THREE-PARAMETRIC LOGNORMAL DISTRIBUTION AND ESTIMATING ITS PARAMETERS USING THE METHOD OF L-MOMENTS

Transcription

1 RELIK ; Paha 5. a 6.. THREE-PARAMETRIC LOGNORMAL DISTRIBUTION AND ESTIMATING ITS PARAMETERS USING THE METHOD OF L-MOMENTS Daa Bílová Abstact Commo statstcal methodology fo descpto of the statstcal samples s based o usg covetoal momets o cummulats. A alteatve appoach s based o usg dffeet chaactestcs whch ae called the L-momets. The L-momets ae a aalogy to the covetoal momets but they ae based o lea combatos of the ode statstcs.e. L- statstcs. Paamete estmatos usg the L-momets ae especally the case of small samples ofte moe pecse tha estmates calculated usg the mamum lelhood method. Ths tet coces wth the applcato of the L-momets the case of lage samples ad wth the compaso of the pecso of the method of L-momets wth the pecso of othe methods momet quatle ad mamum lelhood method) of paamete estmatmato the case of lage samples. We used two types of data amely data sets of dvdual data ad data odeed to the fom of teval fequecy dstbuto. Thee-paametc logomal dstbuto was used as the theoetcal model. Key wods Logomal dstbuto L-momets L-momet method of paamete estmato come dstbuto wage dstbuto JEL Code C C6 Itoducto Fom the statstcal lteatue t s well-ow the use of L-momets coecto wth the data fom the feld of hydology ad meteoology fo eample afall). I such cases thee ae geeally elatvely small data sets. Ths pape deals wth the use of L-momets the case of lage data sets. Thee ae the data of two types amely dvdual data o et aual household come pe capta CZK yeas ad statstcal suvey Mcocesus ad yeas ad 8 statstcal suvey SILC) ad secod data soted to a fom of teval fequecy dstbuto these data efe to goss mothly wage CZK fom offcal web page of the Czech Statstcal Offce). I both cases we

2 RELIK ; Paha 5. a 6.. compae the accuacy of the method of L-momets wth a accuacy of othe methods of paamete estmato. Icome data come fom the statstcal suveys Mcocesus ad SILC of the Czech Statstcal Offce whle the wage data come fom offcal webste of the Czech Statstcal Offce. Thee-paametc logomal dstbuto was used as the basc paametc dstbuto. Accuacy of the method of L-momets was compaed wth the accuacy of othe methods of paamete estmato such as momet method quatle method mamum lelhood method. The questo of come dstbuto s statstcal lteatue athe etesvely teated see fo eample Batošová 6) o Bílová 8). The methods of estmato paametes cludg the thee-paametc logomal dstbuto ae descbed the statstcal lteatue see fo eample Bílová 8). Thee-paametc logomal dstbuto s dscussed detal fo eample Batošová Bía 9) o Bílová 8) momet method of paamete estmato Bílová 8) quatle method Bílová 8) o Spová Sodomová 9) mamum lelhood method Bílová 8). Fo eample pape Msolcz Laghamová ) s based te ala o the owledge of come dstbuto. The cocept of L-momets ad the use of these quattes the estmato of paametes of pobablty dstbuto ae gve Bílová ) Hosg 99) o Hosg Wales 997). Methodology The essece of logomal dstbuto s teated detal fo eample Atchso ad Bow 957). Use of logomal dstbuto coecto wth wage o come dstbutos s descbed Batošová 6) o Bílová 8). The methods of paamete estmato momet method quatle method mamum lelhood method) ae descbed detal fo eample Bílová 8).. L-Momet Method of Paamete Estmato Questo of L-momet s descbed detal fo eample Hosg ad Wales 997). We wll assume that X s a eal adom vaable wth the dstbuto fucto F) ad quatle fucto F) ad X X X ae the ode statstcs of the adom sample of the sze selected fom the dstbuto X. The the -th L-momet of the adom vaable X s defed as

3 RELIK ; Paha 5. a 6.. λ = ) =... EX ) = The lette L the ame L-momets s to stess the fact that -th L-momet λ s a lea fucto of the epected ode statstcs. Natual estmate of the L-momet λ based o the obseved sample s futhemoe a lea combato of the odeed values.e. the so called L-statstcs. The epected value of the ode statstc s of the fom EX! [ )] F [ F )] = d F ). )! )! ) The fst fou L-momets ae of the fom λ = EX F) d F ) λ = E X X ) = F) F ) d F 4) λ = E X X + X ) = F) 6 F 6 F + ) d F 5) ) ) λ4 = E X 44 X 4 + X 4 X 4 = F F F + F ) d F. 4 6) Detals about the L-momets ca be foud Guttma 99) o Hosg 99). We get the fst thee L-momets of the thee-paametc logomal dstbuto LNµ σ ξ) whch s descbed e.g. Hosg 99). The followg elatos ae vald fo these L- momets ep σ λ = ξ + µ + 7)

4 RELIK ; Paha 5. a 6.. ef ep σ σ µ + = λ 8) d ) ep ef ef 6 σ π = τ σ 9) whee efz) s the so called eo fucto defed as. d ) ef t e z z t π = ) Now we wll assume that s a adom sample ad s the odeed sample. The -th sample L-momet s defed as.... ) l = = = ) We ca wte specfcally fo the fst fou sample L-momets = l ) ) l = ) ) l + = 4) ) l l l + = 5) Let us deote the dstbuto fucto of the stadad omal dstbuto as Φ the Φ epesets the quatle fucto of the stadad omal dstbuto. The followg elato holds fo the dstbuto fucto of the thee-paametc logomal dstbuto LNµ σ ξ)

5 RELIK ; Paha 5. a 6.. F l ξ) µ = Φ. σ 6) The estmates of the thee-paametc logomal dstbuto ca the be calculated as 8 + t z = Φ 7) + σˆ 999 8z 6 8 z 7 z 8) 5 µ ˆ = l l σˆ 9) σˆ ef ˆ θ = epµ + σˆ l ˆ. ) Moe o L-momets s fo eample Kyselý ad Pce 7).. Appopateess of the Model I assessg the appopateess of the costucted model we eed to use ay of the cteos whch may be fo eample the sum of all absolute devatos of the obseved ad theoetcal fequeces S evetually ow cteo χ. The questo of the appopateess of the cuve as a model of the come o wage dstbuto these lage sample szes such ae the case of the come ad wage dstbutos ecouteed s eplaed fo eample Bílová 8). Gaph epesetg the developmet of the sample meda ad of the meda of a theoetcal dstbuto usg the cocete method of paamete estmato may bg some sght tems of accuacy of the method of paamete estmato too. Aalyss ad Results Tabs. to peset the estmated paametes of thee-paametc logomal cuves usg two vaous methods of pot estmato of paametes method of L-momets ad mamum lelhood method) ad the sample L-momets o the bass of them the paametes wee estmated. We ca see fom Tab. that the value of the paamete θ theoetcal begg

6 RELIK ; Paha 5. a 6.. Tab. Sample L-momets ad paamete estmatos of thee-paametc logomal dstbuto obtaed usg the L-momet method Icome Sample L-momets Paamete estmato Yea l l l µ σ θ Souce Ow eseach Tab. Sample L-momets ad paamete estmatos of thee-paametc logomal dstbuto obtaed usg the L-momet method Wage Sample L-momets Paamete estmato Yea l l l µ σ θ Souce Ow eseach Tab. Paamete estmatos of thee-paametc logomal dstbuto obtaed usg the mamum lelhood method Icome Wage Yea µ σ θ Yea µ σ θ Souce Ow eseach

7 RELIK ; Paha 5. a 6.. Tab. 4 Sum of absolute devatos of the obseved ad theoetcal fequeces fo all tevals et aual household come pe capta Method Yea L-momet Momet Quatle Mamum lelhood Souce Ow eseach Tab. 5 Sum of absolute devatos of the obseved ad theoetcal fequeces fo all tevals goss mothly wage Method Yea L-momet Momet Quatle Mamum lelhood Souce Ow eseach of the dstbuto) s some cases egatve. Ths meas that logomal cuve gets to egatve tetoy at the begg of ts couse. Because of a vey tght cotact of the lowe tal of the logomal cuve wth the hozotal aes ths fact does ot have to be a poblemfo a good ft of the model. The advatage of the logomal models s that the paametes have a easy tepetato. Also some paametc fuctos of these models have staght tepetato. Because the estmated value of ths paametes s egatve we ca ot eally tepet ths value.

8 RELIK ; Paha 5. a 6.. Fg. Developmet of sample aveage et aual come pe capta ad the theoetcal epected value CZK) Fg. Developmet of sample meda of et aual come pe capta ad the theoetcal meda CZK) Athmetc mea CZK) L-momet Momet 4 Quatle Mamum lelhood Sample Souce Ow eseach Yea Fg. Developmet of sample aveage goss mothly wage ad the theoetcal epected value CZK) Meda CZK) L-momet Momet 4 Quatle Mamum lelhood Sample Souce Ow eseach Yea Fg. 4 Developmet of sample meda of goss mothly wage ad the theoetcal meda CZK) Athmetc mea CZK) L-momet Momet Quatle Mamum lelhood Sample Yea Meda CZK) 5 5 L-momet Momet Quatle 5 Mamum lelhood Sample Yea Souce Ow eseach Souce Ow eseach

9 RELIK ; Paha 5. a 6.. Cocluso Tabs. 4 ad 5 povde moe accuate fomato about the used methods of paamete estmato. These tables cota the sum of absolute devatos of the obseved ad theoetcal fequeces fo all tevals ad theefoe they seve as a obectve cteo fo evaluatg the accuacy of used methods of paamete estmato. It should be oted hee that the case of come dstbuto o the oe had ad the case of wage dstbuto o the othe had we used the same umbe of tevals whose wdth s epaded tme due to the sg level of the dstbutos. As ca be see fom Tabs. 4 ad 5 the method of L-momets povdes the most accuate esults whch ae eve moe accuate tha esults obtaed usg the mamum lelhood method. Aleady metoed mamum lelhood method eded tems of accuacy of the estmatos as the secod best. Quatle method of paamete estmato follows as the thd best secod wost). As epected momet method of paamete estmato povdes the least accuate esults. Fgues to 4 also povde appomate fomato about the accuacy of the used methods of paamete estmato. Fgues ad epeset the developmet of the sample athmetc mea ad the developmet of theoetcal epected values of thee-paametc logomal dstbuto wth paametes estmated usg dffeet methods of paamete estmato. Fgues ad 4 epeset the developmet of the sample meda ad the developmet of theoetcal medas of thee-paametc logomal dstbuto wth paametes estmated usg dffeet methods of paamete estmato. Acowledgmet The pape was suppoted by gat poect IGS 4/ called Aalyss of the Developmet of Icome Dstbuto the Czech Republc sce 99 to the Facal Css ad Compaso of Ths Developmet wth the Developmet of the Icome Dstbuto Tmes of Facal Css Accodg to Socologcal Goups Gede Age Educato Pofesso Feld ad Rego fom the Uvesty of Ecoomcs Pague. Refeeces Atchso J. Bow J.A.C. 957). The Logomal Dstbuto wth Specal Refeece to Its Uses Ecoomcs. Cambdge Uvesty Pess Cambdge. Batošová J. 6). Logathmc-Nomal Model of Icome Dstbuto the Czech Republc. Austa Joual of Statstcs Vol. 5 Iss. pp. 5. ISSN

10 RELIK ; Paha 5. a 6.. Batošová J. Bía V. 9) Modellg of Icome Dstbuto of Czech Households Yeas Acta Oecoomca Pagesa. Vol. 7 No. 4 pp. 8. ISSN Bílová D. 8). Applcato of Logomal Cuves Modelg of Wage Dstbutos. Joual of Appled Mathematcs Vol. Iss. pp ISSN Bílová D. ). Use of the L-Momets Method Modelg the Wage Dstbuto. Batslava Febuay 4 Poceedgs. I th Iteatoal Cofeece APLIMAT Batslava Slova Uvesty of Techology Batslava pages ISBN Guttma N. B. 99). The Use of L-momets the Detemato of Regoal Pecptato Clmates. Joual of Clmate Hosg J. R. M. 99). L-momets Aalyss ad Estmato of Dstbutos Usg Lea Combatos of Ode Statstcs. Joual of the Royal Statstcal Socety Sees B) Vol. 5 No. pp ISSN Hosg J. R. M. Wales J. R. 997). Regoal fequecy aalyss A Appoach Based o L-momets. st ed. New Yo Cambdge Uvesty Pess 9 p. ISBN Kyselý J. Pce J. 7). Regoal Gowth Cuves ad Impoved desg Value Estmates of Etéme Pecptato Evets the Czech Republc. Clmate Reseach Vol. pp ISSN Msolcz M. Laghamová J. ). Labo maet ad smultaeous equatos solved by TSLS. Batslava Febuay 4 Poceedgs. I th Iteatoal Cofeece APLIMAT Batslava Slova Uvesty of Techology Batslava pages ISBN Spová L`. Sodomová E. 9). Icome dstbuto model the Slova Republc used the household SILC data. Wspólczese poblemy modelowaa pogozowaa zaws spoleczo-gospodaczych. Uwesytet Eoomczego w Kaowe pp Cotact Daa Bílová Uvesty of Ecoomcs Pague Faculty of Ifomatcs ad Statstcs Depatmet of Statstcs ad Pobablty blova@vse.cz