COMPARISON OF ESTIMATORS OF PARAMETERS FOR THE RAYLEIGH DISTRIBUTION

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1 COMPARISON OF ESTIMATORS OF PARAMETERS FOR THE RAYLEIGH DISTRIBUTION Eldesoky E. Affy. Faculy of Eg. Shbee El kom Meoufa Uv. Key word : Raylegh dsrbuo, leas squares mehod, relave leas squares, leas absolue devaos, rdge regresso, robus rdge regresso, mome esmaors, oal devaos, mea squared errors, correlao coeffce. Absrac: Ths paper s cocered wh esmao of parameers of he wo parameer Raylegh dsrbuo. We use he leas squares mehod (LS, relave leas squares mehod (RELS., leas absolue devao (, rdge regresso (RR ad robus rdge regresso (RRR, mome (ME ad modfed mome esmaors (MME for esmag he wo parameer Raylegh dsrbuo. To compare bewee dffere mehods of esmao, we use he oal devaos (TD, mea squared error (MSE ad probably plo correlao coeffce (R. Numercal examples are worked ou. Iroduco Raylegh dsrbuo, whch s a specal case of Webull dsrbuo has wde applcao, such as, lfe esg, Palovko (978, clcal sudes dealg wh cacer paes, Gross ad Clark (975 ad Lee (980. Aryawasa ad Templeo (984 have also dscussed some of s applcaos. Lalha ad Aad (996 used he modfed maxmum lkelhood o esmae he scale parameer of he Raylegh dsrbuo Cohe ad Whe (98b used he mome ad modfed mome esmaors for he Webull dsrbuo, Sama ad Mohamed (993 used he mome ad modfed mome esmaors for he Pareo dsrbuo. Meas ad Ilopoulos (003 proposed a class of goodess of f ess for he Raylegh dsrbuo. The ess are based o a weghed egral volvg he emprcal --

2 Laplace rasform. The probably dsrbuo fuco of Raylegh dsrbuo of T s gve by ( γ f ( ; γ, δ = exp[ ( γ /δ ] γ < <, δ δ > 0 ( I hs paper, we use he leas squares mehod, relave leas squares, leas absolue devaos, rdge regresso ad robus rdge regresso, mome ad modfed mome esmaors o esmae he wo parameer of he Raylegh dsrbuo. Al Fawza (000 used he oal devao (TD, mea squared errors (MSE of he resduals, Abd Karm ad Chowdhury (995 ad Probably plo correlao coeffce (R are used o compare bewee hese mehods. Leas squares Mehod (LSM: The cumulave dsrbuo fuco of ( s gve by. = exp[ ( exp[ ( ( γ γ = δ = γ + δ -γ /δ ] γ / δ = log[ ] [ log[ ] { log[ ]} The las equao ca be represeed by / δ ] = Y =a +bx.( Where Y =, a = γ, b = δ, ad s he sample sze. Le S ( a, b = ( Y a bx = Dffereag S w.r.. a ad b he equae o zero, we oba he followg wo ormal equaos { log[ ]}, =, X =..., Y = a + b X = = X Y = ax + b = = = X --

3 Solvg he las wo equaos for a ad b, we oba he leas square esmaes (LSE of a ad b as : aˆ = ( log( ( ( log( ( log( ( log( ( log( (3 ( ( ( ˆ ( log( ( log( b = (4 log( log( All summaos are from o. Relave Leas squares Mehod (RELS: The relave leas squares esmaors of a ad b ca be obaed by mmzg The sum squares of he relave resduals, Pablo ad Bruce(99, w.r.. a ad b as followg Y S = ( = = a bx Y = ( aw bz (5 we ge w z z w w z z = = = z = = = = aˆ = (6 ( w w z w w w z z = = = z = = = = bˆ = (7 ( w where w = /, z [ log( ] = / 3 Rdge regresso mehod, (RR: The rdge regresso esmaors roduced by Roald ad Raymod (978 s gve -3-

4 by ˆ B Rd / / = ( X X + ki X Y (8 where 0< k< s he rdge coeffce, I s he pxp dey marx ad p s he umber of parameers. 4 Leas absolue devaos mehod, (: Wager (959 defed he leas absolue devaos ( esmaors, B as he soluo o he problem M Y X = / B The problem ca be se up as a lear programmg problem as follows: m mze Subjec o = ( + d + d / + Y X B + d d = 0, =,,, d, 0 + d where + d d ad are respecvely, he posve ad egave resduals assocaed wh -h observao. 5 Robus rdge regresso esmaors, (RRR: Ths esmaor was suggesed by Pfaffeberger ad Delma (984,985,990. Ths esmaor combes properes of he esmaors ad he rdge regresso esmaors ad wll be referred o as he R esmaors. The R esmaor ca be wre as Bˆ / * / R = ( X X + k I X Y The value of k * s deermed from he daa usg k * = ˆ / B ˆ B ps where S = = e p -4-

5 Bˆ s he esmaes of B ad e s he resduals from mehod. Cohe ad Whe(98b ad Sama ad Mohamed(993 defed he followg mome (ME ad frs modfed mome (MME-I ad secod modfed mome (MME-II esmaors. 6 Mome Esmaors (ME: The frs wo momes of Raylegh dsrbuo are gve by µ = δ π / + γ µ / = δ + πδγ + γ (0 (9 The varace of x s gve by / var( x = µ µ = δ ( π / ( By equag he sample mome m ad sample varace S o he rgh had sde of (9 ad (, we ge m = δ π / + γ ( S = δ ( π / (3 From (3 he esmae of δ s gve by ˆ δ = S /(4 π (4 By subsue abou δˆ o ( he esmae of γ s gve by 7 Frs Modfed Mome Esmaors (MME-I: Equao ( wll be used ad equao ( wll be replaced by E ( ( = (, ha s δ π / +γ = ( (5 Where ( s he frs order sasc, soluo of ( ad (5 gves he esmae of δ ˆ δ = ( m ( /[ π / - π/ ] (6 Subsue abou δˆ from (6 o ( he esmae of γ wll be -5-

6 ˆ γ = m ˆ δ π / 8 Secod Modfed Mome Esmaors (MME-II: We use equao (, whle equao ( wll replaced by E [ ( = ( /( + = exp[ ( ( ( = δ [log (( + / ] -γ /δ ] + γ or (7 The soluo of ( ad (7 gvg he esmae of δ as m ˆ ( δ = π + log (8 from ( ad (8 we ca oba he esmae of γ as ˆ γ = m ˆ δ π / Goodess of F Aalyss: a Toal devaos (TD: Some mehods of goodess of f aalyss were employed here, hey are oal devaos for each mehod as follows ˆ γ γ ˆ δ δ TD = + γ δ Where γ ad δ are he rue values ad γˆ ad δˆ are he esmaed parameers by ay mehod, he bes mehod whch yelds he mmum oal devao. b Mea square errors (MSE:The MSE ca be calculaed as below: MSE = [ Fˆ ( ] = Where = exp( ( γ / δ ad Fˆ ( = exp( ( ˆ γ ˆ / δ c Correlao coeffce (R : The adequacy of a fed dsrbuo ca be evaluaed by he correlao coeffce, a value of R ear.0 suggess ha he observaos could have bee draw from he fed dsrbuo, he correlao -6-

7 coeffce s gve by ( Y Y ( Yˆ Yˆ = R = ( 0.5 ˆ ˆ ( Y Y ( Y Y = = Where Y s he observed value, Y s average value ad Ŷ s he compued value, Yˆ s average value. 4 Numercal Resuls: A smulao s use order o compare he performace of he proposed esmao mehods. Accordg o lmaos of he compuer me, we carry ou hs comparso akg he sample szes as = 0, 0 ad 40 wh pars of (δ,γ=(,, (,, (3,. The resuls are based o 000 smulao rus. We geeraed radom samples of dffere szes by observg ha f U s uform (0,, he T=γ+δ[-log(-U] / s Raylegh of (δ,γ. Such geeraed daa have bee used o oba esmaes of he ukow parameers. The resuls are gve ables (, ( ad (3. Table ( The esmaes for =0 Mehod True Values Esmaed Values δ γ δˆ γˆ TD MSE R x LSM x

8 x RELS x x x RR x x0-6 x x0-6 3x x0-7 3x x0-0.0 R x x x ME x x x MME-I x x x MME-II x x Table ( The esmaes for =0 Mehod True value Esmaed values δ γ δˆ γˆ TD MSE R x0-6 x LSM x

9 x x RELS x x x RR x x x x x R E E MME x x x MME-I x x MME-II x x Table (3 The esmaes for =40 Mehod True Values Esmaed Values δ γ δˆ γˆ TD MSE R x0-5 64x LSM x

10 x0-9 33x x0-6 3x RELS x x0-9 8x x RR x x0-7 3x x x0-6 75x x0-9 4x x R x0-8 4x x0-9 33x x ME x x MME-I x x x MME-II x x Resuls ad cocluso: All he resuls are lsed able (, (ad (3. From hese ables, we see ha he esmaes of parameers are oo close o he rue values followed by R esmaes, LS esmaes, RELS esmaes ad RR. Also he TD ad MSE ad R from ad R are very small, he mome ad modfed mome esmaors -0-

11 ca be calculaed que easly from he sample daa bu values of TD ad MES ad R are larger ha he correspodg values obaed from he oher mehods. For hese reasos he ad R are preferable. REFERENCES - Abdul Karm, M.D. ad Chowdhury, J.U.(995 A comparso of four dsrbuo used flood frequecy aalyss Bagladesh.Hydrl.Sc.J. 40(, Aryawasa,K.A. ad Templeo,J.G.C.(984 Srucural ferece o he parameer of he Raylegh dsrbuo from doubly cesored samples.sas.hefe,5, Cohe,A.C. ad Whe,B.J.(98b Modfed maxmum lkelhood ad mome esmao of hree parameers Webull dsrbuo.commu. Sas.,heory ad Mehod., ( Gross, A.J. ad Clark, V.A.(975.Survval dsrbuos, Relably Applcaos Bomedcal Sceces. Wley, New York. 5- Lalha, S. ad Aad, M. (996. Modfed maxmum lkelhood esmao for Raylegh dsrbuo. Commu. Sas., Theory Meh., 5 (, Lee, E. T.(980.Sascal Mehods for survval daa aalyss. Lfeme, learg Publcaos. Ic., Belmou. 7- Meas,S.ad Ilopoulos,G.(003 Tess of f for he Raylegh dsrbuo based o he emprcal Laplace rasform. Aals of he Isue of Sasscal Mahemacs, Vol.55, No., pp Mohamed A.Al Fawza (000 Algorhms for esmag he parameers of Webull dsrbuo. Iersa, Oc Pablo,B.S. ad Bruce, E.R. (99. Model parameer esmao usg leas squares. Waer Res.Vol.6, No.6, PP Palovko, A.M (968. Fudameals of relably heory. Academc Press. New --

12 York. - Pfaffeberger, R.C. ad Delma,T.E.(984.A modfed rdge regresso esmaor usg he leas absolue value crero mulple lear regresso model. Proceedg of he Amerca Isue for Decso Sceces. Troo, PP Pfaffeberger, R.C. ad Delma,T.E.(985. A comparso of robus rdge esmaors. Proceedg of he Amerca Sascal Assocao Busess ad Ecoomc Sascs Seco, Las Vegas, Nev., PP Pfaffeberger, R.C. ad Delma,T.E..(990. A comparso of regresso esmaors whe boh mul colleary ad oulers are prese. Ed by: Arhur, Marcel Dekker, Ic. New York ad Basel. 4- Roald, E, W. ad Raymod, H. M.(978. Probably ad sascs for egeers ad scess. Secod. edo. McMlla publshg Co., Ic., New York. 5- Sama A. S. ad Mohamed M.M.(993 Modfed mome esmaors for he hree parameers Pareo dsrbuo. ISSR, Caro Uversy, Vol.(8, Par (. 6- Wager, H.(959. Lear programmg echques for regresso aalyss. J. Am. Sa. Assoc.,54,

Chapter 8. Simple Linear Regression

Chapter 8. Simple Linear Regression Chaper 8. Smple Lear Regresso Regresso aalyss: regresso aalyss s a sascal mehodology o esmae he relaoshp of a respose varable o a se of predcor varable. whe here s jus oe predcor varable, we wll use smple