18 research outputs found
Nonparametric estimation in a regression model with additive and multiplicative noise
In this paper, we consider an unknown functional estimation problem in a
general nonparametric regression model with the feature of having both
multiplicative and additive noise.We propose two new wavelet estimators in this
general context. We prove that they achieve fast convergence rates under the
mean integrated square error over Besov spaces. The obtained rates have the
particularity of being established under weak conditions on the model. A
numerical study in a context comparable to stochastic frontier estimation (with
the difference that the boundary is not necessarily a production function)
supports the theory
Investigating swimming technical skills by a double partition clustering of multivariate functional data allowing for dimension selection
Investigating technical skills of swimmers is a challenge for performance
improvement, that can be achieved by analyzing multivariate functional data
recorded by Inertial Measurement Units (IMU). To investigate technical levels
of front-crawl swimmers, a new model-based approach is introduced to obtain two
complementary partitions reflecting, for each swimmer, its swimming pattern and
its ability to reproduce it. Contrary to the usual approaches for functional
data clustering, the proposed approach also considers the information of the
residuals resulting from the functional basis decomposition. Indeed, after
decomposing into functional basis both the original signal (measuring the
swimming pattern) and the signal of squared residuals (measuring the ability to
reproduce the swimming pattern), the method fits the joint distribution of the
coefficients related to both decompositions by considering dependency between
both partitions. Modeling this dependency is mandatory since the difficulty of
reproducing a swimming pattern depends on its shape. Moreover, a sparse
decomposition of the distribution within components that permits a selection of
the relevant dimensions during clustering is proposed. The partitions obtained
on the IMU data aggregate the kinematical stroke variability linked to swimming
technical skills and allow relevant biomechanical strategy for front-crawl
sprint performance to be identified
Estimation of the Order of Non-Parametric Hidden Markov Models using the Singular Values of an Integral Operator
We are interested in assessing the order of a finite-state Hidden Markov
Model (HMM) with the only two assumptions that the transition matrix of the
latent Markov chain has full rank and that the density functions of the
emission distributions are linearly independent. We introduce a new procedure
for estimating this order by investigating the rank of some well-chosen
integral operator which relies on the distribution of a pair of consecutive
observations. This method circumvents the usual limits of the spectral method
when it is used for estimating the order of an HMM: it avoids the choice of the
basis functions; it does not require any knowledge of an upper-bound on the
order of the HMM (for the spectral method, such an upper-bound is defined by
the number of basis functions); it permits to easily handle different types of
data (including continuous data, circular data or multivariate continuous data)
with a suitable choice of kernel. The method relies on the fact that the order
of the HMM can be identified from the distribution of a pair of consecutive
observations and that this order is equal to the rank of some integral operator
(\emph{i.e.} the number of its singular values that are non-zero). Since only
the empirical counter-part of the singular values of the operator can be
obtained, we propose a data-driven thresholding procedure. An upper-bound on
the probability of overestimating the order of the HMM is established.
Moreover, sufficient conditions on the bandwidth used for kernel density
estimation and on the threshold are stated to obtain the consistency of the
estimator of the order of the HMM. The procedure is easily implemented since
the values of all the tuning parameters are determined by the sample size
Semaine d'Etude Mathématiques et Entreprises 1 : Géométrie des matrices de covariance pour le traitement de signaux radars
Les radars Doppler permettent de détecter des objets volants petits ou de faible signature radar. ThalÚs propose ici de réfléchir aux techniques permettant de faire ressortir de la masse des données radar celles qui sont "aberrantes" afin de repérer parmi le bruit de fond dû aux milieux environnants (nuages de pluie,...) la trace d'un objet volant
Parametric estimation of hidden stochastic model by contrast minimization and deconvolution: application to the Stochastic Volatility Model
We study a new parametric approach for particular hidden stochastic models such as the Stochastic Volatility model. This method is based on contrast minimization and deconvolution. After proving consistency and asymptotic normality of the estimation leading to asymptotic confidence intervals, we provide a thorough numerical study, which compares most of the classical methods that are used in practice (Quasi Maximum Likelihood estimator, Simulated Expectation Maximization Likelihood estimator and Bayesian estimators). We prove that our estimator clearly outperforms the Maximum Likelihood Estimator in term of computing time, but also most of the other methods. We also show that this contrast method is the most robust with respect to non Gaussianity of the error and also does not need any tuning parameter.
Analysis, detection and correction of misspecified discrete time state space models
Misspecifications (i.e. errors on the parameters) of state space models lead to incorrect inference of the hidden states. This paper studies weakly nonlin-ear state space models with additive Gaussian noises and proposes a method for detecting and correcting misspecifications. The latter induce a biased estimator of the hidden state but also happen to induce correlation on innovations and other residues. This property is used to find a well-defined objective function for which an optimisation routine is applied to recover the true parameters of the model. It is argued that this method can consistently estimate the bias on the parameter. We demonstrate the algorithm on various models of increasing complexity
Contrast estimation for noisy observations of diffusion processes via closed-form density expansions
International audienceWhen a continuous-time diffusion is observed only at discrete times with measurement noise, in most cases the transition density is not known and the likelihood is in the form of a high-dimensional integral that does not have a closed-form solution and is difficult to compute accurately. Using Hermite expansions and deconvolution strategy, we provide a general explicit sequence of closed-form contrast for noisy and discretely observed diffusion processes. This work allows the estimation of many diffusion processes. We show that the approximation is very accurate and prove that minimizing the sequence results in a consistent and asymptotically normal estimator. Monte Carlo evidence for the Ornstein-Uhlenbeck process reveals that this method works well and outperforms the Euler expansion of the transition density in situations relevant for financial models