Uniqueness of the NPML mixing measure via total positivity

Establish uniqueness results for the NPML mixing measure associated with exponential and Gaussian covariance mixtures by determining whether total positivity of the kernels $K(\alpha,h)=\exp(-h/\alpha)$ and $K(\alpha,h)=\exp(-h^2/(2\alpha^2))$ is sufficient to guarantee uniqueness, thereby strengthening the finite-support representation result.

Background

The paper proves that at least one NPML covariance representation can be generated by a discrete mixing measure with finite support. This result does not establish that the mixing measure itself is unique; alternative representations may have more support points or may be nondiscrete.

The authors observe that total positivity has yielded uniqueness and sharper support-size results in related mixture models. Because the exponential and Gaussian kernel families are totally positive, determining whether this theory applies to the present covariance-mixture setting could substantially strengthen the theoretical characterization of the NPML estimator. The approach is not universally applicable, since some valid covariance kernels, such as the sinc kernel, are not totally positive.

References

It may be possible to leverage total positivity to establish uniqueness results for the NPMLE mixing measure for mixtures of exponential and Gaussian covariances, since the exponential and Gaussian kernel families $K(\alpha,h)=\exp(-h/\alpha)$ and $K(\alpha,h)=\exp(-h2/(2\alpha2))$ are totally positive.

— Mixture-based Nonparametric Estimation of Spatial Covariance Functions with Applications to HIV Key Population Size Estimation across Sub-Saharan Africa  (2609.10646 - Siriwardana et al., 9 Sep 2026) in Remark: When the base function K(α,h) is a totally positive kernel, Section 2.3.3 (Theoretical Properties of the NPML estimator)