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

Lisandro Fermin, Ricardo Rios, Luis-Ángel Rodríguez

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.

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.

Paper Structure

This paper contains 21 sections, 5 theorems, 77 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Overview of the literature
  3. Contributions
  4. Preliminary
  5. Stability and existence of moments
  6. Probability density
  7. Strong mixing
  8. Identifiability
  9. Main results
  10. Assumption of the estimators
  11. Asymptotic normality: fully observed data case
  12. Asymptotic normality: partially observed data case
  13. Numerical studies
  14. Fully observed data case
  15. Partially observed data case
  16. ...and 6 more sections

Key Result

Theorem 3.1

Assume that the model MS-NAR satisfies the conditions E1-E4, E6-E7, D1, B1-B2, M2, S1, and R2-R3 on a compact set $C$. Then, for each fixed $y\in C$, whenever $nh_n\to \infty$. $\blacktriangleleft$$\blacktriangleleft$

Figures (13)

  • Figure 1: Simulated sample path $Y_{0:n}$ from an MS-NAR model with $m = 3$ states and $n = 3000$.
  • Figure 2: Scatterplot $Y_{k}$ versus $Y_{k-1}$ with real state classification. The points are labeled with respect to the real state of $X_k$: cross for $X_k=1$, triangles for $X_k=2$, and circles for $X_k=3$.
  • Figure 3: Non-parametric kernel regression function estimates in the fully observed data case. The real functions $r_i$ are shown with solid lines and the Nadaraya-Watson kernel estimators with: line with cross for $\hat{r}_{1,n}$, line with triangles for $\hat{r}_{2,n}$, line with circles for $\hat{r}_{3,n}$.
  • Figure 4: Non-parametric kernel density function estimate $\hat{p}_0(y)$ and the frequency histogram of $Y_{0:n}$.
  • Figure 5: Non-parametric kernel regression function estimates in the fully observed data case. The Nadaraya-Watson kernel estimators are shown with: line with cross for $\hat{r}_{1,n}$, line with triangles for $\hat{r}_{2,n}$, line with circles for $\hat{r}_{3,n}$. The real functions $r_i$ are shown with solid lines, and the fill areas correspond to the $95\%$ pointwise confidence bands.
  • ...and 8 more figures

Theorems & Definitions (10)

  • Remark 2.1
  • Remark 3.1
  • Theorem 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.2
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.3