Joural of Moder Appled Statstcal Methods Volume Issue Artcle 8 --03 Comparso of Parameters of Logormal Dstrbuto Based O the Classcal ad Posteror Estmates Raja Sulta Uversty of Kashmr, Sragar, Ida, hamzasulta8@yahoo.com S. P. Ahmad Uversty of Kashmr, Sragar, Ida, sprvz@yahoo.com Follow ths ad addtoal works at: http://dgtalcommos.waye.edu/jmasm Part of the Appled Statstcs Commos, Socal ad Behavoral Sceces Commos, ad the Statstcal Theory Commos Recommeded Ctato Sulta, Raja ad Ahmad, S. P. (03) "Comparso of Parameters of Logormal Dstrbuto Based O the Classcal ad Posteror Estmates," Joural of Moder Appled Statstcal Methods: Vol. : Iss., Artcle 8. DOI: 0.37/jmasm/3837940 Avalable at: http://dgtalcommos.waye.edu/jmasm/vol/ss/8 Ths Regular Artcle s brought to you for free ad ope access by the Ope Access Jourals at DgtalCommos@WayeState. It has bee accepted for cluso Joural of Moder Appled Statstcal Methods by a authorzed edtor of DgtalCommos@WayeState.
Joural of Moder Appled Statstcal Methods November 03, Vol., No., 304-33. Copyrght 03 JMASM, Ic. ISSN 538 947 Comparso of Parameters of Logormal Dstrbuto Based O the Classcal & Posteror Estmates Raja Sulta Uversty of Kashmr Sragar, Ida S. P. Ahmad Uversty of Kashmr Sragar, Ida Logormal dstrbuto s wdely used scetfc feld, such as agrcultural, etomologcal, bology etc. If a varable ca be thought as the multplcatve product of some postve depedet radom varables, the t could be modelled as logormal. I ths study, mamum lkelhood estmates ad posteror estmates of the parameters of logormal dstrbuto are obtaed ad usg these estmates we calculate the pot estmates of mea ad varace for makg comparsos. Keywords: Logormal dstrbuto, mamum lkelhood estmato, posteror estmates & R software Itroducto Atchso & Brow (957) have gve a very comprehesve treatmet of logormal dstrbuto. The logormal dstrbuto arses varous dfferet cotets such as physcs (dstrbuto of partcles due to pulversato); ecoomcs (come dstrbutos); bology (growth of orgasms), etc. Epste (947), Browlee (949), Delaporte (950), Moroey (95) descrbes varous applcatos of logormal dstrbuto to physcal ad dustral processes, tetle research ad qualty cotrol. I the cotet of lfe testg ad relablty problems, the logormal dstrbuto aswers a crtcsm sometmes rased agast the use of ormal dstrbuto (ragg from - to +) as a model for the falure tme dstrbuto whch must rage from 0 to. A radom varable X s sad to have a logormal dstrbuto f U log e X has ormal dstrbuto wth mea µ ad varace. Thus, the pdf of logormal dstrbuto s gve by Dr. Sulta s a research scholar the Departmet of Statstcs. Emal hm at: hamzasulta8@yahoo.com. Dr. Ahmad s a Assstat Professor of the Departmet of Statstcs. Emal hm at: sprvz@yahoo.com. 304
SULTAN & AHMAD f ( ) ep( (log ) ),, 0, 0 µ µ π < < > < () The lkelhood fucto of the radom sample (,, 3,..., ) T would be L( µ, ) ep( (log µ ) ) π () The mea ad varace of the logormal dstrbuto are gve by ad EX α ep µ + (3) V( X) ep µ + ep( ) (4) Mamum Lkelhood Estmators Mamum Lkelhood s a popular estmato techque for may dstrbutos because t pcks the values of the dstrbuto's parameters that make the data more lkely" tha ay other values of the parameters would make them. Ths s accomplshed by mamzg the lkelhood fucto of the parameters gve the data. Cosder the estmato of the parameters α ad. Let U log,,,,. The usg the fact that (U, U,..., U ) s a radom sample from Normal dstrbuto wth parameters (µ, ). The mle of µ ad frst are gve by ˆ µ U U (5) ad 305
COMPARISON OF PARAMETERS OF LOGNORMAL DISTRIBUTION ˆ (6) U U The mle of α ad are gve by ˆ ˆ α ˆ ep µ + (7) ad ˆ ep ˆ µ + ˆ ep ˆ (8) Posteror estmato of the parameter Aga, cosder the estmato of the parameters α ad. Frst obta the posteror estmates of µ ad ad the smultaeously the posteror estmates for α ad wll be obtaed. Laplace (774) foud that t worked eceptoally well to smply always choose the pror probablty dstrbuto for the parameter(s) of the model to be costat o the parameter space. The jot pror pdf for µ ad cosdered s P µ, (9) by Accordg to Bayes theorem, Jot posteror desty of µ ad Posteror desty pror desty* lkelhood (, ) P,. P, π µ µ µ s gve From equato () ad (9) the jot posteror desty of µ ad s gve by (, ) ep ( log ) π µ µ π 306
SULTAN & AHMAD log c π ( µ, ) ep ep µ ( ) (0) where ( log ) log ad c s a ormalzg costat. Ldley (96) eplaed f P ( θ ) be the pror ad P( θ ) posteror pdf P( θ ) s gve by P ( θ ) cp ( θ). P ( θ) be the lkelhood, the, where c s the ormalzg costat. The the value of c s obtaed by c P. P( ) θ θ dθ Therefore, c ca be obtaed by (, ) c π µ dµ d 0 log c d d o ep ep µ µ / Usg the trasformato t µ log 307
COMPARISON OF PARAMETERS OF LOGNORMAL DISTRIBUTION c ep π d / o ( ) c 3 Γ π 3 c ( ) 3 3 π 3 Γ () From the equato (0) 3 log π ( µ, ) ep ep µ 3 π 3 / Γ ( ) () Margal posteror destes of µ ad The margal desty of µ s obtaed by tegratg out from () ad s gve as ( ) (, ) π µ π µ d 0 308
SULTAN & AHMAD 309 / 0 log ep c d π µ µ + log c π µ µ Γ + log 3, B π µ µ + (3) The margal desty of s obtaed by tegratg the jot posteror desty of µ ad gve () over the rage of µ. It s gve as / log ep ep c d π µ µ / ep log ep c d π µ µ
COMPARISON OF PARAMETERS OF LOGNORMAL DISTRIBUTION ( ) π ep π c / ( ) ( ) π 3 ep 3 / 3 Γ ( ) (4) Posteror estmates of µ ad The margal desty of µ s gve (3) s a studet s t pdf. Thus the posteror estmates of µ s gve as ( ) * µ dµ µ E µ 3 B, log µ + log Usg the trasformato t µ 3 log * dt µ 3 3 B, t + 3 30
SULTAN & AHMAD * µ log (5) whch s the posteror estmate for µ uder uform pror. Now the posteror estmate of ca be obtaed from equato (4) as 3 ep * d 0 3 3 ( ) Γ ep 3 * d 3 0 3 3 Γ ( ) 5 * (6) Thus, the posteror estmates of α ad are gve by * log * * α ep µ + ep + ( 5) (7) ad * * * * ep µ + ep 3
COMPARISON OF PARAMETERS OF LOGNORMAL DISTRIBUTION log ep + ep 5 5 * (8) Smulato study ad dscusso The estmates of the mea ad varace usg MLE ad Bayesa estmato was obtaed above. Net to obta s the umercal relatoshp of pot estmates usg true value of the parameters, MLE ad Bayesa estmato. I ths study, samples of 0, 0, 30, 40 ad 50 observatos were geerated from logormal pdf wth parameters µ ad. The smulatos were doe R Software. The mea ad varace were calculated to compare the methods of estmato. The results are preseted Table. I Table, whe pot estmates of logormal dstrbuto are compared usg true values of parameters wth MLE ad Bayesa estmato (by usg uform pror), the best estmator s the Mamum Lkelhood (MLE) because t has the mmum varace. Table. Pot estmates of logormal dstrbuto compared usg true values of parameters wth MLE ad Bayesa estmato True values MLE Posteror estmates Mea Varace Mea Varace Mea Varace ˆ ) ( α ) ( ) ( α ) ( ) ( ˆα ) ( 0 9.9004.336 9.5 6.358 0.695 7.7 9.8447 30.857 30.85 55.0.6655 5.804 0.593 70.663 40.645 56.937 0.974 7.6757 50 0.4039 67.46.5356 356.339 * * Refereces Atchso, J., & Brow, J. A. C. (957). The logormal dstrbuto: Wth specal referece to ts uses ecoomcs. Cambrdge: Uversty Press. 3
SULTAN & AHMAD Browlee, K. A. (949). Idustral epermetato. d ed. Brookly: Chemcal Pub. Co. Delaporte, P. (950). Etude statstque sur les propretes des fotes. Revue De L'sttut Iteratoal De Statstque / Revew Of The Iteratoal Statstcal Isttute, (3/4), 6. do:0.307/40035 Epste. B. (947). The mathematcal descrpto of certa breakage mechasm leadg to the logarthmc-ormal dstrbuto. Joural of Frakl Isttute, 44, 47-477. Laplace, P. S. (774). Mémore sur la probablté des causes par les évéemes. Académe royale des sceces (Frace), 6, 6-656. Ldley, D. V. (96). Itroducto to probablty ad statstcs from a Bayesa vewpot: Part, Iferece, Aberystwyth: Uversty College of Wales. Moroey, M. J. (95). Facts from fgures. Baltmore, MD: Pegu Books. 33