An Autoregressive Model for Time Series of Random Objects: Identifability

Table of Links

Abstract and 1. Introduction

2. Preliminaries

3. The GAR(1) Model

3.1. Model and Stationary Solution

3.2. Identifability

4. Estimation of model parameters and 4.1. Fréchet mean

4.2. Concentration parameter

5. Testing for the absence of serial dependence

6. Numerical experiments

6.1. R with multiplicative noise

6.2. Univariate distributions with a density

6.3. SPD Matrices

7. Application

8. Acknowledgement

Appendix A. General results in Hadamard spaces

Appendix B. Proofs

Reference

3.2. Identifability

Under the conditions of Theorem 3.3, Equation (5) has a stationary solution and the model features two quantities of interest: the time-invariant Fréchet mean of the time series µ ∈ Ω, and the concentration parameter φ ∈ [0, 1]. Before considering the estimation of these quantities, we show that both are identifiable. The identifiability of the Fréchet mean follows directly from the stationarity of the time series and the definition and existence of the Fréchet mean in a Hadamard space.