Estimation of Parameters of Three Parameter Esscher Transformed Laplace Distribution

Transcription

1 Iteratioal Joural of Statistics ad Systems ISSN Volume 1, Number (017), pp Research Idia Publicatios Estimatio of Parameters of Three Parameter Esscher Trasformed Laplace Distributio Dais George 1 ad Rimsha H 1 (Departmet of Statistics, Catholicate College/ MG Uiversity, Pathaamthitta, Kerala, Idia) Project Fellow (UGC Major Project, Catholicate College/ Research Scholar (Bharathiar Uiversity), Pathaamthitta, Kerala, Idia) Abstract I this article, we cosider the estimatio of parameters of three parameter Esscher trasformed Laplace distributio which is a alterative to various types of asymmetric Laplace distributios. The three parameter Esscher trasformed Laplace distributios is the locatio scale family of the oe parameter Esscher trasformed Laplace distributio itroduced by Sebastia George ad Dais George (01). Keywords: Asymmetric distributio, Esscher trasformed Laplace distributio, Heavy-tailed distributio, Maximum Likelihood Estimatio Method. I. INTRODUCTION I the last several decades various forms of asymmetric, heavy-tailed ad asymmetric heavy-tailed distributios have appeared i the literature, see McGill (196), Holla ad Bhattacharya (1968), Balakrisha ad Ambagaspitiya (199), Ghitay(005, 007), Kozumbowski ad Podg'orski (000) ad Poiraud-Casaova ad Thomas Agam (000), Jayakumar ad Kuttykrisha (006), ] VaidaBartkute ad LeoidasSakalauskas (007).

2 670 Dais George ad Rimsha H I this paper, we cosider the estimatio of three parameter Esscher trasformed Laplace distributio, which is the locatio scale family of Esscher trasformed Laplace distributio itroduced by Sebastia George ad Dais George (01). It is a alterative to the existig heavy-tailed ad asymmetric distributios. Esscher trasformed Laplace distributio satisfies may importat statistical properties like ifiite divisibility, geometric iifiite divisibility, stability with respect to geometric summatio, maximum etropy, fiiteess of momets, simplicity etc. ad subclasses of geometric stable distributios. This model provides more flexibility, allowig for asymmetry, peakedess ad tail heaviess tha the ormal model, which are commo features of may data sets, see Sebastia George ad Dais George (01). The probability desity fuctio of ad the distributio fuctio of oe parameter Esscher traformed Laplace distributio, deoted by ETL( ) is give by ad f(x; ) ={ (1 θ ) (1 θ ) e [x(1+θ)], x < 0, θ < 1 e [ x(1 θ)], x 0, θ < 1, (1) F(x) = { (1 ) (1 ) e [x(1+θ)], x < 0, θ < 1 + (1+θ) (1 e [ x(1 θ)] ), x 0, θ < 1. () Here θ is the parameter of the distributio. Graphs of the p.d.f of ETL( ) for various values of are give i Figure 1. Figure1. Desities of Esscher trasformed Laplace distributio for (a) (-1, 0), (b) = 0 (classical Laplace) ad (c) (0, 1).

3 Estimatio of Parameters of Three Parameter Esscher Trasformed Laplace Distributio 671. THREE PARAMETER ESSCHER TRANSFORMED LAPLACE DISTRIBUTION Itroducig the locatio parameter(µ) ad scale parameter(σ) i the ETL( ) distributio, the p.d.f ad d.f of the ETL(,µ,σ) distributio are as follows: ad f(x;,µ,σ) ={ F(x;,µ,σ) ={ (1 θ ) σ (1 θ ) (1 ) σ 1 (1+ ) e[(x μ σ )(1+θ)], x < μ e [(μ x σ )(1 θ)], x μ, e [(x μ σ )(1+θ)], x < μ e [(μ x σ )(1 θ)], x μ, θ < 1, σ > 0, (3) θ < 1, σ > 0, () Graphs of the desity fuctio of ETL(,µ,σ) for µ = 5 ad for various values Of ad σ i Figure. Figure : Desities of Esscher Trasformed Laplace Distributio for (a) σ = 0. ad (-1; 0), (b) σ = 0. ad (0; 1) (c) = -0:6 ad Various Values of σ ad (d) = 0:6 ad Various Values of σ.

4 67 Dais George ad Rimsha H The mea, variace, momet geeratig fuctio, characteristic fuctio ad th momet of the Esscher trasformed Laplace distributio are give by Mea = µ+ σ (1 ), Variace = σ (1+ ) (1 ), Momet Geeratig Fuctio = e tµ 1 t σ (1 ) + t σ (1 ), ad Characteristic Fuctio = e itµ 1 it σ (1 ) + t σ (1 ) th momet of y about μ =! σ [ (1 θ)+1 +(1+θ) +1 (1 θ ) ]. 3. PARAMETER ESTIMATION I this sectio, we study the problem of estimatig the parameters of three parameter Esscher trasformed Laplace distributio. Note that our distributios are essetially covolutios of expoetial radom variables of differet sigs, ad commo estimatio procedures for mixtures of positive expoetial distributios are ot applicable i this case. We shall focus o the method of maximum likelihood method of estimatio. 3.1 Maximum Likelihood Estimatio Let X1,X,,X be a i.i.d radom sample from a ETL(,µ,σ) distributio with desity g,µ,σ give by (3) ad let x1, x, x be their particular realizatio. Our goal is to fid the MLEs of the parameters. The likelihood fuctio ca be as where L(X,,µ,σ) = (1 θ ) (x i μ) + = { x i μ, x i μ 0, x i < μ The the log-likelihood fuctio is e[ (1+θ σ ) i=1 (x i μ) ( 1 θ σ ) i=1 (x i μ) + ] (σ) ad, (5) (x i μ) = { μ x i, x i μ 0, x i < μ. logl(x,,µ,σ ) = log(1 - θ ) - log logσ- D σ where D = D(,µ) = (1+ ) i=1(x i μ) + (1 ) i=1 (x i μ) +

5 Estimatio of Parameters of Three Parameter Esscher Trasformed Laplace Distributio 673 The estimates are give i the followig Table 1. TABLE 1 Cases Parameters Estimates Asympt.variace 1. μ is ukow (σ,θ kow). σ is ukow (μ,θ kow) σ μ = Xj():. Where j() = (1 θ) + 1 [[x]]deote the itegral part of x = 1 [(1 θ) (x i μ) + + (1 i=1 + θ) (x i μ) ] i=1 1 θ σ σ 1 θ 3. θ is ukow (μ,σ kow) θ is uique solutio g(y,, β, )= Log(1-y )-(1-y)α +(1+y) = 0 1 θ (1 θ) + 1 α(μ ) = = 1 [ (x i=1 i μ) + ] (μ )= 1 [ i=1 ] (x i μ). μ,σ ukow θ is kow σ μ = Xj():. = 1 [(1 θ) (x i μ) + + (1 i=1 + θ) (x i μ) ] = [ σ (1 θ ) 0 0 ] σ (1 θ ) i=1 5. θ,σ ukow μ is kow σ = α(μ) = β(μ) θ β(μ) α(μ) [ α(μ) + β(μ) ] = σ (1 + (1 8 θ) + 1) a c [ b ] where a = 1 σ, b= (1+θ)( θ ), (1 θ) c= (1+θ) 1 (1 θ) σ θ θ 6. θ,μ ukow σ is kow α(μ) = 1 σ 1 (x i μ) + i=1 ad = σ (1 θ) [a c b ] Where

6 67 Dais George ad Rimsha H (μ) = 1 σ 1 (x i=1 i μ) R J 1, R J, R J (0, 1 1 ]; J=[-1, ) (μ 1,θ 1 ) (μ,θ ) (μ,θ ) b= a= θ 1 θ σ (1+θ) c= 1 σ 7. μ, θ,σ is kow Fid 1< r such that h(x r:) h(x j:) for j= 1,, where h(μ) = log( α(μ) + (μ) + α(μ) β(μ) μ = X r: = β(μ) θ σ = α(μ) Where α(μ) β(μ) [ α(μ) + β(μ) ] α(μ ) = = 1 [ (x i=1 i μ) + ] (μ )= 1 [ i=1 ] (x i μ) a b c = σ [ d f] g e= Where a = b = 1 σ (1+θ) c = θ ( θ) d= σ (1+θ) σ(1+θ) 1 θ. CONCLUSION I this paper, we studied the estimatio problem of three parameter Esscher trasformed Laplace distributio, which is the locatio scale family of the oe parameter Esscher trasformed Laplace distributio itroduced by Sebastia George ad Dais George (01). Beig heavy-tailed ad asymmetric, this distributio act as a alterative to various asymmetric distributios ad ca be used for modelig real data sets of that ature which are commoly occurrig from diverse fields such as busiess, telecommuicatios, physical ad biological scieces ad hece the estimatio of parameters holds its importace.

7 Estimatio of Parameters of Three Parameter Esscher Trasformed Laplace Distributio 675 REFERENCES [1] N. Balakrisha, ad R. S. Ambagaspitiya. O skewed Laplace distributios. Report, McMaster Uiversity, Hamilto, Otario, Caada., (199). [] Ghitay, M. E., Al-Hussaii, E. K., Al-Jarallah, R. A. (005) MarshallOlki exteded Weibull distributio ad its applicatio to cesored data,j.appl. Stat.,3, [3] Ghitay, M. E., Al-Awadhi F. A., AlkhalfaL, A. (007) Marshall-Olki exteded lomax distributio ad its applicatio to cesored data, Com mu. Statist. Theory Methods, 36, [] F.Esscher. O the probability fuctio m the collective theory of rtsk. Scadmaoza Actuarial Joural, 15, (193), [5] M. S Holla ad S. K.,. Bhattacharya. O a compoud Gaussia distributio. A. Ist. Statist. Math., 0, (1968), [6] Jayakumar, K. ad Kuttykrisha, A.P. A ew asymmetric Laplace autoregressive process. Joural of Statistical Theory ad Applicatios 7, (006),: [7] S. Kotz, T.J. Kozubowski ad Podg\'{o}rski K. The Laplace distributio ad Geeralizatios: A Revisit with Applicatios to Commuicatios, Ecoomics,Egieerig ad Fiace. Birkhauser. Bosto., Techical, (000). [8] T.J Kozubowski ad K. Podg\'{o}rski, Asymmetric Laplace distributio, Math.Sci., 5, (000), [9] W. J. McGill, Radom uctuatios of respose rate. Psychometrica, 7, (196), [10] S.Poiraud-Casaova, ad C. Thomas-Aga,. About mootoe regressio quatiles. Statist. Probab. Lett., 8, (000). [11] Sebastia George ad Dais George. Esscher Trasformed Laplace Distributio ad Its Applicatios. Joural of Probability ad Statistical Sciece, 10(), (01), [1] VaidaBartkute ad Leoidas Sakalauskas. Three Parameter Estimatio of the Weibull distributio by Order Statistics /publicatio/887810, (015),DOI: 10.11/ _001.

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