Existence theory for the unbounded-support NPML estimator

Establish existence of the nonparametric maximum likelihood estimator for mixture covariance models when the candidate mixing-measure support is unbounded, including the necessary spectral conditions for the kernel covariance matrix in the flat limit as the range parameter tends to infinity.

Background

The paper proves existence of the NPML covariance estimator when the mixing-measure support is restricted to a compact interval [0,U] with finite U. The proof relies on a uniform strictly positive lower bound for the eigenvalues of the kernel covariance matrices over the candidate support.

For unbounded support, the authors consider compactifying the parameter space by adjoining infinity. In the Gaussian-kernel example, the covariance matrix converges to an all-ones matrix as the range parameter tends to infinity, producing a rank-one flat-limit covariance. Although compactification can restore closedness of the set of mixture covariances, the compact-support eigenvalue argument no longer applies. A complete existence theory therefore requires analysis of the kernel spectrum in this flat-limit regime.

References

However, our existence proof for the NPMLE would still require modification, since our proof uses strict lower boundedness of the eigenvalues of $K(\alpha,D)$ over $\alpha\in[0,U]$, for $U<\infty$. In the unbounded setting, it seems necessary to consider the spectral properties of $K(\alpha,D)$ in the so called ``flat limit'' as $\alpha\to\infty$, as in \citet{barthelme2021spectral}.

— 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: Extension to unbounded support, Section 2.3.3 (Theoretical Properties of the NPML estimator)