Asymptotic behavior of power and shape parameter CRBs

Determine the asymptotic behavior, as the sample size tends to infinity, of the Cramér–Rao bounds for the power parameter and shape parameters in the specified parameterization of the mean and covariance sequence of a stationary compound Gaussian process when the texture distribution is purely continuous.

Background

The paper considers a stationary compound Gaussian process whose mean is parameterized by θ1\boldsymbol\theta_1 and whose covariance sequence has the form Rx(k)=θ2R(k,θ3)\mathbf R_x(k)=\theta_2\mathbf R(k,\boldsymbol\theta_3), with θ2\theta_2 a power factor and θ3\boldsymbol\theta_3 a shape factor. For a purely continuous texture distribution, the limiting asymptotic Fisher information matrix has a degenerate block structure for the power and shape parameters.

Because the limiting Fisher information matrix contains zero entries in the relevant power-parameter block, the paper does not establish whether the corresponding finite-sample Cramér–Rao bounds converge, diverge, or exhibit another asymptotic behavior. The authors state that no general conclusion can be drawn from their limiting expression.

References

Consequently, applying eq:Whittle's formula to a purely continuous ($\beta=\gamma=0$) distribution of the texture $\tau$, the asymptotic FIM rate~eq:Whittle's formula exhibits the block structure $\begin{bmatrix} \times & {\bf 0} & {\bf 0}\ {\bf 0}T & 0 & {\bf 0}T\ {\bf 0} & {\bf 0} & \times \end{bmatrix}$, and hence no general conclusion can be drawn regarding the asymptotic behavior of ${\rm CRB}{\bf y_n}(\theta_2)$ and ${\rm CRB}{\bf y_n}(\boldsymbol{\theta}_3)$ as $n \to \infty$.

— On the parametric and semiparametric Fisher information matrix for non-zero mean stationary spherical invariant random processes  (2609.20469 - Delmas et al., 17 Sep 2026) in Section 6, Remark 6 of Result 1 (specific parameterization of $(\boldsymbol\mu_x,(\mathbf R_x(k))_{k\in\mathbb Z})$)