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Nonparametric density estimation for stationary processes under multiplicative measurement errors

Duc Trong Dang, Van Ha Hoang, Phuc Hung Thai

Abstract

This paper focuses on estimating the invariant density function $f_X$ of the strongly mixing stationary process $X_t$ in the multiplicative measurement errors model $Y_t = X_t U_t$, where $U_t$ is also a strongly mixing stationary process. We propose a novel approach to handle non-independent data, typical in real-world scenarios. For instance, data collected from various groups may exhibit interdependencies within each group, resembling data generated from $m$-dependent stationary processes, a subset of stationary processes. This study extends the applicability of the model $Y_t = X_t U_t$ to diverse scientific domains dealing with complex dependent data. The paper outlines our estimation techniques, discusses convergence rates, establishes a lower bound on the minimax risk, and demonstrates the asymptotic normality of the estimator for $f_X$ under smooth error distributions. Through examples and simulations, we showcase the efficacy of our estimator. The paper concludes by providing proofs for the presented theoretical results.v

Nonparametric density estimation for stationary processes under multiplicative measurement errors

Abstract

This paper focuses on estimating the invariant density function of the strongly mixing stationary process in the multiplicative measurement errors model , where is also a strongly mixing stationary process. We propose a novel approach to handle non-independent data, typical in real-world scenarios. For instance, data collected from various groups may exhibit interdependencies within each group, resembling data generated from -dependent stationary processes, a subset of stationary processes. This study extends the applicability of the model to diverse scientific domains dealing with complex dependent data. The paper outlines our estimation techniques, discusses convergence rates, establishes a lower bound on the minimax risk, and demonstrates the asymptotic normality of the estimator for under smooth error distributions. Through examples and simulations, we showcase the efficacy of our estimator. The paper concludes by providing proofs for the presented theoretical results.v
Paper Structure (18 sections, 9 theorems, 86 equations, 2 tables)

This paper contains 18 sections, 9 theorems, 86 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Construction of the estimators
  3. Mellin transform
  4. The proposed estimators
  5. Assumptions
  6. Preliminary results
  7. Main results
  8. Asymptotic normality
  9. Bounds of the estimators
  10. Examples and Simulations
  11. Cox-Ingersoll-Ross process
  12. The $m$-Dependent stationary process
  13. Proofs
  14. Proof of Theorem \ref{['th:variance']}
  15. Proof of Theorem \ref{['th: converge-a.s-estimators']}
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $f_X\left(x\right) \in \mathcal{F}_{x,r}\left(A,s\right)$ and $\widehat{f}_{n}(x)$ be given in eq:23. Let $K$ be the kernel function $K$ satisfying Assumption A with $m \ge \left\lfloor s\right\rfloor + 1$, then for any $x > 0$ and $n$ large enough where $\mathop{\mathrm{bias}}\nolimits \left[ {{\widehat{f} }_{n}(x)} \right] = \left| \mathbb{E}{{\widehat{f} }_{n}(x)} - f_X (x) \right|$.

Theorems & Definitions (11)

  • Remark 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Remark 4.1
  • Corollary 4.2
  • Theorem 4.2
  • Theorem 4.3
  • Lemma 6.1
  • ...and 1 more