Estimation of Hilbert Space Valued Parameters by the Method of Sieves
NORTH CAROLINA UNIV AT CHAPEL HILL DEPT OF STATISTICS
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By extending the ideas of Ibragimov Hasminski in the finite dimensional parameter estimation a large deviation inequality for a sieve estimator estimating a Hilbert space valued parameter is obtained. This sieve estimator corresponds to a sieve which consists of finite dimensional, compact, convex sets. The inequality suggests a procedure of consistent estimation of Hilbert space valued parameters and naturally provides the convergence rates of the resultant estimators. The usefulness of this approach is demonstrated by applying it to two examples the first one deals with the estimation of the drift function in a linear stochastic differential equation and the second problem is of the intensity estimation of a nonstationary Poisson process. A detailed discussion of the convergence rates of our estimators and how they compare with the other estimators proposed in the literature is given in both cases.
- Statistics and Probability