Convergence Of Non-standard Form And Applications In Statistical Estimations

As a sparse representation of operators, the non-standard form (NSF) origi-nates from fnding solutions of a linear inverse problem y(t)=Kf(t)+ε(t). In1990's, Beylkin, Coifman and Rokhlin introduce the non-standard form by waveletbasis, which leads to fast numerical algorithms and explicit expression. This disser-tation discusses the convergence of NSF for diferential operators over Besov spaceand Sobolev space with integer exponents. Moreover, we study the optimality ofwavelet estimates for density derivatives.First, we extend the applied spaces of NSF for diferential operators by usingwavelet characterizations for functional spaces. More precisely, it is consideredthat convergence rate in uniform and Lpnorm sense over Besov space B_p,q^s(R) andSobolev space W_p^N(R) with integer exponents.Second, we construct the wavelet thresholding estimator for diferential opera-tors based on wavelet expansion and thresholding method, because wavelet thresh-olding method plays important roles in image processing, statistical estimationsand so on. With the help of maximal function theorem, we give the convergencerate of estimators in uniform and Lpnorm sense over the Besov space B_p,q^s(R) andSobolev space W_p^N(R) with integer exponents. All above results generalize Taoand Vidakovic, as well as Chen and Meng's work in some sense.Finally, we study nonparametric statistical estimations based on the aboveconvergence results. Wavelet estimators for density derivatives are defned by us-ing NSF of diferential operators and wavelet estimators of density. It turns out that our linear estimator performs better than classical kernel estimate. In fact,when r≥p, the linear one for f(m)∈Bsr,q(R)(or WNr(R)) attains the optimalconvergence rate in Lprisk sense. Since the linear one doesn't provide optimalestimate for density derivatives when r <p, we construct nonlinear wavelet esti-mator, which attains the optimal convergence rate for1≤r <(p/2(s+m)+1). Whenr=p/(2(s+m)+1), it is sub-optimal (optimal up to a logarithmic factor). An openproblem is how to defne an optimal (or sub-optimal) wavelet estimator for den-sity derivatives, if p/(2(s+m)+1)<r <p?...