Rates of Convergence of Maximum Smoothed Log-Likelihood Estimators for Semi-Parametric Multivariate Mixtures
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| Format: | Preprint |
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2025
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| _version_ | 1866908633081577472 |
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| author | de Chaumaray, Marie Du Roy Levine, Michael Marbac, Matthieu |
| author_facet | de Chaumaray, Marie Du Roy Levine, Michael Marbac, Matthieu |
| contents | Theoretical guarantees are established for a standard estimator in a semi-parametric finite mixture model, where each component density is modeled as a product of univariate densities under a conditional independence assumption. The focus is on the estimator that maximizes a smoothed log-likelihood function, which can be efficiently computed using a majorization-minimization algorithm. This smoothed likelihood applies a nonlinear regularization operator defined as the exponential of a kernel convolution on the logarithm of each component density. Consistency of the estimators is demonstrated by leveraging classical M-estimation frameworks under mild regularity conditions. Subsequently, convergence rates for both finite- and infinite-dimensional parameters are derived by exploiting structural properties of the smoothed likelihood, the behavior of the iterative optimization algorithm, and a thorough study of the profile smoothed likelihood. This work provides the first rigorous theoretical guarantees for this estimation approach, bridging the gap between practical algorithms and statistical theory in semi-parametric mixture modeling. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_04226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rates of Convergence of Maximum Smoothed Log-Likelihood Estimators for Semi-Parametric Multivariate Mixtures de Chaumaray, Marie Du Roy Levine, Michael Marbac, Matthieu Statistics Theory 62G05 Theoretical guarantees are established for a standard estimator in a semi-parametric finite mixture model, where each component density is modeled as a product of univariate densities under a conditional independence assumption. The focus is on the estimator that maximizes a smoothed log-likelihood function, which can be efficiently computed using a majorization-minimization algorithm. This smoothed likelihood applies a nonlinear regularization operator defined as the exponential of a kernel convolution on the logarithm of each component density. Consistency of the estimators is demonstrated by leveraging classical M-estimation frameworks under mild regularity conditions. Subsequently, convergence rates for both finite- and infinite-dimensional parameters are derived by exploiting structural properties of the smoothed likelihood, the behavior of the iterative optimization algorithm, and a thorough study of the profile smoothed likelihood. This work provides the first rigorous theoretical guarantees for this estimation approach, bridging the gap between practical algorithms and statistical theory in semi-parametric mixture modeling. |
| title | Rates of Convergence of Maximum Smoothed Log-Likelihood Estimators for Semi-Parametric Multivariate Mixtures |
| topic | Statistics Theory 62G05 |
| url | https://arxiv.org/abs/2511.04226 |