Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStrong mixing condition for Hawkes processes and application to Whittle estimation from count data
185
0
0.0
(
0
)
Added by
Felix Cheysson
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
Authors
Felix Cheysson
Ask ChatGPT about the research
No Arabic abstract
This paper focuses on the time series generated by the event counts of stationary Hawkes processes. When the exact locations of points are not observed, but only counts over time intervals of fixed size, existing methods of estimation are not applicable. We first establish a strong mixing condition with polynomial decay rate for Hawkes processes, from their Poisson cluster structure. This allows us to propose a spectral approach to the estimation of Hawkes processes, based on Whittles method, which provides consistent and asymptotically normal estimates under common regularity conditions on their reproduction kernels. Simulated datasets and a case-study illustrate the performances of the estimation, notably of the Hawkes reproduction mean and kernel when time intervals are relatively large.
rate research
Read More
Suppose that particles are randomly distributed in $bR^d$, and they are subject to identical stochastic motion independently of each other. The Smoluchowski process describes fluctuations of the number of particles in an observation region over time. This paper studies properties of the Smoluchowski processes and considers related statistical problems. In the first part of the paper we revisit probabilistic properties of the Smoluchowski process in a unified and principled way: explicit formulas for generating functionals and moments are derived, conditions for stationarity and Gaussian approximation are discussed, and relations to other stochastic models are highlighted. The second part deals with statistics of the Smoluchowki processes. We consider two different models of the particle displacement process: the undeviated uniform motion (when a particle moves with random constant velocity along a straight line) and the Brownian motion displacement. In the setting of the undeviated uniform motion we study the problems of estimating the mean speed and the speed distribution, while for the Brownian displacement model the problem of estimating the diffusion coefficient is considered. In all these settings we develop estimators with provable accuracy guarantees.
The cross-classified sampling design consists in drawing samples from a two-dimension population, independently in each dimension. Such design is commonly used in consumer price index surveys and has been recently applied to draw a sample of babies in the French ELFE survey, by crossing a sample of maternity units and a sample of days. We propose to derive a general theory of estimation for this sampling design. We consider the Horvitz-Thompson estimator for a total, and show that the cross-classified design will usually result in a loss of efficiency as compared to the widespread two-stage design. We obtain the asymptotic distribution of the Horvitz-Thompson estimator, and several unbiased variance estimators. Facing the problem of possibly negative values, we propose simplified non-negative variance estimators and study their bias under a super-population model. The proposed estimators are compared for totals and ratios on simulated data. An application on real data from the ELFE survey is also presented, and we make some recommendations. Supplementary materials are available online.
Coarse structural nested mean models are used to estimate treatment effects from longitudinal observational data. Coarse structural nested mean models lead to a large class of estimators. It turns out that estimates and standard errors may differ considerably within this class. We prove that, under additional assumptions, there exists an explicit solution for the optimal estimator within the class of coarse structural nested mean models. Moreover, we show that even if the additional assumptions do not hold, this optimal estimator is doubly-robust: it is consistent and asymptotically normal not only if the model for treatment initiation is correct, but also if a certain outcome-regression model is correct. We compare the optimal estimator to some naive choices within the class of coarse structural nested mean models in a simulation study. Furthermore, we apply the optimal and naive estimators to study how the CD4 count increase due to one year of antiretroviral treatment (ART) depends on the time between HIV infection and ART initiation in recently infected HIV infected patients. Both in the simulation study and in the application, the use of optimal estimators leads to substantial increases in precision.
This paper studies nonparametric estimation of parameters of multivariate Hawkes processes. We consider the Bayesian setting and derive posterior concentration rates. First rates are derived for L1-metrics for stochastic intensities of the Hawkes process. We then deduce rates for the L1-norm of interactions functions of the process. Our results are exemplified by using priors based on piecewise constant functions, with regular or random partitions and priors based on mixtures of Betas distributions. Numerical illustrations are then proposed with in mind applications for inferring functional connec-tivity graphs of neurons.
We consider high-dimensional measurement errors with high-frequency data. Our focus is on recovering the covariance matrix of the random errors with optimality. In this problem, not all components of the random vector are observed at the same time and the measurement errors are latent variables, leading to major challenges besides high data dimensionality. We propose a new covariance matrix estimator in this context with appropriate localization and thresholding. By developing a new technical device integrating the high-frequency data feature with the conventional notion of $alpha$-mixing, our analysis successfully accommodates the challenging serial dependence in the measurement errors. Our theoretical analysis establishes the minimax optimal convergence rates associated with two commonly used loss functions. We then establish cases when the proposed localized estimator with thresholding achieves the minimax optimal convergence rates. Considering that the variances and covariances can be small in reality, we conduct a second-order theoretical analysis that further disentangles the dominating bias in the estimator. A bias-corrected estimator is then proposed to ensure its practical finite sample performance. We illustrate the promising empirical performance of the proposed estimator with extensive simulation studies and a real data analysis.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
