Research Article U-Statistic for Multivariate Stable Distributions

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1 Hinawi Probability an Statistics Volume 2017, Article ID , 12 pages Research Article U-Statistic for Multivariate Stable Distributions Mahi Teimouri, Saei Rezakhah, an Ael Mohammapour Department of Statistics, Faculty of Mathematics an Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 424HafezAve,Tehran15914,Iran Corresponence shoul be aresse to Ael Mohammapour; Receive 5 December 2016; Revise 9 February 2017; Accepte 19 February 2017; Publishe 3 April 2017 Acaemic Eitor: Steve Su Copyright 2017 Mahi Teimouri et al This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite A U-statistic for the tail inex of a multivariate stable ranom vector is given as an extension of the univariate case introuce by Fan (2006) Asymptotic normality an consistency of the propose U-statistic for the tail inex are prove theoretically The propose estimator is use to estimate the spectral measure The performance of both introuce tail inex an spectral measure estimators is compare with the known estimators by comprehensive simulations an real atasets 1 Introuction In recent years, stable istributions have receive extensive use in a vast number of fiels incluing physics, economics, finance, insurance, an telecommunications Different sorts of ata foun in applications arise from heavy taile or asymmetric istribution, where normal moels are clearly inappropriate In fact, stable istributions have theoretical unerpinnings to accurately moel a wie variety of processes Stable istribution has originate with the work of Lévy [1] There are a variety of ways to introuce a stable ranom vector In the following, two efinitions are propose for a stable ranom vector; see Samoronitsky an Taqqu [2] Definition 1 AranomvectorX =(X 1,,X ) T is sai to be stable in R if for any positive numbers A an B there are apositivenumberc an a vector D R such that AX 1 +BX 2 =CX + D, (1) where X 1 an X 2 are inepenent an ientical copies of X an C=(A α +B α ) 1/α Definition 2 Let 0<α<2ThenX is a non-gaussian αstable ranom vector in R if there exist a finite measure Γ on the unit sphere S ={x =(x 1,,x ) T R x, x =1} an a vector μ =(μ 1,,μ ) T R such that t, s α [1 i sgn t, s tan ( πα )] Γ (s) +i t, μ, α { S φ X (t) = log E(exp (i t,x )) = 2 =1, { t, s [1 + i sgn t, s 2 log t, s ]Γ(s) +i t, μ, α=1, { S π (2) where t, s = i=1 t is i for t =(t 1,,t ) T, s =(s 1,,s ) T, i 2 = 1,ansgn( ) enotes the sign function The pair (Γ, μ) is unique The parameter α, in Definitions 1 an 2, is calle tail inex AranomvectorX is sai to be a strictly α-stable ranom vector in R if μ = 0 for α = 1; see Samoronitsky an Taqqu[2]WenotethatX is strictly α-stable, in the sense of Definition 1, if D = 0 Throughout we assume that X is strictly α-stable an α =1 The probability ensity function of a stable istribution has no close-form expression an moments with orers greater than or equal to α are not

2 2 Probability an Statistics finite for the members of this class The two aforementione ifficulties make statistical inference about the parameters of a stable istribution har However, a series of contributions has permitte inference about the parameters of univariate an multivariate stable istributions For example, in the univariate case, maximum likelihoo (ML) estimation was stuie first by DuMouchel (1971) an then by Nolan [3] Although the ML approach leas to an efficient estimate for samples of large size, it involves numerical complexities A program, calle STABLE uses a cubic spline interpolation of stable ensities for this purpose; see Nolan [4] STABLE estimates all four parameters of a stable istribution for α 04 Sample quantile (SQ) technique is another approach propose by McCulloch [5] The results are simple an consistent estimators of all four parameters base on five sample quantiles The empirical characteristic function (CF) is suggeste by Kogon an Williams [6] The CF an SQ methos work well but are not as efficient as the ML metho As the last approach consiere here, U-statistics for the tail inex an scale parameters of a univariate strictly stable istribution are introuce by Fan [7] In multivariate case, the focus of interest is the spectral measure estimation Among them, we refer to Nolan et al [8], Pivato an Seco [9], Ogata [10], an Mohammai et al [11] The structure of the paper is as follows In Section 2, new estimators for the tail inex an spectral measure of a strictly stable istribution are presente which is an extension of the U-statistic propose by Fan [7] for the univariate case A comprehensive simulation stuy is performe in Section 3 to compare the performance of the introuce estimators an theknownestimatorstworealatasetsareanalyzeinthis sectiontoillustratetheperformanceoftheproposemetho 2 New Estimators This section consists of two subsections Firstly, we propose an estimator for the tail inex Seconly, an estimator for the spectral measure is given 21 Estimation of Tail Inex The main result of this section is given in Theorem 4, which gives U-statistic for the inverse of tail inex of a strictly stable istribution We present the main result in the light of Lemma 3 given as follows The proofs are given in the Appenix Lemma 3 Let X =(X 1,,X ) T be a -imensional strictly stable ranom vector Then, Var log X is finite, where enotes the Eucliean norm Theorem 4 Let x 1,,x n be a sequence of n observations from a -imensional strictly stable ranom vector Then U n =( n 1 2 ) H(x i, x j ), (3) 1 i<j n where H (x i, x j ) = log x i + x j log 2 is the U-statistic for 1/α log x i + log x j 2 log 2 (4) As it is seen, from Theorem 4, the introuce U-statistic is an unbiase estimator for 1/α Hereafter,wewrite α MU = 1/U n as introuce estimator for α Here,subscriptMU inicates that α MU is constructe base on multivariate Ustatistic efine in Theorem 4 It shoul be note that when the true value of α is near two, the kernel given in (4) coul be less than 05 So, α MU is greater than two In this case, we set α MU =2 22 Spectral Measure Estimation We use α MU to estimate an m-point iscrete approximation to the exact spectral measure of the form Γ ( ) = m j=1 γ j I sj ( ), (5) where γ j is a mass at point s j in the unit sphere S an I sj ( ) is an inicator function at point s j ;forj = 1,,m, see Byczkowski et al [12] To estimate Γ( ), we replace Definition2 for a strictly -imensional stable ranom vector with φ X (t) = = m j=1 m j=1 t, s α j [1 i sgn t, sj G(α,t, s j )] γ j, ψ(t, s j,α)γ j, where G(α, t, s j )=tan(πα/2) for α =1an 2/π log t, s j for α=1define Λ ψ(t 1, s 1,α) ψ(t 2, s 1,α) =( ψ(t 1, s 2,α) ψ(t 1, s m,α) ψ(t 2, s 2,α) ψ(t 2, s m,α) ), ψ(t m, s 1,α) ψ(t m, s 2,α) ψ(t m, s m,α) (6) V =( log φ X (t 1 ),, log φ X (t m )) T, (8) where t j =(t j1,,t j ) T S,forj = 1,,m Using (7) an (8), both sies of (6) are connecte together through the following linear system: (7) V = Λγ, (9) where γ =(γ 1,,γ m ) T Assuming that Λ in (9) is nonsingular, then γ = Λ 1 VHence,weestimatethevectorofthe masses as where Λ γ = Λ 1 V, (10) ψ(t 1, s 1, α MU ) ψ(t 1, s 2, α MU ) ψ(t 1, s m, α MU ) ψ(t 2, s 1, α MU ) ψ(t 2, s 2, α MU ) ψ(t 2, s m, α MU ) =( ), ψ(t m, s 1, α MU ) ψ(t m, s 2, α MU ) ψ(t m, s m, α MU )

3 Probability an Statistics 3 V n =( log 1 exp {i t n 1, x i },, log 1 exp {i t i=1 n m, x i }), i=1 n T (11) in which x i is i-th vector observation in ranom sample of size n Due to the stanar error of V, wehavetwoproblems with irect use of (10) Firstly, γ may be complex, an seconly, its real part may be quite negative Since Λ an V are complex while gamma is constraine to be real (an nonnegative), the Eucliean norm use by McCulloch [13] an Nolan et al [8] must be replace with the complex moulus to solve both problems in a novel way For this, we use the nnls( ) library in the R package In the next section, the estimate spectral measure γ,baseon α MU,isshownby γ MU We note that another estimator of γ canbeconstructe by separating both of the real an imaginary parts in the structure of V Butsimulationresultsshowthatconstructe estimator gives the same performance 3 Simulation Stuy This section is in three parts Firstly, we stuy the performance of the propose estimator with the known ones for estimating the tail inex Seconly, we compare the performance of the spectral measure estimator evelope through the introuce tail inex estimator with the known approaches In the last subsection, we give a real ata example to illustrate the efficiency of the propose estimators 31 Performance Analysis of the Tail Inex Estimators Here, we perform a simulation stuy to compare the performance of α MU an four other estimators for α,incluing(1) α ML,(2) α SQ,(3) α CF,an(4) α MM ThefirstthreecompetitorsareML, SQ, an CF estimations for the tail inex, respectively Each of three competitors is obtaine as α PROJ =1/m m j=1 α(u j) after projecting the -imensional stable ranom vector using u j, X Here,m isthenumberofmasses,u j =(u j1,,u j ) T is an arbitrary unit vector, an X is the -imensional stable ranom vector It is worth noting that the first three competitors are compute by the help of STABLE software after projecting The fourth estimator, that is, α MM,isthesecon estimator for tail inex propose by Mohammai et al [11] We compare both the bias an root mean-square error () of estimators for 500 replications of samples of size n = 500 an 5000 of a bivariate stable ranom vector generate by the metho given in Moarres an Nolan [14] We use two settings for iscrete spectral measure with m=8masses, incluing γ 1 = (01, 02, 03, 04, 01, 02, 03, 04) T an γ 2 = (0, 01, 07, 03, 07, 03, 07, 01) T In both cases, masses are concentrate on points s j = (cos(2π(j 1)/m), sin(2π(j 1)/m)) T for j = 1,,8 In the first case that ata are coming from a stable istribution with γ 1, we generate t j from a uniform istribution on the unit sphere S For the secon case, we set t j = s j Biases an s for α = (01 : 01 : 09, 095, 105, 11 : 01 : 19, 195, 2) are showninfigures1an2asfigure1shows,whenγ 1 = (01, 02, 03, 04, 01, 02, 03, 04) T,weobservethat α MU is more efficient than α SQ for n = 5000 Also,itworksbetter than α MM in terms of (for α 18) Base on Figure 2, when γ 2 = (0, 01, 07, 03, 07, 03, 07, 01) T,weobservethat α MU is more efficient than other methos when α < 14 an n = 5000 in the sense of Also, when α < 17 an n = 500, α MU is more efficient than α SQ, α CF,an α MM with respect to 32 Performance Analysis of the Spectral Measure Estimators Here, we compare the performance of the estimator for masses of spectral measure γ = (γ 1,,γ m ) T constructe base on U-statistic, γ MU with the other four known estimators for the spectral measure The competitors are three types of estimators for γ base on empirical characteristic function metho: (1) γ MLE-cf ;(2) γ SQ-cf ;(3) γ CF-cf ;an(4) Mohammai et al [11] estimator for γ, γ MM For computing γ MLE-cf, γ SQ-cf,an γ CF-cf,weusecomman mvstablefit(x, nspectral, metho1, metho2, param) in the STABLE program, where x is ata vector, nspectral is number of spectral measure masses, metho1 is the metho to use for estimating parameters of univariate stable istribution, that is, MLE, SQ, an CF (corresponing coes in STABLE are 1, 2, an 3, respectively), metho2 is the metho to use for estimating parameters of bivariate stable istribution (we set metho2 = 2 which correspons to empirical characteristic function approach, cf), an param refers to kin of parameterization Here, we set param =1 since we are using the characteristic function in (2) More information about the first three competitors is given in Robust Analysis Inc [15] The estimators γ MLE-cf, γ SQ-cf, γ CF-cf, an γ MM are obtaine by substituting α ML, α SQ, α CF,an α MM into (7) an then solving linear system (10), respectively Comparisons are base on the of γ j,for j=1,,m,whichisefineas 1/N N i=1 ( γ ij γ ij ) 2,where N is the number of iterations an γ ij is the estimation of jth component of γ at ith iteration We consier five scenarios for the structure of iscrete spectral measure as follows (1) Inepenent case: γ = (1/4, 0, 1/4, 0, 1/4, 0, 1/4, 0) T (2) Symmetric case: γ = (01, 02, 03, 04, 01, 02, 03, 04) T (3) Uniform case: γ = (1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/ 8) T (4) Triangle case: γ = (0, 01, 07, 03, 07, 03, 07, 01) T (5) Exchangeable case: γ = (01, 02, 01, 04, 03, 02, 03, 04) T We note that the first an the thir scenarios above are similar to Examples 2 an 1 of Nolan et al [8], respectively The fourth scenario is calle Triangle since corresponing ensity contour plot is similar to a triangle For each of the above five scenarios, we arrange the settings of simulation as m=8, α =125;175;n = 2000; 5000 (n is sample size),

4 4 Probability an Statistics Bias MU ML MU ML (a) (b) Bias MU ML MU ML (c) () Figure 1: Biases an s of estimators when ata are generate from a strictly stable istribution with iscrete spectral measure γ 1 = (01, 02, 03, 04, 01, 02, 03, 04) T (a) Bias when n = 500, (b) when n = 500, (c) bias when n = 5000, an () when n = 5000 an N = 500 It shoul be note that masses are locate at s j =(cos(2π(j 1)/m), sin(2π(j 1)/m)) T,forj = 1,,m, an components of t j = (t j1,,t j ) T are generate from a uniform istribution on the unit sphere S Theresultsof simulations are given in Figures 3 6 As it is seen, γ MU shows better performance than γ MM 33 Real Data Analysis Here,wegivetwoexamplesInthe first example, ajuste aily log-return (in percent) for the 30 stocks at the Dow Jones inex is collecte between January 3, 2000, an December 31, 2004 The log-return percent of 1247 closing prices has been compute for AXP (American Express Company) an MRK (Merck & Co Inc) stocks after multiplying the aily log-return by 100; see Nolan [16] The scatter plot of AXP an MRK stocks log-return percent values, X = (AXP, MRK) T, is shown in Figure 7 We use a bivariate α-stable istribution with m = 12 points of masses for spectral measure aresse by s j =(cos(2π(j 1)/m), sin(2π(j 1)/m)) T,forj = 1,,12Weestimatethe

5 Probability an Statistics Bias MU ML MU ML (a) (b) Bias MU ML MU ML (c) () Figure 2: Biases an s of estimators when ata are generate from a strictly stable istribution with iscrete spectral measure γ 2 = (0, 01, 07, 03, 07, 03, 07, 01) T (a) Bias when n = 500, (b) when n = 500, (c) bias when n = 5000, an () when n = 5000 location parameter as μ ML-cf = ( 3438E 07, 2402E 07) T So,astrictlyα-stable istribution is fitte to the Y = (X μ ML-cf ) T Forthis,wesett j = s j,forj = 1,,12 Table1showstheresultsformoellingatathroughfive methos We note that estimate tail inices are α MU = 1581, α MM = 1734, α ML-cf = 1618, α SQ-cf = 1493, an α CF-cf = 1723 Asitisseen,estimatetailinices through estimators α MU an α ML-cf are closer together than the other estimators In the secon example, we focus on the cubic-root of the monthly average of river ischarge We choose ischarge of the Ora an Wisla rivers in Polan uring 1901 to 1986 (raw ata are in m 3 /s They are available at The scatter plot for cubic-root of Ora river ischarge versus cubic-root of Wisla river ischarge is shown in Figure 8 Setting m=8, s j =(cos(2π(j 1)/m), sin(2π(j 1)/m)) T,an

6 6 Probability an Statistics n = 2000 n = Inepenent Symmetric Uniform Figure 3: s of γ uner ifferent scenarios when α = 125 We use the following symbol scheme: for γ UM, I for γ MM,+for γ ML-cf, for γ SQ-cf,an for γ CF-cf t j = s j,forj=1,,8,weobtain μ ML-cf = (79837, 99947) T After fitting a strictly α-stable istribution to the shifte ata, results for estimating spectral measure are given in Table 2 Estimate tail inices are α MU = 1860, α MM = 1312, α ML-cf = 1813, α SQ-cf = 1936, an α CF-cf = 1962 Baseon results given in Table 2, estimate masses through estimators γ MU, γ ML-cf,an γ SQ-cf are closer together than the other estimators We compare here γ MU with γ ML-cf an γ SQ-cf since the latter estimators are among the best estimators for the masses as shown in the previous subsection

7 Probability an Statistics 7 n = 2000 n = Inepenent Symmetric Uniform Figure 4: s of γ uner ifferent scenarios when α = 175 We use the following symbol scheme: for γ UM, I for γ MM,+for γ ML-cf, for γ SQ-cf,an for γ CF-cf

8 8 Probability an Statistics n = 2000 n = Triangle Exchangeable Figure 5: s of γ uner ifferent scenarios when α = 125 We use the following symbol scheme: for γ UM, I for γ MM,+for γ ML-cf, for γ SQ-cf,an for γ CF-cf 4 Conclusion We compare the performance of the introuce U-statistic for the tail inex with the well-known methos, incluing maximum likelihoo, empirical characteristic function, sample quantile, an that introuce in Mohammai et al [11] through a simulation stuy In the sense of root meansquare error, it is prove that propose tail inex estimator always outperforms Mohammai et al [11] an SQ methos when α 14 This is while ML an CF methos show better performance than the propose estimator for large α, say α > 14 in terms of root mean-square error Simulation stuies for estimating the iscrete spectral measure γ uner five scenarios prove that estimator of γ baseonintrouce U-statistic shows, in terms of root mean-square error, better performance than Mohammai et al [11] estimator Analysis of two sets of real ata reveals that estimator of the tail inex an γ base on U-statistic shows expeient performance As some possible future works, firstly, we aim to introuce a U-statistic for the case of a nonzero location parameter Seconly, we look for methoology possibly base on a Ustatistic, to estimate tail, masses, an location parameters simultaneously Finally, recalling that the approach employe in this work is base on characteristic function, the iscrete spectral measure using α MU can be estimate through projection approach Appenix Proof of Lemma 3 We show that E(log 2 X ) < Suppose =2an p = P( X 1), p + =1 p, p = P(X 1 < 0, X 2 <0), p + = P(X 1 <0,X 2 >0), p + = P(X 1 >0,X 2 < 0),anp ++ = P(X 1 >0,X 2 >0)So E(log 2 X ) =E(log 2 X X <1)p +E(log 2 X X 1)p +

9 Probability an Statistics 9 n = 2000 n = Triangle Exchangeable Figure 6: s of γ uner ifferent scenarios when α = 175 We use the following symbol scheme: for γ UM, I for γ MM,+for γ ML-cf, for γ SQ-cf,an for γ CF-cf Table 1: Estimation results after fitting a strictly bivariate α-stable istribution to AXP an MRK stocks ata Estimator γ 1 γ 2 γ 3 γ 4 γ 5 γ 6 γ 7 γ 8 γ 9 γ 10 γ 11 γ 12 γ MU γ MM γ ML-cf γ SQ-cf γ CF-cf Table 2: Estimation results after fitting a strictly bivariate α-stable istribution to Ora an Wisla ischarge ata Estimator γ 1 γ 2 γ 3 γ 4 γ 5 γ 6 γ 7 γ 8 γ MU γ MM γ ML-cf γ SQ-cf γ CF-cf

10 10 Probability an Statistics +E(log 2 X X 1,X 1 <0,X 2 >0)p + MRK aily log-return percent E(log 2 X X 1,X 1 >0,X 2 <0)p + +E(log 2 X X 1,X 1 >0,X 2 >0)p ++ E(log 2 X 1 +X 2 X 1,X 1 <0,X 2 <0) p +E(log 2 X 1 +X 2 X 1,X 1 <0,X 2 >0) p + +E(log 2 X 1 X 2 X 1,X 1 >0,X 2 <0) AXP aily log-return percent Figure 7: Scatter plot for AXP versus MRK aily log-return percent, X =(AXP, MRK) T Cubic-root of Wisla river ischarge Cubic-root of Ora river ischarge Figure 8: Scatter plot for cubic-root of Ora an Wisla rivers ischarge p + +E(log 2 X 1 +X 2 X 1,X 1 >0,X 2 >0) p ++ 2E(log 2 X 1 +X 2 ) +2E(log 2 X 1 +X 2 ) Thus, E(log 2 X ) E(log 2 X )+2E(log 2 X 1 +X 2 ) +2E(log 2 X 1 X 2 ) Generally, for 2,onecanwrite E(log 2 X ) E(log 2 X ) i + ( i=0 i )E(log2 j +X i+1 + +X ), j=1x (A2) (A3) (A4) E(log 2 X X <1)p +E(log 2 X X 1)p + E(log 2 X ) +E(log 2 X X 1), (A1) where X, in the above, enotes one of the components of vector XItshoulbenotethatinequalityE(log 2 X X < 1) E(log 2 X X <1)hols irrespective of Onthe other han, E(log 2 X X 1) =E(log 2 X X 1,X 1 <0,X 2 <0)p where we aopt this convention that 0 j=1 X j = 0Let S(α, β, σ, μ = 0) stans for a univariate strictly stable ranom variable with tail inex α, scale parameter σ, an skewness parameter β ItiswellknownthatifX = (X 1,,X ) T is an α-stable ranom vector, then any linear combination of its components such as b i, X = j=1 b ijx j,for i i b i =( 1,, 1, 1,,1) T,followsastableistributionwith tail inex α, σ i =( S b i, s α 1/α Γ (s)), β i = 1 σ α i ( S b i, s [α] 1/α Γ (s)), (A5)

11 Probability an Statistics 11 where Γ( ) is spectral measure an x [r] = sgn(x) x r ;see Samoronitsky an Taqqu [2] It follows, from Kuruoglu [17], that if X S(α, β, σ, 0),then e(α,β,σ)=e(log 2 X ) = 7π2 6θ 2 +6[log (σ/ cos (θ)) +γ(1 α)] 2 π 2 α 2 (A6) 6α 2, where θ = arctan(β tan(πα/2))also, e(α,β i,σ i )=E(log 2 Z ) = 7π2 6θ 2 i +6[log (σ i /cos (θ i )) + γ (1 α)] 2 π 2 α 2 (A7) 6α 2, where Z= i j=1 X j +X i+1 + +X,fori = 0,,an θ i = arctan(β i tan(πα/2)) Parameters σ i an β i are efine in (A5) Finally, E (log 2 X ) e(α, β, σ) + i=0 ( i ) e (α, β i,σ i ) (A8) The proof is complete since all terms on the right-han sie of (A8) are finite Proof of Theorem 4 We rewrite Definition 1 as AX 1 +BX 2 =CX1 + D Setting A=1, B=1,anD = 0 in (A9), it yiels X 1 + X 2 =2 1/α X 1 (A9) (A10) By applying log-transformation, after taking the Eucliean norm,tobothsiesof(a10),wehave 1 α = log X 1 + X 2 log X 1 (A11) log 2 The right-han sie of (A11) can be use to efine a symmetric kernel of the form H(X 1, X 2 )= log X 1 + X 2 log 2 log X 1 + log X 2 2 log 2 (A12) To guarantee the asymptotic normality of the introuce Ustatistics for 1/α with kernel (A12), we nee to check that E(H(X 1, X 2 )) 2 < ItsufficestoshowthatVarH(X 1, X 2 )< For this, the result of Lemma 3 shows that Var log X 1 is finite On the other han, from (A10) it turns out that Var log X 1 + X 2 = Var log X 1 = Var log X 2 (A13) Weuseproperty(A13)tocalculatevarianceoftheright-han sie of (A12) as Var H(X 1, X 2 )= Var log X 1 log 2 2 K log 2 2, + Var log X 1 2 log 2 2 (A14) where K=Cov (log X 1 + X 2, log X 1 + log X 2 ) 2 Var log X 1 + X 2 Var log X 1 (A15) Applying property (A13) again on the right-han sie of (A15), we have Var H(X 1, X 2 ) 7 Var log X 1 2 log 2, (A16) 2 whereweusetheresultoflemma3togettheright-han sie of (A16) This means that λ=var (E (H (X 1, X 2 ) X 1 )) Var H(X 1, X 2 ) < (A17) Therefore, E(H(X 1, X 2 )) 2 < Now,weefineU-statistic for 1/α with kernel given in (A12) as U n = ( n 1 2 ) H (x i, x j ) 1 i<j n (A18) By efinition, given U-statistic in (A18) is unbiase estimator for 1/α Conflicts of Interest The authors eclare that they have no conflicts of interest References [1] P Lévy, Théorie es erreurs La loi e Gauss et les lois exceptionnelles, Bulletin De La Société Mathématique De France,vol 52, pp 49 85, 1924 [2] G Samoronitsky an M S Taqqu, Stable Non-Gaussian Ranom Processes: Stochastic Moels an Infinite Variance, Stochastic Moeling, Chapman & Hall, New York, NY, USA, 1994 [3] J P Nolan, Maximum likelihoo estimation of stable parameters, in Lévy Processes: Theory an Applications,OEBarnorff- Nielsen, T Mikosch, an I Resnick, Es, pp , Birkhöuser, Boston,, USA, 2001 [4] J P Nolan, Numerical calculation of stable ensities an istribution functions, Communications in Statistics Stochastic Moels, vol 13, no 4, pp , 1997 [5] J H McCulloch, Simple consistent estimators of stable istribution parameters, Communications in Statistics Simulation an Computation,vol15,no4,pp ,1986 [6] SMKogonanDBWilliams, Characteristicfunctionbase estimation of stable parameters, in A Practical Guie to Heavy Taile Data,RAler,RFelman,anMTaqqu,Es,pp , Birkhäuser, Boston,, USA, 1998 [7] Z Fan, Parameter estimation of stable istributions, Communications in Statistics Theory an Methos, vol35,no1-3,pp , 2006 [8] JPNolan,AKPanorska,anJHMcCulloch, Estimation of stable spectral measures, Mathematical an Computer Moelling, vol 34, no 9 11, pp , 2001

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