Explicit Parametric Density Estimators

• Explicit parametric density estimators are statistical models that approximate unknown densities using fully specified, finite-dimensional functional forms such as exponential families, mixture models, and polynomial expansions.
• They employ estimation methods like maximum likelihood, minimum distance, and moment matching, which offer analytic tractability, strong consistency, and computational efficiency.
• These estimators find applications in classical inference, signal processing, and generative modeling by balancing theoretical rigor with practical performance in diverse settings.

Explicit parametric density estimators are statistical models that represent unknown probability densities by finite-dimensional, fully-specified functional forms determined by a set of explicit parameters. The estimation procedure involves selecting a parametric family and optimizing parameters to fit observed data under rigorous, often likelihood-based or distance-based, criteria. These estimators are characterized by analytic tractability, well-defined statistical properties, and computational efficiency. Explicit parametric density estimation encompasses classical approaches (such as maximum likelihood estimation in exponential families), mixture models with tractable likelihoods, projection-based estimators, and optimization-driven constructions such as those based on divergence minimization, Stein discrepancies, and polynomial or moment-based expansions.

1. Formalism and Classes

Explicit parametric density estimators approximate a true density $f(x)$ by a parameterized family $\{f_\theta(x): \theta \in \Theta\}$, where $\theta$ is a finite-dimensional vector. The functional form of $f_\theta$ is specified a priori, ranging from simple exponential families, finite mixtures of known distributions, polynomial or Fourier bases, to explicit rational forms derived from moment constraints. For example, the Bernstein polynomial estimator models the density on $[0,1]$ as a weighted sum of beta densities,

$$f_m(x; p) = \sum_{i=0}^m p_{m,i} \cdot \text{Beta}(x; i+1, m-i+1), \quad p_{m,i} \geq 0, \quad \sum_{i=0}^m p_{m,i} = 1$$

which is an explicit mixture model with parameters $p_{m,i}$ (Guan, 2014).

Recent advances include estimators that use explicit projections in orthonormal bases (polynomial, Fourier, Hermite), rational forms matching sample moments in squared Hellinger distance, and minimum distance approaches (e.g., energy distance, $L^q$-norm) for both normalized and non-normalized models (Wu et al., 2022, Duda, 2017, Betsch et al., 2019).

2. Estimation Methodologies

The principal estimation strategies for explicit parametric models include:

• Maximum Likelihood Estimation (MLE): Parameters are chosen to maximize the likelihood of the observed data,

$$\hat{\theta}_{MLE} = \arg\max_\theta \sum_{j=1}^n \log f_\theta(x_j),$$

often solved via convex optimization or efficient algorithms (e.g., EM for mixtures) (Guan, 2014).

• Minimum Distance Estimation: Minimizing discrepancies between the empirical distribution and the model using divergences or norms. For instance, the minimum $L^q$-distance estimator for $x > 0$ families uses

$$\hat{\theta}_{n,q} = \arg\min_{\theta \in \Theta} \|\eta_n(\cdot, \theta)\|_{L^q}$$

where $\eta_n$ computes a Stein-type discrepancy involving sample data (Betsch et al., 2019).

• Moment Matching via Convex Optimization: Constructing densities that exactly match the first $2n$ sample moments and minimize squared Hellinger distance to a fixed prior,

$p(x) = \frac{r(x)}{[1 + F(x)^\top \Omega F(x)]^2},$

with parameters $\Omega$ solving a finite-dimensional convex program (Wu et al., 2022).

• Linear Projection (L² Fit): Expanding the density in an orthonormal basis $\{\phi_k\}$, coefficients are estimated by empirical averages,

$a_k = \frac{1}{n} \sum_{i=1}^n \phi_k(x^i).$

Normalization and nonnegativity may require explicit corrections or restrictions on the basis and parameter domain (Duda, 2017).

• Robust Divergence-based Methods: Estimation by minimizing density power divergence or related criteria, particularly when the likelihood is intractable but the first two moments are available (Felipe et al., 2023).

3. Statistical Properties and Consistency

Explicit parametric estimators attain diverse asymptotic and robustness properties, depending on the family and criterion:

• Consistency: Under regularity conditions, MLEs in tractable parametric families achieve strong consistency ($\hat{\theta}_n \to \theta_0$) and asymptotic normality (Guan, 2014). Minimum $L^q$ distance estimators are consistent under mild conditions, though rates and asymptotic distributions are generally model-dependent (Betsch et al., 2019).
• Error Rates:
  • Bernstein polynomial estimators on smooth densities with $k$ derivatives attain $\mathrm{MISE} = O(n^{-2k/(2k+1)})$, matching the kernel rate for $k=1$ and becoming nearly parametric for analytic densities ($\mathrm{MISE} = O((\log n) / n)$) (Guan, 2014).
  • Closed-form projections in orthonormal bases yield coefficient variances decaying as $O(1/n)$ under the Central Limit Theorem (Duda, 2017).
  • GAN-style perceptron/energy distance minimizers for Sobolev-regular classes achieve (minimax) total-variation rates of $n^{-\beta/(2\beta + d)}$ (Gerber et al., 2023).
• Robustness: Density power divergence-based estimators with tuning parameter $\alpha > 0$ achieve bounded influence (B-robustness), with performance well-maintained under contamination, unlike standard likelihood methods (Felipe et al., 2023).
• Identifiability/Uniqueness: Explicit moment-constrained rational models (e.g., squared Hellinger-minimizing densities) yield unique solutions due to strict convexity of the optimization functional (Wu et al., 2022).

4. Representative Methods and Examples

Explicit parametric density estimation spans multiple concrete methodologies, with archetypes summarized in the table below:

Family / Method	Parameterization	Fitting Criterion
Bernstein polynomial mixtures (Guan, 2014)	$p_{m,i}$ in $\sum p_{m,i} \beta_{m,i}(x)$	Likelihood / EM or convex optimization
Hellinger moment matching (Wu et al., 2022)	$\Omega$ in $p(x) = r(x)/[1+F^\top\Omega F]^2$	Convex minimization (squared Hellinger)
Polynomial/Fourier/Hermite expansion (Duda, 2017)	$a_k$ in $\sum a_k \phi_k(x)$	L² projection / empirical averages
Minimum $L^q$-distance (Betsch et al., 2019)	$\theta$ in $f(x;\theta)$	Minimize $L^q$ Stein discrepancy
DPD/MDPD Gaussian estimator (Felipe et al., 2023)	$(\mu, \Sigma)$ of $N(\mu, \Sigma)$	Minimize DPD subject to constraints
Perceptron/energy distance ERM (Gerber et al., 2023)	$\theta$ in explicit model class	Minimize perceptron/energy discrepancy

Applications range from microarray p-value FDR estimation (Guan, 2014), robust hypothesis testing (Felipe et al., 2023), to real-time filtering and multimodal noise modeling (Wu et al., 2022).

5. Computational and Algorithmic Aspects

Explicit parametric estimators are favored for computational tractability:

• Closed-form solutions: Projection-based estimators yield immediate coefficient estimates via sample averages; normalization requires only basic linear algebra (Duda, 2017).
• Convex optimization: Squared Hellinger or divergence-based models reduce to finite-dimensional strictly convex programs. Fast convergence is ensured, and the per-iteration cost is $O(n^3 + N_q n^2)$ for $n$ moments and $N_q$ quadrature nodes (Wu et al., 2022).
• EM and gradient-based methods: Mixture models are typically fit by EM; energy distance-based ERMs admit SGD by backpropagating through empirical and model samples (Guan, 2014, Gerber et al., 2023).
• Grid or covering-net methods: For models where direct optimization is infeasible, random search or covering-net arguments suffice, especially when the parameter space’s entropy is low (Gerber et al., 2023).

Efficiency extends to high-dimensional data when basis expansion is restricted, or parametric classes (e.g., Gaussian mixtures within a bounded region) are employed. However, curse-of-dimensionality constraints persist for generic basis expansions, necessitating sparsity, low-rank structure, or sample-efficient embeddings.

6. Robustness, Regularization, and Model Selection

Robustness and regularization enter explicitly via:

• Divergence Tuning: Density power divergence and $L^q$-distance estimators allow explicit control of bias-variance and robustness through parameters $\alpha$ or $q$, with practical bias-variance tradeoff recommendations (moderate $a$ for $L^2$ is often optimal) (Betsch et al., 2019, Felipe et al., 2023).
• Model Complexity: Nestedness of parametric families (e.g., polynomial/Fourier order, mixture component count) motivates model selection via change-point heuristics, cross-validation, or likelihood-based penalties. For Bernstein polynomials, the optimal degree $m$ is identified by a likelihood-increment change-point detection algorithm (Guan, 2014).
• Nonnegativity and Normalization: Not all explicit forms preserve nonnegativity inherently. Polynomial/Fourier expansion densities may require post hoc clipping or basis restriction; exponential-family or rational-form models can enforce positivity directly through parameter constraints (Duda, 2017, Wu et al., 2022).

7. Practical Implications and Application Scope

Explicit parametric density estimators underpin a wide range of applications:

• Classical Inference: Gaussian, exponential, and non-normalized models are directly amenable to explicit parametric estimators, supporting robust testing and estimation where likelihood forms are tractable or surrogates (moment-based Gaussians) suffice (Felipe et al., 2023, Betsch et al., 2019).
• Signal Processing and Filtering: Moment-matching estimators with closed-form rational densities enable real-time state propagation in Bayesian filters without kernel complexity (Wu et al., 2022).
• Multiple Testing/FDR: Beta-mixture forms (Bernstein polynomials) provide low-bias density estimates at the domain boundary, notably outperforming kernel methods for p-value density estimation in genomics (Guan, 2014).
• Density Estimation for Generative Models: Perceptron/energy-distance ERMs yield minimax-close estimators in high-dimensional generative settings and inform the design of GAN discriminators (Gerber et al., 2023).
• Clustering and Classification: Projection-based expansions with signed or complex weights realize density-based clustering or multi-class discrimination by interpreting the sign or argument of the fitted function (Duda, 2017).

The explicit parametric paradigm balances analytic flexibility, computational feasibility, and rigor in density estimation, supporting contemporary inference tasks and integrating seamlessly with both classical and modern statistical workflows.