Product Moments of Sample Variances and Correlation for Variables with Bivariate Normal Distribution

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1 Joural of Matheatis ad Statistis Origial Researh Paper Produt Moets of Saple Variaes ad Correlatio for Variales with Bivariate Noral Distriutio Jua Roero-Padilla LNPP-Coayt, Ceter for Researh ad Teahig i Eoois (CIDE, Mexio City, Mexio Artile history Reeived: -- Revised: 8--6 Aepted: --6 Eail: jua.roero@ide.edu Astrat: A geeral result to otai the produt oets of two saple variaes ad the saple orrelatio whe the data follow a ivariate oral distriutio is derived; the result is expressed i ters of the hypergeoetri futio. As orollaries, two geeral equatios are stated, oe to otai the oets of the orrelatio saple ad oe to otai the oets of the ratio of two saple variaes. To evaluate the produt oets i short losed fors, three theores have ee estalished. The results are used to otai the expetatio ad variae for the ratio of two orrelated saple variaes. Fially, soe exaples of partiular produt oets are provided ad soe validatios were arried out. Keywords: Wishart Distriutio, Produt Moets, Hypergeoetri Futio, Saple Correlatio Coeffiiet, Saple Variae Itrodutio We are iterested i the produt oets of the two saple variaes ( S, S ad saple orrelatio oeffiiet (R of the ivariate oral distriutio, so we a E S S R for fiite a,,. Oe wat to derive ( approah to otai the produt oets of the two saple variaes ad the orrelatio oeffiiet was disussed y Joarder (6, however his results ivolves a ifiite series that does ot osider a iportat ter, without the issig ter, the result of Joarder (6 does ot wor to get soe produt oets, for istae, we are ot ale to get the first oet of R. Here we expad the result of Joarder (6 to derive a ore geeral result. There are differet expressios for oets of R, Ghosh (966 ad Rady et al. ( otaied the first four oets of R, here we derived a geeral equatio for ay oet of R that do ot ivolves a ifiite su ad it is expressed i ters of the well-ow hypergeoetri futio. Let X,,X e iid N p (µ,σ where Σ is positive defied ad > p, the sus of squares ad ross produt atrix is give y: A ( X ( j j X X j X A, is said to have a Wishart distriutio with paraeters p, - ad Σ(pxp, A W p (,Σ ad the T Proaility Desity Futio (PDF is give y (Aderso, : ( p A tr A exp Σ f ( A p p( p p π Σ Γ + i ( i With, > p ad A positive defiite. For p we have the ivariate ase with: Where: A a a a a a S, a S, a S rs S, S a R S S a a / ( ( S is the saple ovariae ad the joit pdf of a, a ad a is give y: ( ( σσ ( / (,, f a a a a a a π ΓΓ a a a exp + + ( σ ( σ ( σσ ( 6. Jua Roero-Padilla. This ope aess artile is distriuted uder a Creative Coos Attriutio (CC-BY. liese.

2 Jua Roero-Padilla/ Joural of Matheatis ad Statistis 6, (:. DOI:.8/jssp.6.. Produt Moets of Saple Variaes ad Saple Correlatio Throughout the paper, we will use the geeralized hypergeoetri futio (e.g., Bailey, 96: (,, :,, ; F a a z p q p q ( a ( a ( a z p ( ( (! q + ( a a E S S R σ σ π Γ + + a Γ Γ Γ Γ + + a ,, ;, ; F ( With: Γ( a a a Γ( a ( ;( ( ( ( We use the joit distriutio of S, S ad R give y Joarder (6: Soe properties of the hypergeoetri futio (Oerhettiger, 97: a (, ; ; ( ;; (, a ( ( F a z F a z z F a, ; ; z ( z F a, ; ; z, d dz a F a z F a z (, ; ; ( +, + ; + ; f S S R (,, ( S S R ( ( σσ ( π Γ S S RSS exp + + ( σ ( σ ( σσ ( Ad the followig results: π / x+ π Γ x si θdθ, x Γ + x yt Γ( x+ t e dt x+, y ( ( ( π! π! + Γ!! where, Γ(. is the gaa futio. Theore For ay fiite a, ad. If is eve, the produt oets of the saple variaes ad saple orrelatio, a E( S S R, are give y: + ( a a σ σ E S S R π Γ + a a F Γ Γ Γ Γ,, ;, ; If is odd are give y: ( The produt oets, for ay a,, are give y: a a ( (,, E S S R ( ( ( Γ! S ( ( ( ( + + a exp ds σ S + + exp ds σ S S R f S S R ds ds dr σσ S S + ( / R R dr + ( / + R ( R dr ( ( π σσ (6 The first two itegrals of Equatio 6 a e expressed as: S + + a S exp ds σ ( S S ( + + Γ + aγ exp ds σ

3 Jua Roero-Padilla/ Joural of Matheatis ad Statistis 6, (:. DOI:.8/jssp a + ( σ ( σ + + Γ + aγ + ( σ σ a ( To otai the last two itegrals of Equatio 6 we use the eta futio: + + ( + ( + ( + ( + + ( ( + w w dw ( B, + R R dr R R dr + R r dr The produt oets are give y the expressio: + ( a a σ σ E S S R π Γ + ( + ( Γ a (7 + +! Γ + ΓΓ a To express E S S R i ters of hypergeoetri futio we have two ases, if is eve redefie j the ( j+ + ad: + ( a a σ σ E S S R ( j ( + π Γ π Γ + j+ aγ + j+ j j! + + Γ + jγ + jγ + j a σ σ Γ a + a π Γ + ad: + + Γ + ΓΓ ( j a + j! j j j j j ( + a tσ σ Γ a + a Γ + π + + Γ + ΓΓ + a F,, ;, ; If is odd redefie j- the ( j + ( a a σ σ E S S R π Γ ( j j Γ + j+ aγ + j+ j ( j! + Γ + Γ + Γ + ( + + ( j ( j j j j j! Γ ( j + σ σ a π! + + Γ j + + aγ j + + ( j! Γ j + Γ j + Γ j + Let h j-, the: + ( a a E S S R σ σ Γ Γ + aγ + Γ + + Γ Γ ( h a + h! h h h h hh

4 Jua Roero-Padilla/ Joural of Matheatis ad Statistis 6, (:. DOI:.8/jssp ( a σ σ Γ π Γ + aγ + ΓΓ + + a F,, ;, ; The followig expressio will e used i the ext two theores, let: + ( + +, ( + ΓΓ, >, > (8! Theore For > ad - < <, we have: F ;; ( If is eve, we have:, π Γ( + +, π Γ If is odd, we have: ( + + +,, Γ F, ; ; + + π ΓΓ + + F, ; ; ( F, ; ; Note that: ( ( E S S r ( F ;; The: F ;; ( If is eve redefie j:, ( j ( j ( j ( ( j F ( ( + Γ Γ! Γ j+ Γ j+ j ( j! π Γ j+! π Γ! j π Γ ;;, ( j ( j + + ( ΓΓ! π Γ + π Γ ( ( ( ( + { } π Γ + π Γ If is odd redefie j-:, j ( ( ( j h ( ( j ( j ( j Γ + j j! j j + Γ( j Γ ( j! π! j π Γ +! ( j + j j + π Γ Γ Γ + Γ + + h h h! h h + + Γ F, ; ;, j ( ( j ( j + j! j + j + ( Γ ( j! j + Γ Γ j π! ( j

5 Jua Roero-Padilla/ Joural of Matheatis ad Statistis 6, (:. DOI:.8/jssp.6.. j j ( Γ( j π + +! Γ j + Γ j + ( π Γ + j ( j j ( ( j + Γ j + Γ j +! + π ΓΓ ( j + + F, ; ; F, ; ; a To evaluate the produt oets E S S R of soe speial ases, Joarder (6 derived a result expressed i his paper y Theore., whih oly holds whe is eve ad the expressio for, is differet (Equatio 8. The ext Theore, osiders the ase whe is ot eve. Theore For > ad -< < ad, defied i (8, we have for eve: π (i, Γ ( (ii, ( + ( L(, (iii, w(, L(, (ix (x (xi (xii (xiii, 6Γ A A A { A, Γ A 6 A A A A 7, Γ A A A { }, + + Γ + A + A { }, + + Γ A + 6 A + A 7 (iv, w(, L(, (v, ( L(, ( ( {} + + {} (vi, w{} (, L(, {} (vii, w{} (, L(, For odd, we have: { } + + +, 8Γ (xiv A + 8 A + 96 A Where: L(, π Γ( ( ( { } { } +, (viii + +, Γ A + A w 6 ((, [( ( + 6 ] ( 6

6 Jua Roero-Padilla/ Joural of Matheatis ad Statistis 6, (:. DOI:.8/jssp { }(, ( ( 6 6 ( 6 ( w ( ( 6 w (, (, ( ( 6 8 ( 6 ( w , + z Γ + z F, ; ; z z F, ; ; z z F, ; ; z z F, ; ; z + A F (,( + / ; / ; ; + A F, ; ; ; + 7 A F, ; ; ; A F, ; ; ; + 9 A F, ; ;, + + z Γ z F, ; ; z z F, ; ; z ( z F, ; ; z Differeig ( with respet to z, we otai (x: Whe is eve, we have ( + +, ΓΓ, >, > ad uder this! seario the proof was provided y Joarder (6. If is odd, let z, the: + + z z Γ F, ; ; z (9 / /, Differeig idetity (9 with respet to z, we got (viii: + + z Γ z F, ; ; z, + + ( + z F, ; ; z, + + z Γ z F, ; ; z z F, ; ; z ( Differeig ow ( with respet to z, we otai (ix:, + + z Γ z F, ; ; z z F, ; ; z z F, ; ; z z F, ; ; z z F, ; ; z z F, ; ; z 7, + + z Γ z F, ; ; z z F, ; ; z z F, ; ; z z F, ; ; z 7 ( Differeig ow ( with respet to z, we otai (xi: 7

7 Jua Roero-Padilla/ Joural of Matheatis ad Statistis 6, (:. DOI:.8/jssp.6.., + + z Γ z F, ; ; z z F, ; ; z z F, ; ; z z F, ; ; z z F, ; ; z z F, ; ; z z F, ; ; z z F, ; ; z, + + z Γ z F, ; ; z z F, ; ; z z F, ; ; z 7 F z, ; ; z z F, ; ; z 7 9 Fially, (xii, (xiii ad (xiv a e otaied y otie the followig: {} {}, ( +, {} ( 6 + 6,,,,, Next Theore, will e useful to evaluate soe produt oets. Theore For a fiite a, : Note that: (, ; ; z F a ( + Γ( + ( a z Γ( Γ( +! Γ( If we let j -, the aove hypergeoetri futio a expressed as: F( a, ; ; z ( ( a z + li! x Γ( x Γ( j Γ( + j z Γ( + z j Γ( ( j! ( a Now osider that (Ngo, : li Γ( x li Γ( x γ x x x x where, γ is the Euler s ostat: ( x γ e l x dx.7766 So the gaa futio ehaves alost exatly lie x whe x gets lose to zero ad ehaves alost Γ( x exatly lie zero whe x gets lose to zero, so we have: az F a z F a z F a z (, ; ; ( ;; + ( + ;; ( Soe exaples of produt oets are give elow For a, ad we have: ( + + σσγ Γ E[ S S R] π Γ F σ σ,, ;, ; az F ( a, ; ; z ( z + ( z ( a ( ( For a, ad we have: 8

8 Jua Roero-Padilla/ Joural of Matheatis ad Statistis 6, (:. DOI:.8/jssp ( + E S S R σσ Γ Γ π Γ F ( + F + ;; + ( σσ ( + σ σ,, ;, ; For a, ad we have: + ( σσ + + E S S R Γ Γ Γ π Γ Γ F,, ;, ; + ( + σσ F, ; ; + ( σσ ( + ( + ( + + σσ + ( + { } For a, ad we have: + ( + E S S R σσ Γ Γ π Γ F,, ;, ; + ( + σ σ F, ; ; + ( σσ + ( + ( + + σσ ( + ( + ( σ + + E S S R σ ΓΓ π Γ ΓΓ F,, ;, ; + ( ( + σ σγ ( + Γ F,, ;, ; Corollary The oets of the saple orrelatio, if is eve, are give y: ( + + E R ΓΓΓ π + + F,, ;, ; If is odd, are give y: ( + + Γ ΓΓ E R π Γ + + F,, ;, ; ( (6 Use a, i Equatio ad, the replae -. The first oet of R, is otaied usig, i Equatio 6: ( + E[ R] Γ Γ Γ π Γ + F,, ;, ; ( + ΓΓ F, ; ; Γ Γ F, + ; ;. (7 For a, ad we have: Equatio 7 is a well-ow equatio for the first oet of R (Ghosh,

9 Jua Roero-Padilla/ Joural of Matheatis ad Statistis 6, (:. DOI:.8/jssp.6.. The seod, third ad four oets of R, are otaied usig,, ad, depedig the ase: ( + E R ΓΓΓ π + F,, ;, ; ( + F,, ;, ; ( ( + E R Γ Γ Γ πγ +,, ;, ; F ( + Γ ( + F,, ;, ; Γ ( + E R ΓΓΓ π + F,, ;, ; ( + F,, ;, ; ( ( + (8 (9 ( I the literature there are expressios for the first four oets of R (Ghosh, 966; Soper et al., 97. The seod, third ad four oet of R ay e expressed as: ( ( + E R F,; ; [ ] ( ( ( ( E R E R + + ΓΓ F, ; ; ( ( ( + E R + F,; ; ( ( ( + F,; ; ( ( ( The equatio of the third oet give y Ghosh (966 is ot orret. Equatio oes fro the equatio reported y Soper et al. (97, whih is Where: We ote that: Ad: q [ ] ( E R E R q q + ( (!(8( + ( + ( + 7 ( + ( ( ( + ( + q π / si ( ( ( + ( + θdθ + F, ; ; + + ( (!(8( + ( + ( + 7 π Γ + + Γ + Γ π Γ + Γ Γ + q ΓΓ ( + ( The expressios for the seod, third ad four oet of R, that we derived, are shorter tha existig equatios. Algeraially it is ot easy to verify if Equatio 8- are equal to Equatio - respetively, as a exerise we wrote a progra usig the software R to get values of the aove expetatios for soe paraeters, the results otaied do ot show differees etwee the equatios derived here ad the equatios reported i the literature. I Tale, we show soe data reported y Soper et al. (97 p. 7 ad what we got with Equatio 7-.

10 Jua Roero-Padilla/ Joural of Matheatis ad Statistis 6, (:. DOI:.8/jssp.6.. Tale. Copariso of the first four oets of R, etwee data reported y Soper et al. (97 ad values otaied with Equatio 7-, whe.6 ad Values reported y Soper et al. (97 Values otaied y Equatio 7- E[R ] E[R ] E[R ] E[R ].8.88 Let S W, e the ratio of two saple variaes. S The statisti W is widely used to test the hoogeeity of two variaes ad soeties is useful to ow the ea ad variae of W. With the ext orollary, we a otai the ea ad variae of W. Corollary For > a, the oets of the ratio of two saple S variaes, W, are give y: S ( a σ σ a + a a E W ΓΓ Γ + a a F, ; ; Use -a ad i Equatio. The expetatio of the ratio of two saple variaes, W, is give y: ( σ σ + E[ W] ΓΓ Γ + F, ; ; ( σ ( + ( + ( ( + + ( + + σ ( ( ( ( ( + σ ( ( The seod oet of the ratio W, is give y: ( σ σ + E W ΓΓ Γ + F, ; ; ( + ( + F, ; ; ( ( σ The variae of W is give y: [ ] Var W ( + ( + F, ; ; ( ( σ ( + σ ( ( ( ( + + F, ; ; σ ( ( ( + ( ( I the ase of, we have: S E S σ ( ( ( ( ( + ( ( Var[ W] σ ( σ If furtherore, the two variaes are equal the expetatio ad variae of W eoe the expetatio ad variae of a etral F-distriutio with paraeters d d. I the future, it will e useful to study the variae of W to test hoogeeity of variaes for two orrelated saples.

11 Jua Roero-Padilla/ Joural of Matheatis ad Statistis 6, (:. DOI:.8/jssp.6.. Colusio We derive a geeral result to get ay produt oets of the saple variaes ad saple orrelatio oeffiiet whe the data oe fro a ivariate oral distriutio, the fial expressio is give i ters of the hypergeoetri futio whih is a well-ow futio ad there exists oputatioal routies to e evaluated. A geeral expressio to get the oets of the orrelatio saple is otaied ad a validatio of the result was arried out. Fially ad equatio to get the oets of the ratio of two saple variaes was derived ad as a partiular ase, the expetatio ad variae of the ratio of two saple variaes were otaied. Ethis This artile is origial ad otais upulished aterial. There are o ethial issues ivolved i ay aspet of this paper. Referees Aderso, T.W.,. A Itrodutio to Multivariate Statistial Aalysis. st Ed., Wiley-Itersiee, ISBN-: 769, pp: 7. Bailey, W.N., 96. Geeralized Hypergeoetri Series. st Ed., Hafer, New Yor, pp: 8. Ghosh, B.K., 966. Asyptoti expasios for the oets of the distriutio of orrelatio oeffiiet. Bioetria, : 8-6. DOI:.7/76 Joarder, A.H., 6. Produt oets of a ivariate Wishart distriutio. J. Proaility Stat. Si., : -. Ngo, T.M.,. O the ehavior of the gaa futio o the egative side. Seior Hoors Theses, Uiversity of New Orleas, Louisiaa, USA. Oerhettiger, F., 97. Hypergeoetri Futios. I: Hadoo of Matheatial Futios with Forulas, Graphs ad Matheatial Tales, Araowitz, M. ad I. Stegu (Eds., Dover puliatios, I. New Yor, ISBN-: 8667, pp: -66. Rady, E.A., H.A. Fergay ad A.M. Edress,. A easy way to alulate the oets of saple orrelatio oefiiets. Cou. Fa. Si. Uiv. A. Series A. : -8. Soper, H.E., A.W. Youg, B.M. Cave, A. Lee ad K. Pearso, 97. O the distriutio of the orrelatio oeffiiet i sall saples. Appedix II to the papers of "studet" ad R. A. fisher. Bioetria, : 8-. DOI:.7/8