A Characterization of Skew Normal Distribution by Truncated Moment

Transcription

1 Availabl at Appl Appl Math ISSN: Vol 9 Iss Jn 04 pp 8-38 Applications and Applid Mathmatics: An Intrnational Jornal AAM A Charactrization o Skw Normal Distribtion by rncatd Momnt M Shakil Dpartmnt o Mathmatics Miami Dad Collg Hialah Camps Hialah Florida 330 USA mshakil@mdcd M Ahsanllah Dpartmnt o Managmnt Scincs Ridr Univrsity Lawrncvill Nw Jrsy USA ahsan@ridrd BM Golam Kibria Dpartmnt o Mathmatics and Statistics Florida Intrnational Univrsity Univrsity Park Miami Florida 3399 USA kibriag@id Abstract A probability distribtion can b charactrizd throgh varios mthods his papr discsss a nw charactrization o skw normal distribtion by trncatd momnt It is hopd that th indings o th papr will b sl or rsarchrs in dirnt ilds o applid scincs Kywords: Charactrization skw normal trncatd momnt AMS Sbjct Classiications: 6E0 6E5 Introdction Bor a particlar probability distribtion modl is applid to it th ral world data it is ncssary to conirm whthr th givn probability distribtion satisis th ndrlying rqirmnts by its charactrization hs charactrization o a probability distribtion plays an 8

2 AAM: Intrn J Vol 9 Iss Jn 04 9 important rol in probability and statistics A probability distribtion can b charactrizd throgh varios mthods s or ampl Ahsanllah t al 04 among othrs In rcnt yars thr has bn a grat dal o intrst in th charactrization o probability distribtions by trncatd momnts For ampl th dvlopmnt o th gnral thory o th charactrization o probability distribtions by trncatd momnt bgan with th work o Galambos and Kotz 978 Frthr dvlopmnt in this ara contind with th contribtions o many othr athors and rsarchrs among thm Kotz and Shanbhag 980 Glänzl and Glänzl t al 984 ar notabl Howvr most o ths charactrizations ar basd on a simpl rlationship btwn two dirnt momnts trncatd rom th lt at th sam point Many athors hav also stdid charactrizations o th skw normal distribtion SND For ampl Gpta t al 004 stdid th charactrization rslts or th skw normal distribtion basd on qadratic statistics For dtaild drivations o ths rslts th intrstd radrs ar rrrd to Gpta t al 004 and rrncs thrin S also Arnold and Lin 004 whr th athors hav shown that th skw normal distribtions and thir limits ar actly th distribtions o ordr statistics o bivariat normally distribtd variabls Frthr sing gnralizd skw normal distribtions Arnold and Lin 004 hav bn abl to charactriz th distribtions o random variabls whos sqars oby th chi sqar distribtion with on dgr o rdom For mor on charactrizations w rr th intrstd radrs to Ahsanllah t al 04 among othrs It appars rom th litratr that no attntion has bn paid to th charactrizations o th skw normal distribtion sing trncatd momnt As pointd ot by Glänzl 987 ths charactrizations may also srv as a basis or paramtr stimation In this papr w prsnt a nw charactrization o th skw normal distribtion sing th trncatd momnt by considring a prodct o rvrs hazard rat and anothr nction o th trncatd point h organization o this papr is as ollows Sction discsss th skw normal distribtion SND and som o its proprtis In Sction 3 charactrization o th skw normal distribtion by trncatd momnt is prsntd h conclding rmarks ar providd in Sction 4 Distribtional Proprtis o a Skw Normal Distribtion his sction discsss th skw normal distribtion SND and som o its distribtional proprtis Dinition: A continos random variabl X is said to hav a skw normal distribtion with paramtrs dnotd by Y ~ SN i its probability dnsity nction and cmlativ distribtion nction ar rspctivly givn by

3 30 M Shakil t al and F whr 0and ar to b rrrd as th location scal and shap paramtrs rspctivly; and dnot th probability dnsity nction and cmlativ distribtion nction o th standard normal distribtion rspctivly; and dnots Own s 956 nction as givn by / d 0 In particlar i in th abov dinitions 0 thn w hav a standard skw normal distribtion dnotd by X ~ SN with probability dnsity nction as givn by whr X ; 3 and t dt dnot th probability dnsity nction and cmlativ distribtion nction o th standard normal distribtion rspctivly h continos random variabl X is said to hav a skw normal distribtion sinc th amily o distribtions it rprsnts inclds th standard N 0 distribtion as a spcial cas bt in gnral with mmbrs having skwd dnsity his is also vidnt rom th act that X ~ Chi-sqar distribtion with on dgr o rdom or all vals o th paramtr As pointd ot by Azzalini 985 th skw normal dnsity nction X ; has th ollowing charactristics:

4 AAM: Intrn J Vol 9 Iss Jn 04 3 a 0 N0 SN b I X ~ SN thn SN c I X ~ and Z ~ N 0 thn SN Z ~ HN 0 that is th hal-normal distribtion d I SN X ~ thn h MGF o X ~ t X is givn by M t t t It is asy to s that E X and X g h charactristic nction o whr Var t X is givn by t i h t t SN tnds to whr h 0 y dy or 0 and h h h By introdcing th ollowing linar transormation Y X that is X Y whr 0 0 w obtain th skw normal distribtion with th probability dnsity nction givn by Som charactristic vals o th random variabl Y ar as ollows: E Y Man: Varianc: Var Y and 4 E X Skwnss: Var X whr / /

5 3 M Shakil t al Not: h skwnss is limitd in th intrval 4 EX Krtosis: 3 Var X 4 3 / MGF t t t h shap o th skw normal probability dnsity nction givn by dpnds on th vals o th paramtr For som vals o th paramtrs th shaps o th pd o SNλ ar providd in Figrs and blow Figr : Plot o th pd o SNλ or 0 5 Figr : Plot o th pd o SNλ or 3 0 h skw normal distribtion rprsnts a paramtric class o probability distribtions rlcting varying dgrs o skwnss which inclds th standard normal distribtion as a spcial cas h trm skw normal distribtion SND was introdcd by Azzalini who gav a

6 AAM: Intrn J Vol 9 Iss Jn systmatic tratmnt o this distribtion h skwnss paramtr involvd in this class maks it possibl or probabilistic modling o th data obtaind rom skwd poplation h skw normal distribtions ar also sl in th stdy o th robstnss and as priors in Baysian analysis o th data For rcnt dvlopmnts on th skw normal distribtion th intrstd radrs ar rrrd to Pwsy 000 Gpta t al 00 Nadarajah & Kotz 003 Dalla Vall 004 Gnton 004 Gpta & Gpta 004 Azzalini 006 Nadarajah & Kotz 006 Chakraborty and Hazarika 0 Azzalini and Rgoli 0 and vry rcntly Ahsanllah t al 04 among othrs 3 A Nw Charactrization o th Skw Normal Distribtion In what ollows w will prsnt a nw charactrization o th skw normal distribtion in a dirnt dirction W shall do this by sing trncatd momnt For this w nd th ollowing assmptions and Lmma Lmma 3 Lt th random variabl X b a random variabl having absoltly continos with rspct to Lbsg masr cmlativ distribtion nction cd F and th probability dnsity nction pd W assm = in{ F > 0} and sp{ F } W din F and g is a dirntiabl nction with rspct to or all ral Lmma 3 Sppos that X has an absoltly continos with rspct to Lbsg masr cd Fwith corrsponding pd and EX X < ists or all ral hn EX X< = g whr g is a dirntiabl nction and or all ral F i c g' d g whr c is dtrmind sch that d = Not: Sinc cd F is absoltly continos with rspct to Lbsg masr thn by g ' Radon-Nikodym horm th pd ists and hnc d ists g

7 34 M Shakil t al Proo: W hav d g F F hs d g Dirntiating both sids o th qation with rspct to w obtain g' g ' On simpliication w gt ' g' g Intgrating th abov qation w obtain g' d c g whr c is dtrmind sch that d his complts th proo o Lmma 3 W now hav th ollowing charactrization thorm horm 3 A Charactrization horm horm 3: Sppos that X has an absoltly continos with rspct to Lbsg masr cd F pd / and EX X < ists or all in α β W assm and E X and ist or all hn whr EX X < =g

8 AAM: Intrn J Vol 9 Iss Jn F H g d thown 956 nction as givn by is d 0 / and H λ = d i and only i any ral or Proo: Sppos thn g F F d

9 36 M Shakil t al ] [ H Sppos that H g thn ] [ ' H g g hs ' ' X g g On intgrating th abov qation w hav / c whr / / d c his complts th proo o horm3 4 Conclding Rmarks As pointd ot abov bor a particlar probability distribtion modl is applid to it th ral world data it is ncssary to conirm whthr th givn probability distribtion satisis th ndrlying rqirmnts by its charactrization hs charactrization o a probability distribtion plays an important rol in probability and statistics A probability distribtion can b charactrizd throgh varios mthods his papr considrs a nw charactrization o th skw normal distribtion sing trncatd momnt by considring a prodct o rvrs hazard rat and anothr nction o th trncatd point In this rgard som distribtional proprtis o th skw

10 AAM: Intrn J Vol 9 Iss Jn normal distribtion ar also providd W bliv that th indings o this papr wold b sl or th practitionrs in varios ilds o stdis and rthr nhancmnt o rsarch in distribtion thory and its applications Acknowldgmnt h athors wold lik to thank th ditor and iv anonymos rviwrs or hlpl sggstions which improvd th prsntation o th papr Rrncs Ahsanllah M Kibria B M G and Shakil M 04 Normal and Stdnt s t Distribtions and hir Applications Atlantis Prss Paris Franc Arnold B C and Lin G D 004 Charactrizations o th skw-normal and gnralizd chi distribtions Sankhyā Azzalini A 985 A class o distribtions which inclds th normal ons Scand J Statist 7-78 Azzalini A 986 Frthr rslts on a class o distribtions which inclds th normal ons Statistica Azzalini A 006 Atti dlla XLIII Rinion dlla Socità Italiana di Statistica volm Rinioni plnari spcializzat pp5-64 Azzalini A and Rgoli G 0 Som proprtis o skw-symmtric distribtions Annals o th Institt o Statistical Mathmatics Chakraborty S and Hazarika P J 0 A Srvy on th hortical Dvlopmnts in Univariat Skw Normal Distribtions Assam Statistical Rviw Dalla Vall A 004 "h skw-normal distribtion" in Skw-Elliptical Distribtions and hir Applications: A Jorny Byond Normality Gnton M G Ed Chapman & Hall / CRC Boca Raton FL pp 3-4 Gnton M G Ed 004 Skw-lliptical distribtions and thir applications: a jorny byond normality Chapman & Hall/CRC Boca Raton FL Galambos J and Kotz S 978 Charactrizations o probability distribtions A niid approach with an mphasis on ponntial and rlatd modls Lctr Nots in Mathmatics 675 Springr Brlin Glänzl W 987 A charactrization thorm basd on trncatd momnts and its application to som distribtion amilis Mathmatical Statistics and Probability hory Bad atzmannsdor 986 Vol B Ridl Dordrcht Glänzl W 990 Som consqncs o a charactrization thorm basd on trncatd momnts Statistics Glänzl W lcs A and Schbrt A 984 Charactrization by trncatd momnts and its application to Parson-typ distribtions Z Wahrsch Vrw Gbit Gpta R C and Gpta R D 004 Gnralizd skw normal modl st

11 38 M Shakil t al Gpta A K Chang F C and Hang W J 00 Som skw-symmtric modls Random Oprators Stochastic Eqations Gpta A K Ngyn and Sanqi J A 004 Charactrization o th skw-normal distribtion Annals o th Institt o Statistical Mathmatics Kotz S and Shanbhag DN 980 Som nw approachs to probability distribtions Advancs in Applid Probability Nadarajah S and Kotz S 003 Skwd distribtions gnratd by th normal krnl Statistics and Probability Lttrs Nadarajah S and Kotz S 006 Skw distribtions gnratd rom dirnt amilis Acta Applicanda Mathmatica 9-37 Own D B 956 abls or compting bivariat normal probabilitis h Annals o Mathmatical Statistics Pwsy A 000 Problms o inrnc or Azzalini's skwnormal distribtion Jornal o Applid Statistics DOI: 0080/