62 research outputs found

    Deconvolution for an atomic distribution: rates of convergence

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    Let X1,...,XnX_1,..., X_n be i.i.d.\ copies of a random variable X=Y+Z,X=Y+Z, where Xi=Yi+Zi, X_i=Y_i+Z_i, and YiY_i and ZiZ_i are independent and have the same distribution as YY and Z,Z, respectively. Assume that the random variables YiY_i's are unobservable and that Y=AV,Y=AV, where AA and VV are independent, AA has a Bernoulli distribution with probability of success equal to 1−p1-p and VV has a distribution function FF with density f.f. Let the random variable ZZ have a known distribution with density k.k. Based on a sample X1,...,Xn,X_1,...,X_n, we consider the problem of nonparametric estimation of the density ff and the probability p.p. Our estimators of ff and pp are constructed via Fourier inversion and kernel smoothing. We derive their convergence rates over suitable functional classes. By establishing in a number of cases the lower bounds for estimation of ff and pp we show that our estimators are rate-optimal in these cases.Comment: 27 page

    √n-consistent parameter estimation for systems of ordinary differential equations : bypassing numerical integration via smoothing

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    We consider the problem of parameter estimation for a system of ordinary differential equations from noisy observations on a solution of the system. In case the system is nonlinear, as it typically is in practical applications, an analytic solution to it usually does not exist. Consequently, straightforward estimation methods like the ordinary least squares method depend on repetitive use of numerical integration in order to determine the solution of the system for each of the parameter values considered, and to find subsequently the parameter estimate that minimises the objective function. This induces a huge computational load to such estimation methods. We propose an estimator that is defined as a minimiser of an appropriate distance between a nonparametrically estimated derivative of the solution and the right-hand side of the system applied to a nonparametrically estimated solution. Our estimator bypasses numerical integration altogether and reduces the amount of computational time drastically compared to ordinary least squares. Moreover, we show that under suitable regularity conditions this estimation procedure leads to a vn-consistent estimator of the parameter of interest

    Conservation laws for invariant functionals containing compositions

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    The study of problems of the calculus of variations with compositions is a quite recent subject with origin in dynamical systems governed by chaotic maps. Available results are reduced to a generalized Euler-Lagrange equation that contains a new term involving inverse images of the minimizing trajectories. In this work we prove a generalization of the necessary optimality condition of DuBois-Reymond for variational problems with compositions. With the help of the new obtained condition, a Noether-type theorem is proved. An application of our main result is given to a problem appearing in the chaotic setting when one consider maps that are ergodic.Comment: Accepted for an oral presentation at the 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2007), to be held in Pretoria, South Africa, 22-24 August, 200

    Non-parametric Bayesian drift estimation for stochastic differential equations

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    We consider non-parametric Bayesian estimation of the drift coefficient of a one-dimensional stochastic differential equation from discrete-time observations on the solution of this equation. Under suitable regularity conditions that are weaker than those previosly suggested in the literature, we establish posterior consistency in this context. Furthermore, we show that posterior consistency extends to the multidimensional setting as well, which, to the best of our knowledge, is a new result in this setting.Comment: 27 page
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