Matrix of General Stable Distributions Close to the Normal Distribution

We investigate behavior of the Fisher information matrix of general stable distributions. Du-Mouchel (1975, 1983) proved that the Fisher information I αα of characteristic exponent α diverges to infinity as α approaches 2. Nagaev and Shkol'nik (1988) made more detailed analysis of I αα and derived asymptotic behavior of I αα diverging to infinity as α approaches 2 in the symmetric case. Extending their work in this paper we have obtained behavior of the Fisher information matrix of general stable distributions as α approaches 2 by detailed study of behavior of the corresponding density and its score functions. We clarify the limiting values of the 4 × 4 Fisher Information matrix with respect to the location µ, the scale σ, the characteristic exponent α and the skewness parameter β. The family of stable distributions has enjoyed great interest of researchers in many fields e.g., mathematics , physics, cosmology and even economics. These applications are summarized in Uchaikin and Zolotarev (1999). In statistical inference estimation of stable parameters has been of great interest. In recent years, maximum likelihood estimation of stable distributions has become feasible (see Brorsen ans Yang (1990), Nolan (2001) or Matsui and Takemura (2004)). Even in time series models like GARCH using general stable distributions, maximum likelihood estimation is possible owing to recent development of global algorithm (see Liu and Brorsen (1995)). Since the Fisher information matrix gives useful criteria for the accuracy of estimation, it is indispensable to analyze that of general stable distributions. Near Gaussian distribution (α = 2), the information of α, I αα diverges to ∞ and asymptotic behavior of I αα as α ↑ 2 is of great interest. Nagaev and Shkol'nik (1988) have solved this problem excellently for symmetric stable distributions. However for general stable distributions Nagaev and Shkol'nik (1988) stated " We note the problems under study are as yet unresolved for non-symmetric stable distributions. " In this paper we investigate and obtain asymptotic behavior of I θθ , θ = µ, σ, α, β, as α ↑ 2 under general stable distributions. Let Φ(t) = Φ(t; µ, σ, α, β) = exp −|σt| α 1 + iβ(sgn t) tan πα 2 (|σt| 1−α − 1) + iµt 1 denote the characteristic function of general stable distribution (α = 1) with parameters For the standard case (µ, σ) = (0, 1) we simply write the characteristic function as (1.1) Φ(t; α, …