On Asymptotic Expansions of the Scale Mixtures of Stable Distribution

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1 On Asymptotic Epansions of the Scale Mitures of Stable Distribution Dhaifalla K. Al-Mutairi and Vu Kim Tuan Department of Statistics and Operations Research and Department of Mathematics and Computer Science, College of Science, Kuwait University, Kuwait Abstract The method of Shimizu and Fujikoshi (1997) is modified to obtain an estimate for the distribution of a scale miture of any distribution so that a reduction in the resulting error is achieved. General conditions for the convergence of this method are given. A particular condition on the method convergence for scale miture of stable distribution is also given. Keywords: Asymptotic epansion; Cauchy distribution; Normal distribution; Scale miture; Stable distribution. 1 Introduction Suppose that Z is a real-valued random variable (r.v.) with distribution function G(z) for < z <, and that Y is a positive r.v., independent of Z, with distribution function H(y) for 0 < y <. For δ > 0 and ρ = ±1, let X = Y δρ Z. Then the corresponding distribution of X F () = = 0 0 P (X Y = y) dh(y) G(y δρ ) dh(y) = E y [G(y δρ )], (1) is a scale miture of G with miing distribution H. Fujikoshi (1993) provides an epository description of the scale miture of some well-known distributions and gives error bounds for the asymptotic approimations. It can be seen that model (1) specifies F () in terms of the distribution function of Z evaluated at y δρ and the moments of Y. When a scale miture setup is appropriate, the goal is to find an estimate of F () in terms of a quantity depending on G and the moments of Y. Shimizu and Fujikoshi (1997) developed an approimation method 1

2 for F (). An overview of the method is as follows. It is based on a Taylor series epansion of G(y δ ρ ) about y = 1 that gives an estimate for F () as ] k F () = E y [G(y δ ρ i ) G() + y i G(y δ ρ ) E y [(y 1) i ] y=1 i! with an error i=1 (2) E y ( R k (y) ) β δ, k E y ( y 1 k+1 ). (3) Without a doubt, this method is a very useful tool and is an ecellent machinery for estimating F (). This method has a uniform error estimate for F () composed of the product of two terms: (a) β δ, k term that contains all information regarding the first k 1 derivative of g(), and (b) the k + 1st moment of Y about 1 denoted by E y ( y 1 k+1 ). It is noted that in general the moment E y ( y 1 k+1 ) increases as k increases. So for the error term to be small, β δ, k as a function of k must decrease. However, a close eamination of the error components β δ, k for scale miture of normal and Gamma distributions reveals that it may go to as more information on the density of Z is used in the analysis. In fact, the obtained error of this method increases as k, the number of derivatives of G being used, increases. Tables 3, 7, and 8 of Shimizu and Fujikoshi (1997) demonstrate the fact that the error size increases eponentially as k increases. So it is advisable to use this method only for k = 0, since the method fails to incorporate vital information regarding the derivatives of G as it drastically increases the error. Moreover, there is no general guidelines on the use of this method. In this paper we propose a modification to the method of Shimizu and Fujikoshi (1997) to provide an approimation for F () with an error β δ, k () that goes to 0 for each as k. We will specify conditions on which this method is useful and other conditions when the method is not applicable. In general, the proposed modified method of Shimizu and Fujikoshi (1997) converges if sup gk () E y ( y 1 k ) 0. (4) Both methods fall short of successfully handling the moments of Y about 1. In this paper, we impose conditions on the distribution of Y so that a reduction in the resulting error is achieved. 2 Main Results Suppose that h(y) is the density function of Y. If the support of h(y) contains y = y 0 > 2, then E y ( y 1 k ) = 0 y 1 k dh(y) C(y 0 1) k (5) 2

3 eponentially. Formula (5) shows that the selection of a beta distribution or any distribution defined over the interval (0, 2] as a miing distribution is appropriate, since the moments E y ( y 1 k+1 ) are bounded, while a gamma or chi-squared distribution as a miing distribution is not completely appropriate, since the error term goes to. The following approimation formula for the distribution function of Z evaluated at the point y is necessary to prove the main results of this paper. Suppose that the probability density function g is k time continuously differentiable over its support. Lemma 1 For each y > 0, G(y) is given by and G(y) = G() + where g (0) () = g(), and 0 (, y). i=0 g (i) () i+1 (y 1) i+1 (i + 1)! + R k (y), (6) R k (y) s 0 up g k ( 0 ) k+1 (y 1) k+1, (7) (k + 1)! Proof. Unlike Shimizu and Fujikoshi (1997), we use a Taylor series epansion of g(t) about so that y G(y) = G() + = G() + y g(t) dt [g() + i=1 g (i) () (t )i i! + g (k) ( t ) (t )k ] dt, t (, t). (8) Evaluate the integral in (8) to obtain i+1 (y 1) i+1 G(y) = G() + (i + 1)! i=0 g (i) () + R k (y), with g (0) () = g(), and R k (y) = y g k ( t ) (t )k dt, t (, t) (, y). So R k (y) y s 0 g k ( t ) (t )k dt up g k ( 0 ) k+1 (y 1) k+1, 0 (, y). (k + 1)! 3

4 From Lemma 1, we construct an approimating function to F () as with an approimating error bound So, the method converges if F () G() + g (i) () i=0 i+1 (i + 1)! E y[(y 1) i+1 ], (9) E y [ R k (y) ] s 0 up g (k) k+1 ( 0 ) (k + 1)! E y[(y 1) k+1 ]. (10) s up gk () (k + 1)! E y ( y 1 k+1 ) 0, (11) as k increases. The function G follows a standard symmetric stable distribution if its characteristic function has the following form ψ(u) = e u α, where 0 < α 2, and α is called the characteristic eponent. When α = 2, the stable distribution reduces into the normal distribution. When α = 1, the stable distribution reduces into the Cauchy distribution. The net theorem shows the approimating error bound for a scale miture of a standard symmetric stable distribution. Theorem 1 If Z follows a stable distribution with eponent α then an approimating error bound of F () is given by E y [ R k (y) ] k+1 ( k + 1 (k + 1)! α π Γ α ) E y [ y 1 k+1 ]. (12) Proof. The inversion formula for the characteristic function of the stable density function g() is eploited to determine the supremum of g k (). Thus, sup g (k) () = sup 1 d k 2π d k 1 u k e uα du π 0 = 1 ( ) k + 1 απ Γ. α eiu u α du Hence, Equation (10) yields that E y [ R k (y) ] k+1 ( k + 1 (k + 1)! α π Γ α ) E y [ y 1 k+1 ]. (13) As a result of Theorem 1, the net lemma gives the asymptotic behavior of E y [ R k (y) ] for different values of α. 4

5 Lemma 2 For a scale mitures of stable distribution with eponent α and h(y) C (y + 1) λ e yβ, β 2, and C > 0, E y [ R k (y) ] 0 if 1 < α 2, 0 if 0 < α < 1, 0 if E y [(y 1)] < 1 and α = 1, 0 if E y [(y 1)] 1 and α = 1. (14) Proof. Equation (13) can be written as E y [ R k (y) ] k+1 E y [(y 1) k+1 ] α π Stirling s approimation asserts that Γ( + α) 2π e +α 1 2, and the error bound in (15) is approimated by E y [ R k (y) ] k+1 E y [(y 1) k+1 ] α π Γ( k+1 α ) Γ(k + 2). (15) e k( 1 α 1) k (k+1)( 1 α 1) α 1 2 k+2 α, (16) which converges to 0 if 1 < α 2 and diverges if 0 < α < 1. When α = 1, we obtain E y [ R k (y) ] k+1 E y [(y 1) k+1 ], which goes to 0 if k+1 E y [(y 1) k+1 ] < 1, and goes to if k+1 E y [(y 1) k+1 ] 1. For 0 < α < 1, Lemma 3 shows that the stable density is not analytic at 0 and the density tail is very thick, thus the distribution of the scale miture of stable distribution cannot be estimated using k derivatives of g(). 3 Concluding Remarks From Formula (5), we observe that the method of Shimizu and Fujikoshi (1997) can be used directly, without any modifications, for scale mitures of any distribution accompanied with beta distribution or any distribution defined over the (0, 2] interval as a miing distribution. Otherwise, we can use the modification of this paper. First, we must check if the general conditions given in (4) are satisfied. If all the methods described in this paper fail to be applicable, we recommend the direct computation of the double integrals in (1). The problem of estimating F () is amenable to deterministic and Monte Carlo methods as thoroughly eplained by Demidovich and Maron (1987). 5

6 References [1] Demidovich, B.P., Maron, I.A. Computational Mathematics, MIR Publishers, Moscow, Soviet Union, [2] Fujikoshi, Y., Error bounds for asymptotic approimations of some distribution functions, Multivariate Analysis: Future Directions, (ed. C. R. Rao), , North-Holland, Amestrdam, [3] Shimizu, R., Fujikoshi, Y., Sharp error bounds for asymptotic epansions of the distribution functions for scale mitures, Annals of the Institute of Statistical Mathematics, 49, , [4] Nolan, J. Stable Distributions: Models for Heavy-Tailed Data, Birkhuser, Boston, [5] West, M., On scale mitures of normal distributions, Biometrics,74, ,

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