Authors:
(1) Matthieu Bult´e, Department of Mathematics Sciences, University of Copenhagen and Faculty of Business and Economy Administration, Bielefeld University;
(2) Helle Sørensen, Department of Mathematics Sciences, University of Copenhagen.
Ties
Summary and 1. Introduction
2. Preliminaries
3. The Gar model (1)
3.1. Stationary model and solution
3.2. Identification
4. Estimate of the model parameters and 4.1. Fréchet means
4.2. Concentration parameter
5. Test of the absence of series dependence
6. Digital experiences
6.1. R with a multiplicative noise
6.2. Univariate distributions with density
6.3. SPD matrices
7. Application
8. Thanks
Annex A. General results in Hadamard spaces
Annex B. tests
Reference
Abstract
The random variables in metric spaces indexed by time and observed at also spaced times receive increased attention due to their large applicability. However, the absence of a structure inherent in metric spaces has resulted in a literature which is mainly non -parametric and without model. To fill this gap in the models of chronological series of random objects, we introduce an adaptation of the classic linear self -regressive model adapted to the data located in a Hadamard space. The parameters of interest for this model are the average Fréchet and a concentration parameter, which we both prove can be systematically estimated from the data. In addition, we offer a test statistic and establish its asymptotic normality, thus allowing hypothesis tests for the absence of series dependence. Finally, we introduce a bootstrap procedure to obtain critical values for test statistics under the null hypothesis. The theoretical results of our method, including the convergence of estimators as well as the size and power of the test, are illustrated by simulations, and the utility of the model is demonstrated by an analysis of a chronological series of consumer inflation expectations.
1. Introduction
Random variables in general metric spaces, also called random objects, have received increasing attention in recent statistical research. The configuration of the metric space of generality does not require any algebraic structure and is only based on the definition of a distance function. This allows the methods developed to be applied in fields ranging from conventional configurations to more complex use cases on non -standard data. This includes the study of functional data (Ramsay and Silverman (2005)), data on Riemannian collectors, correlation matrices and these applications to IRMF data (Petersen and Muller (2019)) or adjacence and social networks (Dubey and Muller (2020)), among others.
An example of particular interest because of its wide range of applications is that of data including probability density functions. Probability distributions are a difficult example of a space that is both functional, and therefore with infinite dimension, but also non -Euclidean in the constraints characterizing the density functions. This leads to a certain number of different approaches to study these objects: they have been studied as the image of Hilbert's spaces under transformations (Petersen and M¨uller (2016)), as specific Hilbert spaces with operators of addition and scalar multiplication (van den Boogaart et al. (2014)), as well as metric spaces of different distances (Panaretos and Zemel (2020); Klassen (2016)). See Petersen et al. (2022) for a review of these methodologies. Distributions can be found in many applications; Considering the distribution of socio-economic factors within a population such as income (Yoshiyuki (2017)), fertility (Mazzuco and Scarpa (2015)) or mortality data (Chen et al. (2021)). They are also useful when examining the distributions of economic factors beliefs (Meeks and Monti (2023)), allowing economic analyzes to consider entire distributions rather than empirical expectations.
The study of random objects has received recent attention with work in standard statistical questions (Dubey and Muller (2019, 2020); McCormack and Hoff (2023, 2022); Köstenberger and Stark (2023)) as well as various approaches to regression (Petersen and Muller (2019); Bult´e). Since the configuration of general metric spaces offers very little structure, part of the literature supposes an additional structure on the space so that the standard statistical quantities are well defined. This is generally done by assuming that metric space is a Hadamard space, see for example Sturm (2003) for a detailed examination of results in Hadamard and Bacak spaces (2014) for the calculation of frequency means in such spaces.
In many applications mentioned above, the data can be naturally observed several times at a regular interval and for a temporal series. In this case, observations may not be independent and models and analyzes require additional care to take this dependence into account. This work was mainly carried out in a non -parametric framework, with a conventional low dependence hypothesis. This was done, for example, to test series dependence (Jiang et al. (2023)) or to prove the consistency of the average estimator Fréchet (Caner (2006)).
Although this work line can be widely applied, they rely on non -parametric hypotheses rather than offering a specific model for data generation. However, models of chronological series have been developed for specific random objects by exploiting the structure of the space for study. A popular class of models is that of self -regressive models, which have been defined using the linear structure of functional spaces (Bosq (2000); Caponera and Marinucci (2021)) or exploit a tangent space structure (ZHU and M¨ULLER (2022); Xavier and Manton (2006); Ghodrati and Panaretos (2023); To name only a few.
Inspired by existing self -regressive models, we offer a self -regressive model for random objects. Based on an interpretation of iteration in the linear self -regressive model as a noisy weighted sum at the average, we define a model configured by an average and concentration parameters. For this to be possible, we assume an additional structure and demand that space be a Hadamard space, and exploits the geometry of space to define the iteration of the chronological series by geodesy. We develop the associated methodology and theory for estimation and hypothesis tests in this model. This includes estimators of the average and concentration parameters, and we offer a test statistics to test without autocorrelation, corresponding to the observation of an IID sample.
The article is organized as follows: Section 2 gives a presentation of useful concepts and results in Hadamard spaces for the rest of the article. In section 3, we present our self -regressive model and present a theorem providing a sufficient condition for the existence of a stationary solution of the iterated system of equations associated with the model, and prove the identifiability of the model parameters. We propose in section 4 estimators of these parameters and prove the convergence results for these estimators. In section 5, we propose a test for a zero hypothesis of independence based on a test statistic whose asymptotic behavior we characterize under the zero hypothesis and the alternative of a non -zero concentration parameter. Finally, we illustrate our theoretical results in section 6 with a digital study.
