A Note on Parametric Estimation of Lévy Moving Average Processes

A Minimal Contrast Estimator for the Linear Fractional Stable Motion

“…Indeed, the choice of the two evaluation points for the empirical characteristic function in [18] is rather ad hoc and we will show in the empirical study that the minimal contrast estimator exhibits better finite sample properties and robustness in various settings. Similarly to [18], we will show that the weak limit theory for our estimator has a normal and a stable regime, and the asymptotic distribution depends on the interplay between the parameters α and H. At this stage we remark that the minimal contrast approach has been investigated in [16] in the context of certain Lévy moving average models, which do not include the lfsm or its associated noise process, but only in the asymptotically normal regime. Another important contribution of our paper is the subsampling procedure, which provides confidence regions for the parameters of the model irrespectively of the unknown asymptotic regime.…”

“…for all s, t ∈ [0, T ]. (5.18) This is performed in a similar fashion as in [16,Section 4.1]. First, we reduce the problem.…”

“…Furthermore, we propose several ideas to obtain feasible confidence regions in various parameter settings. Our work is mainly related to [16,18], in which parameter estimation for the linear fractional stable motion and related Lévy moving average processes has been studied.…”

Power variations for fractional type infinitely divisible random fields

“…Obviously, the proposed estimation procedure assumes prior knowledge of the parameter β, since we need to choose p ∈ (0, β). In the case d = 1 the papers [17,27,28,29] have suggested to use negative powers p ∈ (−1, 0) to estimate the parameter H for unknown β. A similar idea should apply in the random field setting, although negative power variations are beyond the scope of our paper.…”

“…We remark that the aforementioned probabilistic results are of immense importance for statistical applications. Indeed, they have been applied in [27,28,29] to obtain complete parametric estimation of the linear fractional stable models and related processes in low and high frequency settings. Earlier studies on similar estimation problems, which are mainly concerned with estimation of the self-similarity parameter, can be found in [3,17,34,40].…”