Kig Fhd Uiversity of Petroleum & Mierls DEPARTMENT OF MATHEMATICAL CIENCE Techicl Report eries TR 434 April 04 A Direct Proof of the Joit Momet Geertig Fuctio of mple Me d Vrice Awr H. Jorder d A. Lrdji DHAHRAN 36 AUDI ARABIA www.kfupm.edu.s/mth/ E-mil: c-mth@kfupm.edu.s
A Direct Proof of the Joit Momet Geertig Fuctio of mple Me d Vrice Awr H. Jorder d A. Lrdji Deprtmet of Mthemtics d ttistics Kig Fhd Uiversity of Petroleum d Mierls Dhhr 36, udi Arbi Emils: jstt@gmil.com d lrdji@kfupm.edu.s Abstrct. Without ssumig idepedece of smple me d vrice, or without usig y coditiol distributio, we preset direct derivtio of the joit momet geertig fuctio for smple me d vrice for idepedetly, ideticlly d ormlly distributed rdom vribles. Key Words: mple me, smple vrice, idepedece, momet geertig fuctio MC (00). 6E0, 6H0, 6G35.. Itroductio The idepedece of smple me d vrice of idepedetly, ideticlly d ormlly distributed vribles is essetil i the bsic deftio of tudet t -sttistic, d lso i the developmet of my sttisticl methods. It is usully proved usig the idepedece of X d ( X X, X X), (see e.g. Theorem, p.340, Rohtgi d leh, 00), but this requires bckgroud o idepedece of fuctios of rdom vribles (Theorem, p., Rohtgi d leh, 00) tht my ot be esily ccessible to begg udergrdute studets. There re severl proofs of the idepedece of smple me d vrice, the simplest of which seems to be the oe due to huster (973) which uses momet geertig fuctio. It should however be oted tht proofs ccessible to udergrdute studets hve bee issue of discussio. ee for exmple, Zeh (99) d lso Americ ttistici, 99, Volume 46, No., pp. 7-75. I this ote we give ew proof of this idepedece tht lso uses momet geertig fuctio, but, ulike huster s proof, voids the use of coditiol distributios d seems to be more suitble for udergrdute studets.. ome Prelimiries Let, X ( =,3, ) hve rbitrry -dimesiol joit distributio. We defie the smple me x d vrice s by x = x i d ( ) s = ( x x ), i respectively. The smple vrice c lso be represeted by (.) ( ) = ( ). i j( i) = i j s x xx
Also for ideticlly distributed observtios, X with commo me µ, we deote µ EX ( ), the -th momet of X d µ EX ( µ ), the cetered momet of X order. The me µ d vrice µ V( X) will be simply deoted by µ d σ = µ µ respectively. The momet geertig fuctio of X ~ N ( µσ, ) is give by M t t t X ( ) = exp µ + σ, < t <. Let A= [ ] be positive defte symmetric mtrix. The the followig itegrl is kow: j= / ( π ) exp xix j + bi xi dxdx dx = exp b A b, / j= A (.) where b= ( b, b, b ). fuctio M X ( b) = exp b Σb, The bove expressio usully ppers s prt of the momet geertig (.3) of rdom vrible X hvig multivrite orml distributio N(0, Σ). Aderso (984, p. d p.47). ee for exmple, For the proof of the followig theorem delig with the properties of ptter mtrix, see Ro (973, p.67). Theorem. Let the mtrix Γ= [ ] where = α, for i =,,,, d = β, for i =,,,, d j( i) =,,,. The the followig hold:. Γ= + ( ) α β [ α ( ) β], b. Γ exists if d oly if α β d α ( ) β. Moreover, if the d re the etries of Γ, α + ( ) β =, [ α + ( ) β]( α β) β d =, [ α + ( ) β]( α β) i j. Corollry. Let the mtrix Γ= [ ] where = α, for i =,,,, d = β, for i =,,,, d j( i) =,,,. Further if α + ( ) β =, the we hve the followig:. Γ= ( α β),
b. j= j= =, 3 c. j= j= =. Proof.. It is obvious from prt () of Theorem.. b. ice α + ( ) β =, it follows tht = j= + j( i) = which equls. c. ice α + ( ) β =, it follows from Theorem. tht β =, ( i =,,,, j( i) =,,, ). The α β β =, α β ( i =,,, ) d = j= + which equls. j( i) = 3. The Joit M.G.F. of mple Me d Vrice Without usig y coditiol distributio or ssumig idepedece of X d direct proof of the joit momet geertig fuctio of X d ideticlly d ormlly distributed rdom vribles., we preset, bsed o idepedetly, Theorem 3. Let the rdom vribles, X, ( ) be idepedetly, ideticlly d ormlly distributed with EX ( ) = µ d Vr( X) = σ. The the joit momet geertig fuctio of the smple me X d vrice is give by ( )/ σ t σ t M ( t,, t) = ex µ t p, X +. (3.) I prticulr, X d re idepedet, d Proof. The joit momet geertig fuctio of X d ( ) M ( t, t ) = exp t x + t s f ( x ), i dxi which equls ( ) / σ ~ χ. is give by
= / ( ) i µ ( π) σ + σ M ( t, t ) exp t x t s exp ( x ) dx. i 4 Usig the represettio of s give i (.) i the bove itegrl, d the trsformtio x = µ + σz, ( i =,,, ) yields i i M t t t z z z z z dz t t (, ) = exp( / σ )exp exp. i i j i i ( π ) i ( ) = j( i) = σ t Let b be the -vector [ ], d Γ= ( ) be the mtrix give by σ t =, if i = σ t, if i j. ( ) j exp( tµ ) M ( t,, t ) = exp b exp. X / izi ziz j dzi ( π ) (3.) j= The equtio (3.) c the be writte s exp( tµ ) M ( t,, t ) = exp z z exp( b z) dz, X / ( π ) Γ which, by (.3), c be evluted to be M t t t b b / (, ) = exp( µ ) Γ exp Γ. (3.3) It is esy to check tht σ t σ t b Γ b = trγ bb = = j=, (3.4) where the lst step follows by Corollry.(c). Also by Corollry.(), we hve ( ) / Γ = α β where ( )/ / σ t Γ =. σ t α = d σ t β =. ( ) σ t ice α β =, we obti (3.5)
Pluggig (3.4) d (3.5) i (3.3), we obti (3.). 5 Ackowledgemets The uthors grtefully ckowledges the excellet reserch fcilities provided by Kig Fhd Uiversity of Petroleum d Mierls, udi Arbi through the Project IN305. They lso express grtitude to their collegues Professor H. Azd, Dr. M. Hfidz Omr d Dr. M. Riz for my fruitful discussios. Refereces Aderso, T.W. (984). A Itroductio to Multivrite ttisticl Alysis. New York: Joh Wiley. Ro, C.R. (973). Lier ttisticl Iferece d Its Applictios. d ed. New York: Joh Wiley. Rohtgi, V.K. d leh, A.K.M.E. (00). Probbility d ttistics. Wiley. huster, J. (973). A simple method of techig the idepedece of X d ttistici, 7(), 9-30. Zeh, P.W. (99). O Provig tht X d 45(), -.. The Americ re idepedet. The Americ ttistici,