Tests of fit for symmetric variance gamma distributions

Transcription

1 Tests of fit for symmetri variane gamma distributions Fragiadakis Kostas UADPhilEon, National and Kapodistrian University of Athens, 4 Euripidou Street, Athens, Greee. Keywords: Variane Gamma Distribution, Charateristi Funtion, EM - type algorithm, Goodness of Fit Abstrat New goodness of fit tests for the family of symmetri Variane Gamma distributions are onstruted. A Generalized EM type algorithm for the parameters estimation have been used. The proposed tests are based on a weighted integral inorporating the empirial harateristi funtion.. Introdution The aim of this paper is to provide goodness-of-fit tests for the symmetri normal variane gamma distribution SNVG. The NVG distribution is defined as a mixture of a normal distribution with a Gamma distribution. Speifially, if V Γ, λ distribution where Γα, β denotes the Gamma distribution with density fx = xα β α Γα e βx, α, β, x > 0. and if Z V N0, V, where Nµ, σ denotes the normal distribution with mean µ and variane σ, then the distribution of Y where Y = δ + Z is the SNVG distribution with density funtion fy = Γλ π y δ λ Kλ y δ where K r x denotes the modified Bessel funtion of order r evaluated at x. Therefore, the SNVG distribution is a three - parameter model, denoted by SNV Gδ,, λ whih depends on a loation parameter δ R, a sale parameter > 0, and a shape parameter λ > 0. Suppose that on the basis of independent opies X, X,..., X n, of a random variable X we wish to test the null hypothesis H 0 : The law of X is SNV Gδ,, λ for some δ R, > 0 and λ > 0. The motivation for onsidering the SNVG is that this distribution, due to the heavier tails with respet to normality, beomes a ompetitive alternative hoie of model for appliations in Eonomis, and partiularly for modelling finanial data. September 0-4, 007 Castro Urdiales Spain

2 . Estimation of Parameters A ritial issue is the estimation of the parameters. sample X, X,..., X n, the log likelihood take the form lδ,, λ = n logγλ π + λ log xi δ + log K λ xi δ Given a random + whih is rather ompliated to be maximized. Instead a Generalized EM approah will be used. Firstly an EM type algorithm is desribed. Based on the mixture representation we need to augment the observed data x, x,..., x n, and the unobserved data v, v,..., v n. At the E step we need the onditional expetations of some funtions of v i. The onditional distribution of V i X i = x i an be easily seen to be a Generalized Inverse Gaussian distribution of the form V X = x GIG λ, x i δ, We know that if X GIGλ, δ, γ distribution it holds that r δ EX r K λ+r δγ = γ K λ δγ The log likelihood of the omplete data X i, V i, i =,,..., n fatorizes in two parts and hene one an derive that = n x i δ, δ = v i and λ is the solution of the equation Ψλ = n logv i x i v i v i This implies that the onditional expetations needed for the M step have the form EV i X i and Elog V i X i. Despite the fat that EV i X i = K xi δ λ 3 x i δ K λ xi δ September 0-4, 007 Castro Urdiales Spain

3 3 unfortunately the onditional expetation Elog V i X i annot be written in a useful losed form formula and hene it is hard to be derived. There are two distint solutions for this problem. The first one is to estimate this expetation using Monte Carlo, resulting to a MCEM algorithm and the seond one is to use a GEM Generalized EM. The seond approah will be used. The idea is instead of solving Ψλ = n logv i we just seek a λ that inreases the log likelihood. The algorithm has the following steps E Step Calulate with the urrent estimates s i = EV i X i = K xi δ λ 3 x i δ M Step Update the parameters using = n K λ s i x i δ, δ = xi δ x is i s i Find a new λ by a grid searh in the neighborhood of the urrent value. It suffies to find a value for λ that improves the log likelihood and not neessarily the maximum one. Sine for given λ the other two estimates improve the log likelihood, a new λ that provides better log likelihood ensures the monotoni property of the EM. 3. Test Statistis At this setion we study a new family of omnibus tests of H 0 based on the empirial harateristi funtion CF. Despite the fat that the density funtion of X is ompliated, the CF, φt = Ee itx of X is simply φt; δ,, λ = e iδt + t λ. It is natural to onstrut a test statisti based on the standardized data i.e. δ = 0 and =. Hene, the harateristi funtion of the standardized data is φt; λ = By taking the first derivative we have + t λ + t φ t + λtφt = 0 Therefore if Y, Y,..., Y n follows the SNVG distribution, equation Dt; λ = 0 where Dt; λ + t φ t + λtφt must be satisfied for the standardized data September 0-4, 007 Castro Urdiales Spain

4 4 X j = Yj ˆδ n ĉ n, j =,,..., n. Taking into aount the empirial analogue of CF we have D n t; ˆλ n = + t φ nt + ˆλ n tφ n t where φ n t = n Speifially I suggest to rejet the null hypothesis H 0 for large values of ˆT n,w = n j= e itxj D n t; ˆλ n wtdt, 3 with wt denoting a non negative weight funtion. Notie that the test statisti remains invariant under the linear transformation X δ + X, for eah δ R and > 0. From 3 we have by straightforward algebra ˆT n,w n where j,k= [ 4ˆλ ni X j X k + X j X k I 4 X j X k + I X j X k + +I 0 X j X k + ˆλ n X j X k I s 3 X j + X k I s 3 X j X k + ] +I s X j + X k I s X j X k I m b + I p s b = + t m osbtwtdt, m = 0,, 4, t p sinbtwtdt, p =, 3. Although theoretial properties of the test statisti remain qualitatively invariant, provided that wt satisfies some general onditions, partiular appeal lies with weight funtions that render the test statisti in a losed formula suitable for omputer implementation. 4. Bootstrap Proedure In this setion a bootstrap proedure for the new test given by 3 is proposed. To atually implement the test ritial points are required. However, the null distribution of the test statisti depends on the value of the shape parameter λ, whih is unknown. Therefore we resort to a parametri bootstrap proedure in order to obtain the ritial point p α of the test as follows:. Conditionally on the observed value of Y j, j =,,..., n, ompute the estimates ˆδ n, ĉ n, ˆλ n and then the observations ˆX j = Y j ˆδ n /ĉ n, j =,,..., n. September 0-4, 007 Castro Urdiales Spain

5 5.a. Calulate the value of the test statisti, say ˆT, based on ˆX j and ˆλ n..b.. Generate a bootstrap sample Y j, j =,,..., n, from SNV G0,, ˆλ n..b.. On the basis of Y j, j =,,..., n, ompute the estimates ˆδ n, ĉ n, ˆλ n and then the observations ˆX j = Y j ˆδ n/ĉ n, j =,,..., n..b.3. Calulate the value of the test statisti, say ˆT, based on ˆX j ˆλ n. and 3. Repeat steps.b. -.b.3., and alulate M values of ˆT, say ˆT j, j =,,..., M. 4. Obtain p α as T M αm, where T j, j =,,..., M denotes the ordered T j - value. Aknowledgements: The author wishes to sinerely thank S. Meintanis and D. Karlis for their strong support and guidane during this work. 5. Bibliography [] N. Gürtler, and N. Henze, Goodness of fit tests for the Cauhy distribution based on the empirial harateristi funtion, Ann. Instit. Statist. Math. 5, [] B. Klar, and S. Meintanis, Tests for normal mixtures based on the empirial harateristi funtion, Comput. Statist. Dat. Anal. 49, [3] I. A. Koutrouvelis, and S. G. Meintanis, Testing for stability based on the empirial harateristi funtion with appliations to finanial data, J. Statist. Comput. Simulation 64, [4] D. Karlis, An EM Type Algorithm for ML estimation for the Normal - Inverse Gaussian Distribution, Statistis and Probability, 57, September 0-4, 007 Castro Urdiales Spain