Randomly Weighted Averages on Order Statistics

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Apple Mathematcs 3 4 34-346 http://oog/436/am3498 Publshe Ole Septembe 3 (http://wwwscpog/joual/am Raomly Weghte Aveages o Oe Statstcs Home Haj Hasaaeh Lela Ma Ghasem Depatmet of Statstcs aculty of

1 Apple Mathematcs Publshe Ole Septembe 3 ( Raomly Weghte Aveages o Oe Statstcs Home Haj Hasaaeh Lela Ma Ghasem Depatmet of Statstcs aculty of Mathematcal Sceces Uvesty of Tab Tab Ia Emal: home@tabuac Receve July 3 ; evse Jauay 4 3; accepte Jauay 3 Copyght 3 Home Haj et al Ths s a ope access atcle stbute ue the Ceatve Commos Attbuto Lcese whch pemts uestcte use stbuto a epoucto ay meum pove the ogal wo s popely cte ABSTRACT We stuy a well-ow poblem coceg a aom vaable ufomly stbute betwee two epeet aom vaables Two ffeet etesos aomly weghte aveage o epeet aom vaables a aomly weghte aveage o oe statstcs have bee touce fo ths poblem o the seco metho two-se powe aom vaables have bee efe By usg classc metho a powe techcal metho we stuy some popetes fo these aom vaables Keywos: Two-Se Powe; Momet; Weghte Aveages; Powe Dstbuto Itoucto Va Asch [] touce the oto of a aom vaable ufomly stbute betwee two epeet aom vaables a whch aose stuyg the stbuto of poucts of aom matces fo stochastc seach of global mama By lettg a to have etcal stbuto he eve that: fo a o [ ] s ufom o [ ] f a oly f a have a Acse stbuto; a possesses the same stbuto as a f a oly f a ae egeeate o have a Cauchy stbuto Solta a Home [] followg Johso a Kot [3] etee Va Asch s esults They put to be epeet a cosee S R+ R + + R + R whee aom popotos R U U ( ( ( R R U U ae oe statstcs fom a ufom stbuto o [ ] a U ( These aom popotos ae ufomly stbute ove the ut smple They employe Steltjes tasfom a that: S possesses the same stbuto as f a oly f ae egeeate o have a Cauchy stbuto; a Va Asch s esult fo Acse hols fo oly I ths pape we touce two famles of stbutos suggeste by a aoymous efeee of the atcle to whom the autho epesses hs eepest gattue We say that s a aom vaable betwee two epeet aom vaables wth powe stbuto f the cotoally stbuto of gve at s ma( < < ( ( < < ma( The stbuto ( wll be sa to follow a cotoally ecte powe stbuto Whe s a tege o the stbuto gve by ( smplfes to the stbuto that was touce befoe Also we use Steltjes methos fo moe o the Steltjes tasfom see aye [4] o we call ecte tagula aom vaable o futhe geealg Va Asch esults we touce a seemgly moe atual cotoally powe stbuto We call two-se powe (TSP aom vaable f the cotoally stbuto of gve at s y y y y < < ( y y y Copyght 3 ScRes

2 H HAJIR ET AL 34 The stbuto wll be sa to follow a co- toally uecte powe stbuto whe y m y m ( y ma ( a s a tege Aga fo the stbuto gve by ( smplfes to the stbuto that was touce by Va Asch The ma am of ths atcle s povg a geealato of oto to the esults of Va Asch fo some othe values of (othe tha Ths atcle s ogae as follows We touce pelmaes a pevous wos Secto I Secto 3 we gve some Chaacteatos fo stbuto of gve ( whe I Secto 4 we f stbuto of gve ( by ect a powe metho a gve some eamples a Chaacteatos of such stbutos by use of Solta a Home esults [5] Pelmaes a Pevous Wos I ths secto we fst evew some esults of Va Asch [] a the mofy them a lttle Bt to ft ou famewo to be touce the fothcomg sectos Usg the Heavse fucto ( U( < we coclue that fo ay gve stct values a the cotoal stbuto ( ( s ( U ( U( Lemma o stct eal s a tege we have ( + ( ( (! ( ( D α Poof It follows fom the Leb fomula α Let h D ( I whee I s a teval a ( I s the set of all eal fuctos f that ae α - g fo some Tmes ffeetable o I If costats c a { } The P M g( h( g( { h( g ( } P + { h( g( } We use the Lebt fomula fo the ( th evatve of a pouct amely { h( g ( } N h( Let whee h g h g M + N { } { } + M h ( g ( + N h ( g (! It follows that + Sce g( g ( g( g ( Cosequetly N g( { h ( g( } P whee afte some al- gebac wo P M g( h( Theefoe M + N g h + h g { } Ths completes the poof Aothe tool fo povg ou ma theoem s the followg fomula tae fom the Schwat Dstbuto theoy amely [ ] ϕ Λ ( ϕ! Λ ( [ ] whee Λ s a stbuto ucto a Λ s the -th stbutoal evatve of Λ The cotoal stbuto ( gve by ( leas us to a lea fuctoal o comple Value fuctos f: efe o the set of eal umbes : f f ( ( (!( It easly follows that f af + bg a f + b g (3 o ay comple-value fuctos f g a comple f costats a b We ote that Wheeve f ( ( U( a ( f g f ( ( f ( (!( Copyght 3 ScRes

3 34 H HAJIR ET AL Also we ote that ca be vewe as: ( U f ( Thus! ( ( P U (! f ( f (4 Theefoe by usg (3 alog wth (4 a a staa agumet the tegato theoy we obta that (! f f (5 o ay ftely ffeetable fuctos f fo whch the coespog tegals ae fte Now (5 togethe wth ( lea us to ( f f (6 o the above metoe fuctos f whee s the (-th stbutoal evatve of the stbuto of Let us eote the Steltjes tasfom of a stbuto H by S( H H( o evey the set of comple umbes whch oes ot belog to the suppot of H e ( supph c The followg lemma cates how the Steljes tasfom of a ae elate Lemma Let be a aom vaables that satsfes ( Suppose that the aom vaables a ae epeet a cotuous wth stbuto fuctos a espectvely The ( ( S ( ( S S c supph Poof It follows fom (6 that A ( S g fo g ( a S g! ( ( Now t follows that ( g ( (!( A by usg Lemma we have Theefoe ( g ( ( ( ( ( ( S ( (! ( ( S ( ( S S c supph Ths fshes the poof Note that Va Asch s lemma s the case of : ( S S S (7 We also ote that the Steltjes tasfom of Cauchy S + c satsfes (7 stbuto e 3 Chaacteatos Now we apply Lemma fo some chaacteatos whe a ae ot etcally stbute Theoem 3 Let a be epeet aom vaables a be a aomly weghte aveage gve ( o we have a f has ufom stbuto o [ ] the has semccle stbuto o [ ] f a oly f has Acs stbuto o [ ]; b f has ufom stbuto o [ ] the has powe semccle stbuto o [ ] f a oly f has powe semccle stbuto e ( 3 f ( 4 c f has Beta ( stbuto o [ ] the has 3 3 Beta stbuto f a oly f has Beta Copyght 3 ScRes

4 H HAJIR ET AL 343 stbuto; f has ufom stbuto o [ ] the has Beta ( stbuto f a oly f has Beta ( stbuto Poof o the f pat we ote that the aom vaable has ufom stbuto a has Acs stbuto o [ ]; the S( ( l + l A S( om Lemma a substtutg the coespog Steltjes tasfoms of stbutos we get S ( ( 3 The soluto S( ( Whch s the Steltjes tasfom of the semccle stbuto o [ ] o the oly f pat we assume that the aom vaable has semccle stbuto The t follows fom Lemma that ( S ( 3 The poof s complete By a agumet smla to that gve a solvg the followg ffeetal equatos ( ( 6( ( l l 6 S ( (fo the f pat a +3 S 6 l l (fo the oly f pat The poof ca be complete 3 By Lemma ( we have S ( (fo the f pat a S( ( (fo the oly f pat The poof ca be complete by solvg the above ffeetal equatos 4 By Lemma ( we have ( ( 6( ( l l 6+3 S ( (fo the f pat a S 6 l l (fo the oly f pat Solvg the ffeetal equatos ca complete the poof 4 TSP Raom Vaables I Secto 3 we use a poweful metho base o the use of Steltjes tasfoms to obta the stbuto of gve ( It seems that oe ca ot use that metho to f stbuto of gve ( So we employ a ect metho to f the stbuto of Let us follow Lemma 4 to f a smple metho to get the stbuto of followg [] a the wo of them leas us to the followg lemma Lemma 4 Suppose W has a powe stbuto wth paamete s a tege a let y m ( y ma ( whee epeet aom vaables ae Let Y + W Y Y The s a TSP aom vaable ca be equvaletly efe by ( + + W Poof ( + ( PY W Y Y Poof y P( y+ W ( y y y y U m ma a also m ( ( ( W ma m ( U + m ( + + ma [ ] Copyght 3 ScRes

5 344 H HAJIR ET AL the so + W ( + + W 4 Momets of TSP Raom Vaables The followg theoem poves equvalet cotos o μ E Theoem 4 Suppose that s a TSP aom vaable satsfyg ( If a ae aom vaables a E fo all teges the E Γ + Γ + E ( y y Γ + + Γ + E E W E + ( E E y ( y y Poof By usg Lemma we obta that ( ( ( EW W EY Y EW ( ( W EY ( Y Γ + Γ + E y y Γ + + Γ + ( ( Poof Ths ca be easly pove by Lemma 4 Poof 3 ( E ( W y ( y y E Y + W Y Y EW E y y y ( ( ( E E y y y + Let us cose epectato a vaace of st we suppose that EY μ EY μ VaY σ VaY a Cov Y Y The σ a also f E E σ E μ+ μ + the E EY EY EY + + ( By + Y+ Y We have E( E( Y + ( EY EY + + (4 It ca easly follow fom (4 that the Acs esult of Va Asch [] s oly tue fo about the vaace we have Va ( μ μ + ( + σ + ( + ( σ + σ ( + ( + ollowg the computato of epectato a vaace we evaluate them fo some well-ow stbutos If a have staa omal stbutos the fom Theoem 3 a the fact that a + ae epeet t follows that the fst seco a th oe momets ae equal espectvely to E π + a E 3 E + + ( + ( π ( + 3( + ( + Also case a have ufom stbutos Theoem 4 mples that a E Γ + Γ + Γ + + Γ Va E + 3 ( ( + ( + Theoem 4 Suppose that s a TSP aom vaable satsfyg (4 the s locato vaat; If a have symmetc stbuto aou μ the has symmetc stbuto aou μ oly whe Poof Is mmeate We ca assume wthout loss of geealty that μ If has a symmetc stbuto aou eo the Copyght 3 ScRes

6 H HAJIR ET AL 345 We ote that Y+ W( Y Y Y+ W( Y Y ( Y+ W Y Y Y+ W Y Y m ma Sce ( A we have + ( + ( Y W Y Y Y W Y Y (43 By equatg the cotoal stbutos gve at a (33 we coclue that It ca also easly follow fom Theoem (4 that the Cauchy esult of Va Asch [] s tue oly fo 4 Dstbutos of TSP Raom Vaables I ths subsecto we vestgate computg stbutos by the ect metho We wll gve two eamples of evato base o (4 Ths metho may be complcate some cases but we have chose some easy to f eamples We use aomly weghte aveage o oe statstcs to f the stbuto of Gauss hype geometc fucto ( abc ; whch s a well-ow specal fucto that we use ths way Eample 4 Let a W be epeet aom vaables such that a ae ufomly stbute ove [ ] a W has a powe fucto stbuto wth paamete We f the value f ( ; by meas of f ; W( w theefoe < < w w f W( w (44 ( w< < w By usg the stbuto of W the esty fucto f ( ; ca be epesse tems of the Gauss hype geometc fucto ( abc ; whch s a well-ow specal fucto Iee accog to Eule s fomula the Gauss hype geometc fucto assumes the tegal epesetato ( ; Γ( c ( b ( c b abc b t t t cb ( ( a Γ Γ t whee a b c ae paametes subject to < a < c> b> wheeve they ae eal a s the vaable ( ; f + + ( ( ( (45 whee > a Thee ae some mpotat fuctos as a Gauss hype geometc fucto ( log + ;; e lm b a b; b; a a ( ( a ;; Whe smla calculatos lea to the followg stbuto ( ( f ( log log < < Whe s a tege we obta the followg stbuto f ( ( ( ( (( < < The pobablty esty fucto f was touce by Johso a Kot [3] fo the fst tme ue the ttle ufomly aomly mofe te So f ( ; ca be see as a eteso of the above metoe stbuto We ote that fom (4 a a smple Mote Calo poceue usg oly smulate ufom vaables oe ca to smulate the stbuto (45 Theoem 4 Let be a uecte tagula aom vaable that satsfes ( Suppose that the aom vaables a ae epeet a cotuous wth the stbuto uctos a espectvely The ( S ( S ( ( + S S whee ( S ( ( ( ( Poof By usg a agumet smla to that gve secto 3 we ca coclue that So ( f f S g o g ( ( ( ( om Copyght 3 ScRes

7 346 H HAJIR ET AL ( ( ( ( ( g + A by usg patal factoal ule we have ( g + ( ( ( ( ( Theefoe S ( + ( a b I ( a b t ( t t ( a b> B a b 5 Cocluso ( We have escbe how ecte methos coul be use fo obtag the stbutos Chaacteatos a popetes of the aom mtue of vaables efe ( The TSP aom vaable whe a have ufom stbutos le us to a ew famly of stbuto whch ca be egae as some geealato of ufomly aomly mofe te The popose moel the ect metho ca easly lea to stbuto geealatos though ths s ot possble fo the Steltjes metho but hee the chaactestcs ca be easly compute A 6 Acowlegemets S ( S ( S ( ( + S The autho s eeply gateful to the aoymous efeee fo eag the ogal mauscpt vey caefully a fo Ths fshes the poof mag valuable suggestos It s woth metog that the peset metho yels othe etesos too; the followg s such a eample Eample 43 Suppose that W ae epeet aom vaables If a have Ufom stbutos o [ ] a W has Beta ( stbuto the has the same stbuto as W If the pouct momets of oe statstcs ae ow those of W ca be eve fom that of By usg Theoem 4 The the stbuto of W s chaactee by that of By a agumet smla to the oe gve eample 4 whe W has a Beta stbuto wth Paametes f m ; as a m we f the stbuto B( m ( I ( m B( m B( m + ( I ( m B( m < < whee I ( ab s complete Beta fucto: REERENCES [] W Va Asch A Raom Vaable Ufomly Dstbute betwee Two Iepeet Raom Vaables Sahaya Vol 49 No 987 pp 7- o:8/ [] A R Solta a H Home Weghte Aveages wth Raom Popotos That Ae Jotly Ufomly Dstbute ove the Ut Smple Statstcs & Pobablty Lettes Vol 79 No 9 9 pp 5-8 o:6/jspl99 [3] N L Johso a S Kot Raomly Weghte Aveages The Ameca Statstca Vol 44 No 3 99 pp o:37/68535 [4] A I aye Haboo of ucto a Geeale ucto Tasfomatos CRC Pess Loo 996 [5] A R Solta a H Home A Geealato fo Two- Se Powe Dstbutos a Ajuste Metho of Momets Statstcs Vol 43 No 6 9 pp 6-6 o:8/ Copyght 3 ScRes

Professor Wei Zhu. 1. Sampling from the Normal Population

Professor Wei Zhu. 1. Sampling from the Normal Population AMS570 Pofesso We Zhu. Samplg fom the Nomal Populato *Example: We wsh to estmate the dstbuto of heghts of adult US male. It s beleved that the heght of adult US male follows a omal dstbuto N(, ) Def. Smple