Victor Patrangenaru

Locally homogeneous Riemannian manifolds and Cartan triples

AbstractAnn-dimensional Cartan triple is a triple (g, Γ,

Edgeworth Expansions and the Bootstrap

\bar \Omega

Large Sample Theory of Estimation in Parametric Models

) consisting of a Lie subalgebra g of so(n) (endowed with the Killing form), a linear map Γ:ℝn → g⊥ and a bilinear antisymmetric map Ω ε Λ2(ℝn, g), which together satisfy (1.25)–(1.28) of Section 1. LetMn be the set ofmaximal n-dimensional Cartan triples, and letAn be thenatural action of the orthogonal group O(n) onMn (Section 3). One shows that there is a bijective mapping from the set of local isometry classes ofn-dimensional locally homogeneous Riemannian manifolds to the set of orbits ofAn (Theorem 3.1(a)). Under this bijection, the classes of homogeneous Riemannian manifolds correspond to orbits ofclosed Cartan triples.

Tests in Parametric and Nonparametric Models

This chapter provides an introduction to nonparametric estimations of densities and regression functions by the kernel method.

Consistency and Asymptotic Distributions of Statistics

This chapter outlines the proof of the validity of a properly formulated version of the formal Edgeworth expansion, and derives from it the precise asymptotic rate of the coverage error of Efron’s bootstrap. A number of other applications of Edgeworth expansions are outlined.

Markov Chain Monte Carlo (MCMC) Simulation and Bayes Theory

This chapter introduces Efron’s nonparametric bootstrap, with applications to linear statistics, and semi-linear regression due to Bickel and Freedman.

Introduction to General Methods of Estimation

The main focus of this chapter is the asymptotic Normality and optimality of the maximum likelihood estimator (MLE), under regularity conditions. The Cramer–Rao lower bound for the variance of unbiased estimators of parametric functions is shown to be achieved asymptotically by the MLE. Also derived are the asymptotic Normality of M-estimators and the asymptotic behavior of the Bayes posterior.

Fréchet Means and Nonparametric Inference on Non-Euclidean Geometric Spaces

The asymptotic theory of tests in parametric and nonparametric models and their relative efficiency is presented here. In particular, livelihood ratio, Wald’s test and chisquare tests are derived in parametric models. The nonparametric tests discussed include two-sample rank tests and the Kolmogorov–Smirnov tests. Also presented are goodness-of-fit tests and inference for linear time series models.