THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 6, Number 2/2005, pp

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1 THE PULISHING HOUSE PROEEDINGS OF THE RONIN DEY Series OF THE RONIN DEY Volume 6 Numer 5 pp. - ON THE RTIO XY FOR THE ELLIPTILLY SYETRI ESSEL DISTRIUTION Srlees NDRJH Smuel OTZ Deprtment of Sttistics Universit of Ners Lincoln NE Deprtment of Engineering ngement nd Sstems Engineering The George Wshington Universit Wshington D.. 5 orresponding uthor: Srlees NDRJH e-mil: sndrj@unlserve.unl.edu The distriution of distriution. X Y is derived when Y X hs the ellipticll smmetric ot tpe e words: essel function ellipticll smmetric essel distriution hpergeometric functions.. INTRODUTION For ivrite rndom vector X Y the distriution of the rtio XY is of interest in prolems in iologicl nd phsicl sciences econometrics nd rning nd selection. Exmples include endelin inheritnce rtios in genetics mss to energ rtios in nucler phsics trget to control precipittion in meteorolog nd inventor rtios in economics. The distriution of XY hs een studied severl uthors especill when X nd Y re independent rndom vriles nd come from the sme fmil. For instnce see rsgli [] nd orhonen nd Nrul [] for norml fmil Press [3] for Student's t fmil su nd Lochner [4] for Weiull fmil Shcolnic [5] for stle fmil Hwins nd Hn [6] for non-centrl chisqured fmil nd Provost [7] for gmm fmil. However there is reltivel little wor of this ind when X nd Y re correlted rndom vriles. Some of the nown wor include Hinle [8] for ivrite norml fmil ppenmn [9] for ivrite t fmil nd Lee et l [] for ivrite gmm fmil. In this pper we stud the distriution of XY when X Y hs the ellipticll smmetric essel distriution given the joint proilit densit function pdf f x { xα β xα β } xα β xα β for < x < < < < α < < β < > > nd < <. is the modified essel function of the third ind defined for n with x lim x n ν n ν x nd the normliing constnt stisfies I π x { I x I x } sin π x! Γ Recommended rius IOSIFESU memer of the Romnin cdem

2 Srlees NDRJH Smuel OTZ π Γ For detils on the theor nd pplictions of the essel functions see orenev []. When nd σ reduces to the ellipticll smmetric Lplce distriution given the joint proilit densit function pdf f x πσ x α. β x α β σ for < x < < < < α < < β < σ > nd < <. is the modified essel function of the third ind of order ero. The prmeter is the correltion coefficient etween the x nd components. For detils on properties of these distriutions see Jensen [] nd Fng et l. [3]. essel rndom vriles re generlitions of Lplce rndom vriles nd hve found pplictions in vriet of res tht rnge from imge nd speech recognition nd ocen engineering to finnce. The re rpidl ecoming distriutions of first choice whenever something with hevier thn Gussin tils is oserved in the dt. In mn of the ppliction res one would e interested in the correltion or dependence etween two essel rndom vriles. Some exmples re: in communiction theor X nd Y could represent the rndom noise corresponding to two different signls. in ocen engineering X nd Y could represent distriutions of nvigtion errors. in finnce X nd Y could represent distriutions of log-returns of two different commodities. in imge nd speech recognition X nd Y could represent input distriutions. In ech of the ove exmples the ellipticll smmetric essel distriution given could e used to model the dependence etween X nd Y. The im of this pper is to clculte the distriution of the rtio X Y when X Y hs the joint pdf. The clcultions of this pper involve the hpergeometric functions defined nd α x G α; β γ ; x β γ! α β H α β; γ δ η; x γ δ η c c c! c denotes the scending fctoril. We lso need the following lemms. Lemm Eqution.6..3 Prudniov et l [4] volume For > β > nd α >ν x! x β x cx dx ν ν ν ν αβν ν αν αν αβν αβν c ν c Γ ν β αν H ; ν ;. 4

3 3 On the rtio XY for the ellipticll smetric essel distriution Lem Eqution.6..4 Prudniov et l [4] volume. For > nd >ν x x ν cx x β d ν ν ν ν αβν ν αν αν αβν αβν c ν c Γ ν β αβν H ; ν ; 4 ν αβν ν αβν αβν β β 3αβν 3ναβ c ν c Γ Γ H ; ; 4 nd αβ4 αβ αβν αβν β 3β 3 ναβ ναβ c ν c β Γ Γ H ; ; 4 Lemm 3 Eqution.6... Prudniov et l [4] volume. For c > nd α>ν β α α x x ν cx dx c αν αν Γ Γ. Further properties of the ove specil functions cn e found in Prudniov et l [4] nd Grdshten nd Rhi [5]. PDF Theorem derives n explicit expression for the pdf of Z X Y in terms of the hpergeometric functions. Theorem Suppose X nd Y re jointl distriuted ccording to nd let sinθ β α cosθ α β sinθ 3 α β αβ 4 nd Furthermore define nd D α sinθ β cosθ. g θ Ω g θ Λ

4 Srlees NDRJH Smuel OTZ Γ G; ; I Γ Γ 4 8 nd 4 Γ 3 Ω G ; ; 4 4 Γ 3 G ; ; Γ 3 Λ G ; ; 4 4 Γ 3 G ; ; 4 4 Γ D 3 D G ; ; 4 4 Γ D 3 D G ; ;. 4 If then the pdf of Z X Y cn e expressed s On the other hnd if < then grctn g π rctn g f grctn g π rctn g f π rctn. rctn. Proof: Set X Y T sinθ T cosθ. Under this trnsformtion the Jcoin is T nd so one cn express the joint pdf of T θ s g t θ t t t nd re given 3 nd 4 respectivel. Set t t t t t nd note tht 3 d t ± dt. Note further tht t is n incresing function of t with if. On the other hnd if < then t decreses etween t D efore incresing for ll t D D is given 5. Thus the mrginl pdf of θ cn e expressed s g θ g θ if nd s g θ g θ g θ if <

5 5 On the rtio XY for the ellipticll smetric essel distriution c d g θ 4 nd D d g. θ 5 These two expressions ctull reduce to those given 6 nd 7 respectivel s shown elow. onsider 4. Note tht for ll. Thus using the series expnsion x x one cn expnd s d d d d d g θ 6 the lst step follows sustituting. ppliction of Lemm nd properties of the hpergeometric functions shows tht the two integrls in 6 reduce s d nd d Ω respectivel nd Ω re given 8 nd 9 respectivel. Sustituting these into 6 one notes tht 4 reduces to 6. using Lemm one cn similrl show tht 5 reduces to 7. The result of the theorem follows noting tht the pdf of tn Z θ cn e expressed s

6 Srlees NDRJH Smuel OTZ 6 grctn g π rctn f nd tht g θ g θ if nd g θ g θ g θ if <. Figures nd illustrte possile shpes of the pdfs - for rnge of vlues of α β nd υ. Note the diminishing scle of the densities with incresing vlues of nd how their loction depends on the reltive mgnitudes of α nd β. 7 Fig. Plots of the pdfs - for nd : α nd β ; : α nd β 3; c: α 3 nd β ; d: α 3 nd β 3. The four curves in ech plot re: the solid curve for.; the curve of dots for.4; the curve of dots for.6; nd the curve of dots nd dots for.8.

7 7 On the rtio XY for the ellipticll smetric essel distriution Fig. Plots of the pdfs - for α β nd : ; :.5; c: ; nd d:. The four curves in ech plot re: the solid curve for.; the curve of lines for.4; the curve of dots for.6; nd the curve of lines nd dots for.8. orollr considers prticulr form for the pdf of Z for the cseα β. Note tht the resulting pdf is elementr nd depends onl on. orrolr : If X nd Y re jointl distriuted ccording to nd if α β then the pdf of Z reduces to grctn f 8 for θ [ π. g θ π{ sinθ }

8 Srlees NDRJH Smuel OTZ 8 Proof: Note tht in this cse 3 tes the form g t θ t t is given. Integrting this over t <ields the form for g. given in the corollr. The result of the corollr follows noting tht the pdf of Z tnθ cn e expressed s grctn g π rctn f 9 nd tht grctn θ g π rctn θ for the form for g.. REFERENES. RSGLI G. Rtios of norml vriles nd rtios of sums of uniform vriles Journl of the mericn Sttisticl ssocition 6 pp ORHONEN P. J. ND NRUL S.. The proilit distriution of the rtio of the solute vlues of two norml vriles Journl of Sttisticl omputtion nd Simultion 33 pp PRESS S. J. The t rtio distriution Journl of the mericn Sttisticl ssocition 64 pp SU. P. ND LOHNER R. H. On the distriution of the rtio of two rndom vriles hving generlied life distriutions Technometrics 3 pp SHOLNI S.. On the rtio of independent stle rndom vriles In: Stilit Prolems for Stochstic odels Uhgorod 984 Lecture Notes in themtics 55 Springer erlin pp HWINS D.. ND HN. -P. ivrite distriutions noncentrl chi-squre rndom vriles ommunictions in Sttistics---Theor nd ethods 5 pp PROVOST S.. On the distriution of the rtio of powers of sums of gmm rndom vriles Pistn Journl Sttistics 5 pp HINLEY D. V. On the rtio of two correlted norml rndom vriles iometri 56 pp PPENN R. F. note on the multivrite $t$ rtio distriution nnls of themticl Sttistics 4 pp LEE R. Y. HOLLND. S. ND FLUE J.. Distriution of rtio of correlted gmm rndom vriles SI Journl on pplied themtics 36 pp ORENEV. G. essel Functions nd Their pplictions Tlor & Frncis London.. JENSEN D. R. ultivrite distriutions In: Encclopedi of Sttisticl Sciences 6editors S. ot N. L. Johnson nd.. Red John Wile nd Sons New Yor pp FNG. -T. OTZ S. ND NG. W. Smmetric ultivrite nd Relted Distriutions hpmn nd Hll London PRUDNIOV. P. RYHOV Y.. ND RIHEV O. I. Integrls nd Series volumes nd 3 Gordon nd rech Science Pulishers msterdm GRDSHTEYN I. S. ND RYZHI I.. Tle of Integrls Series nd Products sixth edition cdemic Press Sn Diego. Received Septemer 3 4