Mixed Spectra for Stable Signals from Discrete Observations

This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial kernel to build a periodogram which we then smooth by two spectral windows taking into account the width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing often encountered in the case of estimation from discrete observations of a continuous time process.

Simple discrete-time self-exciting models can describe complex dynamic processes: A case study of COVID-19

Hawkes processes are a form of self-exciting process that has been used in numerous applications, including neuroscience, seismology, and terrorism. While these self-exciting processes have a simple formulation, they can model incredibly complex phenomena. Traditionally Hawkes processes are a continuous-time process, however we enable these models to be applied to a wider range of problems by considering a discrete-time variant of Hawkes processes. We illustrate this through the novel coronavirus disease (COVID-19) as a substantive case study. While alternative models, such as compartmental and growth curve models, have been widely applied to the COVID-19 epidemic, the use of discrete-time Hawkes processes allows us to gain alternative insights. This paper evaluates the capability of discrete-time Hawkes processes by modelling daily mortality counts as distinct phases in the COVID-19 outbreak. We first consider the initial stage of exponential growth and the subsequent decline as preventative measures become effective. We then explore subsequent phases with more recent data. Various countries that have been adversely affected by the epidemic are considered, namely, Brazil, China, France, Germany, India, Italy, Spain, Sweden, the United Kingdom and the United States. These countries are all unique concerning the spread of the virus and their corresponding response measures. However, we find that this simple model is useful in accurately capturing the dynamics of the process, despite hidden interactions that are not directly modelled due to their complexity, and differences both within and between countries. The utility of this model is not confined to the current COVID-19 epidemic, rather this model could explain many other complex phenomena. It is of interest to have simple models that adequately describe these complex processes with unknown dynamics. As models become more complex, a simpler representation of the process can be desirable for the sake of parsimony.

Simple discrete-time self-exciting models can describe complex dynamic processes: a case study of COVID-19

Hawkes processes are a form of self-exciting process that has been used in numerous applications, including neuroscience, seismology, and terrorism. While these self-exciting processes have a simple formulation, they are able to model incredibly complex phenomena. Traditionally Hawkes processes are a continuous-time process, however we enable these models to be applied to a wider range of problems by considering a discrete-time variant of Hawkes processes. We illustrate this through the novel coronavirus disease (COVID-19) as a substantive case study. While alternative models, such as compartmental and growth curve models, have been widely applied to the COVID-19 epidemic, the use of discrete-time Hawkes processes allows us to gain alternative insights. This paper evaluates the capability of discrete-time Hawkes processes by retrospectively modelling daily counts of deaths as two distinct phases in the progression of the COVID-19 outbreak: the initial stage of exponential growth and the subsequent decline as preventative measures become effective. We consider various countries that have been adversely affected by the epidemic, namely, Brazil, China, France, Germany, India, Italy, Spain, Sweden, the United Kingdom and the United States. These countries are all unique concerning the spread of the virus and their corresponding response measures, in particular, the types and timings of preventative actions. However, we find that this simple model is useful in accurately capturing the dynamics of the process, despite hidden interactions that are not directly modelled due to their complexity, and differences both within and between countries. The utility of this model is not confined to the current COVID-19 epidemic, rather this model could be used to explain many other complex phenomena. It is of interest to have simple models that adequately describe these complex processes with unknown dynamics. As models become more complex, a simpler representation of the process can be desirable for the sake of parsimony.

Meta-analysis of lagged regression models: A continuous-time approach

In science, the gold standard for evidence is an empirical result which is consistent across multiple studies. Meta-analysis techniques allow researchers to combine the results of different studies. Lagged effects models based on longitudinal data are increasingly the target for meta-analysis: However, in current practice, little attention is paid to the unique challenges of meta-analyzing these lagged effects. Namely, it is well-known that lagged effects estimates change depending on the time that elapses between measurement waves. This means that studies that use different uniform time intervals between observations (e.g., 1 hour vs 3 hours or 1 month vs 2 months) can come to very different parameter estimates, and seemingly contradictory conclusions, about the same underlying process. In this article, we introduce, describe, and illustrate a new meta-analysis method (CTmeta) which assumes an underlying continuous-time process, thereby offering a potential solution to the time-interval problem. We illustrate this method and compare it with the current best-practice in dealing with time-interval dependency in the meta-analysis literature.

Deviations for Jumping Times of a Branching Process Indexed by a Poisson Process

Consider a continuous time process {Yt=ZNt, t≥0}, where {Zn} is a supercritical Galton–Watson process and {Nt} is a Poisson process which is independent of {Zn}. Let τn be the n-th jumping time of {Yt}, we obtain that the typical rate of growth for {τn} is n/λ, where λ is the intensity of {Nt}. Probabilities of deviations n-1τn-λ-1>δ are estimated for three types of positive δ.

LAMN property for the drift and volatility parameters of a sde driven by a stable Lévy process

This work focuses on the local asymptotic mixed normality (LAMN) property from high frequency observations, of a continuous time process solution of a stochastic differential equation driven by a truncated α-stable process with index α ∈ (0, 2). The process is observed on the fixed time interval [0,1] and the parameters appear in both the drift coefficient and scale coefficient. This extends the results of Clément and Gloter [Stoch. Process. Appl. 125 (2015) 2316–2352] where the index α ∈ (1, 2) and the parameter appears only in the drift coefficient. We compute the asymptotic Fisher information and find that the rate in the LAMN property depends on the behavior of the Lévy measure near zero. The proof relies on the small time asymptotic behavior of the transition density of the process obtained in Clément et al. [Preprint HAL-01410989v2 (2017)].

Large deviations for Lotka-Nagaev estimator of a randomly indexed branching process

Consider a continuous time process {Yt=ZNt, t ? 0}, where {Zn} is a supercritical Galton-Watson process and {Nt} is a renewal process which is independent of {Zn}. Firstly, we study the asymptotic properties of the harmonic moments E(Y-rt) of order r > 0 as t ? ?. Then, we obtain the large deviations of the Lotka-Negaev estimator of offspring mean.