Minimax rate of estimation for invariant densities associated to continuous stochastic differential equations over anisotropic Holder classes
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We study the problem of the nonparametric estimation for the density π of the stationary distribution of a d-dimensional stochastic differential equation (X_t)_t ∈ [0, T]. From the continuous observation of the sampling path on [0, T], we study the rate of estimation of π(x) as T goes to infinity. One finding is that, for d ≥ 3, the rate of estimation depends on the smoothness β = (β_1, ... , β_d) of π. In particular, having ordered the smoothness such that β_1 ≤ ... ≤β_d, it depends on the fact that β_2 < β_3 or β_2 = β_3. We show that kernel density estimators achieve the rate (log T/T)^γ in the first case and (1/T)^γ in the second, for an explicit exponent γ depending on the dimension and on β̅_3, the harmonic mean of the smoothness over the d directions after having removed β_1 and β_2, the smallest ones. Moreover, we obtain a minimax lower bound on the 𝐋^2-risk for the pointwise estimation with the same rates (log T/T)^γ or (1/T)^γ, depending on the value of β_2 and β_3.
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