Nonparametric statistics for scalar ergodic diffusion processes

This thesis is directed towards a twofold aim concerning a statistical problem and its probabilistic foundations. We consider the question of estimating the drift and the invariant density for a large class of scalar, ergodic diffusion processes based on continuous observations in supremum-norm loss. Concentration inequalities and moment bounds or continuous time analogues of classical empirical processes driven by diffusions are provided. These serve as the central probabilistic device for the statistical analysis of the sup-norm risk. The unknown drift is supposed to belong to a nonparametric class of smooth functions of unknown order. We suggest an adaptive approach which allows to construct data driven drift estimators attaining minimax optimal sup-norm rates of convergence. In addition, we prove a Donsker theorem for the classical kernel estimator of the invariant density and establish its semiparametric efficiency. Finally, both results are combined to propose a fully data-driven bandwidth selection procedure which simultaneously yields both a rate-optimal drift estimator and an asymptotically efficient estimator of the invariant density of the diffusion. Crucial tool for our investigation are uniform exponential inequalities for empirical processes and related stochastic integrals driven by scalar, ergodic diffusion processes. Providing these probabilistic tools, we lay the foundation typically required for the study of sup-norm properties of estimation procedures for a large class of diffusion processes. The idea originates in the classical i.i.d. context where Talagrand-type concentration inequalities are a key device for the statistical sup-norm analysis. Aiming for a parallel substitute in the diffusion framework, we present a systematic, self-contained approach to such uniform concentration inequalities via martingale approximation and moment bounds obtained by the generic chaining method

Concentration analysis of multivariate elliptic diffusion processes

We prove concentration inequalities and associated PAC bounds for continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approach via the Poisson equation allowing us to consider a very broad class of subexponentially ergodic processes. These results add to existing concentration inequalities for additive functionals of diffusion processes which have so far been only available for either bounded functions or for unbounded functions of processes from a significantly smaller class. We demonstrate the power of these exponential inequalities by two examples of very different areas. Considering a possibly high-dimensional parametric nonlinear drift model under sparsity constraints, we apply the continuous-time concentration results to validate the restricted eigenvalue condition for Lasso estimation, which is fundamental for the derivation of oracle inequalities. The results for discrete additive functionals are used to investigate the unadjusted Langevin MCMC algorithm for sampling of moderately heavy-tailed densities $\pi$. In particular, we provide PAC bounds for the sample Monte Carlo estimator of integrals $\pi(f)$ for polynomially growing functions $f$ that quantify sufficient sample and step sizes for approximation within a prescribed margin with high probability

In vivo effect of immobilisation of bone morphogenic protein 2 on titanium implants through nano-anchored oligonucleotides

The aim of the present study was to test the hypothesis that immobilisation of bone morphogenic proteins on the surface of titanium implants through nano-anchored oligonucleotides can enhance peri-implant bone formation. Non-coding 60-mer DNA oligonucleotides (ODN) were anchored to the surface of custom made sandblasted acid etched (SAE) titanium screw implants through anodic polarisation, gamma-sterilised with a standard dose of 25 kGy, and were hybridised with complementary 30-mer strands of DNA oligonucleotides conjugated to rhBMP2. Blank SAE implants, SAE implants with nano-anchored ODN and SAE implants with nano-anchored ODN and non-conjugated rhBMP2 served as controls. The implants were inserted into the tibiae of 36 Sprague Dawley rats. Perforations at the head and the tip of the implants allowed for bone ingrowth. Bone ingrowth into perforations and bone implant contact (BIC) as well as bone density (BD) at a distance of 200 µm from the implant surface were assessed after 1 , 4 and 13 weeks. Implants with nano-anchored ODN strands hybridised with conjugated rhBMP2 exhibited enhanced bone ingrowth into the perforations and increased BIC after 1 week as well as increased BIC after 4 weeks compared to controls. No difference was seen after 13 weeks. Bone density around the outer implant surface did not differ significantly at any of the intervals. It is concluded that rhBMP2 immobilised on the surface of titanium implants through nano-anchored oligonucleotide strands can enhance bone implant contact. The conditions of sterilisation tested allowed for handling under clinically relevant conditions