Consistency of the Whittle pseudo-likelihood estimator for Matérn parameters

Establish the consistency, under fixed-domain (infill) asymptotics with known smoothness ν, of the profile Whittle pseudo-likelihood estimator for the Matérn microergodic parameter m = σ^2 α^{2ν} and the range parameter α. Consider observations of a stationary Matérn Gaussian random field on the bounded domain D = [0, 2π)^4 on the regular lattice {0, …, n_obs−1}^4 with spacing h = 2π/n_obs. Define the discrete Fourier coefficients Z_k and periodogram I_k = |Z_k|^2, the model-based variances u_α(k) = h^8 ∑_{r∈Z^4} c_α(r) A(r) exp(−2π i ⟨k, r⟩/n_obs), where c_α(r) is the lattice covariance for σ^2 = α^{−2ν} (i.e., m = 1) and A(r) = ∏_{ℓ=1}^4 (n_obs − |r_ℓ|)_+. Define the Whittle pseudo-likelihood Q_N(m, α) = ∑_{k∈K} [ log(m u_α(k)) + I_k/(m u_α(k)) ], profile m̂(α) = (1/|K|)∑_{k∈K} I_k/u_α(k), and estimators α̂ = argmin_α Q_N(m̂(α), α), m̂ = m̂(α̂). Prove that (α̂, m̂) converges to the true (α, m) as n_obs → ∞.

Background

The paper resolves the critical dimension d = 4 case for identifiability of Matérn parameters under fixed-domain asymptotics, showing that microergodic matching still yields mutually singular measures when range parameters differ. Building on this, the author proposes using discrete Fourier coefficients to construct a Whittle-type diagonal Gaussian pseudo-likelihood for parameter estimation on a bounded domain in R^4 with known smoothness ν.

The section defines a concrete Whittle pseudo-likelihood Q_N(m, α) based on exact grid Fourier coefficients and their model-implied variances u_α(k). Simulations indicate that the profile estimator can recover the range parameter α. The explicit open question is to provide a rigorous proof of consistency for the joint estimation of the microergodic scale m and the range α in this fixed-domain setting.