Ratio of Two Random Variables: A Note on the Existence of its Moments

Transcription

1 Metodološki zvezki, Vol. 3, o., 6, -7 Ratio of wo Rado Variables: A ote o the Existece of its Moets Ato Cedilik, Kataria Košel, ad Adre Bleec 3 Abstract o eable correct statistical iferece, the kowledge about the existece of oets is crucial. he obective of this aer is to study the existece of the oets for the ratio Z = /, where ad are arbitrary rado variables with the additioal assutio P ( = ) =. We reset three existece theores showig that secific behaviour of the distributio of i the eighbourhood of zero is essetial. Sile coseueces of these theores give evidece to the existece of the oets for articular rado variables; soe of these results are well kow fro stadard robability theory. However, we obtai the i a sile way. Itroductio he ratio of two orally distributed rado variables occurs freuetly i statistical aalysis. Fro stadard robability literature, see for exale Johso et al. (994), it is kow that the ratio of two cetred oral variables Z = / is a o-cetred Cauchy variable. Marsaglia (965) ad Hikley (969) discussed the geeral situatio: [ ] ~ ( µ, µ, σ, σ, ρ ± ). Cedilik et al. (4) studied the geeral situatio as well, they followed the sae rocedure as Hikley did. hey showed that the desity of the ratio of two arbitrary oral variables ca be exressed very eatly as a roduct of two factors. Biotechical Faculty, Uiversity of Lublaa, Jaikareva, Lublaa, Sloveia; ato.cedilik@bf.ui-l.si Biotechical Faculty, Uiversity of Lublaa Jaikareva, Lublaa, Sloveia; kataria.kosel@bf.ui-l.si 3 atioal Istitute of Biology, Uiversity of Lublaa, Veča ot, Lublaa, Sloveia; adre.bleec@ib.si

2 Ato Cedilik, Kataria Košel, ad Adre Bleec he first factor, the Cauchy art, is the desity for a o-cetred Cauchy variable, σ σ C a = ρ, b = ρ, which is ideedet of the exected values µ σ σ ad µ. he secod factor, the deviat art, is a colicated fuctio of z (see Cedilik et al., 3). For illustratio, let us cosider the ( µ, µ, σ = σ =, ρ =.5). We vary µ fro -4 to +4 with the ste of, ad µ fro to with the ste of.5. he Cauchy art is the sae for all these cases. he deviat art, however, takes differet shaes. Figure a dislays the Cauchy ad the deviat art, Figure b the desity which is their roduct. It should be oited out that the ratio of two orally distributed rado variables has o oets due to the fact that the asytotic behaviour of the desity is the sae as that of the Cauchy art. o eable correct statistical iferece the kowledge about the existece of the oets of the ratio is crucial. he obective of this aer is to study the existece of the oets of the ratio for the geeral settig. ρ= Figure a: he Cauchy art ad the deviat art for the ratio /, where [ ] ~ ( µ, µ, σ = σ =, ρ =.5). µ varies fro -4 to +4 with the ste of (horizotally) ad µ fro to with the ste of.5 (vertically). he Cauchy art is costat for all these cases. For µ = µ = the deviat art euals.

3 Ratio of wo Rado Variables 3 ρ= Figure b: he desity for the ratio /, where [ ] ~ ( µ, µ, σ = σ =, ρ =.5). µ varies fro -4 to +4 with the ste of (horizotally) ad µ fro to with the ste of.5 (vertically). Existece of the oets of the ratio I what follows, we cosider the rado vector [ ], where ad are arbitrary rado variables with the additioal assutio P ( = ) =. Coseuetly, the ratio Z = / is a well defied rado variable. We reset three theores o the existece of the oets ad soe coseueces. he roofs are based o the well kow Hölder s ieuality which we reset i Aedix to ut o view the otatios used i the text. heore. Suose that there exists such a ε >, that P ( < ε ) =. If has the oets of order, oegative (ossibly ot iteger), the Z = / has the oets of the sae order. he followig relatioshi holds: E ( Z ) E ( ) ε (.)

4 4 Ato Cedilik, Kataria Košel, ad Adre Bleec Proof. It is obvious. But ust for the sake of uifor treatet, cosider Hölder s ieuality for P( ε ε ) =, U = ad ess su V =. Sice P ( ε ε ) = ad ε. QED A sile coseuece of this theore is the fact that ay ratio /, where is discrete ad does ot have as a adheret oit, iherits the existece of the oets fro. heore. Let, be oegative (ossibly ot iteger) ubers, >, ad let have the oets of order. Further assue there exist two ositive real ubers ε ad C, such that for ay δ withi the iterval < δ < ε, the followig holds: ( + ε ) < C δ ( δ ) P (.) he Z = / has all the oets of order. Proof. For =, the roof is trivial. For > assuig U =, V =, ad =, = : [ dp] Z dp dp., use Hölder s ieuality Sice has the oets of order, the first factor o the right is fiite. Let us cosider the secod factor, deotig =. Estiatio gives (.): P P ( + ε ) δ = > < δ = δ P ( < δ ) C Hece, ε P ( > t) C t for ay t which is large eough. We show further that uder (.) the secod factor is fiite dp = dp + dp = < P ( ) + = + + P( + < + + )

5 Ratio of wo Rado Variables 5 + = + + P ( > + ) + = + + for large eough. QED C ( + )(+ ε ) <, he exressio () is very geeral because it holds for a arbitrary rado variable. Exale resets its for for cotiuous rado variable, Exale rovides the discrete case. Exale. Assue that has the oets of order cotiuously distributed with the desity a v( y) A y ( A >, a > ) for ε < y < ε, else it is arbitrary. he: P δ δ a A + a ( < δ ) = v( y) dy Ay dy = δ + a δ Fro (.) we get =. + ε + +a herefore, Z has all the oets of order less tha. + + a., ad is I the light of this exale soe results o the existece of the oets for articular rado variables aear which are well kow fro stadard robability theory. For illustratio we reset two of the. Exale a. As a sile coseuece of Exale cosider which is orally distributed. he a = ; coseuetly, < ; therefore o oets of Z exist. A secial case is etioed i Itroductio: the ratio of two orally distributed rado variables has o oets. + +a Exale b. Cosider ~ (, ), U ~ ( r χ ) ad the ratio = r, a U Studet variable with r degrees of freedo. It is well kow that = U has a chi-distributio with r degrees of freedo with the robability desity r y v( y) = y ex (for y ) r / r Γ ( ) ( ) which ca be estiated i light of Exale :

6 6 Ato Cedilik, Kataria Košel, ad Adre Bleec r v ( y) y (for ay real y ). r / Γ r ( ) he has all oets of order less tha (where is arbitrary), which is + r as close to r as oe wats. Hece, has all the oets of order less tha r, as it is well kow fro stadard robability theory. Exale. Let have a discrete distributio with a strictly decreasig ifiite seuece ( a ) of values, ad P ( = a ) = v. Assue that there exist two ositive real ubers ε ad C such that if = + ha (.) is valid. (+ε ) a < ε the followig holds: v C ( a ) (.3) he ext theore describes the reverse situatio: we study the existece of the oets of give the oets for Z ad. heore 3. If has the oets of order ad Z the oets of order, ad are ositive (ossibly ot itegers), the has the oets of order. he followig relatioshi holds: + E / / ( ) E( ) E( Z ), (.4) where >, / + / = ad i{, }. Proof. he case = is trivial. Let us use Hölder s ieuality for V = Z for soe U = ad >. he estiatio has sese if ad. Hece, {, } ax i{ } i, = ; the axiu is reached at + + =.

7 Ratio of wo Rado Variables 7 Aedix Hölder s ieuality: Let (, F, P) be a robability sace, where is a set of outcoes g, F a Borel s σ -algebra of evets, ad P the robability easure. Further, let U ad V be rado variables o. he / U ( g) V ( g) dp U ( g) dp V ( g) dp / for ay air of ositive,, where / + / =, if the itegrals o the right coverge. If = ad =, the U ( g) V ( g) dp U ( g) dp ess su V ( g) g Refereces [] Cedilik, A., Košel, K., ad Bleec, A. (4): he Distributio of the ratio of oitly oral variables. Metodološki zvezki., [] Hikley, D.V. (969): O the ratio of two correlated oral rado variables. Bioetrika, 56, 3, [3] Johso,.L, Kotz, S. ad Balakrisha,. (994): Cotiuous Uivariate Distributios.. Joh Wiley ad Sos. [4] Marsaglia, G. (965): Ratios of oral variables ad ratios of sus of uifors variables. JASA, 6, 63-4.