868 research outputs found
Nonparametric estimation of scalar diffusions based on low frequency data
We study the problem of estimating the coefficients of a diffusion (X_t,t\geq
0); the estimation is based on discrete data X_{n\Delta},n=0,1,...,N. The
sampling frequency \Delta^{-1} is constant, and asymptotics are taken as the
number N of observations tends to infinity. We prove that the problem of
estimating both the diffusion coefficient (the volatility) and the drift in a
nonparametric setting is ill-posed: the minimax rates of convergence for
Sobolev constraints and squared-error loss coincide with that of a,
respectively, first- and second-order linear inverse problem. To ensure
ergodicity and limit technical difficulties we restrict ourselves to scalar
diffusions living on a compact interval with reflecting boundary conditions.
Our approach is based on the spectral analysis of the associated Markov
semigroup. A rate-optimal estimation of the coefficients is obtained via the
nonparametric estimation of an eigenvalue-eigenfunction pair of the transition
operator of the discrete time Markov chain (X_{n\Delta},n=0,1,...,N) in a
suitable Sobolev norm, together with an estimation of its invariant density.Comment: Published at http://dx.doi.org/10.1214/009053604000000797 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Kalikow-type decomposition for multicolor infinite range particle systems
We consider a particle system on with real state space and
interactions of infinite range. Assuming that the rate of change is continuous
we obtain a Kalikow-type decomposition of the infinite range change rates as a
mixture of finite range change rates. Furthermore, if a high noise condition
holds, as an application of this decomposition, we design a feasible perfect
simulation algorithm to sample from the stationary process. Finally, the
perfect simulation scheme allows us to forge an algorithm to obtain an explicit
construction of a coupling attaining Ornstein's -distance for two
ordered Ising probability measures.Comment: Published in at http://dx.doi.org/10.1214/12-AAP882 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Diffusion Approximations for Online Principal Component Estimation and Global Convergence
In this paper, we propose to adopt the diffusion approximation tools to study
the dynamics of Oja's iteration which is an online stochastic gradient descent
method for the principal component analysis. Oja's iteration maintains a
running estimate of the true principal component from streaming data and enjoys
less temporal and spatial complexities. We show that the Oja's iteration for
the top eigenvector generates a continuous-state discrete-time Markov chain
over the unit sphere. We characterize the Oja's iteration in three phases using
diffusion approximation and weak convergence tools. Our three-phase analysis
further provides a finite-sample error bound for the running estimate, which
matches the minimax information lower bound for principal component analysis
under the additional assumption of bounded samples.Comment: Appeared in NIPS 201
Privacy-preserving parametric inference: a case for robust statistics
Differential privacy is a cryptographically-motivated approach to privacy
that has become a very active field of research over the last decade in
theoretical computer science and machine learning. In this paradigm one assumes
there is a trusted curator who holds the data of individuals in a database and
the goal of privacy is to simultaneously protect individual data while allowing
the release of global characteristics of the database. In this setting we
introduce a general framework for parametric inference with differential
privacy guarantees. We first obtain differentially private estimators based on
bounded influence M-estimators by leveraging their gross-error sensitivity in
the calibration of a noise term added to them in order to ensure privacy. We
then show how a similar construction can also be applied to construct
differentially private test statistics analogous to the Wald, score and
likelihood ratio tests. We provide statistical guarantees for all our proposals
via an asymptotic analysis. An interesting consequence of our results is to
further clarify the connection between differential privacy and robust
statistics. In particular, we demonstrate that differential privacy is a weaker
stability requirement than infinitesimal robustness, and show that robust
M-estimators can be easily randomized in order to guarantee both differential
privacy and robustness towards the presence of contaminated data. We illustrate
our results both on simulated and real data
Maximum likelihood estimation by monte carlo simulation:Toward data-driven stochastic modeling
We propose a gradient-based simulated maximum likelihood estimation to estimate unknown parameters in a stochastic model without assuming that the likelihood function of the observations is available in closed form. A key element is to develop Monte Carlo-based estimators for the density and its derivatives for the output process, using only knowledge about the dynamics of the model. We present the theory of these estimators and demonstrate how our approach can handle various types of model structures. We also support our findings and illustrate the merits of our approach with numerical results
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