J. Sci. Res. 11 (1), (2019) A Bayesian Approach for Estimating Parameter of Rayleigh Distribution

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1 Publatos Avalable Ole J. S. Res. (), 3-39 (09) JOURNAL OF SCIENTIFIC RESEARCH A Bayesa Approah for Estmatg Parameter of Raylegh Dstrbuto J. Mahata *, M. B. A. Talukdar Departmet of Statsts, Uversty of Chttagog, Chttagog-433, Bagladesh Reeved 0 Jue 07, aepted fal revsed form 6 September 08 Abstrat Ths paper s oered wth estmatg the parameter of Raylegh dstrbuto (speal ase of two parameters Webull dstrbuto) by adoptg Bayesa approah uder squared error (SE), LINEX, MLINEX loss futo. The performaes of the obtaed estmators for dfferet types of loss futos are the ompared. Better result s foud Bayesa approah uder MLINEX loss futo. Bayes rsk of the estmators are also omputed ad preseted graphs. Keywords: Lear epoetal (LINEX) loss futo; Modfed lear epoetal (MLINEX) loss futo; Squared error loss futo (SE). 09 JSR Publatos. ISSN: (Prt); (Ole). All rghts reserved. do: J. S. Res. (), 3-39 (09). Itroduto Raylegh dstrbuto s amed after the Brtsh Nobel prze wer physst Lord Raylegh (84-99). Ths dstrbuto has got remarkable atteto the feld of relablty theory ad survval aalyss, probablty theory ad operatos researh. Apart from ths, ommuato theory to model multple paths of dese sattered sgals reahg a reever ad the physal sees to model wd speed, wave heghts [8], soud/lght radato, rado sgals ad wd power, the applato of ths dstrbuto s oteable. I addto to, egeerg, to measure the lfetme of a objet, where the lfetme depeds o the objet s age. For eample, resstors, trasformers, ad apators arraft radar sets. Raylegh dstrbuto s also used mture models. Most of the researhers worked o the lassal ad Bayesa aalyss of two ad three ompoets mture models. Saleem ad Aslam [9] dsussed the use of the formatve ad the o-formatve prors for Bayesa aalyss of the two ompoet mture of Raylegh dstrbutos. Aslam et al. [] studed Bayesa aalyss of a three ompoet mture of Raylegh dstrbutos wth ukow mg proportos. Boudjerd et al. [4] appled Bayesa estmato of the Raylegh dstrbuto uder dfferet loss futo. Sdhu et al. [0] employed Bayesa * Correspodg author: johstat@u.a.bd

2 4 Estmatg Parameter of Raylegh Dstrbuto feree of mture of two Raylegh dstrbutos. After aalyzg ts sgfae, ths paper has bee teded to get the best estmate of the parameter of ths dstrbuto osderg vared loss futos usg smulated ad real data. A otuous radom varable X s sad to have a Raylegh [] dstrbuto f ts probablty desty futo s gve by f, e ; 0, 0 () =0 ; otherwse. Where, s the oly oe sale parameter of ths dstrbuto. Replag, we get f, e ; 0, 0 =0 ; otherwse Now Mote Carlo tehques for geeratg sample from Raylegh dstrbuto s appled wth the help of verse trasform method.. Sample Geerato from Raylegh Dstrbuto Let X be a Raylegh varate havg the p.d.f ; e f ; 0, 0 =0 ; otherwse The umulatve dstrbuto futo of ths dstrbuto s F F 0 0 f d e d Let p Now from equato (3) we have F e. Now by verse trasformato method we have U e lu Where, U s uformly dstrbuted over the rage [0,]. Usg ths relatoshp a sample of () (3)

3 J. Mahata et al. J. S. Res. (), 3-39 (09) sze a easly be geerated from Raylegh dstrbuto. 3. Dfferet Estmators of Parameter ad Ther Bayes Rsk Bayesa approah uder dfferet loss futo has bee appled for estmatg the parameters of the Raylegh dstrbuto. 3.. Pror ad posteror desty futo We get from () the p.d.f of Raylegh dstrbuto s, f e ; 0, 0 =0 ; otherwse Let the pror desty futo of s a gamma dstrbuto defed as g e 0,, ; (4) Where, α ad β s the shape ad sale parameter respetvely. Now the posteror desty futo of for the gve sample s f d g f g f,,. e f Therefore, f ~ G, () 3.. Bayes estmator uder squared error (SE) loss futo Now let the loss futo be squared [] error defed as ; L (6) Hee uder SE loss futo Bayes estmator of θ s (7)

4 6 Estmatg Parameter of Raylegh Dstrbuto Therefore,, s the Bayes estmator of uder SE loss. Squared error loss futo s applable whe the loss s symmetr ature Bayes estmator uder LINEX loss futo Let us osder the followg LINEX [3] loss futo whh s appled real estate assessmet 0 0, ; ) ( k e k D L. (8) Where, D represets the estmato error ad s the shape parameter of the loss futo. For LINEX loss futo, the Bayes estmator of s gve by e E l Now, d f e e E e E Therefore, l 3 3 e E So, we egletg the 3 d ad hgher order term, the we get, Hee, (9) s the Bayes estmator uder LINEX loss futo.

5 J. Mahata et al. J. S. Res. (), 3-39 (09) Bayes estmator uder MLINEX loss futo Whe the loss s asymmetr, the the MLINEX loss futo s appled. MLINEX loss futo s the modfato of LINEX loss futo. Now, let us osder the MLINEX loss [6] futo defed as 0 0, ; log ; L (0) For MLINEX loss futo Bayes estmator of s obtaed from. E Here. d f E Hee, ) ( ) (, se () whh s the Bayes estmator uder MLINEX loss futo. 3.. Bayes rsk of the estmator Let X X X X...,,,... be a radom sample of sze draw from a desty futo ) / ( f. Let, L be the loss futo estmatg by wth desty futo () g. The rsk of the estmator s defed by,, L E R ad the Bayes rsk [7] of estmator wth respet to the loss futo, L ad pror desty ) ( g, deoted by, g R s defed as otuous s θ ;f g θ dθ θ,θ R θ dsrete s θ ;f g θ θ,θ R,, R E R g g

6 8 Estmatg Parameter of Raylegh Dstrbuto 4. Empral Study The estmated values of the parameter ad Bayes rsk of the estmators are omputed by Mote-Carlo smulato method from Raylegh dstrbuto. A short algorthm of the above smulato s gve below: ) Geerate a radom sample of sze usg the followg formula l U, where U s uformly dstrbuted over the rage [0,] ad spef value of. ) Obta Bayes estmator uder dfferet loss futo. ) Repeat the above steps 000 tmes ad deote the Bayes estmates of as j, j,,..., v) Smulated Bayes estmate of 000 j. v) Calulate the smulated Bayes rsk (SBR) of f R. The umeral results ad ther graphs are as follows. Table. Estmated value ad Bayes rsk of dfferet estmators of the parameter dstrbuto whe,, ad. j of Raylegh Crtera Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Fg. depts that, for,,, ad dfferet values of, the Bayes estmator uder MLINEX loss futo provdes better estmate tha the other loss futos eept =.

7 Bayes Rsk J. Mahata et al. J. S. Res. (), 3-39 (09) 9 Bayes Rsk of dfferet estmator of parameter Fg.. Graph of Bayes rsk of dfferet estmators of the parameter whe,, ad. Table. Estmated value ad Bayes rsk of dfferet estmators of the parameter dstrbuto whe 3,, ad. of Raylegh dstrbuto of the Raylegh Crtera Estmate Bayes rsk Sample sze Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk For 3,,,, Bayes estmator uder MLINEX loss futo provde better estmate tha other loss futos eept =.

8 Bayes Rsk Estmatg Parameter of Raylegh Dstrbuto Bayes Rsk of dfferet estmator of parameter Fg.. Graph of Bayes rsk of dfferet estmators of the parameter whe 3,, ad Sample sze of the Raylegh dstrbuto Table 3. Estmated value ad Bayes rsk of dfferet estmators of the parameter dstrbuto whe,, ad. of the Raylegh Crtera Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk

9 Bayes Rsk J. Mahata et al. J. S. Res. (), 3-39 (09) 3 Bayes Rsk of dfferet estmator of parameter Fg. 3. Graph of Bayes rsk of dfferet estmators of the parameter whe,, ad. of the Raylegh dstrbuto Fg. 3 shows that the Bayes rsk of the Bayesa approah uder MLINEX loss futo s lower tha others loss futo eept for sample sze =. Table 4. Estmated value ad Bayes rsk of dfferet estmators of the parameter dstrbuto whe, 4,.ad. of the Raylegh Crtera Estmate Bayes rsk Estmate Bayes rsk Sample sze Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk

10 Bayes Rsk Estmatg Parameter of Raylegh Dstrbuto Bayes Rsk of dfferet estmator of parameter Fg. 4. Graph of Bayes rsk of dfferet estmators of the parameter whe, 4,.ad. of the Raylegh dstrbuto Fg. 4 represet that, Bayes estmator uder MLINEX loss futo have smaller rsk tha other loss futos for all ases of sample sze. Table. Estmated value ad Bayes rsk of dfferet estmates of the parameter dstrbuto whe, 3,.ad. of the Raylegh Crtera Estmate Bayes rsk Estmate Bayes rsk Sample sze Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk

11 Bayes Rsk J. Mahata et al. J. S. Res. (), 3-39 (09) 33 Bayes Rsk of dfferet estmator of parameter Fg.. Graph of Bayes rsk of dfferet estmators of the parameter whe, 3,.ad. of the Raylegh dstrbuto It s see from the Fg., for, 3,., ad dfferet values of, the Bayes estmator uder MLINEX loss futo provdes better estmate tha the other loss futos. Table 6. Estmated value ad Bayes rsk of dfferet estmators of the parameter dstrbuto whe 0,, ad. of the Raylegh Crtera Estmate Bayes rsk Estmate Bayes rsk Sample sze Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk

12 Bayes Rsk Estmatg Parameter of Raylegh Dstrbuto Bayes Rsk of dfferet estmator of parameter Fg. 6. Graph of Bayes rsk of dfferet estmators of the parameter whe 0,, ad Alpha of the Raylegh dstrbuto Although for varyg ad the sample sze moderately large, we get the same result.e. the Bayes estmator uder MLINEX loss futo gve better result tha the other loss futos eept =. Table 7. Estmated value ad Bayes rsk of dfferet estmators of the parameter dstrbuto whe 0, 3, ad. of the Raylegh Crtera Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk

13 Bayes Rsk J. Mahata et al. J. S. Res. (), 3-39 (09) 3 Bayes Rsk of dfferet estmator of parameter Fg. 7. Graph of Bayes rsk of dfferet estmators of the parameter whe 0, 3, ad Beta of the Raylegh dstrbuto Fg. 7 dsplays that, Bayes estmator uder MLINEX loss futo gve smaller Bayes rsk tha Bayes estmator uder SE ad LINEX loss futo eept & 6. Table 8. Estmated value ad Bayes rsk of dfferet estmators of the parameter of the Raylegh dstrbuto whe 0,, ad. Crtera 0. Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk

14 Bayes Rsk Estmatg Parameter of Raylegh Dstrbuto Bayes Rsk of dfferet estmator of parameter Fg. 8. Graph of Bayes rsk of dfferet estmators of the parameter of the Raylegh dstrbuto whe 0,, ad. Fg. 8 show for varyg ad moderately large; the Bayes estmator uder MLINEX loss futos gves the better result tha the other loss futos. The varato the performae of the estmators for varous sample szes ad vared parameters are observed from Tables -8. From Fgs. -8, the Bayes rsk of the estmators of dfferet loss futo, the MLINEX loss futo s mmum. Therefore, t a be oluded that, Bayes estmator uder MLINEX loss futo s better tha all other estmators the study.. Real Study For fttg the dstrbuto, weather data have bee used ths paper. Wd speed (kph) data have bee hose the perod of of Dhaka Arport. Table 9. Estmated value ad Bayes rsk of dfferet estmators of the parameter value of. for dfferet Crtera Estmate Bayes rsk Estmate Bayes rsk Lamda Square Estmate Bayes rsk Estmate...38 Bayes rsk

15 J. Mahata et al. J. S. Res. (), 3-39 (09) 37 Crtera Estmate Bayes rsk Estmate Bayes rsk Table 9 shows that for all ases Bayes rsk of Bayesa approah uder MLINEX loss futo s the smallest tha other approahes. Table 0. Estmated value ad Bayes rsk of dfferet estmators of the parameter for dfferet value of. Crtera Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Estmate Bayes rsk Table 0 represets that for dfferet values of α, Bayes estmate uder MLINEX loss futo have the smallest Bayes rsk tha other Bayesa approahes. If we wat to predt about the wd speed o spef rego the Raylegh dstrbuto has bee used ad for fttg the dstrbuto f we use Bayesa approah uder MLINEX loss futo the t wll gve better result. Beause MLINEX loss futo shows the smallest Bayes rsk. 6. Coluso I ths study, we have osdered the Bayesa estmato approah to estmate the parameter of Raylegh dstrbuto. I Bayesa approah, squared error (SE), lear epoetal (LINEX) ad modfed lear epoetal (MLINEX) loss futos have bee used. We oduted a omprehesve smulato ad real data to judge the relatve performae of the Bayes estmator uder dfferet loss futos at dfferet sample szes ad vared parameters of pror dstrbuto. From smulated results ad real study, smallest rsk has bee observed by Bayesa approah uder MLINEX loss futo tha other loss futos about all ases. Also Fgs. -8, dowward shape has bee dsplayed by

16 38 Estmatg Parameter of Raylegh Dstrbuto MLINEX loss futo tha SE ad LINEX loss futos. That meas, Bayesa approah uder MLINEX loss futo gves better results tha other loss futos. Therefore, Bayesa approah uder MLINEX loss futo a be suggested to estmate the parameter of Reylegh dstrbuto. Referees. S. A. Ahmed, ad H. O. Mahammed, Jorda J. Meh. Id. Eg. 6, 3 (0).. M. Aslam, M.Tahr, Z. Hussa, ad B. Al-Zahra, PLOS, 0, (0). 3. S. C. Bswas, J. Mahata, ad M. K. Roy, It. J. Stat. Med. Res. 7, 7 (08). 4. K. Boudjerda, A.Chadl, A. Merradj, ad H. Fellag, Eletro J. Appl. Stat. Aal. 0, 0 (07). M. R. Hasa ad A. R. Bazd, J. S. Res. 9, 67 (06) J. Mahata ad M. B. A. Talukder, Ope S. J. Stat. Applat., 40 (08). 7. A. M. Mood, F. A. Graybll, ad D. C. Boes, Itroduto to the Theory of Statsts, 3 rd Edto (MGraw-Hll Book Compay, USA, 974). 8. W. Rattaaptko, Oea Eg. 37, 4 (00) M. Saleem, ad M. Aslam, J. Appl. Stat. S. 6, 493 (008). 0. T. N. Sdhu, N. Feroze, M. Aslam, ad A. Shafq, Pujab Uv. J. Math. 48, 49 (06).. C. Walk, Had-Book o Statstal Dstrbutos for Epermetalsts (Uversty of Stokholm, 007). Apped A. R ode for estmatg dfferet estmators ad ther Bayes rsk by smulated data Ray<-futo(,,a,b,m) { u<-ruf() <-(-m*log(-u))^(/) bse<-(sum(^)+b)/(+a) rbse<-((sum(^)+b)^)/((+a)*(+a+))-bse^ bl<-bse-*(sum(^)+b)*bse/ k<-(gamma(+a+)/gamma(+a))*((sum(^)+b)^) rbl<-ep(bl)*k-*bl+*bse- bml<-((gamma(+a-)/gamma(+a))^(/))*(sum(^)+b) rbml<-bml*(gamma(+a-)/gamma(+a))*((sum(^)+b)^)-*log(bml)- lst(bse,rbse,bl,rbl,bml,rbml) } Ray(00,,,,.) Apped B. R ode for estmatg dfferet estmators ad ther Bayes rsk by real data lbrary(foreg)

17 J. Mahata et al. J. S. Res. (), 3-39 (09) 39 data<-read.table("d:/d.tt") Ray<-futo(,a,b) { <-data[,] <-legth() bse<-(sum(^)+b)/(+a) rbse<-/(((sum(^)+b)^)/((+a)*(+a+))-bse^) bl<-bse-*(sum(^)+b)*bse/ k<-(gamma(+a+)/gamma(+a))*((sum(^)+b)^) rbl<-/(ep(bl)*k-*bl+*bse-) bml<-((gamma(+a-)/gamma(+a))^(/))*(sum(^)+b) rbml<-/(bml*(gamma(+a-)/gamma(+a))*((sum(^)+b)^)-*log(bml)-) lst(bse,rbse,bl,rbl,bml,rbml) } Ray(,,)

ENGINEERING ECONOMICS

ENGINEERING ECONOMICS Factor Name Coverts Symbol Formula Sgle Paymet Compoud Amout to F gve P (F/P, %, ) ( + ) Sgle Paymet Preset Worth to P gve F (P/F, %, ) ( + ) Uform Seres to A gve F (A/F, %, ) Skg