Digit analysis for Covid-19 reported data

The coronavirus which appeared in December 2019 in Wuhan has spread out worldwide and caused the death of more than 280,000 people (as of May, 11 2020). Since February 2020, doubts were raised about the numbers of confirmed cases and deaths reported by the Chinese government. In this paper, we examine data available from China at the city and provincial levels and we compare them with Canadian provincial data, US state data and French regional data. We consider cumulative and daily numbers of confirmed cases and deaths and examine these numbers through the lens of their first two digits and in particular we measure departures of these first two digits to the Newcomb-Benford distribution, often used to detect frauds. Our finding is that there is no evidence that cumulative and daily numbers of confirmed cases and deaths for all these countries have different first or second digit distributions. We also show that the Newcomb-Benford distribution cannot be rejected for these data

Normalized information-based divergences

This paper is devoted to the mathematical study of some divergences based on the mutual information well-suited to categorical random vectors. These divergences are generalizations of the "entropy distance" and "information distance". Their main characteristic is that they combine a complexity term and the mutual information. We then introduce the notion of (normalized) information-based divergence, propose several examples and discuss their mathematical properties in particular in some prediction framework.Comment: 36 page

Variational approach for spatial point process intensity estimation

We introduce a new variational estimator for the intensity function of an inhomogeneous spatial point process with points in the $d$-dimensional Euclidean space and observed within a bounded region. The variational estimator applies in a simple and general setting when the intensity function is assumed to be of log-linear form $\beta+{\theta }^{\top}z(u)$ where $z$ is a spatial covariate function and the focus is on estimating ${\theta }$. The variational estimator is very simple to implement and quicker than alternative estimation procedures. We establish its strong consistency and asymptotic normality. We also discuss its finite-sample properties in comparison with the maximum first order composite likelihood estimator when considering various inhomogeneous spatial point process models and dimensions as well as settings were $z$ is completely or only partially known.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ516 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Expectiles for subordinated Gaussian processes with applications

In this paper, we introduce a new class of estimators of the Hurst exponent of the fractional Brownian motion (fBm) process. These estimators are based on sample expectiles of discrete variations of a sample path of the fBm process. In order to derive the statistical properties of the proposed estimators, we establish asymptotic results for sample expectiles of subordinated stationary Gaussian processes with unit variance and correlation function satisfying $\rho(i)\sim \kappa|i|^{-\alpha}$ (\kappa\in \RR) with $\alpha>0$. Via a simulation study, we demonstrate the relevance of the expectile-based estimation method and show that the suggested estimators are more robust to data rounding than their sample quantile-based counterparts

Asymptotic properties of the maximum pseudo-likelihood estimator for stationary Gibbs point processes including the Lennard-Jones model

This paper presents asymptotic properties of the maximum pseudo-likelihood estimator of a vector \Vect{\theta} parameterizing a stationary Gibbs point process. Sufficient conditions, expressed in terms of the local energy function defining a Gibbs point process, to establish strong consistency and asymptotic normality results of this estimator depending on a single realization, are presented.These results are general enough to no longer require the local stability and the linearity in terms of the parameters of the local energy function. We consider characteristic examples of such models, the Lennard-Jones and the finite range Lennard-Jones models. We show that the different assumptions ensuring the consistency are satisfied for both models whereas the assumptions ensuring the asymptotic normality are fulfilled only for the finite range Lennard-Jones model

Poisson intensity parameter estimation for stationary Gibbs point processes of finite interaction range

We introduce a semi-parametric estimator of the Poisson intensity parameter of a spatial stationary Gibbs point process. Under very mild assumptions satisfied by a large class of Gibbs models, we establish its strong consistency and asymptotic normality. We also consider its finite-sample properties in a simulation study

The median of a jittered Poisson distribution

Let $N_\lambda$ and $U$ be two independent random variables respectively distributed as a Poisson distribution with parameter $\lambda >0$ and a uniform distribution on $(0,1)$. This paper establishes that the median, say $M$, of $N_\lambda+U$ is close to $\lambda +1/3$ and more precisely that $M-\lambda-1/3=o(\lambda^{-1})$ as $\lambda\to \infty$. This result is used to construt a very simple robust estimator of $\lambda$ which is consistent and asymptotically normal. Compared to known robust estimates, this one can still be used with large datasets ($n\simeq 10^9$)