Minimum Contrast Estimators on Sieves: Exponential Bounds and Rates of Convergence

Exponential bounds for minimum contrast estimators

The paper focuses on general properties of parametric minimum contrast estimators. The quality of estimation is measured in terms of the rate function related to the contrast, thus allowing to derive exponential risk bounds invariant with respect to the detailed probabilistic structure of the model. This approach works well for small or moderate samples and covers the case of a misspecified par...

Sieve Estimators : Consistency and Rates of Convergence

The first term is the estimation error and measures the performance of the discrimination rule with respect to the best hypothesis in H. In previous lectures, we studied performance guarantees for this quantity when ĥn is ERM. Now, we also consider the approximation error, which captures how well a class of hypotheses {Hk} approximates the Bayes decision boundary. For example, consider the hist...

Bounds on achievable convergence rates of parameter estimators via universal coding

In many SPECT and PET tomographic geometries, the m X n (m 2 n) system response (($,,)) is a sparse matrix, i.e., its number of nonzero elements is only O(n) as compared to O(n2) for the nonsparse case. Note, however, that even when the system response matrix is sparse, the matrix A (25) is not generally sparse, and it would appear that the recursive algorithm (IO) of Carollary 1 requires O(n2)...

Last Passage times of Minimum Contrast Estimators

Reconstruction on Trees: Exponential Moment Bounds for Linear Estimators

Consider a Markov chain (ξv)v∈V ∈ [k] on the infinite b-ary tree T = (V,E) with irreducible edge transition matrix M , where b ≥ 2, k ≥ 2 and [k] = {1, . . . , k}. We denote by Ln the level-n vertices of T . Assume M has a real second-largest (in absolute value) eigenvalueλ with corresponding real eigenvector ν 6= 0. Letting σv = νξv , we consider the following root-state estimator, which was i...