The Exponentiated Lomax Distribution: Different Estimation Methods

Transcription

1 Ameca Joual of Appled Mathematcs ad Statstcs 4 Vol. No Avalable ole at Scece ad Educato Publshg DOI:.69/ajams--6- The Expoetated Lomax Dstbuto: Dffeet Estmato Methods Hamdy M. Salem * Depatmet of Statstcs Faculty of Commece Al-Azhe Uvesty Egypt & Commuty College Buadah Qassm Uvesty Saud Aaba *Coespodg autho: d.hamdysalm@yahoo.com Receved Septembe 9 4; Revsed Novembe 7 4; Accepted Novembe 4 Abstact Ths pape coces wth the estmato of paametes fo the Expoetated Lomax Dstbuto ELD. Dffeet estmato methods such as maxmum lkelhood quas-lkelhood Bayesa ad quas-bayesa ae used to evaluate paametes. Numecal study s dscussed to llustate the optmal pocedue usg MATHCAD pogam (. A compaso betwee the fou estmato methods wll be pefomed. Keywods: Expoetated Lomax Dstbuto maxmum lkelhood estmato quas-lkelhood estmato bayesa estmato quas-bayesa estmato Cte Ths Atcle: Hamdy M. Salem The Expoetated Lomax Dstbuto: Dffeet Estmato Methods. Ameca Joual of Appled Mathematcs ad Statstcs vol. o. 6 (4: do:.69/ajams Itoducto Authos of the statstcal dstbutos feld have cotuous motvatos fo developg a vaety dstbutos to become moe flexble ad moe fttg fo eal data sets. These ew statstcal dstbutos ae used to descbe ad tepet the pheomea. The dea of expoetated dstbutos was utlzed to ceate ew dstbutos. Codeo & Casto ( exteded may kow dstbutos as omal Webull gamma Gumbel ad vese Gaussa dstbutos. Gupta et al. (998 toduced a class of expoetated dstbutos based o cumulatve dstbuto fucto CDF fo the expoetal dstbuto. I a smla mae Nadaajah ad Kotz (6 poposed the expoetated gamma ad expoetated Gumbel dstbutos. Gauss & Codeo (3 poposed a ew method of addg two paametes to a cotuous dstbuto that exteds the dea of Nadaajah ad Kotz (6. Alzaateh et al. (3 poposed aothe ew method fo geeatg may ew dstbutos. Ths method s called the T-X famly of dstbutos. It has a coecto betwee the hazad fuctos ad each geeated dstbuto as a weghted hazad fucto of the adom vaable X. Alzaateh et al. (3 fouded seveal kow cotuous dstbutos to be specal cases of these ew dstbutos. Weddebum (974 toduced a mpotat exteso of maxmum lkelhood estmato to get the optmal paamete estmato. Ths method s called Quas- Lkelhood. It s equed assumptos about meas ad vaace fuctos cotast to the full dstbutoal assumptos of oday lkelhood. Quas-Lkelhood fo a obsevato X wth mea µ ad vaace V ( µ takes ths fom: x ( ; µ x µ = µ V ( µ µ x µ o Q( x; µ = dµ + fucto of X V ( µ whee µ s E( X ad V ( µ s ( V X. (. Fo a sample of sze the quas-bayesa estmato s depeded o eplacg the lkelhood fucto by the atual expoetal of the quas-lkelhood fucto. Ths pape s ogazed as follows: I Secto The ELD dstbuto wll be defed. I secto 3 dffeet estmato methods wll be used such as maxmum lkelhood quas-lkelhood Bayesa ad quas-bayesa to obta the estmatos of paametes. Secto 4 coces wth compag pocedues of the estmatos ad compaes the pefomaces though umecal smulatos.. The Expoetated Lomax Dstbuto The CDF ad the pobablty desty fucto pdf of the ELD espectvely ae: ( = ( + ; > > F x x x ad ( + (. f( x = ( + x ( + x ; (. x > ad >

2 Ameca Joual of Appled Mathematcs ad Statstcs 365 Note that whe = the pdf of the ELD educes to the Expoetated Paeto dstbuto wth paamete. Also whe = = the pdf of the ELD educes ( to the stadad Lomax dstbuto wth oe paamete. The suvval fucto ad the hazad fucto of the ELD espectvely take the followg foms: ( = ( + ; > > S x x x ad ( + ( + x ( + x hx ( = ; ( + x x > ad >. (.3 (.4 Fgue. The pdf ad CDF caves of the ELD at dffeet values of the ad paametes ( The th momets = ( µ = E x = µ of the ELD ae: ( + x ( + x ( + x dx = ( B ( ; =. whee B( ab Τ( a Τ( b =. Γ ( a + b (.5 Thus the mea ad vaace of the ELD espectvely ae: µ = µ = B B( (.6 va( x = B B (.7 3. Dffeet Estmato Methods 3.. Maxmum Lkelhood Estmatos The lkelhood fucto of the ELD based o the samples X X X s: L( = ( ( + ( + x ( + x (3.. Ad the log-lkelhood fuctos fo ad ae espectvely: [ ] ( l( + l( + l( (3.. + ( l ( + ( + l( + x x = = The devatves of (3.. wth espect to ad espectvely ae as follows: = + ( a ( + x l( + x = ( + x = = ( + x = + l ( + x = + ( a ( + ( + ( + = ( + x = x x x ( + x (3..3 (3..4 (3..5 The maxmum lkelhood estmatos of the paametes ad ca be obtaed by solvg equatos (3..3 (3..4 ad (3..5 afte equatg them to zeo. Ufotuately thee s o closed fom fo the estmatos ˆ ˆ ad ˆ. So Newto Raphso method s usg to solve these equatos umecal aalyss see Salem (3. Now the log lkelhood fucto whch (3.. ca be used to costuct Fshe fomato matx I has the fom: whee I = (3..6

3 366 Ameca Joual of Appled Mathematcs ad Statstcs l( + x = + ( a ( + x = ( + x (3..7 l( + x ( + x ( + x ( = ( x = ( + ( x ( + x ( = + ( a = ( + x ( + x ( + ( + ( + ( + x x x x ( + ( + x (3..8 (3..9 l( + x = ( + x (3.. = ( + x = ] [ ( + x ] = ( + x x ( + x = ( a = ( + + ( x ( + x ( + ( + x l( + x x( + x ( + x ( + x ( + l( + x x ( + x x ( + x = = ( Quas-Lkelhood Estmatos (3.. (3.. Let the pdf of the ELD Ex ( = µ ad va( x of the adom vaable X whch s take fom the ELD as (. (.6 ad (.7 espectvely the Ex ( = µ = B B( va( x = µ B B = V ( µ B B (3.. (3.. whee V (. s assumed to be kow ad the paamete µ may be ukow. So the Quas-Lkelhood fucto (. gves: Qx ( = B B( l B B( l ( x = (3..3 The devatves of Qx ( µ wth espect to ad espectvely ae: whee = Ψ( Ψ( k. Γ ( + Γ ( + x Γ( k (3..4 = Γ ( + Γ( = [ + Ψ( Ψ( k. ] Γ ( + Γ( x Γ( k (3..5 = Γ ( + Γ( x = ( = + Γ. s y + k = ( Γ y = Γ ( + a (3..6 ( Γ s = Γ ( k = ad Ψ (. s the Ps-gamma. It's ofte called Polly-gamma fucto. Fo detals see (Amos (983. The equatos (3..4 (3..5 ad (3..6 wll be solved usg the same umecal aalyss whch used pevous maxmum lkelhood estmato method Bayesa Estmatos Let X X X be depedet adom samples daw fom the ELD as equatos (. (.. The cojugate gamma po dstbutos fo wth ηε ae employed espectvely as follows: δ β δ + β g( = e > δ β > Γ( δ (3.3. paametes ( δ β ( ε η ε + η g( = e > εη > Γ( ε (3.3. The o-fomatve po dstbuto of wth paamete ρ s: g ( = ρ < ρ> (3.3.3 So the jot po dstbuto fo ad s:

4 Ameca Joual of Appled Mathematcs ad Statstcs 367 β η δ + ε + η β g = ρ e.(3.3.4 Γ Γ ( δ ε ( δ ( ε The posteo desty of ad Based o the samples X X X ad lkelhood fucto s: π ( Ω X X X = δ + ε + η β e L( X X X ρ δ + ε + η β ρ e (3.3.5 whee Ω s a vecto of the paametes ad ad ( L X X X s the lkelhood fucto. Now the Bayes estmatos of the paametes ad ude symmetc squae loss fucto ca be obtaed by gettg o the expectato of the magal dstbuto of these paametes. I addto the magal dstbuto h(. X X X of ay paamete ca be obtaed by tegato of the posteo dstbuto π ( Ω X X X wth espect to othe paametes. So the posteo dstbuto of the paamete ad espectvely ae: ( = π Ω X X X dd ( = π Ω X X X d d ( = π Ω X X X d d (3.3.6 (3.3.7 (3.3.8 Cosequetly the Bayes estmatos of the paametes ad ude symmetc squae loss fucto espectvely ae: ( = E h X X X = h X X X d ( = E h X X X = h X X X d ( = E h X X X = h X X X d (3.3.9 (3.3. (3.3. The Bayes sk of the paametes ad based o squae eo loss fucto espectvely ae: ( π X X X d d va = d (3.3. ( δ + ε + η β ρ e δ + ε + η β ρ e ( π X X X d d va = d (3.3.3 va ( δ + ε + η β ρ e ( π X X X d d = d ( Quas-Bayesa Estmatos ( The quas Bayesa estmato s smla to the quaslkelhood estmato bath of them use the lkelhood fucto howeve the eale s dffeet because t uses atual expoetal of the quas lkelhood fucto. Fo a sample of sze whch s take fom the ELD the atual expoetal of the quas lkelhood fucto s gve by: Q*( x = e B B x B B ( = ( (3.4. By usg the thee po dstbutos whch dscussed (3.3. (3.3. ad (3.3.3 fo the paametes ad espectvely the the posteo dstbuto s: ( π * Ω X X X = δ + ε + η β ρ e Q*( x δ + ε + η β ρ e Q*( x dω (3.4. Thus the same techque fo Bayesa estmato method fom (3.3.6 to (3.3.4 ad wth the help of compute facltes wll be used to evaluate the magal dstbuto h*. ( X X X quas-bayesa estmatos * * ad * ude symmetc squae loss fucto ad Bayes sk fo the paametes ad espectvely. 4. Smulato Study The compute pogam MATHCAD ( s used to obta umecal llustato fo the last theoetcal esults fo small medum ad lage sample szes. A compaso betwee the fou estmato methods wll be pefomed. samples geeated fom ELD wth paametes ae used at dffeet values of these paametes. (

5 368 Ameca Joual of Appled Mathematcs ad Statstcs Mea squae eos (MSE of the thee paametes wll be calculated. Table dcates to that the quas-lkelhood ad quas- Bayesa estmatos fo the two paametes ad ae bette tha the maxmum lkelhood ad Bayesa - ude symmetc squae loss fucto estmatos at all sample szes espectvely. Also the pefomace of the quaslkelhood ad quas-bayesa estmatos fo ae vey close to the pefomace of the maxmum lkelhood ad Bayesa - ude symmetc squae loss fucto estmatos at all sample szes espectvely. Though the esults we ca see the mea squae eos MSE of all estmatos ae deceasg as the sze of sample s lage. The quas-bayesa estmato s closest method because t goes to the eal paamete values. Table. paamete MLE MSE QMLE MSE Bayes MSE QBayes MSE Cocluso Ths pape studed the estmato of paametes fo the Expoetated Lomax Dstbuto va fou estmato method. These methods wee maxmum lkelhood quaslkelhood Bayesa ude symmetc squae loss fucto ad quas-bayesa estmatos. Numecal study was vestgated to llustate the optmal pocedue. Whe the sample szes ae ceasg the mea squae eos MSE of all estmatos ae deceasg. Refeeces [] Alzaateh A. Lee C. & Felx F. (3. "A ew method fo geeatg famles of cotuous dstbutos". METRON [] Amos D. (983 "A potable Fota suboute fo devatves of Ps fucto". ACM Tasactos o Mathematcal softwae 9 ( [3] Codeo M. & Casto M. (. "A ew famly of geealzed dstbutos". Joual of Statstcal. Computato ad Smulato. 8 ( [4] Gauss M. & Codeo M (3. "The Expoetated Geealzed Class of Dstbutos". Joual of Data Scece.. -. [5] Gupta C. Gupta P. & Gupta D. (998. "Modelg falue tme data by Lehma alteatves". Commucatos Statstcs- Theoy ad Methods [6] Nadaajah S. & Kotz S. (6. "The Expoetated Type Dstbutos". Acta Applcadae Mathematca. 9 ( [7] Salem H. (3. "Ifeece o Stess-Stegth Relablty fo Weghted Webull Dstbuto". Ameca Joual of Mathematcs ad Statstcs. 3 (4: -6. [8] Weddebu R. (974 "Quas-Lkelhood Fuctos Geealzed Models ad the Gauss-Newto Method". Bometka. 6 (