Asymptotically Efficient Estimation of Location for a Symmetric Stable Law

Volatility estimators for discretely sampled Lévy processes

“…Recall that a way of estimating a parameter for i.i.d. variables X j is to use the empirical characteristic function j ∈J exp(iwX j ), or j ∈J cos(wX j ) in the symmetrical case, for some given w (or several w's at once), where J is the index set; in the Lévy process setting, see, for example, [8,10,18,19] and Chapter 4 in [23].…”

“…If, further, this variable has a density which depends smoothly on the parameter of interest, η, the Fisher information at stage n has the form I n, (η) = nI (η), where I (η) > 0 is the Fisher information of the model based on the observation of the single variable X , we have the LAN property with rate √ n, the asymptotically efficient estimators η n are those for which √ n( η n − η) converges in law to the normal distribution N(0, I (η) −1 ) and the MLE solves the problem (see, e.g., [5][6][7]). In this setting, a variety of other methods have been proposed in the literature: using the empirical characteristic function as an estimating equation (see, e.g., [8,10,18,19] and Chapter 4 in [23]), maximum likelihood by Fourier inversion of the characteristic function (see [9]), a regression based on the explicit form of the characteristic function (see [14]) or other numerical approximations (see [16,17]). Some of these methods were compared in [3].…”

Spectral estimation of the fractional order of a Lévy process

“…The only way to draw inference is to use the fact that the increments form independent realizations of infinitely divisible probability distributions. In this setting, a variety of methods have been proposed in the literature: standard maximum likelihood estimation DuMouchel (1973aDuMouchel ( , 1973bDuMouchel ( , 1975, using the empirical characteristic function as an estimating equation [see, e.g., Press (1972), Fenech (1976), Feuerverger and McDunnough (1981a), Singleton (2001)], maximum likelihood by Fourier inversion of the characteristic function Feuerverger and McDunnough (1981b), a regression based on the explicit form of the characteristic function Koutrouvelis (1980) or other numerical approximations Nolan (1997). Some of these methods were compared in Akgiray and Lamoureux (1989).…”