Fitting the Generalized Logistic Distribution. by LQ-Moments

Transcription

1 Applied Mathematical Scieces, Vol. 5, 0, o. 54, Fittig the Geealized Logistic Distibutio by LQ-Momets Ai Shabi Depatmet of Mathematic, Uivesiti Teologi Malaysia Abdul Aziz Jemai Scieces Mathematic Studies Cete, Uivesiti Kebagsaa Malaysia Abstact The method of LQ-momets (LQMOM) fo estimatig paametes ad quatiles of the Geealized Logistic (GL) distibutio ae itoduced. We exploe ad exted class of LQMOM with cosideatio combiatios of p ad values i the age 0 ad 0.5. The popula quatile estimato amely the weighted eel quatile (WKQ) estimato is poposed to estimate the quatile fuctio. A compaiso of these methods is doe by simulatio. The pefomaces of the poposed estimatos of the GL distibutio was compaed with the estimatos based o L-momets fo vaious sample sizes ad etu peiods. The oveall esults show the LQMOM povides bette esults oly fo small o modeate sample size. Keywods: The Weighted Keel Quatile; LQ-momets; L-momets; Quic estimato Itoductio Exteme values fom obseved data ae of special impotace i may applicatios. Fo this pupose, the theoy of statistical extemes, i.e. exteme values of a adom vaiable, ofte povides a basis fo developig the egieeig pedictio. A commo poblem i hydology is to choice the tue distibutio of the exteme value. Oe of the exteme value distibutios fo maximum values is

2 664 A. Shabi ad A. A. Jemai the geealized logistic (GL) distibutio. The GL distibutio is sewed distibutios ae playig a impotat ole i vaious eseach studies (Balaisha ad Hossai, 007). The model is ecommeded as the stadad fo flood fequecy aalysis i the UK, the use of this distibutio has iceased i populaity i hydology. This model is also used i suvival aalysis as a paametic model fo evets ad i ecoomics as a simple model of wealth o icome (Alasasbeh ad Raqab, 009). This distibutio is pefeed to the geealized exteme value distibutio fo flood estimatio i the couty togethe with L-momets (LMOM) method as the paamete estimatio method (IH, 999). The GL distibutio is i fact a geealizatio of the -paamete log-logistic distibutio, which had ealy bee examied by Ahmad et al., (998) i a at-site ad egioal study ivolvig Scottish flood data. They compaed the thee paamete log-logistic (LL) distibutio to geealized exteme value (GEV), thee-paamete logomal, ad Peaso Type distibutios ad foud the LL distibutio pefomed exteme well compaed with these othe distibutios. May paamete estimatio methods have bee poposed to fit statistical distibutio to fit statistical distibutio (Sigh ad Ahmad, 004), (Gupta ad Kudu, 004), (Alasasbeh ad Raqab, 009). Now-a-days, the method of LMOM is widely applied i hydology (e.g., Hosig ad Wallis, 997), (Saaasubamaia ad Siivasa, 999), (Kavae, 005) ad it seems to be gadually displacig othe methods of paamete estimatio. The mai advatages of usig the method of LMOM ae that the paamete estimates ae moe eliable (i.e., smalle mea-squaed eo of estimatio) ad ae moe obust agaist outlies tha momets ad ae usually computatioally moe tactable tha maximum lielihood method (Hosig, 990). The LMOM have foud wide applicatios i such fields of applied eseach as civil egieeig, meteoology, ad hydology. The method of LMOM has become a stadad pocedue i hydology fo estimatig the paametes of cetai statistical distibutios. Mudola ad Hutso (998) exteded LMOM to ew momet lie etitles called LQ-momets (LQMOM). The method of LQMOM was oigially itoduced as a extesio of the LMOM method fo paamete estimatio. They foud the LQMOM always exists, ae ofte easie to compute tha LMOM, ad i geeal behave similaly to the LMOM. The LQMOM ae costucted by usig fuctioal defiig the quic estimatos, whee the paametes of quic estimato tae the values p = 0, = fo the media, p = / 4, = / 4 fo the timea ad p = 0., = / fo the Gastwith, i places of expectatios i LMOM. The LQMOM method was develop impoved fo the Exteme Values Type (EV) (Ai & Jemai, 009), Geealized exteme value (GEV) distibutio (Ai & Jemai, 007 (a),(b) ) ad Log Nomal distibutio (Ai & Jemai, 006), ad the pefomaces of the poposed method wee compaed with the estimatos based o method of LMOM. The simulatios studies show that the method of LQMOM always pefom bette tha the LMOM ad momets methods ove the etie sample size cosideed.

3 Fittig the geealized logistic distibutio 665 The objective of this pape is to develop impoved LQMOM that do ot impose estictios o the value of p ad such as the media, timea o the Gastwith but we exploe a exteded class of LQMOM with cosideatio combiatios of p ad values i the age 0 ad 0.5. The popula quatile estimato amely the weighted eel quatile (WKQ) estimato will be poposed to estimate the quatile fuctio. Mote Calo simulatios ae coducted to illustate the pefomace of the poposed estimatos of the GL distibutio wee compaed with the estimatos based o LMOM fo vaious sample sizes ad etu peiods. The Thee-Paamete Geealized Logistic Distibutio The GL has the pobability desity fuctio (pdf) x ε ( ) x ε f x = + () δ δ δ whee ε, δ ad ae paametes of locatios, scale ad shape, espectively. The cumulative distibutio fuctio of GL is ( ) x ε F x = + δ Quatiles fuctio of GL distibutio is give by Q ( F ) = ε + δ Q 0 ( F ) () whee F Q0 ( F) = (4) F () Defiitio Of LQ-Momets Estimatos Let,...,, be a adom sample fom a cotiuous distibutio fuctio Q( u) = F ( u, ad let : :... : F (.) with quatile fuctio ) deote the coespodig ode statistics. The the th LQMOM ξ is give by = ξ ( ) τ : ), =,,... (5) = 0

4 666 A. Shabi ad A. A. Jemai whee 0 /, 0 p /, ad τ p, : ) pq ( ) + ( p) Q (/ ) + pq ( ) ( = : : : = [ B: ( )] + ( p) Q[ B: (/ )] + pq[ B: ( pq )] (6) is the quic estimato of locatio ad B : ( ) is the quatile of a beta adom vaiable with paamete ad +, ad Q (.) deotes the quatile estimato. The fist fou LQMOM of the adom vaiable ae defied as ξ = ( ) (7) τ p, : { τ : ) τ : )} { τ : ) τ : ) τ : )} 4 4 { τ 4:4 ) τ :4 ) + τ :4 ) τ : 4 ξ = (8) ξ = + (9) ξ = )} (0) The LQ-sewess is defied as ξ η = () ξ. Estimatio of LQ-momets Fo samples of size, the th sample LQMOM is give by ˆ = ξ ( ) ˆ τ : ), =,,... () = 0 whee ˆ ) pqˆ ( ) + ( p) Qˆ (/ ) + pqˆ ( τ = : : : : ˆ [ ( )] ( ) ˆ[ (/ )] ˆ = Q B [ : + p Q B: + pq B: ( p )] () is the quic estimato of locatio ad B : ( ) is the quatile of a beta adom vaiable with paamete ad +, ad ˆQ (.) deotes the quatile estimato sample. The fist fou sample LQMOM of the adom vaiable ae defied as ˆ ξ = ˆ ( ) (4) τ p, ˆ ξ = {ˆ τ ) ˆ τ ( )} (5) : : )

5 Fittig the geealized logistic distibutio 667 ˆ ξ = {ˆ τ ) ˆ τ ) + ˆ τ ( )} (6) : : : ˆ ξ = {ˆ τ ) ˆ τ ) + ˆ τ ) ˆ τ ( )} (7) 4 4 4:4 :4 :4 :4 The sample LQ-sewess is defied as ˆ ξ ˆ η = (8) ˆ ξ. The Quatile Estimatos Let : :... : be the coespodig ode statistics. The populatio quatiles estimatos of a distibutio is defied as Q ( q) = F ( q) = if{ x : F( x) q}, 0 < q < (9) whee F (x) is the distibutio fuctio [7]. The popula class of L quatile estimatos is called eel quatile estimatos has bee widely applied. The L quatile estimatos is give by i Q ˆ / ( q) K h ( t q) dt i: i = ( i ) / = (0) whee K is a desity fuctio symmetic about 0 ad K h ( ) = (/ h) K( / h) () The appoximatio of the L quatile estimato is called as the weighted eel quatile estimato (WKQ) (Huag ad Bill, 999) is give by =, 0 < q < () i Q ˆ ( q) K h w j, q i: i= j= ad the data poit weights ae ( ), i =,, ( ) wi, = (), i =,,...,. ( ) whee K( t) = (π ) / exp( t / ) is the Gaussia Keel, is a optimal badwidth (Sheathe & Mao, 990). h = [ q( q) / ] /

6 668 A. Shabi ad A. A. Jemai. Method of LQ-Momets The LQMOM estimatos fo the GL distibutio behave similaly to the LMOM. Fom equatios (4)-(7) ad equatio (4), the expessios fo the LQMOM of the GL distibutio ca be summaized as follow: ξ = ε + δr p, : ) (4) ξ = δ R ) R )} (5) ξ 4 { : : = δ R ) R ) + R )} (6) { : : : 4 δ{ R, 4:4 ) R, :4 ) R, :4 ) R, p p + p p : 4 ξ = )} (7) ad LQ-sewess of GL distibutio is { R :) R :) + R : )} η = (8) { R ) R )} whee : : R ) = pq { B ( )} + ( p) Q { B (/ )} + pq { B ( )}] (9) j: 0 j: 0 j: 0 j: The LQMOM estimatos δˆ, εˆ ad ˆ of the paametes ae the solutios of equatios (4)-(8) fo the δ, ε ad, whe ξ ae eplaced by thei estimatos ξˆ. Equatios (8) ca be expessed as [{ pc + ( p) C + pc } { pc4 + ( p) C5 + pc6 } + { pc7 + ( p) C η = { pd + ( p) D + pd } { pd4 + ( p) D5 + pd6 } (0) The coefficiets used i the equatio (0) ae give i Table. 8 + pc 9 }] Table : Coefficiets of the Equatio (0) C = [ B ( )] C = [ B (/ )] C = [ B ( )] : 4 : 7 = [ B: ( )] = [ B: ( )] 4 = [ B: ( )] : 5 : 8 = [ B: (/ )] = [ B: (/ )] 5 = [ B: (/ )] : 6 : 9 = [ B: ( )] = [ B: ( )] 6 = [ B: ( )] C = [ B ( )] C = [ B (/ )] C = [ B ( )] C C C D D D D D D Equatios (0) does ot give a explicit solutio fo ad has to be solved

7 Fittig the geealized logistic distibutio 669 umeically usig a iteative method. Fo ease of computatio the followig equatio with good accuacy has bee costucted based o (0) ad (8) = A η + A () 0 η whee the values of coefficiet A 0 ad A fo ad p i the ages 0 to 0.5 ae give i Table. Oce the value of ˆ is obtaied δˆ, ad εˆ ca be estimated successively fom equatios (5) ad (4) as ˆ ˆ ξ δ = () { Rˆ ) Rˆ )} : : ε = ˆ ξ ˆ{ δ ˆ ( )} () ˆ R p, : 4 Defiitio of L-Momets Let,...,, be a adom sample fom a cotiuous distibutio fuctio Q( u) = F ( u, ad let : :... : F (.) with quatile fuctio ) deote the coespodig ode statistics. The the th LMOM λ is give by (Hosig, 990) = λ ( ) E( : ), =,,... (4) = 0 The L i L-momets emphasizes that λ is a liea fuctio of the expected ode statistic. The expectatio of a ode statistic may be witte as E m! j m j [ : m ] = x( F) F ( F) df ( j )!( m j)! 0 j (5)

8 670 A. Shabi ad A. A. Jemai Table : Values of Coefficiets i Equatios () p = 0.05 p = 0.0 p = 0.5 p = 0.0 A 0 A A 0 A A 0 A A 0 A p = 0.5 p = 0.0 p = 0.5 p = 0.40 A 0 A A 0 A A 0 A A 0 A p = 0.45 p = 0.50 A 0 A A 0 A

9 Fittig the geealized logistic distibutio 67 The sample LMOM ae give by (Asquith, 007) j i i ( ) j= 0 j j = j ˆλ i: (6) i= 4. Method of L-momets The LMOM estimatos fo the GL distibutios ae give by ˆ ˆ λ = ˆ λ (7) ˆ ˆ λ δ = Γ( + ˆ) Γ( ˆ) (8) ˆ ˆ δ ˆ ε = λ { Γ( + ˆ) Γ( ˆ)} ˆ (9) 5 Simulatio Study A umbe of simulatio expeimets wee coducted to ivestigate the popeties of LQMOM estimatos fo the GL distibutio. A simulatio study was caied out, with seveal sample sizes, shape paametes of the GL populatio, ad etu peiods T. With o loss of geeality, the scale paamete ε was set equal to ad the locatio paamete δ was set equal to 0. The values = 0, 0, 0, 50, 00 wee chose, alog with values T = 0, 0, 00, 000 yeas ( F(x) = 0.90, 0.95, 0.99, 0.999) ad = -0.05, -0.5, -0.5, The values that ae cosideed costitute, fom the pactical stadpoit, a sample of small, modeate ad lage values. The simulatios was desiged to coside LQMOM with =0.05(0.05)0.5 ad p = 0.05(0.05)0.5. Fo each combiatio of, T ad pais (, p ), we geeated adomly 5000 samples fo = 0 ad 0, ad 000 samples fo 0 fom GL distibutio. The oot meas squae eos (RMSEs) fo the quatile estimates wee the calculated. Focus was o estimatig quatiles of the GL distibutio, sice they costitute the mai iteest i hydology ad may othe applied statistical fields. Fo each combiatio of sample size ad etu peiods, the RMSE of quatile estimatos fo the LMOM ad LQMOM methods wee ecoded ad ae show i Table. Fo estimatig the quatiles of the GL

10 67 A. Shabi ad A. A. Jemai distibutio, the LQMOM method outpefoms the LMOM method whe the sample size is small ( 0 ) ad = -0.05, -0.5, Figue shows the RMSE of 00-yea quatiles estimated by LQMOM ad LMOM fo diffeet values of ad. Fo othe T values, simila esults wee obtaied, but ot epoted. Fom Figue it is also see RMSE fo Q (T ) deceases as icease. The simulatio of the peset study have show that the LQMOM method povides bette esults tha the LMOM methods oly fo small samples sizes ( 0 ) ad = -0.05, -0.5, While fo = -0.5, the LQMOM show simila pefomaces. Figue : RMSE fo estimates of T, T = 00 ( = -0.05, -0.5, -0.5, -0.5) 6 Hydological Example To compae the pefomace of LQMOM ad LMOM methods i a moe ealistic, two sets of aual maximum flood seies fo the Rive Kelvi ad Rive Spey, wee tae fom Ahmad et al.(988). The paamete estimates ad quatiles Q (F) fo T = 00 fo each data set, usig LQMOM ad LMOM ae give i Table 4.

11 Fittig the geealized logistic distibutio 67 Table : RMSE of Quatile Estimatos of GL Quatiles Method T = 0 T = 0 T = 00 T = LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM LQMOM LMOM

12 674 A. Shabi ad A. A. Jemai Table 4: Paametes of the Distibutio Fitted to the aual maximum flood seies fo the Rive Kelvi ad Rive Spey Optimal (, p) Paametes εˆ δˆ ˆ Rives Method Qˆ ( T = 00) Kelvi LQMOM (0.5, 0.5) LMOM Spey LQMOM (0.5, 0.75) LMOM Obseved ad computed fequecy cuves fo the two data sets ae plotted i Figue. The obseved data values ae plotted agaist the coespodig logistic educed vaiates L( F i /( Fi )), i =, K,, whee F i = ( i 0.44) /( + 0.) is the Gigoto (96) plottig positio fo the i th smallest of obsevatios. Fo the ive Kelvi, the fequecy cuves of LQMOM ad LMOM ae i faily close ageemet. Howeve, fo the ive Spey, the fequecy cuves obtaied by the LQMOM method is i close ageemet ad lie much close to the data tha LMOM. This suggests that fom the GL distibutio may easoably be fitted to the aual maximum flood seies, by the LQMOM tha the LMOM method. Figue : The Geealized Logistic Distibutio Fitted to Aual Maximum Flows Data fo the Rive (a) Kelvi ad (b) Spey Based o LQ-Momets ad L-Momets Method 7 Coclusios The LQ-momets ae costucted by usig fuctioal defiig the quic estimatos, such as the media, timea o Gastwith, i places of expectatios i L-momets ae developed fo GL distibutio. The quic estimatos usig

13 Fittig the geealized logistic distibutio 675 weighted eel estimatos ae itoduced fo estimatig the quatile fuctios. A compaiso with competitive estimatio pocedues, amely, the LQMOM ad LMOM methods, was doe. Root mea squae eo was used as pefomace idex. The study evealed that whe optimal combiatios pais ad p ae used, the LQMOM method i may cases pefoms bette tha the LMOM with espect to RMSE i estimatig high quatiles ad shape paamete fo small o modeate sample size. The hydological example showed that the fit of a model such LQMOM depeds sigificatly o the choice of combiatio of ad p that ae used. Aalysis study has demostated that the choice ad p of quic estimato fo LQMOM based o the media, timeas o Gaswith is ot optimal fo the estimatio of GL quatiles. Acowledgmet. The authos thafully acowledged the fiacial suppot povided by Uivesiti Teologi Malaysia ude IRGS Gat (vot. 779). The autho also would lie to tha the Depatmet of Iigatio ad Daiage, Miisty of atual Resouces ad Eviomet, Malaysia fo povidig the data. Refeeces [] M.I. Ahmad, C.D. Siclai ad A. Weitty, A. Log-Logistic flood fequecy aalysis. Joual of Hydology, 98(987), [] M. Alasasbeh ad M.Z. Raqab. Estiamtio of the Geealized Distibutio Paamets: Comapaative Study. Statistical Methodology. 8(009), [] S. Ai ad A.A. Jemai, LQ-Momet: Applicatio to the Log-Nomal Distibutio. Joual of Mathematics ad Statistics, ()(006), [4] S. Ai ad A.A. Jemai, LQ-Momets Fo Statistical Aalysis Of Exteme Evets. Joual of Mode Applied Statistical Methods, 6()(007), 8-8. [5] S. Ai ad A.A. Jemai, LQ-Momet: Applicatio to the Geealized Exteme Value. Joual of Applied Scieces, 7()(007), 5-0. [6] S. Ai ad A.A. Jemai, Estimatio of the Exteme Value Type I by the Method of LQ-Momet. Joual of Mathematics ad Statistics 5(4)(009), W.H. Asquith, L-Momets ad TL-momets of the Geealized Lambda Distibutio. Computatioal Statistics & Data Aalysis. 5(007), N. Balaisha ad A. Hossai, Ifeece fo the Type II geealized logistic distibutio ude pogessive Type II cesoig. Joual of Statistical Computatio ad Simulatio, 77()(007), 0-0. [7] R.D. Gupta ad D. Kudu, Geelaized expoetial distibutio: diffeet methods of estimatio, Joual of Statistical Computatio ad Simulatio, 59(004), 5-7.

14 676 A. Shabi ad A. A. Jemai [8] J.R.M. Hosig, L-momets: Aalysis ad Estimatio of Distibutio Usig Lie Combiatios of Ode Statistics. J. Roy. Statist. Se. B. 5(990), [9] J.R.M. Hosig ad J.R. Wallis, Regioal Fequecy Aalysis: A appoach based o L-Momets Cambidge: Cambidge Uivesity pess (997). [0] M.L. Huag ad P. Bill, A Level Cossig Quatile Estimatio Method. Statistics & Pobability Lettes 45(999), -9. [] IH. The Flood Estimatio Hadboo, Istitute of Hydology. Walligfod UK. (999). [] J. Kavae, Estimatio of quatile mixtues via L-momets ad timmed L-momets. Computatioal Statistics & Data Aalysis, 5(006), [] G.S. Mudhola ad A.D. Hutso, LQ-Momets: Aalogs of L-Momets. Joual of Statistical Plaig ad Ifeece, 7(998), [4] A.Saaasubamaia ad K. Siivasa, Ivestigatio ad Compaiso of Samplig Popeties of L-Momets ad Covetioal Momets. Joual of Hydology, 8(999), -4. [5] S.J. Sheathe ad J.S. Mao, Keel Quatile Estimatos. Joual of the Ameica Statistical Associatio, 85 (990), [6] V.P. Sigh ad M.A. Ahmad, Compaative Evaluatio of The Estimatos of The Thee-Paamete Geealized Paeto Distibutio. Joual of Statistical Computatio ad Simulatio, 74() (004), [7] Q.J. Wag, Diect Sample Estimatos of L Momets. Wate Resouces Reseach, Techical Note, (996), Received: Mach, 0