Some properties of the Pearson type II (power semicircle) distribution

Transcription

1 НАУЧНИ ТРУДОВЕ НА РУСЕНСКИЯ УНИВЕРСИТЕТ - 03 том 5 серия 6 Some propertes of the Pearso type II (power semcrcle dstrbuto Евелина Велева Някои свойства на разпределението на Пирсън от тип II: Разпределението на Пирсън от тип II може да се разглежда като частен случай на разпределението на Пирсън от тип I известно още и като Бета разпределение с 4 параметъра В литературата се среща и под името степенно полукръгово разпределение До интерестни рекурентни свойства на това разпределение се достига при анализа на ковариационни и корелационни матрици на многомерното нормално разпределение Ключови думи: разпределение на Пирсън от тип I и II Бета разпределение степенно полукръгово разпределение INTRODUCTION The system of Pearso dstrbutos was developed by Karl Pearso 894 ad 895 to provde flexble descrptos of the o-ormal dstrbutos ecoutered hs bometrc research The orgal papers are reproduced [6] Apart from the fttg of models for observed frequecy dstrbutos the Pearso dstrbutos have also bee used to provde approxmatos to other theoretcal dstrbutos ad to study the effect of o-ormalty o samplg dstrbutos (see [3] The Pearso type II dstrbuto has probablty desty fucto of the form ([5]: f x = C a x x ( a a ( ( ( b where a > 0 ad b > are parameters; C s a ormalzg costat It s a partcular case of the Pearso type I dstrbuto whch depeds o 4 parameters a b c d bd> a 0 c > 0 wth probablty desty fucto of the form b d f( x = C( a+ x ( c x x ( a c ( where C s a ormalzg costat The Pearso type I dstrbuto s also kow as four parameters Beta dstrbuto because whe a = 0 ad c = ( gves the probablty desty fucto of the classcal Beta dstrbuto The Pearso type II dstrbuto occurs also uder the ame power semcrcle dstrbuto ([] Whe b = / ( gves the desty of the so called semcrcle dstrbuto The other values of b ca be preseted the form b = θ + / thus we obta from ( the desty of the power semcrcle dstrbuto PS( θ a Γ ( θ + fθ ( x; a = ( a x πa Γ ( θ + 3/ θ + / x ( a a (3 a > 0 θ > 3/ The Pearso type II dstrbuto or the power semcrcle dstrbuto s the dstrbuto of the sample correlato coeffcet for a sample of observatos o two depedet radom varables wth a bvarate ormal dstrbuto The dstrbuto ψ ( m m of the sample correlato matrx for a sample of m + observatos o depedet radom varables wth multvarate ormal dstrbuto has probablty desty fucto of the form (see [9]:

2 НАУЧНИ ТРУДОВЕ НА РУСЕНСКИЯ УНИВЕРСИТЕТ - 03 том 5 серия 6 ( Γ( m / ( ( m / detu m / ( f (U = (4 Γ for every postve defte matrx U = ( u j wth uts o the ma dagoal where Γ( s ( /4 the multvarate gamma fucto defed as Γ ( γ = π Γ [ γ + ( j/] j= Whe = from (4 we obta the desty of the sample correlato coeffcet for the observed varables the form ( m / ( ( m 3/ u ( m / ( Γ f( u = π Γ Accordg to (3 ths s the desty of the dstrbuto (( 4/ u ( (5 PS m I ths paper we cosder propertes of the power semcrcle dstrbuto resultg from ts specal role the aalyss of sample correlato ad covarace matrces PROPERTIES Let PR ( be the set of all real symmetrc postve defte matrces of order Let us deote by DR ( the set of all real symmetrc matrces of order wth postve dagoal elemets whch off-dagoal elemets are the terval (- There exst a bjecto h: D( R P( R cosdered [7] - [] The mage of a arbtrary matrx X = ( x j from DR ( by the bjecto h s a matrx Y = ( y j from PR ( such that yj j= xj j j = (6 y j = x j x xj j j = (7 r y j= x xj j xr x r j ( xq ( xq j + xj ( xq ( xqj (8 r= q= q= < j The ext Proposto s proved [8] Proposto Let ξ = ( ξ j be a radom symmetrc matrx of order wth uts o the ma dagoal Suppose that ξ j < j are depedet ad ξ j PS (( m 3/ where m s a teger m Let V be the matrx V = h( ξ where h s the bjecto defed by (6 (8 The the matrx V has dstrbuto ψ ( m Theorem below gves a terestg property of the power semcrcle dstrbuto whch follows from the represetato (8 for a arbtrary elemet of a postve defte matrx Theorem Let ζ η = k be depedet radom varables ζ η PS ( m 3/ where m s a teger m> k > The the radom varable ( ζη + ζ η ( ζ ( η + ζ η ( ζ ( η ( ζ ( η

3 НАУЧНИ ТРУДОВЕ НА РУСЕНСКИЯ УНИВЕРСИТЕТ - 03 том 5 серия 6 s dstrbuted PS (( m 4/ + + ζ η ( ζ ( η ( ζ ( η k k k k + η ( ζ ( η ( ζ ( η k k k Proof Let us have m + observatos o k + depedet radom varables wth multvarate ormal dstrbuto The dstrbuto of the sample correlato matrx V = ( V j s ψ ( k + m ([9] Each elemet V j of V s the sample correlato coeffcet calculated o the bass of m + observatos o the correspodg par of depedet factors Hece V j < j are detcally dstrbuted V j PS (( m 4/ wth desty fucto gve by (5 From Proposto we have that V = h( ξ where ξ = ( ξ j s a radom matrx wth uts o the ma dagoal The off-dagoal elemets of ξ are depedet ad ξ j PS (( m 3/ The the elemet V kk + of V accordg to (8 ca be preseted as k r k Vkk + = ξrk ξrk + ( ξqk ( ξqk + + ξkk + ( ξqk ( ξqk + r= q= q= Now f we deote ζ = ξ k = k η = ξ k + = k the Theorem follows The sample covarace matrx for a sample from a multvarate ormal dstrbuto has Wshart dstrbuto (see [] A radom matrx W wth Wshart dstrbuto W ( m Σ where m ad Σ s a postve defte matrx has probablty desty fucto of the form f (W (det W m Σ = m / m / Γ ( m /(det Σ ( m / tr(w Σ / e (9 for ay real postve defte matrx W where tr( deotes the trace of a matrx Wshart dstrbuto ca be also produced from the bjecto h (see [8] [9] Here the power semcrcle dstrbuto also plays a key role I the proposto below we deote by χ ( m the ch-square dstrbuto wth desty fucto m/ x/ f( x = x e x > 0 m / Γ m / ( Proposto Let ξ = ( ξ j be a radom symmetrc matrx of order Suppose that ξ j j are depedet ξ j PS (( m 3/ for < j ad ξ σ χ ( m = The parameters σ σ are arbtrary postve umbers Let W be the matrx W= h( ξ where h s the bjecto defed by (6 (8 The the matrx W has dstrbuto W ( m Σ Σ= dag( σ σ Ths represetato of the Wshart dstrbuto leads to aother property of the power semcrcle dstrbuto

4 НАУЧНИ ТРУДОВЕ НА РУСЕНСКИЯ УНИВЕРСИТЕТ - 03 том 5 серия 6 Theorem Let ζ = k be depedet radom varables ζ PS (( m 3/ = k ζ k χ ( m where m s a teger m> k > Let ν = k be the radom varables ν = ζ ζk ν = ζ ζk ( ζ ( ζ = k (0 νk = ζk ( ζ ( ζk ( The the radom varables ν = k are depedet ad ν N(0 = k ν k χ ( m k + To proof Theorem we eed the ext Lemma whch ca be easly checked usg the equaltes (6 (8 defg the bjecto h Lemma Let X = ( x j be a matrx from DR ( ad Y = h(x be the correspodg postve defte matrx from PR ( Let U be the lower tragular matrx where U x s x s x s 0 0 = x s x s x s s = sj x j 0 ( = j = (3 s = x ( x ( x < j (4 j j j j The Y= UU t s = ( x ( x j = (5 j j j j j Proof of Theorem Let W be a radom symmetrc matrx wth Wshart dstrbuto W( m I where I s the detty matrx of sze From Proposto we have that W= h( ξ where ξ = ( ξ j s a radom matrx wth depedetly dstrbuted elemets ξ j PS (( m 3/ < j ad ξ χ ( m = The radom varable ζ s dstrbuted as ξ k for = k Let U = ( U j be the lower tragular matrx costructed from the matrx ξ = ( ξ j accordg to Lemma From ( (5 t ca be see that U k = ξ k ξk k U k = ξk ξkk ( ξ k ( ξ k = k Ukk = ξkk ( ξ k ( ξk k Cosequetly the varables ν = k are dstrbuted as the elemets U k U kk of U Bartlett ([4] usg a dfferet approach proves that the elemets U k U kk of a

5 НАУЧНИ ТРУДОВЕ НА РУСЕНСКИЯ УНИВЕРСИТЕТ - 03 том 5 серия 6 t lower tragular matrx U such that W= UU are depedet Uk N(0 = k ad U χ ( m k + kk CONCLUSIONS The Pearso type II ((power semcrcle dstrbuto plays a key role geerato by the bjecto h of postve defte radom matrces wth dfferet dstrbutos (see [7] [] More propertes smlar to Theorem ad ca be derved from the represetatos of the correspodg dstrbutos of postve defte radom matrces It s terestg whether smlar propertes hold for the Pearso type I dstrbuto or partcular for the classcal Beta dstrbuto ЛИТЕРАТУРА [] T W Aderso A Itroducto to Multvarate Statstcal Aalyss Joh Wley & Sos N Y 3ed 003 [] O Arzmed V P erez-abreu O the o-classcal fte dvsblty of power semcrcle dstrbutos Commu Stoch Aal 4 ( hoor of Gopath Kallapur [3] P Armtage Pearso Dstrbutos Ecyclopeda of Bostatstcs 005 [4] MS Bartlett O the theory of statstcal regresso Proc Roy Soc Edb 53 ( [5] L Devroye No-Uform Radom Varate Geerato New York Sprger-Verlag 986 [6] K Pearso Early Statstcal Papers Cambrdge Uversty Press Cambrdge 948 [7] E Veleva Uform radom postve defte matrx geerato Math ad Educato Math 35 ( [8] E Veleva А represetato of the Wshart dstrbuto by fuctos of depedet radom varables Auare Uv Sofa Fac of Eco & Buss Adm 6 ( [9] E Veleva Some New Propertes of Wshart Dstrbuto Appled Mathematcal Sceces Vol 008 o [0] E Veleva Uformly Dstrbuted Postve Defte Matrces Wth Bouded Trace Joural of the Techcal Uversty at Plovdv Fudametal Sceces ad Applcatos Vol [] E Veleva Stochastc represetatos of the Bellma gamma dstrbuto Proc of It Cof o Theory ad Applcatos of Mathematcs ad Iformatcs ICTAMI 009 Alba Iula Romaa За контакти: гласд-р Евелина Велева Катедра Приложна математика и статистика Русенски университет Ангел Кънчев е-mal: eveleva@u-rusebg Докладът е рецензиран - 4 -