Matrix Differential Calculus with Applications in Statistics and Econometrics

Recent years have witnessed an explosion in the volume and variety of data collected in all scientific disciplines and industrial settings. Such massive data sets present a number of challenges to researchers in statistics and machine learning. This book provides a self-contained introduction to the area of high-dimensional statistics, aimed at the first-year graduate level. It includes chapters that are focused on core methodology and theory - including tail bounds, concentration inequalities, uniform laws and empirical process, and random matrices - as well as chapters devoted to in-depth exploration of particular model classes - including sparse linear models, matrix models with rank constraints, graphical models, and various types of non-parametric models. With hundreds of worked examples and exercises, this text is intended both for courses and for self-study by graduate students and researchers in statistics, machine learning, and related fields who must understand, apply, and adapt modern statistical methods suited to large-scale data.

High-Dimensional Statistics: A Non-Asymptotic Viewpoint

Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

Diffusion processes and their sample paths

1. Nonparametric statistical models 2. Gaussian processes 3. Empirical processes 4. Function spaces and approximation theory 5. Linear nonparametric estimators 6. The minimax paradigm 7. Likelihood-based procedures 8. Adaptive inference.

Mathematical foundations of infinite-dimensional statistical models

Let $Y$ be a stochastic process on $[0,1]$ satisfying $dY(t) = n^{1/2} f(t) dt + dW(t)$, where $n \ge 1$ is a given scale parameter (``sample size''), $W$ is standard Brownian motion and $f$ is an unknown function. Utilizing suitable multiscale tests we construct confidence bands for $f$ with guaranteed given coverage probability, assuming that $f$ is isotonic or convex. These confidence bands are computationally feasible and shown to be asymptotically sharp optimal in an appropriate sense.

Optimal Confidence Bands for Shape-Restricted Curves

Let $b(x)$ be the probability that a sum of independent Bernoulli random variables with parameters $p_1, p_2, p_3, \ldots \in [0,1)$ equals $x$, where $\lambda := p_1 + p_2 + p_3 + \cdots$ is finite. We prove two inequalities for the maximal ratio $b(x)/\pi_\lambda(x)$, where $\pi_\lambda$ is the weight function of the Poisson distribution with parameter $\lambda$.

The density ratio of generalized binomial versus Poisson distributions

We analyze four different approaches to estimate a multivariate probability density (or the log-density) and its first and second order derivatives. Two methods, local log-likelihood and local Hyvarinen score estimation, are in terms of weighted scoring rules with local quadratic models. The other two approaches are matching of local moments and kernel density estimation. All estimators depend on a general kernel, and we use the Gaussian kernel to provide explicit examples. Asymptotic properties of the estimators are derived and compared. In terms of rates of convergence, a refined local moment matching estimator is the best.

Local Estimation of a Multivariate Density and its Derivatives

An important tool for statistical research are moment inequalities for sums of independent random vectors. Nemirovski and coworkers (1983, 2000) derived one particular type of such inequalities: For certain Banach spaces $(\B,\|\cdot\|)$ there exists a constant $K = K(\B,\|\cdot\|)$ such that for arbitrary independent and centered random vectors $X_1, X_2, ..., X_n \in \B$, their sum $S_n$ satisfies the inequality $ E \|S_n \|^2 \le K \sum_{i=1}^n E \|X_i\|^2$. We present and compare three different approaches to obtain such inequalities: Nemirovski's results are based on deterministic inequalities for norms. Another possible vehicle are type and cotype inequalities, a tool from probability theory on Banach spaces. Finally, we use a truncation argument plus Bernstein's inequality to obtain another version of the moment inequality above. Interestingly, all three approaches have their own merits.

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Nemirovski's Inequalities Revisited

We derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. An important example covered within this framework is to test for local monotonicity on all scales simultaneously. We investigate the moderately ill-posed setting, where the Fourier transform of the error density in the deconvolution model is of polynomial decay. For multiscale testing, we consider a calibration, motivated by the modulus of continuity of Brownian motion. We investigate the performance of our results from both the theoretical and simulation based point of view. A major consequence of our work is that the detection of qualitative features of a density in a deconvolution problem is a doable task although the minimax rates for pointwise estimation are very slow.

Lutz Duembgen

Papers

Optimal Confidence Bands for Shape-Restricted Curves

The density ratio of generalized binomial versus Poisson distributions

Local Estimation of a Multivariate Density and its Derivatives

Nemirovski's Inequalities Revisited

Multiscale Methods for Shape Constraints in Deconvolution: Confidence Statements for Qualitative Features