ()= [ Qt () ] + λ ad q, q,, the probabltes correspodg to Q, are related to q, q, through q k = qk, k=,,. λ I the usual case whch λ s ukow, t must be estmated. Let λ deote a estmator of λ. The q, q, ca be estmated by q k = q, =,,. k k λ Oe approach to estmate λ s to use a estmator based o equato (4.). Ths yelds the estmator λ = log p. (4.5) 0 A alteratve approach s to use the method descrbed secto 3. I ths case, we assume the exstece of a kow fucto W such that W( θ ) = Var( X ) where θ = EX ( ; θ ). The a estmator λ s gve by the soluto to equato (3.), whch may be wrtte as M M M M q q + W q = λ = = λ = =. λ (4.6) Other estmators of λ ca be used, provded that the followg codto s satsfed.

12 84 Assumpto 7 The estmator λ s a cotuously dfferetable fucto of ( p 0, p,, p M ) ad s a cosstet estmator of λ. Let q=( q, q,, q M ) deote the vector of probabltes correspodg to Q ; smlarly, defe q =( q, q,, q M ) ad p =( p, p,, p M ). A method of calculatg q, q, s gve the followg theorem. Theorem 4. Assume that assumptos 5-7 hold ad that N has a Posso dstrbuto wth mea λ. Defe a M M matrx H as follows: 0, f <, H = =,,..., M. p f =,..., M, Hece ad q = H p q = q. λ Proof: Let α ()= t P () t ad let β()= t Q() t. Therefore M β()= t q t, = ()= α t p t, =0 ad β( t)= log α ( t), usg equato (4.4). It ow follows from Sever (005, lemma 4.) that r + p = q p, r =0,,,. (4.7) r+ + r =0 r +

13 Noparametrc estmato radom sum models 85 Expressg terms of the matrx H ad the vectors q ad p, equato (4.7) ca be wrtte as Hq = p. The result for q ow follows from the relatoshp betwee q ad q. The followg result shows that ( q, q,, q M ) s asymptotcally ormally dstrbuted. Theorem 4. Assume that assumptos 5-7 hold ad that N has a Posso dstrbuto wth mea λ. Therefore, as, q q q q qm q M coverges dstrbuto to a M dmesoal multvarate ormal dstrbuto wth mea vector 0. Proof: It follows from equato (4.7) that ( q, q,, q M ) s a cotuously dfferetable fucto of ( p 0, p,, p M ). Uder assumpto 7, t follows that ( q, q,, q M ) s a cotuously dfferetable fucto of ( p 0, p,, p M ). The result ow follows from the δ -method (e.g., Sever, 005, secto 3.) together wth the fact that ( p 0, p,, p M ) s asymptotcally ormally dstrbuted (e.g., Sever, 005, example.7). The asymptotc covarace matrx of ( q, q,, q M ), whch s ot specfed theorem 4., ca be obtaed usg the δ -method (see, e.g., Sever, 005, theorem 3.). Sce ( q, q,, q M ) s a complcated fucto of ( p 0,, p M ), we do ot gve ts expresso here. The computato of such expresso ca be obtaed va the formula Sever (005, theorem 3.). Note that the estmators q, q,, qm are ot guarateed to be o-egatve or are they guarateed to sum to. Thus, practce, ay egatve estmate should be replaced by 0. It may also be helpful to ormalze the estmates by dvdg them by ther sum.

14 Estmato uder a compoud geometrc model I ths subsecto, we cosder the compoud geometrc dstrbuto whch plays a mportat role relablty, queueg, regeeratve processes, ad surace applcatos. We ow cosder the case whch N has a geometrc dstrbuto wth the parameter λ ad probablty fucto PN ( = )=( λλ ), =0,,, (4.8) where 0 < λ <. The aalyss ths case s very smlar to the aalyss gve secto 4. for the compoud Posso model ad, hece, we keep the treatmet here bref. For the compoud geometrc model, λ Pt ()=, t. λ Qt ( ) Let λ deote a estmator of λ satsfyg assumpto 7 ad defe Q by / λ Qt ()= +. λ Pt ( (4.9) ) A method of calculatg q, q, s gve the followg theorem. Theorem 4.3 Assume that assumptos 5-7 hold ad that the dstrbuto of N s gve by (4.8). Hece k p k k pk qk = q, =,,. k λ p 0 = p (4.0) 0 Proof: Rewrtg (4.9) as λqtpt ()()= Pt () + λ for t, ad dfferetatg ths expresso mples k Q ( ) t P ( k ) t P ( k) t k k λ () ()= (), =,,. =0

15 Noparametrc estmato radom sum models 87 Evaluatg ths expresso at t = 0, ad settg q 0 =0, lead to (4.0). The followg result shows that ( q, q,, q M ) s asymptotcally ormally dstrbuted. The proof s essetally the same as the proof of theorem 4. ad, hece, s omtted. Theorem 4.4 Assume that assumptos 5-7 hold ad that the dstrbuto of N s gve by (4.8). Therefore, as, q q q q qm q M coverges dstrbuto to a M dmesoal multvarate ormal dstrbuto wth mea vector 0. The commets followg theorem 4. ca be appled here as well. Remark 4.. Our approach secto 4. works well wth other evet dstrbutos (e.g. bomal ad egatve bomal) as log as they obey the assumptos ths secto. Also, the results that ca be obtaed to such dstrbutos are very smlar to those already gve subsectos 4. ad 4.3, so t s ot ecessary to clude such results here. Mathematcs Departmet, Faculty of Scece, Tata Uversty, Tata, Egypt Departmet of Statstcs, Northwester Uversty, USA HASSAN S. BAKOUCH THOMAS A. SEVERINI ACKNOWLEDGMENTS The authors thak the referees for ther suggestos ad commets whch have sgfcatly mproved the paper. The secod author gratefully ackowledges the support of the U.S. Natoal Scece Foudato. REFERENCES H. S. BAKOUCH, M. M. RISTIĆ (009), Zero trucated Posso teger-valued AR() model, Metrka, do 0.007/s G. BUCHMANN, R. GRÜBEL (003), Decompoudg: a estmato problem for Posso radom sums, Aals of Statstcs, 3, pp

16 88 G. BUCHMANN, R. GRÜBEL (004), Decompoudg Posso radom sums: Recursvely trucated estmates the dscrete case, Aals of the Isttute of Statstcal Mathematcs, 56, pp J. CAI, G. E. WILLMOT (005), Mootocty ad agg propertes of radom sums, Statstcs & Probablty Letters, 73, pp C. A. CHARALAMBIDES (005), Combatoral methods dscrete dstrbutos, New York: Wley. C. CHATFIELD, C. M. THEOBALD (973), Mxtures ad radom sums, The Statstca,, pp S. A. KLUGMAN, H. H. PANJER, G. E. WILLMOT (004), Loss models: from data to decsos, d ed., Wley-Iterscece. O. LUNDBERG (964), O radom processes ad ther applcato to sckess ad accdet statstcs, d ed., Almqvst ad Wksell (Uppsala). P. MCCULLAGH, J. A. NELDER (989), Geeralzed lear models, d ed., Chapma & Hall/CRC. J. N. MEDHI (97), Stochastc processes, 3rd ed., Joh Wley: New York. H. H. PANJER (006), Operatoal rsk: modelg aalytcs, New York: Wley. E. PEKÖZ, S. M. ROSS (004), Compoud radom varables, Probablty the Egeerg ad Iformatoal Sceces, 8, pp S. M. PITTS (994), Noparametrc estmato of compoud dstrbutos wth applcatos surace, Aals of the Isttute of Statstcal Mathematcs, 46, pp M. M. RISTIĆ, H. S. BAKOUCH, A. S. NASTIĆ (009), A ew geometrc frst-order teger-valued autoregressve (NGINAR()) process, Joural of Statstcal Plag ad Iferece, 39, pp M. SAHINOGLU (99), Compoud Posso software relablty model, IEEE Trasactos o Software Egeerg, 8, pp T. A. SEVERINI (005), Elemets of dstrbuto theory, Cambrdge Uversty Press: Cambrdge. M. WEBA (007), Optmal predcto of compoud mxed Posso processes, Joural of Statstcal Plag ad Iferece, 37, pp SUMMARY Noparametrc estmato radom sum models Let X, X,, XN be depedet, detcally dstrbuted, o-egatve, tegervalued radom varables ad let N be a o-egatve, teger-valued radom varable depedet of X, X,, XN. I ths paper, we cosder two oparametrc estmato problems for the radom sum varable S = X =0. The frst s the estmato of the meas of X ad N based o the secod-momet assumptos o dstrbutos of S N N X = 0 0 =, X ad N. The secod s the oparametrc estmato of the dstrbuto of X gve a parametrc model for the dstrbuto of N. Some asymptotc propertes of the proposed estmators are dscussed.