Ilia Negri

Stationary Distribution Function Estimation for Ergodic Diffusion Process

The problem of nonparametric stationary distribution function estimation by the observations of an ergodic diffusion process is considered. The local asymptotic minimax lower bound on the risk of all the estimators is found and it is proved that the empirical distribution function is asymptotically efficient in the sense of this bound.The problem of nonparametric stationary distribution function estimation by the observations of an ergodic diffusion process is considered. The local asymptotic minimax lower bound on the risk of all the estimators is found and it is proved that the empirical distribution function is asymptotically efficient in the sense of this bound.

Goodness of fit test for ergodic diffusions by discrete time observations: an innovation martingale approach

We consider a nonparametric goodness-of-fit test problem for the drift coefficient of one-dimensional ergodic diffusions. Our test is based on the discrete-time observation of the processes, and the diffusion coefficient is a nuisance function which is estimated in some sense in our testing procedure. We prove that the limit distribution of our test is the supremum of the standard Brownian motion, and thus our test is asymptotically distribution free. We also show that our test is consistent under any fixed alternatives.

On compound Poisson processes arising in change-point type statistical models as limiting likelihood ratios

Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In a previous paper of one of the authors it was established that one of these likelihood ratios, which is an exponential functional of a two-sided Poisson process driven by some parameter, can be approximated (for sufficiently small values of the parameter) by another one, which is an exponential functional of a two-sided Brownian motion. In this paper we consider yet another likelihood ratio, which is the exponent of a two-sided compound Poisson process driven by some parameter. We establish, that similarly to the Poisson type one, the compound Poisson type likelihood ratio can be approximated by the Brownian type one for sufficiently small values of the parameter. We equally discuss the asymptotics for large values of the parameter and illustrate the results by numerical simulations.

Estimating unobservable signal by Markovian noise induction

Abstract.A sub threshold signal is transmitted through a channel and may be detected when some noise - with known structure and proportional to some level - is added to the data. There is an optimal noise level, called of stochastic resonance, that corresponds to the minimum variance of the estimators in the problem of recovering unobservable signals. For several noise structures it has been shown the evidence of stochastic resonance effect. Here we study the case when the noise is a Markovian process. We propose consistent estimators of the sub threshold signal and we solve further a problem of hypotheses testing. We also discuss evidence of stochastic resonance for both estimation and hypotheses testing problems via examples.

Z-process method for change point problems with applications to discretely observed diffusion processes

The aim of this paper is to develop a general, unified approach, based on some partial estimation functions which we call “Z-process”, to some change point problems in mathematical statistics. The method proposed can be applied not only to ergodic models but also to some models where the Fisher information matrix is random. Applications to some concrete models, including a parametric model for volatilities of diffusion processes are presented. Simulations for randomly time-transformed Brownian bridge process appearing as the limit of the proposed test statistics are performed with computer intensive use.

On Gaussian Compound Poisson Type Limiting Likelihood Ratio Process

Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. Recently it was established that one of these likelihood ratios, which is an exponential functional of a two-sided Poisson process driven by some parameter, can be approximated (for sufficiently small values of the parameter) by another one, which is an exponential functional of a two-sided Brownian motion. In this chapter we consider yet another likelihood ratio, which is the exponent of a two-sided compound Poisson process driven by some parameter. We establish that the compound Poisson type likelihood ratio can also be approximated by the Brownian type one for sufficiently small values of the parameter. We equally discuss the asymptotics for large values of the parameter.

Backtesting VaR and Expectiles with Realized Scores

Several statistical functionals such as quantiles and expectiles arise naturally as the minimizers of the expected value of a scoring function, a property that is called elicitability (see Gneiting, 2011 and the references therein). The existence of such scoring functions gives a natural way to compare the accuracy of different forecasting models, and to test comparative hypotheses by means of the Diebold-Mariano test (see e.g. Ziegel and Nolde, 2016). In this paper we suggest a procedure to test the accuracy of a quantile or expectile forecasting model in an absolute sense, as in the original Basel I backtesting procedure of Value-at-Risk. To this aim, we study the asymptotic and finite-sample distributions of empirical scores for Normal and Uniform i.i.d. samples. We compare on simulated data the empirical power of our procedure with alternative procedures based on empirical identification functions (i.e. in the case of VaR the number of violations) and we find an higher power in detecting at least misspecification in the mean. We conclude with a real data example where both backtesting procedures are applied to AR(1)-Garch(1,1) models fitted to SP500 logreturns for VaR and expectiles’ forecasts.

Review on Goodness of Fit Tests for Ergodic Diffusion Processes by Different Sampling Schemes

We review some recent results on goodness of fit test for the drift coefficient of a one-dimensional ergodic diffusion, where the diffusion coefficient is a nuisance function which however is estimated. Using a theory for the continuous observation case, we first present a test based on deterministic discrete time observations of the process. Then we also propose a test based on the data observed discretely in space, that is, the so-called tick time sample scheme. In both sampling schemes the limit distribution of the test is the supremum of the standard Brownian motion, thus the test is asymptotically distribution free. The tests are also consistent under any fixed alternatives.