A Robbins-Monro algorithm for non-parametric estimation of NAR process with Markov-Switching: asymptotic normality

Abstract: This paper is the second part of our study on the non-parametric estimation of MS-NAR processes started with [L. Fermin et al. 2017]. We consider the Nadaraya-Watson type regression function estimator for non-linear autoregressive Markov switching processes. In this context the regression function estimator is interpreted as a solution of a local weighted We have introduced, in the first work, a restoration-estimation Robbins-Monro algorithm to approximate the estimator, and we proved identifiability of model and the consistency of the non-parametric estimator. In this work, we obtain the central limit theorem for the non-parametric estimator, whether the Markov chain is observed or not. Finally, we present a detailed simulation study illustrating the performances of our estimation procedure.

• The paper establishes the asymptotic normality of kernel-based estimators for MS-NAR models using a Robbins-Monro stochastic approximation approach.
• It demonstrates convergence through simulations, validating low misclassification rates and reliable variance estimation even under regime intersection.
• The methodology integrates restoration steps with adaptive averaging to manage latent states, offering practical insights on bandwidth selection and computational efficiency.

Asymptotic Normality for Non-Parametric Estimation in MS-NAR Processes via a Robbins-Monro Algorithm

Introduction and Context

The paper "A Robbins-Monro algorithm for non-parametric estimation of NAR process with Markov-Switching: asymptotic normality" (2603.29440) addresses non-parametric estimation procedures for Markov Switching Non-linear Autoregressive (MS-NAR) models. These models generalize classical autoregressive structures by allowing the regression function at each time step to switch according to a latent Markov chain. This framework captures nonstationarity and regime heterogeneity common in econometrics, signal processing, and environmental statistics.

Parametric estimation for MS-NAR and related HMMs is well-established, with known results on consistency and asymptotic normality under regularity assumptions. However, non-parametric methodology for regime-switching models is less developed, primarily due to the identifiability challenges, curse of dimensionality, and complications arising from latent states. The present work is focused on establishing theoretical properties (specifically, central limit theorems—CLTs) for non-parametric regression estimators, notably those constructed via a Robbins-Monro stochastic approximation algorithm, both with fully and partially observed data sequences.

Model Definition and Identifiability

The MS-NAR model considered is given by:

$Y_k = r_{X_k}(Y_{k-1}) + e_k$

where $(X_k)$ is a finite Markov chain, $r_i(\cdot)$ are regime-specific regression functions, and $(e_k)$ are i.i.d. errors. This structure encompasses threshold models, Markov-switching regressions, and general nonlinear HMMs.

Identifiability conditions are thoroughly detailed, drawing from linear independence of the component densities, separation of regression functions, and the non-singularity of the Markov transition matrix. The analysis embraces both the classical setting (observed regimes) and the challenging hidden regime case.

Non-Parametric Estimation and Robbins-Monro Algorithm

The core estimator for the regression function in regime $i$ at design point $y$ is a Nadaraya-Watson-type kernel smoother. In the completely observed setup, these are standard weighted least-squares estimators with weights depending both on the kernel and the observed regimes.

However, with latent state processes, direct evaluation is infeasible. The authors introduce a restoration-estimation procedure: regimes $(X_k)$ are pseudo-sampled in each iteration (restoration/Monte Carlo step) given the current parameter estimates and observed $(Y_k)$. Then, gradient-based minimization of a local potential function (weighted squared errors) is performed using a Robbins-Monro-type update, leading to stochastic approximations of the minimizers (i.e., the regression functions).

Adaptive averaging (Polyak-Ruppert procedure) is used to enhance statistical efficiency and reduce asymptotic variance. The key technical contributions are in establishing the theoretical limit distribution (CLT) of these averaged estimators and detailing conditions under which the stochastic approximation sequence converges and is asymptotically normal.

Main Theoretical Results

Fully Observed Setting

Under geometric strong-mixing and regularity (moment and bandwidth) conditions, the paper proves that the kernel estimator for each regime is uniformly consistent and, for fixed $y$, satisfies a central limit theorem:

$$\sqrt{n h_n} \big( \hat{r}_{i,n}(y) - r_i(y) \big) \rightarrow N\left( 0, \frac{\sigma^2_i \|K\|_2^2}{f_i(y)} \right)$$

where $(X_k)$0 is the $(X_k)$1 norm, $(X_k)$2 is the design density within regime $(X_k)$3, and the variance constant $(X_k)$4 incorporates the second conditional moment. The CLT proof relies on martingale approximations and covariance bounds for strong-mixing processes, tackling both the "truncated" and "non-truncated" parts of the estimator.

Hidden Markov Chain (Partially Observed) Setting

The Robbins-Monro sequence for the regression parameter $(X_k)$5 (averaged estimator) is shown to satisfy:

$(X_k)$6

where $(X_k)$7 is the limiting minimizer of the conditional expectation of the local potential, and $(X_k)$8 is explicitly constructed in terms of the limiting (empirical) Fisher information and the asymptotic covariance of the martingale differences in the stochastic gradient. All relevant terms, including the necessary derivatives and expectations, are computable in terms of the data and pseudo-trajectory samples generated during iterations.

Numerical Study and Empirical Findings

Simulations are conducted with $(X_k)$9 regimes and nonlinear regression functions demonstrating regime intersection scenarios. Results validate the theoretical CLTs—empirical confidence intervals calibrated from the asymptotic variance formulas reliably cover the true regression functions. The Robbins-Monro algorithm demonstrates rapid convergence of the transition matrix estimates and achieves low misclassification rates for the latent regimes (approximately $r_i(\cdot)$0 in the studied scenario), even in boundary/intersection zones where mislabeling is inherently challenging. Residual variance estimation aligns closely with the data-generating value, confirming the practical efficacy of the variance estimation procedure.

Notable empirical findings include:

• The variance of the estimator increases at the support boundaries, due to scarcity of design points.
• When the conditional densities corresponding to different regimes intersect, misclassification is non-negligible but has limited effect on regression estimation.
• The computational bottleneck is the Forward-Backward restoration step, scaling as $r_i(\cdot)$1, with practical implications for large $r_i(\cdot)$2.

Limitations, Robustness, and Practical Considerations

Three critical aspects are addressed:

• Bandwidth Selection: Fixed, globally chosen bandwidths suffice for theoretical guarantees, but data-driven or adaptive bandwidths would improve performance, especially in regions of data sparsity. However, consistent inference for adaptive selection is a non-trivial extension, both technically and computationally.
• Number of Regimes ($r_i(\cdot)$3): While theoretically all finite $r_i(\cdot)$4 are supported, computational complexity and identifiability degrade for large $r_i(\cdot)$5. Alternative order selection procedures or low-rank penalization strategies merit future investigation.
• Variance Estimation: Plug-in estimators for the asymptotic variance achieve asymptotic validity via Slutsky's theorem, substantiated by empirical coverage probabilities.

The methodology is robust under challenging scenarios (e.g., intersecting regression functions), though inevitably the curse of dimensionality and boundary effects limit performance when $r_i(\cdot)$6 or in small-sample regimes.

Implications and Directions for Future Research

This work rigorously closes the gap between non-parametric HMM and non-parametric Markov-switching regression theory, by obtaining both consistency and asymptotic normality for kernel-based regression estimators under hidden Markovian regime uncertainty. These results underpin valid statistical inference—such as pointwise confidence bands—in real-world problems where distributional assumptions on regression functions are inappropriate or unknown.

Directions for future research include:

• Theory and practice of adaptive bandwidth selection in the non-parametric MS-NAR context.
• Development of computationally efficient alternatives for very large $r_i(\cdot)$7, e.g., combining with particle methods or fast variational approximations.
• Estimation of the number of regimes under minimal assumptions, possibly leveraging spectral or integral operator approaches.
• Extension to higher-dimensional or functional data settings and exploration of minimax rates in this context.

Conclusion

The paper provides a comprehensive analysis of non-parametric estimation for MS-NAR models under both fully and partially observed regimes, leveraging a theoretically grounded Robbins-Monro stochastic approximation framework. Both consistency and asymptotic normality are established, enabling reliable construction of confidence intervals and hypothesis tests. The algorithm demonstrates strong empirical performance, and the analysis clarifies the principal practical and theoretical limitations of these estimators. This work constitutes a significant advancement in the non-parametric inference toolkit for stochastic dynamic models with latent switching structure (2603.29440).