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A trigonometrically approximated maximum likelihood estimation for$\alpha$-stable laws is proposed. The estimator solves the approximatedlikelihood equation, which is obtained by projecting a true score function onthe space spanned by trigonometric functions. The projected score is expressedonly by real and imaginary parts of the characteristic function and theirderivatives, so that we can explicitly construct the targeting estimatingequation. We study the asymptotic properties of the proposed estimator and showconsistency and asymptotic normality. Furthermore, as the number oftrigonometric functions increases, the estimator converges to the exact maximumlikelihood estimator, in the sense that they have the same asymptotic law.Simulation studies show that our estimator outperforms other moment-typeestimators, and its standard deviation almost achieves the Cram\'er--Rao lowerbound. We apply our method to the estimation problem for $\alpha$-stableOrnstein--Uhlenbeck processes in a high-frequency setting. The obtained resultdemonstrates the theory of asymptotic mixed normality.

Trigonometrically approximated maximum likelihood estimation for stable law