UC Riverside UC Riverside Electronic Theses and Dissertations

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1 UC Rversde UC Rversde Electroc Theses ad Dssertatos Ttle Estmato of the Parameters of Skew Normal Dstrbuto by Appromatg the Rato of the Normal Desty ad Dstrbuto Fuctos Permalk Author Dey Debarsh Publcato Date -- Peer revewed Thess/dssertato escholarshp.org Powered by the Calfora Dgtal Lbrary Uversty of Calfora

2 UNIVERSITY OF CALIFORNIA RIVERSIDE Estmato of the Parameters of Skew Normal Dstrbuto by Appromatg the Rato of the Normal Desty ad Dstrbuto Fuctos A Dssertato submtted partal satsfacto of the requremets for the degree of Doctor of Phlosophy Appled Statstcs by Debarsh Dey August Dssertato Commttee: Dr. Subr Ghosh Charperso Dr. Barry Arold Dr. Ama Ullah

3 Copyrght by Debarsh Dey

4 The Dssertato of Debarsh Dey s approved: Commttee Charperso Uversty of Calfora Rversde

5 ACKNOWLEDGEMENTS I would take ths opportuty to epress my scere grattude ad thaks to my advsor Professor Subr Ghosh for hs cotuous ad utrg gudace over the course of the last fve years. Hs scere terest ot oly my academc progress but also my persoal well-beg has bee a source of sustaed motvato for me. I would also lke to eted my scere thaks to Professor Barry Arold of the Departmet of Statstcs UC Rversde ad Professor Ama Ullah of the Departmet of Ecoomcs UC Rversde for gracously acceptg to serve o my PhD Commttee ad for ther valuable tme ad advce. I wsh to thak the etre faculty of the Departmet of Statstcs for erchg me wth ther vast kowledge varous felds of Statstcs. I would lke to thak the etre staff of the Departmet of Statstcs who were always ready wth ther help. My freds here at UCR have also bee a source of major stregth ad support for the last fve years. Your fredshp has made my eperece at Rversde all the more memorable ad fulfllg. v

6 A very specal thaks to my wfe Trupt for beg wth me ad for beg my pllar of stregth. Though she joed me oly a few moths ago her boudless affecto ad mmese support are valuable to me ths accomplshmet. I would lke to thak my parets because t s for them ad ther hard work ad sacrfce that made me acheve whatever lttle I have acheved. Wthout ther costat support ad ther mmese fath me I mght ot have pursued ad persevered. Together wth them my youger brother Tuka ad my Gradmother Da are equally commtted to my success ad have take mmese prde my accomplshmets. Fally I would lke to epress my most humble grattude to my Dve Master Bhagawa Sr Sathya Sa Baba wthout whose Grace ad Beevolece I could ot have acheved aythg. v

7 ABSTRACT OF THE DISSERTATION Estmato of the Parameters of Skew Normal Dstrbuto Usg Lear Appromatos of the Rato of the Normal Desty ad Dstrbuto Fuctos by Debarsh Dey Doctor of Phlosophy Graduate Program Appled Statstcs Uversty of Calfora Rversde August Dr Subr Ghosh Charperso The ormal dstrbuto s symmetrc ad ejoys may mportat propertes. That s why t s wdely used practce. Asymmetry data s a stuato where the ormalty assumpto s ot vald. Aal (985) troduces the skew ormal dstrbuto reflectg varyg degrees of skewess. The skew ormal dstrbuto s mathematcally tractable ad cludes the ormal dstrbuto as a specal case. It has three parameters: locato scale ad shape. I ths thess we attempt to respod to the complety ad challeges the mamum lkelhood estmates of the three parameters of the skew ormal dstrbuto. The complety s traced to the rato of the ormal desty ad dstrbuto fucto the lkelhood equatos the presece of the skewess parameter. Soluto to ths problem s obtaed by appromatg ths rato by lear ad o-lear fuctos. We observe that the lear appromato performs qute v

8 satsfactorly. I ths thess we preset a method of estmato of the parameters of the skew ormal dstrbuto based o ths lear appromato. We defe a performace measure to evaluate our appromato ad estmato method based o t. We preset the smulato studes to llustrate the methods ad evaluate ther performaces. v

9 Cotets. Itroducto. Motvato ad Hstorcal Developmet Normal Dstrbuto ad Smple Lear Regresso Skew Normal Dstrbuto ad Regresso Thess Descrpto v

10 . The Uvarate Skew Normal Dstrbuto. The Uvarate Skew Normal Dstrbuto Momets of the Uvarate Skew Normal Dstrbuto Lkelhood Fucto ad Mamum Lkelhood Estmates Challeges of the Mamum Lkelhood Estmates of the Skew Normal Dstrbuto Lterature revew o challeges Appromatos of the rato of the Stadard Normal Desty ad Dstrbuto Fuctos Itroducto Motvato Fttg the Lear Appromato to the Rato Fttg the No-lear Appromato to the Rato Estmato of the Shape Parameter of the Stadard Skew Normal Dstrbuto 5 4. Itroducto The Estmato Procedure usg A ()

11 4.. Case I : Covarate X Preset Case II : Covarate X Abset Measure of Goodess of Ft The Estmato Procedure usg B () Case I : Covarate X Preset Case II : Covarate X Abset Measure of Goodess of Ft A Smulated Data Case I : Covarate X Preset Case II : Covarate X Abset Estmatg Bas ad Accuracy Appromatos usg Smulatos Estmato of Locato Scale ad Shape Parameter of a Skew Normal Dstrbuto 8 5. Itroducto Relatos Amog Estmated Parameters Estmato Procedure usg A () A smulated Data Estmatg Bas ad Accuracy Appromato usg Smulatos Cocluso 4 Bblography 7

12 Lst of Fgures. The pdf of ~ SN( ) ad Probablty that < for values of ragg from to Plots of R () agast for. 5 ad Plots of A () agast for. 5 ad Plot of R () agast for. 5 (cotuous les) ad A () agast for. 5 (dotted les) Plot of R () agast for (cotuous les) ad A () agast for (dotted les)

13 3.5 Plot of R () agast for (cotuous les) ad A () agast for (dotted les) Plots of B () agast for. 5 ad Plot of R () agast for. 5 (cotuous les) ad B () agast for. 5 (dotted les) Plot of R () agast for (cotuous les) ad B () agast for (dotted les) Plot of R () agast for (cotuous les) ad B () agast for (dotted les) Plots of both R ( ) ad A ( ) agast whe covarate X s preset ˆ 4. Plots of both R ( ) ad B ( ) agast whe covarate X s preset ˆ 4.3 Plots of both R ( ) ad A ( ) agast whe covarate X s abset ˆ 4.4 Plots of both R ( ) ad B ( ) agast whe covarate X s abset ˆ 5. Plot of the estmated log-lkelhood lˆ agast the values of [ ) b Plot of the estmated log-lkelhood lˆ agast the values of b [.35.5]

14 Lst of Tables 4.. The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe ad. 5 A appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe ad. 5 A appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe 3 5. ad. 5 A appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe 3 5. ad. 5 A appromatg (3.) by (3.)

15 4.5. The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe ad. 5 B appromatg (3.) by (3.3) The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe ad. 5 B appromatg (3.) by (3.3) The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe 3 5. ad. 5 B appromatg (3.) by (3.3) The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe 3 5. ad. 5 B appromatg (3.) by (3.3) The true values of the sample se ad the proporto of tmes the sg of s correctly determed The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. 5 appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. 5 appromatg (3.) by (3.) v

16 5.4 The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. 5 appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. 5 appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) v

17 5. The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad 3. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad 3. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad 3. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad 3. appromatg (3.) by (3.). ) v

18 5.8 The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. 5 appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. 5 appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. 5 appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. 5 appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) v

19 5.5 The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad 3. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad 3. appromatg (3.) by (3.) v

20 5.3 The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad 3. appromatg (3.) by (3.) The values of Q Meda Mea Q 3 ad SD for b ˆ ˆ ˆ ˆ ˆ whe = ad 3. appromatg (3.) by (3.)

21 Chapter Itroducto

22 . Motvato ad Hstorcal Developmet I drawg statstcal fereces uder the parametrc framework we assume a dstrbuto that descrbes the data the best possble way. The celebrated Gaussa dstrbuto s the most popular dstrbuto for descrbg a data. Its popularty has bee drve by ts aalytcal smplcty assocated Cetral Lmt Theorem ts multvarate eteso- both the margals ad codtoals beg ormal addtvty ad other propertes. However there are umerous stuatos where the Gaussa dstrbuto assumpto may ot be vald. Alteratvely may ear ormal dstrbutos have bee proposed (Mudholkar ad Hutso () Turer(96) Pretce ( 975) ad Aal ( 985)). These famles descrbe the varatos from ormalty share some desrable propertes of ormal dstrbutos to some etet ad also clude the ormal dstrbuto as a specal case. Uder may stuatos whe the data caot be satsfactorly modeled by a ormal dstrbuto these para metrc dstrbutos provde alteratves drawg fereces. Some of these famles deal wth the devatos from symmetry the ormal dstrbuto. They are aalytcally tractable cota reasoable degrees of skewess ad

23 kurtoss ad clude the ormal dstrbuto as a specal case. Oe such dstrbuto s the skew ormal dstrbuto whch was proposed by Aal (985). Whle the ormal dstrbuto wth ts symmetry has oly locato ad scale parameters the skew ormal dstrbuto has a addtoal shape parameter descrbg the skewess. From practcal stadpot ths s a very desrable property where may real lfe stuatos some skewess s always preset the data. I addto the skew ormal dstrbuto shares may mportat propertes of the ormal dstrbuto: for eample the skew ormal destes are umodal ther support s the real le ad the square of a skew ormal radom varable has the Ch-square dstrbuto wth oe degree of freedom. Arold et al. (993) provded the followg motvato for the skew ormal model. Let X be the GPA ad Y be the SAT score of studets who wat to be admtted colleges. We assume (XY) to follow a bvarate ormal dstrbuto. Ths mples the margal dstrbuto of both X ad Y are ormal. Suppose we cosder a college where the admtted studets have a above average SAT score. If we cosder the X values of the admtted studets that college the dstrbuto of the X values follows a skew ormal dstrbuto. Although the skew ormal dstrbuto has some attractve propertes there are some completes ad challeges preset drawg the ferece o ts parameters. Detaled dscussos o these ssues ca be foud Sartor (6) Dalla Valle (4) Pewsey () Mot (3) ad others. Dfferet methods overcomg these ssues 3

24 are preseted by Aal ad Capattao (999) Lseo ad Loperfdo () Aa Clara Mot (3) Sartor (5). Ths thess detfes a source of the complety the mamum lkelhood estmato of the skew ormal parameter. The source s fact the rato of a ormal desty ad dstrbuto fuctos the presece of locato scale ad shape parameters of the skew ormal dstrbuto. We propose appromatos of ths rato by lear ad o-lear fuctos. We observe that the lear appromato performs qute satsfactorly appromatg the comple fucto. Hece we used the lear appromato for our estmato procedure. We cosder a lear regresso setup where the locato parameter of the skew ormal dstrbuto s a lear fucto of a covarate X. 4

25 . Normal Dstrbuto ad Smple Lear Regresso The ormal dstrbuto s wdely used descrbg data may applcatos. The dstrbuto s symmetrc wth mea ad stadard devato ad appromately 99.7% of the dstrbuto les the rage 3. The probablty dstrbuto fucto of a ormal radom varable X s f ( ; ) ep. (.) We deote X ~ N( ). We cosder a smple lear regresso setup wth Y as the respose varable ad X as the predctor varable. A smple lear regresso model wth the ormalty assumpto s Y (.) where Y s the respose varable s the predctor varable s the tercept parameter 5

26 s the slope parameter ad are..d. N ( ) for.... The ukow parameters are ad. Thus for fed values of the observed Y s are depedet N( ).... The jot probablty dstrbuto fucto of Y Y... Y s the gve by y... y f ( y... y ; ) L f ( ; y ) ep y. (.3) The log-lkelhood fucto s gve by log L y... y log log y. (.4) The lkelhood equatos for ad are obtaed by mamg the log-lkelhood equato gve (.4) wth respect to ad respectvely. 6

27 The lkelhood equatos are gve by: ad y (.5) y (.6) y. (.7) Solvg (.5)-(.7) we get the mamum lkelhood estmators of ad as : ad y y ˆ (.8) ˆ y (.9) ˆ ˆ y ˆ ˆ. (.) The ftted values of y s... are the yˆ ˆ ˆ. (.) The resduals are gve by yˆ.... y 7

28 .3 Skew Normal Dstrbuto ad Regresso We preset the skew ormal dstrbuto ad ts propertes detal Chapter. Here we troduce the skew ormal radom varable ad a outle of our research. A radom varable Y s sad to have a skew ormal dstrbuto wth locato parameter scale parameter ad skewess parameter f ts probablty desty fucto s gve by: y y f ( y; ) y (.) where (.) s the probablty desty fucto ad. fucto. We deote Y ~ SN( ). s the cumulatve dstrbuto We cosder a smple lear regresso setup wth Y as the respose varable ad X as the predctor varable. A smple lear regresso model wth the skew-ormalty assumpto s Y (.3) where Y s the respose varable 8

29 s the predctor varable s the tercept parameter s the slope parameter ad are..d. N ( ) for.... The ukow parameters are ad. Thus for fed values of.... the observed Y s are depedet SN ) ( We have four ukow parameters of terest amely ad. I ths thess we resolve the challeges ad the completes volved wth the mamum lkelhood estmato method for estmatg the parameters ad ad preset a ew method for solvg the mamum lkelhood estmatg equatos. 9

30 .4 Thess Descrpto I Chapter we preset the skew ormal dstrbuto ad ts propertes ts momets lkelhood fucto ad mamum lkelhood estmates ad the assocated Iformato matr. We also dscuss the challeges of the mamum lkelhood estmates ad preset a revew of the lterature o the challeges. I Chapter 3 we detfy the rato of ormal desty ad dstrbuto fuctos the presece of the shape parameter as the source of the complety estmatg the parameters ad propose a lear ad o-lear appromato of the rato. I Chapter 4 we preset a procedure for the estmato of the shape parameter of the skew ormal dstrbuto assumg that the locato ad the scale parameter are kow. We also evaluate the performace of the estmato procedure by a smulato study. I Chapter 5 we preset a procedure for the estmato of the locato scale ad the shape parameter of the skew ormal dstrbuto. Here we use oly the lear appromato of the rato for our estmato procedure. We evaluate the performace of the estmators by a smulato study.

31 Chapter The Uvarate Skew Normal Dstrbuto.

32 . The Uvarate Skew Normal Dstrbuto I ths Secto we preset the uvarate skew ormal dstrbuto troduced by Aal (985). We the dscuss several propertes ad momets of ths dstrbuto. Defto. A radom varable s sad to have a stadard ormal dstrbuto f ts probablty desty fucto (pdf) s gve by / ( ) e We descrbe a stadard ormal radom dstrbuto by ~ N( ).. (.) The cumulatve dstrbuto fucto (cdf) of a stadard ormal radom varable wll be deoted by. where ( ) udu. (.)

33 Defto. [Aal(985)] A radom varable X s sad to have a skew ormal dstrbuto f ts pdf s gve by f ( ) ( ) ( ) (.3) where a real umber s the skewess parameter ormal pdf ad cdf respectvely as gve (.) ad (.).. ad. are the stadard For brevty we wrte ~ SN( ). The cdf of a skew ormal dstrbuto s deoted by ; where f u ; du. (.4) Fgure. shows the shape of the pdf (.3) for three values of the skewess parameter amely ad. We observe that the shape of the pdf becomes creasgly skewed to the rght wth the crease the value of. Whe the shape becomes slghtly skewed to the rght ad whe the shape becomes close to the pdf of a half ormal radom varable. The rght-tals of these dstrbutos for the above three values of become vrtually dstgushable for the values of greater tha. 3

34 Fgure. The pdf of ~ SN( ) ad. We ow state the propertes P-P5 of ~ SN( ) : P. Whe ~ N( ) P. As f () teds to ( ) I whch s the half ormal pdf 4

35 P3. ~ SN( ) P4. ; ; P5. log ( ) f s a cocave fucto of ad hece the pdf f () s a umodal fucto of P6. ~.. Momets of the Uvarate Skew Normal Dstrbuto. I ths Secto we derve the momet geeratg fucto ad the momets of a skew ormal radom varable. Lemma. Let be a stadard ormal radom varable ad let h ad k be real umbers. The k E h k for all h ad k (.5) h where. s the stadard ormal dstrbuto fucto. 5

36 Proof. Let be a stadard ormal radom varable. For ay real h ad k we wrte E h k as h k E h k h k d. (.6) Dfferetatg (.6) wth respect to k we get: h k k h k d. ep h k d. k ( h ) hk ep ep d ( h ) h. Lettg hk u h h we have h k k h k u ep ep ( h ) du. h k. h 6

37 Now tegratg wth respect to k we have k h k. h Ths proves the lemma. Theorem. Whe ~ SN( ) the momet geeratg fucto of s t t M ( t) ep. (.7) Proof: M ep t ep t ( ) ( d ( t) E ) t ep ep ) t ( t dt t ep ep u u tdu t ep E u t where U ~ N( ). From Lemma we get t t M ( t) ep. 7

38 8 The frst momet of a skew ormal radom varable s gve by ep ep '() t t t t t t M E. The secod momet of a skew ormal radom varable s gve by ep ' ep ''() t t t t t M E ep t t t t. We get E (.8)

39 ad Var. (.9) I practce t s commo to work wth a locato ad scale trasformato Y where s a real umber ad. Hece the desty for the radom varable Y dstrbuted as SN ( ) s y y f ( y; ) y. (.) The epectato ad varace of Y are gve by E Y (.) ad Var Y. (.) 9

40 .3 Lkelhood fucto ad mamum lkelhood estmates. Let y y y be a depedet ad detcally dstrbuted sample from ) ( SN where ad are ukow ad ad ) ( are real umbers. The the lkelhood fucto s: y y L ) (. (.3) The log-lkelhood fucto s gve by log L l. log log log log y y (.4) We defe y y y W. (.5)

41 The lkelhood equatos for ad are obtaed by mamg the loglkelhood fucto gve (.4) wth respect to ad respectvely. The lkelhood equatos obtaed by takg partal dervatves of the lkelhood equatos wth respect to ad are: ) ( y W y l (.6) ) ( y W y y l (.7) ) ( y W y l. (.8) Let ˆ ˆ ad ˆ be the solutos for ad of the equatos (.6)-(.8). We wrte ˆ ˆ ˆ ˆ ˆ ˆ ˆ y y y W. (.9) The from (.6)-(.8) we get y W y ˆ ˆ ˆ ˆ (.)

42 y y W ˆ ˆ ˆ (.) y ˆ ˆ. (.) The Fsher Iformato Matr s gve by. l l l l l l l l l L E L E L E L E L E L E L E L E L E I We let p Y ad E a k k k=.

43 3 Now we derve the elemets of the Fsher-Iformato matr. Frst we have l l L E L E Y Y Y E E E d a d a 3 ep. Let u. Hece udu ) d (.

44 4 Thus we have u du ue a L E 3 l. It s kow that the frst momet of a stadard ormal radom varable s ero. Thus l a L E. Net we have l l L E L E l L E Y Y Y E E

45 5 E d p 3 a d d p 3 ep ep a d du u u p 3/ 3 ep ep a du u It s kow that the secod momet of a stadard ormal radom varable s oe. Thus 3/ 3 l a p p p L E

46 6 3/ a p. Net we have l l L E L E l L E Y Y Y E E d a d 3/ a p p

47 7 3/ a p. Net we have Y Y Y Y E L E l E 3 3 Usg the fact that the odd momets of a stadard ormal are ero ad the fact that the secod momet s oe we have l a L E.

48 8 Net we have l l L E L E l L E Y Y Y E E 3 Usg the fact that the odd momets of a stadard ormal are ero ad the fact that the secod momet s oe we have l a L E. Fally we have Y Y Y E L E l

49 9 E a. Hece the Fsher Iformato matr s gve by: 3 / 3 / 3 / 3 / a a a p a a a p a p a p a I. (.)

50 .4 Challeges of the Mamum Lkelhood Estmates of the Uvarate Skew Normal Dstrbuto. We ow dscuss the two ma problems that arse the mamum lkelhood estmato of the parameters of the skew ormal dstrbuto. Frstly the lkelhood fucto wth respect to mght be ubouded. Cosequetly the estmate of becomes fte though realty s fte. Whe the sample se s small ths stuato arses more frequetly. We epla ths stuato for SN ( ). Let be be a radom sample from ~ SN( ). From (.4) we ca wrte the log-lkelhood equato as l( ) l L log (.) 3

51 Whe are all postve l ( ) s a creasg fucto of. Hece the estmate of becomes ubouded. The frequecy of ubouded estmates of decreases as the sample se creases. For stace f 5 ad = we have the probablty.73 of havg all postve observatos. The probablty decreases to. for =. But for large values of the probablty of gettg a ubouded estmate of s stll qute hgh. Fgure. Probablty that < for values of ragg from to 3. 3

52 For ~ SN( ) we calculate P( < ) for ragg from to 3 ad preset them Fgure.. As creases the probablty for the observatos to be all postve a small sample s very hgh. The reverse s true whe s egatve that results a sample wth all egatve values. Secodly the Iformato matr becomes sgular whe. Ths sgularty ca be traced to the parameter redudacy of the parameterato for the ormal case a fact detfed usg the results of Catchpole & Morga (997). They detfy a epoetal famly model as beg parameter redudat f the mea ca be epressed usg a reduced umber of parameters. From equato (.3.7) E(Y) s a fucto of all three parameters ad whereas for t s just a fucto of oe parameter. 3

53 .5 Lterature Revew o Challeges. I ths secto we are revewg the dfferet procedures descrbed the lterature for the stuatos wth ubouded estmates of. I order to deal wth the problem of ubouded estmates of the SN model Aal ad Capattao (999) propose to stop the mamato procedure whe the log-lkelhood value s ot sgfcatly lower tha the mamum. Sartor (5) proposes to reduce the asymptotc bas of the mamum lkelhood estmate by meas of a pealato to the lkelhood fucto. He proposes a two step procedure for estmatg the parameters of SN Dstrbuto. Itally the estmators ˆ ad ˆ of ad are computed. The the secod step ˆ ad ˆ are fed ad the a bas prevetve method proposed by Frth (993) s appled to the score fucto of the skewess parameter order to gve a fte estmate. 33

54 Lseo ad Loperfdo () performs a default Bayesa aalyss for the skew ormal dstrbuto to show that the Jeffrey s pror to s proper. Aa Clara Mot (3) uses the mmum ch-square method proposed by Neyma (949) to estmate parameters of the dstrbuto of dscrete or grouped data. I the SN model the sgularty of the formato matr ca be removed by a certa parameterato of the parameters. (Aal985; Aal ad Captao 999; Choga 997; Pewsey ). 34

55 Chapter 3 Appromatos of the Rato of the Stadard Normal Desty ad Dstrbuto Fuctos 35

56 3. Itroducto. I ths chapter we troduce a lear ad a o-lear appromato of the rato of the stadard ormal desty ad dstrbuto fuctos the presece of a ukow costat represetg the shape of the skew ormal dstrbuto. The purpose of ths appromato s to estmate the skew ormal shape parameter. I (.5) we have defed the rato of the stadard ormal desty ad dstrbuto fuctos W(y) as W y) y y (. We wrte y ad W (y) as R () where R ( ). (3.) 36

57 The umercal value of R () s. Fgure 3. s showg the graphs of R () agast for. 5 ad. The graphs Fgure 3. tersect at. It s see that the slope of R () s postve whe takes egatve values ad egatve whe takes postve values. It s also otced that the magtude of the slope creases for bgger values of. Fgure 3.. Plots of R () agast for. 5 ad. 37

58 For a gve we wat to appromate R () by the followg lear ad olear fucto for 3 3 A ( ) (3.) B ( ) ep (3.3) where ad are ukow costats ad f f ;. These appromatos become weaker for the values of satsfyg Motvato. We cosder a radom varable Y satsfyg where s a real umber ad ad ~ SN( ). The radom varable Y s dstrbuted as SN ( ). Y (3.4) 38

59 Now we cosder two cases:. chages wth a covarate X ad we wrte (3.5) where s a gve value of the covarate X.. No such covarate X s avalable. Case I. Covarate X Preset The radom varable Y ths case s dstrbuted skew ormal wth desty y y f ( y; ) y. (3.6) We deote the dstrbuto wth the desty (3.6) as SN ). ( We ow cosder depedet observatos y from the skew ormal dstrbuto wth desty (3.6). We deote y... ad R ( ). 39

60 4 The Mamum Lkelhood Equatos ca the be epressed as R (3.7) R (3.8) R (3.9). (3.) Case II. Covarate X Abset Ths stuato s dscussed secto.. The mamum lkelhood equatos ca be epressed as: R (3.) R (3.). (3.3)

61 We observe that the complety the equatos (3.7)-(3.) ad (3.)-(3.3) s due to the presece of the comple fucto R (). We propose to deal wth ths complety by appromatg R () by A () ad B () as gve (3.) ad (3.3). We use equatos (3.7)-(3.) ad (3.)-(3.3) to estmate the parameters R () A () ad B () based o the values of Fttg the Lear Appromato to the rato. I Secto (3.) we have see that for a gve R () s appromated by A () as gve (3.). I ths secto we eame how well the lear fucto () appromates R (). I Fgure 3. we plot A () agast for. 5 ad. We cosder to take the followg values: ad. 569 for. 5 ad 4 A

62 respectvely. As Fgure 3. the graphs Fgure 3. tersect at = ad the slope of A () s postve whe takes egatve values ad egatve whe takes postve values. Also the magtude of the slope creases for bgger values of. Fgure 3.. Plots of A () agast for. 5 ad. To compare R () ad A () we plot R () (cotuous le) ad A () (dotted le) agast values for. 5 the same graph. We preset the plots Fgure 3.3. We draw smlar graphs for ad respectvely fgure 3.4 ad 3.5. We 4

63 cosder ad. 569 for. 5 ad respectvely. We observe that all the three graphs the lear fucto A () qute accurately appromates R () for the values cosdered. Fgure 3.3. Plot of R () agast for. 5 (cotuous les) ad A () agast for. 5 (dotted les). 43

64 Fgure 3.4. Plot of R () agast for (cotuous les) ad A () agast for (dotted les). 44

65 Fgure 3.5. Plot of R () agast for (cotuous les) ad A () agast for (dotted les). 45

66 3.4 Fttg the No-Lear Appromato to the rato. I Secto (3.) we have see that for a gve R () s appromated by B () as gve (3.3). I ths secto we eame how well the lear fucto () appromates R (). B I Fgure 3.6 we plot B () agast for. 5 ad. We cosder to take the followg values: ad. 8 for. 5 ad respectvely. As Fgure 3. the graphs Fgure 3.6 tersect at = ad the slope of B () s postve whe takes egatve values ad egatve whe takes postve values. Also the magtude of the slope creases for bgger values of. 46

67 Fgure 3.6. Plots of B () agast for. 5 ad. To compare R () ad B () we plot R () (cotuous le) ad () B (dotted le) agast values for. 5 the same graph. We preset the plots Fgure 3.7. We draw smlar graphs for ad respectvely fgure 3.8 ad 3.9. We cosder ad. 8 for. 5 ad respectvely. We observe that all the three graphs the o-lear fucto B () qute accurately appromates R () for the values cosdered. 47

68 Fgure 3.7. Plot of R () agast for. 5 (cotuous les) ad B () agast for. 5 (dotted les). 48

69 Fgure 3.8. Plot of R () agast for (cotuous les) ad B () agast for (dotted les). 49

70 Fgure 3.9. Plot of R () agast for (cotuous les) ad B () agast for (dotted les). 5

71 Chapter 4 Estmato of the Shape parameter of the Stadard Skew Normal Dstrbuto 5

72 4. Itroducto. I Chapter 3 we have dscussed that the complety solvg the mamum lkelhood equatos arse from the presece of the rato of the ormal desty ad dstrbuto fucto R () the lkelhood equatos. We propose to solve ths complety by appromatg R () frst by a lear fucto () A as gve (3.) ad the by a o-lear fucto B () as gve (3.3). We have see chapter 3 that whe we cosder a covarate X to be preset we have four parameters of terest amely ad ad whe we cosder o covarate to be preset we have three parameters of terest amely ad. I ths chapter we assume ad to be kow. The ukow parameters the are ad whe we cosder the lear appromato A () for () appromato B () for R (). R ad ad whe we cosder the o-lear Whe we cosder a covarate X to be preset we have a data set of pars of observatos ( y )... where s are observatos of the covarate X ad y s are the observed values of the depedet varable Y where Y SN( ). ~ Wth the kow values of we get the observed values of where y... from the observed values of y. Now we have pars of 5

73 observatos ). Whe we cosder o covarate to be preset we have ( observatos y... from radom varable Y where Y ~ SN( ). Assumg ad to be kow we trasform y... to where y. Based o these observatos we dscuss a estmato procedure for estmatg the parameters R () A () ad () B both whe a covarate X s preset ad whe t s abset. From the estmates of ad we obta the estmates of ad. We also preset the performace of our estmato procedure wth smulato results. 4. The estmato Procedure usg A () I ths chapter we assume that the parameters ad are kow. I ths secto we preset the estmato procedure for estmatg the parameters ad usg the lear appromato A () gve (3.) for R ( ) gve (3.) the presece ad absece of a co-varate X. I Secto 4.. we cosder the estmato procedure the presece of a co-varate X ad Secto 4.. we cosder the estmato procedure the absece of ay co-varate. I Secto 4..3 we dscuss about a measure of goodess of ft to observe how the fucto A () performs appromatg R (). 53

74 4.. Case I. Covarate X Preset We have pars of observatos ( y )... where... are observatos of the covarate X ad y... are the observed values of the depedet varable Y where Y ~ SN( ). From the y s we calculate the s where y.... Here we ote that though we assume that the parameters ad are kow we preset the estmatg equatos wth respect to ad so that we ca use them to estmate the ukow parameters ad. We assume R ( ) A ( ) from (3.) ad (3.). The the estmatg equatos wth respect to ad ca be wrtte as: (4.) (4.) (4.3) 54

75 55. (4.4) Smplfyg (4.)-(4.3) ad usg (4.4) we ca wrte them the followg form: (4.5) The equato (4.5) ca be epressed the geeral form W (4.6) where W ad. We assume ) ( W Rak ad the estmates of ad ca be epressed the geeral form (Rao(973)) ' ' ˆ W W W. (4.7) From (4.7) we get the estmates ˆ ad ˆ ˆ from where we ca calculate ˆ ad ˆ ˆ. We deote ) ( A at ˆ ˆ by ) ( ˆ A.

76 4.. Case II. Covarate X Abset Here we have observatos y... from the radom varable Y where Y ~ SN( ). From the y s we calculate the s where y.... Here we ote that though we assume that the parameters ad are kow we preset the estmatg equatos wth respect to ad so that we ca use them to estmate the ukow parameters ad. We assume R ( ) A ( ) from (3.) ad (3.). The the estmatg equatos wth respect to ad ca be wrtte as: (4.8) (4.9). (4.) Smplfyg (4.8)-(4.9) ad usg (4.) we ca wrte them the followg form: (4.) 56

77 The equato (4.) ca be epressed the geeral form W. (4.) where W ad. We assume Rak ( W) ad the estmates of ad ca be epressed the geeral form (Rao(973)) W ' W W' ˆ. (4.3) From (4.3) we get the estmates ˆ ad ˆ ˆ from where we ca calculate ˆ ad ˆ ˆ. We deote A () at ˆ ˆ by Aˆ ( ) Measure of goodess of ft. For evaluatg goodess of ft for appromatg R () gve (3.) by () (3.) for each value of... we defe A gve ) R ( ) Aˆ ( ). (4.4) A ( 57

78 For a partcular value of (4.6) gves us a measure of the absolute dfferece betwee the true value of the rato ) ad the estmated value of the lear appromato Aˆ ( ). R ( To get a measure of how close the estmated lear appromato s to the true rato we calculate Ave the average of A ( ) over all the values of s A.... Ave A s gve by Ave A A ( ). (4.5) A small value of Ave dcates a satsfactory appromato of R () by A (). A 4.3 The estmato Procedure usg B () I ths Secto we preset the estmato procedure for estmatg the parameters ad usg the lear appromato B () defed (3.3) for R () defed (3.) the presece ad absece of a co-varate X. I Secto 4.3. we cosder the estmato procedure the presece of a co-varate X ad Secto 4.3. we cosder the estmato procedure the absece of ay co-varate. I Secto we dscuss 58

79 about a measure of goodess of ft to observe how the fucto B () performs appromatg R (). Here we wll assume hece we ca wrte (3.3) as B ( ) ep. (4.6) 4.3. Case I. Covarate X Preset We have pars of observatos ( y )... where... are observatos of the covarate X ad y... are the observed values of the depedet varable Y where Y ~ SN( ). From the y s we calculate the s where y.... Here we ote that though we assume that the parameters ad are kow we preset the estmatg equatos wth respect to ad so that we ca use them to estmate the ukow parameters ad. 59

80 6 We assume ) ( ) ( B R from (3.) ad (4.6). The the estmatg equatos wth respect to ad ca be wrtte as: ep (4.7) ep (4.8) ep (4.9). (4.) Smplfyg (4.7)-(4.9) ad usg (4.) we ca wrte them the followg form: ep ep ep (4.) The equato (4.) ca be epressed the geeral form W (4.)

81 where W ep ep ep ad. We assume Rak ( W) ad the estmates of ad ca be epressed the geeral form (Rao(973)) W ' W W' ˆ. (4.3) From (4.3) we get the estmates ˆ ad ˆ ˆ from where we ca calculate ˆ ad ˆ ˆ. We deote B () at ˆ ˆ by Bˆ ( ) Case II. Covarate X Abset Here we have observatos y... from the radom varable Y where Y ~ SN( ). From the y s we calculate the s where y.... Here we ote that though we assume that the parameters ad are kow we 6

82 6 preset the estmatg equatos wth respect to ad so that we ca use them to estmate the ukow parameters ad. We assume ) ( ) ( B R from (3.) ad (4.6). The the estmatg equatos wth respect to ad ca be wrtte as: ep (4.4) ep (4.5). (4.6) Smplfyg (4.4)-(4.5) ad usg (4.6) we ca wrte them the followg form: ep ep (4.7) The equato (4.7) ca be epressed the geeral form W (4.8) where W ep ep ad.

83 We assume Rak ( W) ad the estmates of ad ca be epressed the geeral form (Rao(973)) W ' W W' ˆ. (4.9) From (4.8) we get the estmates ˆ ad ˆ ˆ from where we ca calculate ˆ ad ˆ ˆ. We deote B () at ˆ ˆ by Bˆ ( ) Measure of goodess of ft. For evaluatg goodess of ft for appromatg R () gve (3.) by () (4.6) for each value of... we defe B gve ) R ( ) Bˆ ( ). (4.3) B( For a partcular value of (4.3) gves us a measure of the absolute dfferece betwee the true value of the rato ) ad the estmated value of the lear appromato Bˆ ( ). R ( 63

84 To get a measure of how close the estmated lear appromato s to the true rato we calculate Ave the average of B ( ) over all the values of s B.... Ave B s gve by Ave B B ( ). (4.3) A small value of Ave dcates a satsfactory appromato of R () by B (). B 4.4 A smulated Data I order to evaluate the performace of the estmato procedure gve Secto 4. ad 4.3 we preset the estmated values of ad usg a smulated data. We cosder the case whe co-varate X s preset ad the case whe t s abset. Here we assume the parameters ad are kow. We geerate... from SN ( ) where ad keepg e decmal places. The rouded values at the secod decmal place are (Dataset 4.) We ow treat the value of to be ukow. The ukow parameters are the ad whe we cosder A () ad ad whe we cosder B (). 64

85 4.4. Case I. Covarate X preset The... (Eercse.5 Page 5 Medehall Beaver ad Beaver (9)) values are (Dataset 4.) We cosder 3 5 ad. 5. We geerate a radom varable Y from Dataset ad such that y.... We ca the wrte Y SN( X 3 5X.5 ). ~ The y values... obtaed are (Dataset 4.3) We recall here that the values of ad are kow whle s ukow. Hece the values are kow where y.... We ow work wth the dataset y ) trasformed to ).... ( ( 65

86 4.4. (A) Estmato usg lear appromato A (). We ow follow the procedure dscussed Secto 4.. where we appromated () R (3.) by A ( ) gve (3.) to estmate the parameters ad the presece of a covarate X. I the Equato (4.7) we have: 9.5 W ad We obta from (4.7) ad as: ˆ ˆ Hece the estmated lear fucto Aˆ ( ) s Aˆ ( ) The measure of goodess of ft s calculated to be Ave A We observe that the value of Ave A s qute small ad hece we ca coclude that our estmated lear fucto Aˆ ( ) has satsfactorly appromated R (). 66

87 Fgure 4. Plots of both R ( ) ad A ( ) ˆ agast whe covarate X s preset. I Fgure 4. we plot both R ( ) ad A ( ) ˆ agast. From ths fgure we observe that for the data cosdered the appromato A () for R ( ) s qute strog wth the rage of values betwee [-.5] ad moderately strog the rage [--.5]. 67

88 4.4. (B) Estmato usg o-lear appromato B (). We ow follow the procedure dscussed Secto 4.3. where we appromated () R (3.) by B ( ) gve (3.3) to estmate the parameters ad the presece of the covarate X. I the Equato (4.3) we have: 3.34 W ad We obta from (4.3) the estmates of ad as: ˆ ˆ Hece the estmated o-lear fucto Bˆ ( ) s Bˆ ( ) ep. The measure of goodess of ft s calculated to be Ave B We observe that the value of Ave B s qute small ad hece we ca coclude that our estmated lear fucto Bˆ ( ) has satsfactorly appromated R (). 68

89 I Fgure 4. we plot both R ( ) ad B ( ) ˆ agast. From ths fgure ad the data cosdered we fd that the stregth of appromato B ( ) for R ( ) s very strog wth the rage of values betwee [-] ad moderately strog the rage [--] ad []. ˆ Fgure 4. Plots of both R ( ) ad B ( ) agast whe covarate X s preset. ˆ 69

90 4.4. Case II. Covarate X abset We cosder 3 5 ad. 5. We geerate a radom varable Y from Dataset ad such that y.... We ca the wrte Y ~ SN( 3.5 ). The y values... obtaed are (Dataset 4.4) We recall here that the values of ad are kow whle s ukow. Hece the values are kow where y.... We ow work wth the dataset y trasformed to. 7

91 4.4. (A) Estmato usg lear appromato A (). We ow follow the procedure dscussed Secto 4.. where we appromated () R (3.) by A ( ) gve (3.) to estmate the parameters ad the absece of ay covarate. I the Equato (4.7) we have: W ad 9.5. We obta from (4.3) the estmates of ad as: ˆ.776 ˆ Hece the estmated lear fucto Aˆ ( ) s Aˆ ( ) The measure of goodess of ft s calculated to be Ave A We observe that the value of Ave A s qute small ad hece we ca coclude that our estmated lear fucto Aˆ ( ) has satsfactorly appromated R (). 7

92 Fgure 4.3 Plots of both R ( ) ad A ( ) ˆ agast whe covarate X s abset. I Fgure 4.3 we plot both R ( ) ad A ( ) ˆ agast. Ths fgure s very smlar to Fgure 4. ad we obsereve that for the data cosdered the appromato A () for R ( ) s qute strog wth the rage of values betwee [-.5] ad moderately strog the rage [--.5]. 7

93 4.4.(B) Estmato usg o-lear appromato B (). We ow follow the procedure dscussed Secto 4.3. where we appromated () R (3.) by B ( ) gve (4.6) to estmate the parameters ad the absece of ay covarate. I the Equato (4.3) we have: W ad We obta from (4.3) the estmates of ad as: ˆ ˆ Hece the estmated o-lear fucto Bˆ ( ) s Bˆ ( ) ep. The measure of goodess of ft s calculated to be Ave B We otce that the value of Ave B s qute small ad hece we ca coclude that our estmated lear fucto Bˆ ( ) has satsfactorly appromated R (). 73

94 I Fgure 4.4 we plot both R ( ) ad B ( ) ˆ agast. Ths fgure s very smlar to Fgure 4. ad we otce that the stregth of appromato B ( ) for R ( ) s very strog wth the rage of values betwee [-] ad moderately strog the rage [--] ad []. ˆ Fgure 4.4 Plots of both R ( ) ad B ( ) agast whe covarate X s abset. ˆ 74

95 4.5 Estmatg Bas ad Accuracy Appromatos usg smulatos We ow obta datasets by repeatg the smulato method descrbed the earler sectos tmes. We cosder the dfferet values of the parameters ad. However we preset here the outcomes oly for a few sets of the parameter values. For a set of fed values of the parameters we geerate data sets ad from each data set we obta the umercal values of ˆ ˆ ˆ or ˆ ˆ ad ˆ ˆ or ˆ ˆ for the appromato (3.) or (3.3) usg the equato (4.5) or (4.). We also calculate the umercal values ) or ) for twety values A ( each of datasets. We the calculate ther average B ( Ave A or Ave B over ther twety calculated values. Fally from datasets we get such Ave A values ad such Ave B values. We preset the Frst Quartle Q Meda Mea Thrd Quartle Q 3 ad Stadard Devato (SD) for the values of all the cases. I Tables we preset oly four out of may sets of parameter values that we have doe our calculatos. I Tables the umercal values of the dfferece betwee ad Meda ˆ could be cosdered as the estmates of bas ˆ as a estmate of (3.). The 75

96 umercal values of ( - Meda ˆ ) are -.89 Table Table Table 4.3 ad -.5 Table 4.4 whe we estmate appromatg (3.) by (3.). The umercal values of ( - Meda ˆ ) are.56 Table Table Table 4.7 ad -.37 Table 4.8 whe we estmate appromatg (3.) by (3.3). The suffcetly small umercal values of the estmated bases thus calculated from Tables dcate the absece of a alarmg bas the estmates of. We observe that the appromato (3.) provdes a small overestmate of. O the other had the appromato (3.3) provdes a small uderestmate of. We also get a smlar pcture whe we use ( - Mea ˆ ) for estmatg the bas ˆ for estmatg (3.). The meda values of Ave A are.857 Table Table Table 4.3 ad.336 Table 4.4. The meda values of Ave B are.347 Table Table Table 4.7 ad. 59 Table 4.8. The goodess of the two appromatg fuctos (3.) ad (3.3) dcated by the umercal values of Ave A ad Ave B are comparable to each other ad are suffcetly strog appromatg fuctos of (3.). We observe that o-lear appromato gve (3.3) does ot sgfcatly perform better tha the lear appromato gve (3.). Thus our et chapter we wll restrct ourselves oly to cosderg the lear appromato whch s much smpler ad qute satsfactory to develop a ew estmato procedure. 76

97 Table 4.. The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe ad. 5 appromatg (3.) by (3.) A Table 4.. The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe ad. 5 appromatg (3.) by (3.) A Table 4.3. The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe 3 5. ad. 5 appromatg (3.) by (3.) A 77

98 Table 4.4. The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe 3 5. ad. 5 appromatg (3.) by (3.) A Table 4.5. The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe ad. 5 appromatg (3.) by (3.3) B Table 4.6. The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe ad. 5 appromatg (3.) by (3.3) B 78

99 Table 4.7. The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe 3 5. ad. 5 appromatg (3.) by (3.3) B Table 4.8. The values of Q Meda Mea Q 3 ad SD for ˆ ˆ ˆ ˆ ˆ ad Ave whe 3 5. ad. 5 appromatg (3.) by (3.3) B 79

100 Chapter 5 Estmato of Locato Scale ad Shape parameters of a Skew Normal Dstrbuto 8

101 5. Itroducto. I chapter 4 we have dscussed the estmato procedure to estmate R () A () ad B () assumg the parameters ad to be kow cosderg a covarate X to be preset. I ths chapter we assume all the parameters.e. ad to be ukow. To estmate the ukow parameters we appromate R () (3.) by the lear fucto () addtoal ukow parameter. A (3.). Thus we have oe We cosder a data set of pars of observatos ( y )... where s are observatos of the covarate X ad y s are the observed values of the depedet varable Y where Y SN( ). We rewrte the estmatg equatos (3.7)- ~ (3.) usg A () R (). Based o the observatos y )... ( ad the estmatg equatos we dscuss some mportat relatoshps that est betwee the estmated parameter values. Usg these relatoshps we preset a estmato procedure for estmatg the ukow parameters. We also dscuss the performace of our estmato procedure wth smulato results. 8

102 8 5. Relatos Amog Estmated Parameters I ths secto we obta some terestg relatos amog the estmated parameter values. Usg these relatos we preset a estmato procedure for the ukow parameters. By substtutg ) ( ) ( A R the mamum lkelhood equatos gve (3.7)- (3.) we get: y y (5.) y y (5.) y y y y (5.3) ad y. (5.4)

103 We ote that Let ˆ ˆ ˆ ˆ ad ˆ be the solutos for ad the equatos (5.)- (5.4). We defe wˆ y ˆ ˆ. (5.5) From (5.)-(5.4) usg (5.5) we wrte ˆ ˆ wˆ ˆ ˆ (5.6) ˆ ˆ wˆ ˆ ˆ (5.7) w ˆ ˆ wˆ wˆ ˆ (5.8) ˆ ˆ. (5.9) From (5.8) ad (5.9) we get ˆ ˆ ˆ ˆ ˆ ˆ wˆ ˆ w. (5.) Multplyg both sdes of (5.6) by ˆ ad usg (5.) we have ˆ ( ˆ ˆ) ˆ ˆ ˆ ˆ w. (5.) From (5.6)-(5.) we obta Observatos - relatg the estmated parameter values. 83

104 Observato. If ˆ ˆ the the mamum lkelhood estmate of s gve by y y ˆ. (5.) ˆ ˆ Proof: Whe multplyg both sdes of (5.6) by ad subtractg t from (5.7) we obta y ˆ. (5.3) The (5.3) follows mmedately from (5.). Note: We ote that the estmate of gve (5.) for SN( ) s eactly the same as the estmate of for N( ). Observato : We have ˆ ( ˆ ˆ) (5.4) Proof: Combg (5.) ad (5.) we obta (5.4). 84

105 Observato 3: We have ( ˆ ˆ) (5.5) Proof: We kow w wˆ ˆ (5.6) The from (5.9) (5.) ad (5.6) the equalty (5.5) follows. Observato 4: We have ˆ (5.7) Proof: I (3.) ad hece ˆ. Combg (5.4) ad (5.5) we have ˆ. ˆ ˆ ˆ). From (5.4) ( ˆ. Hece Observato 5. We have ˆ ˆ ˆ ˆ ˆ y. (5.8)

106 Proof: From (5.5) we wrte wˆ y ˆ ˆ. (5.9) Dvdg both sdes of (5.9) throughout by we have w y ˆ ˆ. (5.) ˆ Now from (5.) ad (5.) we get (5.8). Observato 6: We observe ) ˆ f ad oly f w ˆ ) ˆ f ad oly f w ˆ. Proof: Sce ˆ the results () ad () are clear from (5.) ad (5.7). Observato 7: Whe we deote ˆ by ˆ ad t s gve by ˆ. (5.) y y ˆ Proof: Whe we kow that y N( ) =. ~ 86

107 From (.5)-(.7) we obta the mamum lkelhood estmates of ad : y y ˆ ˆ y ˆ Hece (5.) follows. a d ˆ ˆ y. ˆ Observato 8. We have ˆ ˆ (5.) ˆ Proof: From (5.9) (5.) ad (5.) we get ˆ ˆ ˆ ˆ ˆ. (5.3) Ths mples ˆ ˆ ˆ ˆ. (5.4) Usg (5.) we obta from (5.4) ˆ ˆ ˆ. (5.5) Hece the equalty of (5.) s true. 87

108 Observato 9. We have ˆ ˆ wˆ. (5.6) Proof: We wrte for... y ˆ ˆ y y ˆ y ˆ. ˆ By squarg both sdes ad addg over... we get y ˆ ˆ y y ˆ y ˆ ˆ. (5.7) Hece (5.6) s true. Observato. We have ˆ ˆ ˆ ˆ ˆ ˆ w. (5.8) Proof: The proof s clear from (5.) ad (5.3). 88

109 89 Observato.We have ˆ Proof: Let y y y be a radom sample from ). ( X SN That s y = where s a radom sample from ) ( SN. Hece y. Now y y ˆ.

110 9 Aga y y ˆ ˆ. From (5.9) we have ˆ ˆ

111 5.3 The estmato Procedure usg A () Here we recall that we have fve ukow parameters ad ad we have oly four estmatg equatos gve (5.7)-(5.). Hece we observe that we do ot have eough estmatg equatos to solve for the estmates uquely. To address ths ssue we troduce a ew varable b ad epress the estmates of ad terms of b. Let us wrte ( ˆ ˆ) b. (5.6) From Observato ad Observato 4 we otce that b. (5.7) From Observato ad (5.6) we have b ˆ. (5.8) From (5.) ad (5.6) we have ˆ ˆ (5.9) b.. where ˆ y y ˆ 9