Statistical inference across time scales

We investigate statistical inference across time scales. We take as toy model the estimation of the intensity of a discretely observed compound Poisson process with symmetric Bernoulli jumps. We have data at different time scales: microscopic, intermediate and macroscopic. We quantify the smooth statistical transition from a microscopic Poissonian regime to a macroscopic Gaussian regime. The classical quadratic variation estimator is efficient in both microscopic and macroscopic scales but surprisingly shows a substantial loss of information in the intermediate scale that can be explicitly related to the sampling rate. We discuss the implications of these findings beyond this idealised framework.Comment: 29 pages, 2 figure

Adaptive directional estimator of the density in R^d for independent and mixing sequences

A new multivariate density estimator for stationary sequences is obtained by Fourier inversion of the thresholded empirical characteristic function. This estimator does not depend on the choice of parameters related to the smoothness of the density; it is directly adaptive. We establish oracle inequalities valid for independent, $\alpha$-mixing and $\tau$-mixing sequences, which allows us to derive optimal convergence rates, up to a logarithmic loss. On general anisotropic Sobolev classes, the estimator adapts to the regularity of the unknown density but also achieves directional adaptivity. In particular, if A is an invertible matrix, if the observations are drawn from X $\in$ R^d , d $\ge$ 1, it achieves the rate implied by the regularity of AX, which may be more regular than X. The estimator is easy to implement and numerically efficient. It depends on the calibration of a parameter for which we propose an innovative numerical selection procedure, using the Euler characteristic of the thresholded areas

Optimal adaptive estimation on R or R+ of the derivatives of a density

In this paper, we consider the problem of estimating the d-order derivative of a density f, relying on a sample of n i.i.d. observations with density f supported on R or R+. We propose projection estimators defined in the orthonormal Hermite or Laguerre bases and study their integrated L2-risk. For the density f belonging to regularity spaces and for a projection space chosen with adequate dimension, we obtain rates of convergence for our estimators, which are proved to be optimal in the minimax sense. The optimal choice of the projection space depends on unknown parameters, so a general data-driven procedure is proposed to reach the bias-variance compromise automatically. We discuss the assumptions and the estimator is compared to the one obtained by simply differentiating the density estimator. Simulations are finally performed and illustrate the good performances of the procedure and provide numerical comparison of projection and kernel estimators

Estimation of the Jump Size Density in a Mixed Compound Poisson Process

International audienceIn this paper, we consider a mixed compound Poisson process, i.e. a random sum of i.i.d. random variables where the number of terms is a Poisson process with random intensity. We study nonparametric estimators of the jump density by specific deconvolution methods. First, assuming that the random intensity has exponential distribution with unknown expectation , we propose two types of estimators based on the observation of an i.i.d. sample. Risks bounds and adaptive procedures are provided. Then, with no assumption on the distribution of the random intensity, we propose two nonparametric estimators of the jump density based on the joint observation of the number of jumps and the random sum of jumps. Risks bounds are provided, leading to unusual rates for one of the two estimators. The methods are implemented and compared via simulations. February 25, 201

Disruption of information processing in schizophrenia: The time perspective

AbstractWe review studies suggesting time disorders on both automatic and subjective levels in patients with schizophrenia. Patients have difficulty explicitly discriminating between simultaneous and asynchronous events, and ordering events in time. We discuss the relationship between these difficulties and impairments on a more elementary level. We showed that for undetectable stimulus onset asynchronies below 20ms, neither patients nor controls merge events in time, as previously believed. On the contrary, subjects implicitly distinguish between events even when evaluating them to be simultaneous. Furthermore, controls privilege the last stimulus, whereas patients seem to stay stuck on the first stimulus when asynchronies are sub-threshold. Combining previous results shows this to be true for patients even for asynchronies as short as 8ms. Moreover, this peculiarity predicts difficulties with detecting asynchronies longer than 50ms, suggesting an impact on the conscious ability to time events. Difficulties on the subjective level are also correlated with clinical disorganization. The results are interpreted within the framework of predictive coding which can account for an implicit ability to update events. These results complement a range of other results, by suggesting a difficulty with binding information in time as well as space, and by showing that information processing lacks continuity and stability in patients. The time perspective may help bridge the gap between cognitive impairments and clinical symptoms, by showing how the innermost structure of thought and experience is disrupted