TL-Moments and L-Moments Estimation for the Generalized Pareto Distribution

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1 Applied Mathematical Sciences, Vol. 3, 2009, no. 1, TL-Moments L-Moments Estimation fo the Genealized Paeto Distibution Ibahim B. Abdul-Moniem Madina Highe Institute fo Management Technology Madina Academy, Giza, Egypt Youssef M. Selim Ministy of Infomation, Caio, Egypt Abstact In this pape, the timmed L-moments (TL-moments) L-moments of the Genealized Paeto distibution (GPD) up to abitay ode will be deived used to obtain the fist fou TL-moments L- moments. TL-sewness, L-sewness, TL-utosis L-utosis ae hled fo the GPD. Using the fist two TL-moments L-moments, the unnown paametes fo the GPD can be estimated. A numeical illustate fo the new esults will be given. Keywods: GPD, TL-moments, L-moments, sewness, utosis, Method of TL-moments L-moments estimation, Beta function, Gamma function, Ode statistics 1 Intoduction The method of L-moment estimatos have ecently appeaed. Hosing (1990) gives estimatos fo log-nomal, gamma genealized exteme value distibutions. L-moment estimatos fo genealized Rayleigh distibution was intoduced by Kundu Raqab (2005). avanen (2006) applied the method of L-moment estimatos to estimate the paametes of polynomial quantile mixtue. He intoduced the mixtue composed of two paametic families, ae the nomal-polynomial quantile Cauchy-polynomial quantile. The stad method to compute the L-moment estimatos is to equate the sample L-moments with the coesponding population L-moments. A population L- moment L is defined to be a cetain linea function of the expectations of

2 44 Ibahim B. Abdul-Moniem Youssef M. Selim the ode statistics Y 1:,Y 2:,..., Y : in a conceptual om sample of size fom the undelying population. Fo example, L 1 = E(Y 1:1 ), which is the same as the population mean, is defined in tems of a conceptual sample of size = 1, while L 2 =(1/2)E(Y 2:2 Y 1:2 ), an altenative to the population stad deviation, is defined in tems of a conceptual sample of size =2. Similaly, the L-moments L 3 L 4 ae altenatives to the un-scaled measues of sewness utosis μ 3 μ 4 espectively. See Silito (1969). Compaed to the conventional moments, L-moments have lowe sample vaiances ae moe obust against outlies. Elami Seheult (2003) intoduced an extension of L-moments called TL-moments. TL-moments ae moe obust than L-moments exist even if the distibution does not have a mean, fo example the TL-moments ae existed fo Cauchy distibution. Abdul-Moniem (2007) applied the method of L-moment TL-moment estimatos to estimate the paametes of exponential distibution. The following fomula gives the th TL-moments (see Elami Seheult (2003)). L (t) = 1 1 =0 ( 1) ( 1 )E(Y +t :+2t ), (1) whee t tae the values 1, 2, 3,... Note that the th L-moments can be obtained by taing t = 0. The GPD is defined by Abd Elfattah et. al (2007). They deived some well now distibutions as a special cases fom GPD. GPD has the following pobability density function fom: f(y; α,, λ, δ) = δα (y λ ) δ 1 [1+( y λ ) δ ] (α+1) ; y λ>0,α,&δ >0 (2) whee is the scale paamete, λ is the location paamete (α, δ) ae the shape paametes. The coesponding cumulative distibution function is F (y; α,, λ, δ) =1 [1+( y λ ) δ ] α. (3) The main aim of this pape is to deive TL-moments L-moments of the GPD up to abitay ode using it to estimate the unnown paametes. This pape is oganized as follows: in Section 2, we intoduced population TLmoments TL-moment estimatos fo the GPD. The population L-moments L-moment estimatos fo the GPD was pesented in Section 3. In Section 4, A numeical illustate fo the new esults will be given. 2 TL-moments fo the GPD In this section, the population TL-moment of ode fo the GPD will be obtained. The sample TL-moments the TL-moments estimatos also discussed.

3 Genealized Paeto distibution Population TL-moments Using fomula (1) two functions (2) (3), the TL-moment of ode fo the GPD taing the following fom L (t) = 1 1 =0 ( 1) ( 1 ( +2t)! ) ( + t 1)!(t + )! (I), whee I = λ y[1 [1+( y λ [1 + y λ )δ ] α ] +t 1 δα ( y λ )δ 1 dy )δ ] α(t++1)+1 By exping [1 [1+( y λ )δ ] α ] +t 1 binomially, we get I = +t 1 ( + t 1 j )( 1) j λ y δα ( y λ )δ 1 [1+( y λ dy )δ ] α(t++j+1)+1 let z =( y λ )δ, this led to y = z 1 δ + λ J = δ( y λ )δ 1, then I = +t 1 ( + t 1 j )( 1) j α 0 [1 + z] z 1 δ + λ α(t++j+1)+1 dy = The L (t) +t 1 becomes ( + t 1 j )( 1) j α[β(1 + 1 δ,α(t + + j +1) 1 δ ) λ + α(t + + j +1) ] L (t) = 1 1 =0 ( 1) ( 1 ( +2t)! ) ( + t 1)!(t + )! +t 1 ( + t 1 j ) ( 1) j α[β(1 + 1 δ,α(t + + j +1) 1 δ )+ λ α(t + + j +1) ] (4) whee, t =1, 2, 3,... Hee, we tae t = 1 (see Elami Seheult (2003)) then equation (4) becomes

4 46 Ibahim B. Abdul-Moniem Youssef M. Selim = 1 1 =0 ( 1) ( 1 ( + 2)! ) ( ) ( )!(1 + )! j ( 1) j α[β(1 + 1 δ,α( + j +2) 1 δ )+ λ α( + j +2) ] (5) whee =1, 2, 3,...; α, λ, δ >0. The fist fou TL-moments can be obtained by taing =1, 2, 3 4 in (5) as follows 1 = Γ(1 + 1)[3Γ(3α)Γ(2α 1) 2Γ(2α)Γ(3α 1)] + λ (6) Γ(2α)Γ(3α) 2 = 3Γ(1 + 1)[Γ(4α)Γ(2α 1) + Γ(2α)Γ(4α 1)] Γ(2α)Γ(4α) 6Γ(1 + 1)Γ(3α 1) Γ(3α) 3 = 10Γ(1 + 1)[Γ(3α)Γ(2α 1) 4Γ(2α)Γ(3α 1)] 3Γ(2α)Γ(3α) + 10Γ(1 + 1)[5Γ(5α)Γ(4α 1) 2Γ(4α)Γ(5α 1)] 3Γ(4α)Γ(5α) 4 = 15Γ(1 + 1)[Γ(4α)Γ(2α 1) + 15Γ(2α)Γ(4α 1)] 4Γ(2α)Γ(4α) + 35Γ(1 + 1 )[Γ(5α)Γ(6α 1 ) 3Γ(6α)Γ(5α 1 )] 2Γ(5α)Γ(6α) 25Γ(1 + 1)Γ(3α 1) Γ(3α) The TL-sewness ( 3 ) TL-utosis ( 4 ) will be (7) (8) (9) 3 = L(1) 3 2 = 10Γ(4α)[Γ(3α)Γ(2α 1 δ ) 4Γ(2α)Γ(3α 1 δ )] 9Ψ(α, δ) + 10Γ(2α)Γ(3α)[5Γ(5α)Γ(4α 1 δ ) 2Γ(4α)Γ(5α 1 δ )] 9Γ(5α)Ψ(α, δ) (10)

5 Genealized Paeto distibution 47 whee 4 = L(1) 4 2 = 5Γ(4α)[3Γ(3α)Γ(2α 1 δ ) 20Γ(2α)Γ(3α 1 δ )] 12Ψ(α, δ) + 15Γ(2α)Γ(3α)[15Γ(5α)Γ(4α 1) 14Γ(4α)Γ(5α 1)] 12Γ(5α)Ψ(α, δ) + 70Γ(2α)Γ(3α)Γ(4α)Γ(6α 1) δ 12Γ(6α)Ψ(α, δ) Ψ(α, δ) = Γ(4α)[Γ(3α)Γ(2α 1 δ ) 2Γ(2α)Γ(3α 1 δ )] +Γ(2α)Γ(3α)Γ(4α 1 δ ) (11) 3 Sample TL-moments TL-moment estimatos TL-moments can be estimated fom a sample as linea combination of ode statistics. Elami Seheult (2003) pesent the following estimato fo sample TL-moments: l (t) = 1 n t i=t+1 1 =0 ( 1) ( 1 ( i 1 )( + t 1 )( n i t + ) x i:n (12) n +2t ) whee a b fo all ( a b ) x i:n denotes the i th ode statistic in a sample of size n. Fom (6), (7) (12) with α δ ae nown t = 1, we can get the TL-moment estimato fo (ˆ TL ) λ(ˆλ TL ) as follows l (1) 1 = n 1 6 n(n 1)(n 2) i=2 (i 1)(n i)x i:n = ˆ TL Γ(1 + 1 δ )[3Γ(3α)Γ(2α 1 δ ) 2Γ(2α)Γ(3α 1 δ )] Γ(2α)Γ(3α) + ˆλ TL, (13)

6 48 Ibahim B. Abdul-Moniem Youssef M. Selim l (1) 2 = n 1 12 n(n 1)(n 2)(n 3) { ( i 1 2 n 2 i=2 ( i 1 1 i=3 )( n i 2 )x i:n } )( n i 1 = 3ˆ TL Γ(1 + 1)[Γ(4α)Γ(2α 1) + Γ(2α)Γ(4α 1)] Γ(2α)Γ(4α) 6ˆ TL Γ(1 + 1)Γ(3α 1) Γ(3α) By solving equations (13) (14), we get )x i:n (14) ˆ TL = l (1) 2 { 3Γ(1 + 1 δ )[Γ(4α)Γ(2α 1 δ ) + Γ(2α)Γ(4α 1 δ )] Γ(2α)Γ(4α) 6Γ(1 + 1)Γ(3α 1) }, (15) Γ(3α) ˆλ TL = l (1) 1 ˆ TL Γ(1 + 1 δ )[3Γ(3α)Γ(2α 1 δ ) 2Γ(2α)Γ(3α 1 δ )] Γ(2α)Γ(3α) (16) 4 L-moments fo the GPD In this section, the population L-moment of ode fo the GPD as a special case fom fomula (4) will be intoduced. Sample L-moments L-moments estimatos also studied. 4.1 Population L-moments Hee, the population L-moment of ode fo the GPD as a special case fom (4) by taing t = 0 will be 1 L = ( 1) ( 1 =0 1 ) 2 ( 1 j )( 1) j α[β(1 + 1 δ,α( + j +1) 1 δ )+ λ α( + j +1) ] (17) The fist fou L-moments can be obtained by taing =1, 2, 3 4 in (17) as follows L 1 = Γ(1 + 1)Γ(α 1) + λ, (18) Γ(α)

7 Genealized Paeto distibution 49 L 2 = Γ(1 + 1)[Γ(2α)Γ(α 1) Γ(α)Γ(2α 1)] Γ(α)Γ(2α) L 3 = Γ(1 + 1)[Γ(2α)Γ(α 1) 3Γ(α)Γ(2α 1)] Γ(α)Γ(2α) + 2Γ(1 + 1)Γ(3α 1) Γ(3α) L 4 = Γ(1 + 1)[Γ(2α)Γ(α 1) 6Γ(α)Γ(2α 1)] Γ(α)Γ(2α) + 5Γ(1 + 1 )[2Γ(4α)Γ(3α 1 ) Γ(3α)Γ(4α 1 )] Γ(3α)Γ(4α) The L-sewness (τ 3 ) L-utosis (τ 4 ) will be (19) (20) (21) τ 3 = L 3 = Γ(2α)Γ(3α)Γ(α 1 ) 3Γ(α)Γ(3α)Γ(2α 1 ) L 2 Γ(3α)[Γ(2α)Γ(α 1) Γ(α)Γ(2α 1)] 2Γ(α)Γ(2α)Γ(3α 1 δ + ) Γ(3α)[Γ(2α)Γ(α 1) Γ(α)Γ(2α 1)] (22) τ 4 = L 4 L 2 = Γ(2α)Γ(α 1 δ ) 6Γ(α)Γ(2α 1 δ ) Γ(2α)Γ(α 1 δ ) Γ(α)Γ(2α 1 δ ) + 5Γ(α)Γ(2α)[2Γ(4α)Γ(3α 1 δ ) Γ(3α)Γ(4α 1 δ )] Γ(3α)Γ(4α)[Γ(2α)Γ(α 1 δ ) Γ(α)Γ(2α 1 δ )] (23) 4.2 Sample L-moments L-moment estimatos Sample L-moments can be estimated fom (12) by taing t = 0 as follows l = 1 n i=1 1 =0 ( 1) ( 1 i 1 )( 1 )( n i ) ( n x i:n (24) ) whee x i:n as above. Fom (18), (19) (24) with α δ ae nown, the L-moment estimato fo (ˆ L ) λ(ˆλ L ) will be l 1 = 1 n n i=1 x i:n = x = ˆ L Γ(1 + 1 δ )Γ(α 1 δ ) Γ(α) + ˆλ L, (25)

8 50 Ibahim B. Abdul-Moniem Youssef M. Selim l 2 = 2 n(n 1) n (i 1)x i:n x i=1 = ˆ L Γ(1 + 1)[Γ(2α)Γ(α 1) Γ(α)Γ(2α 1)] Γ(α)Γ(2α) By solving equations (25) (26), we get (26) ˆ L = l 2 Γ(α)Γ(2α) Γ(1 + 1 δ )[Γ(2α)Γ(α 1 δ ) Γ(α)Γ(2α 1 δ )] (27) ˆλ L = l 1 ˆ L Γ(1 + 1 δ )Γ(α 1 δ ) Γ(α) (28) 5 A numeical illustation By geneating samples of size 10(10)40 with eplications. Applying the pogam of Mathcad (2001), the estimates thei mean squae eo (MSE) of the unnown paametes λ using equations (15), (16), (27) (28) ae computed. Table (1) pesents the estimates of λ its MSEs using the exact value of λ = 2 with diffeent values of =0.2, 0.4, 0.6, 0.8, 1, 1.2, Table(1) Estimates MSEs of λ n=10 n=20 n=30 n= ˆλL (0.0023) (0.0011) (0.0008) (0.0006) ˆλ TL (0.0017) (0.0007) (0.0005) (0.0003) ˆ L (0.0139) (0.0068) (0.0051) (0.0035) ˆ TL (0.0082) (0.0037) (0.0025) (0.0017) 0.4 ˆλL (0.0088) (0.0041) (0.0032) (0.0022) ˆλ TL (0.0072) (0.0028) (0.0018) (0.0013) ˆ L (0.0563) (0.0261) (0.0202) (0.0141) ˆ TL (0.0354) (0.0148) (0.0098) (0.0069) 0.6 ˆλL (0.0199) (0.0101) (0.0065) (0.0049) ˆλ TL (0.0162) (0.0065) (0.0041) (0.0029) ˆ L (0.1266) (0.0637) (0.0418) (0.0317) ˆ TL (0.0796) (0.0342) (0.0221) (0.0159) 0.8 ˆλL (0.0354) (0.0189) (0.0115) (0.0087) ˆλ TL (0.0284) (0.0114) (0.0072) (0.0052) ˆ L (0.2187) (0.1212) (0.0744) (0.0563) ˆ TL (0.1400) (0.0609) (0.0392) (0.0282)

9 Genealized Paeto distibution 51 Table(1) Continued n=10 n=20 n=30 n= ˆλL (0.0553) (0.0256) (0.0183) (0.0152) ˆλ TL (0.0444) (0.0181) (0.0112) (0.0083) ˆ L (0.3418) (0.1647) (0.1215) (0.0973) ˆ TL (0.2188) (0.0952) (0.0631) (0.0463) 1.2 ˆλL (0.1185) (0.0392) (0.0288) (0.0218) ˆλ TL (0.0633) (0.0251) (0.0161) (0.0119) ˆ L (0.7124) (0.2482) (0.1846) (0.1402) ˆ TL (0.3094) (0.1325) (0.0886) (0.0667) 1.4 ˆλL (0.1612) (0.0568) (0.0346) (0.0280) ˆλ TL (0.0862) (0.0355) (0.0222) (0.0161) ˆ L (0.9697) (0.3612) (0.2251) (0.1783) ˆ TL (0.4212) (0.1871) (0.1215) (0.0882) 1.6 ˆλL (0.1476) (0.0683) (0.0482) (0.0388) ˆλ TL (0.1123) (0.0444) (0.0282) (0.0208) ˆ L (0.9117) (0.4397) (0.3081) (0.2451) ˆ TL (0.5492) (0.2408) (0.1523) (0.1126) The MSEs ae epoted within bacets against each estimates. Fom Table (1), one can show that The values of MSEs decease as n inceases. The values of MSEs fo ˆλ TL ˆ TL ae smalle than the coesponding values fo ˆλ L ˆ L. The values of MSEs fo ˆλ TL, ˆ TL, ˆλ L ˆ L incease as the exact value of inceases. Refeences [1] A. M. Abd Elfattah, E. A. Elshepieny E. A.Hussein, A new Genealized Paeto distibution, InteStat Decembe, # 1 (2007). [2] I. B. Abdul-Moniem, L-moments TL-moments estimation fo the Exponential distibution, Fa East J.Theo. Stat. 23 (1)(2007), [3] E. A. Elami, A. H. Seheult, Timmed L-moments, Computational Statistics & Data Analysis, 43 (2003),

10 52 Ibahim B. Abdul-Moniem Youssef M. Selim [4] J. Hosing, L-moments: Analysis estimation of distibutions using linea combinations of ode statistics, Jounal of Royal Statistical Society B 52 (1)(1990), [5] J. Kavanen, Estimation of quantile mixtues via L-moments timmed L-moments, Computational Statistics & Data Analysis, 51(2)(2006), [6] D. Kundu, M. Z. Raqab, Genealized Genealized Paeto distibution: diffeent methods of estimations, Computational Statistics & Data Analysis, 49 (2005), [7] G.P. Silito, Deivation of appoximations to the invese distibution function of a continuous univaiate population fom the ode statistics of a sample, Biometia 56(1969), Received: May 1, 2008