[Show abstract][Hide abstract] ABSTRACT: In this note we prove the following law of the iterated logarithm for the
Grenander estimator of a monotone decreasing density: If $f(t_0) > 0$, $f'(t_0)
< 0$, and $f'$ is continuous in a neighborhood of $t_0$, then \begin{eqnarray*}
\limsup_{n\rightarrow \infty} \left ( \frac{n}{2\log \log n} \right )^{1/3} (
\widehat{f}_n (t_0 ) - f(t_0) ) = \left| f(t_0) f'(t_0)/2 \right|^{1/3} 2M
\end{eqnarray*} almost surely where $ M \equiv \sup_{g \in {\cal G}} T_g =
(3/4)^{1/3}$ and $ T_g \equiv \mbox{argmax}_u \{ g(u) - u^2 \} $; here ${\cal
G}$ is the two-sided Strassen limit set on $R$. The proof relies on laws of the
iterated logarithm for local empirical processes, Groeneboom's switching
relation, and properties of Strassen's limit set analogous to distributional
properties of Brownian motion.
[Show abstract][Hide abstract] ABSTRACT: We present a general law of the iterated logarithm for stochastic processes
on the open unit interval having subexponential tails in a locally uniform
fashion. It applies to standard Brownian bridge but also to suitably
standardized empirical distribution functions. This leads to new
goodness-of-fit tests and confidence bands which refine the procedures of Berk
and Jones (1979) and Owen (1995). Roughly speaking, the high power and accuracy
of the latter procedures in the tail regions of distributions are esentially
preserved while gaining considerably in the central region.
[Show abstract][Hide abstract] ABSTRACT: We present new algorithms for $M$-estimators of multivariate location and
scatter and for symmetrized $M$-estimators of multivariate scatter. The new
algorithms are considerably faster than currently used fixed-point and other
algorithms. The main idea is to utilize a Taylor expansion of second order of
the target functional and devise a partial Newton-Raphson procedure. In
connection with the symmetrized $M$-estimators we work with incomplete
$U$-statistics to accelerate our procedures initially.
[Show abstract][Hide abstract] ABSTRACT: This survey provides a self-contained account of M-estimation of multivariate
location and scatter, with special emphasis on maximum likelihood estimation
for multivariate t-distributions. In particular, we present new proofs for
existence of the underlying M-functionals, and discuss their weak continuity
and differentiability. Moreover, we present M-estimation of scatter in a rather
general framework with matrix-valued random variables. By doing so we reveal a
connection between Tyler's (1987) M-functional of scatter and the estimation of
proportional covariance matrices. Moreover, this general framework allows us to
treat a new class of scatter estimators, based on symmetrizations of arbitrary
order.
[Show abstract][Hide abstract] ABSTRACT: We consider nonparametric maximum-likelihood estimation of a log-concave density in case of interval-censored, right-censored and binned data. We allow for the possibility of a subprobability density with an additional mass at +∞, which is estimated simultaneously. The existence of the estimator is proved under mild conditions and various theoretical aspects are given, such as certain shape and consistency properties. An EM algorithm is proposed for the approximate computation of the estimator and its performance is illustrated in two examples.
[Show abstract][Hide abstract] ABSTRACT: In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive (point) estimation, the construction of adaptive confidence regions is severely limited (cf. Li in Ann Stat 17:1001–1008, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence regions for the best approximating model in terms of risk. One of our constructions is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral χ
2-distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.
Probability Theory and Related Fields 01/2013; 155(3-4). DOI:10.1007/s00440-012-0414-7 · 1.53 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Identifying and comparing different steady states is an important task for clinical decision making. Data from unequal sources, comprising diverse patient status information, have to be interpreted. In order to compare results an expressive representation is the key. In this contribution we suggest a criterion to calculate a context-sensitive value based on variance analysis and discuss its advantages and limitations referring to a clinical data example obtained during anesthesia. Different drug plasma target levels of the anesthetic propofol were preset to reach and maintain clinically desirable steady state conditions with target controlled infusion (TCI). At the same time systolic blood pressure was monitored, depth of anesthesia was recorded using the bispectral index (BIS) and propofol plasma concentrations were determined in venous blood samples. The presented analysis of variance (ANOVA) is used to quantify how accurately steady states can be monitored and compared using the three methods of measurement.
E-Health and Bioengineering Conference (EHB), 2013; 01/2013
[Show abstract][Hide abstract] ABSTRACT: 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.
The Annals of Statistics 07/2011; 41(3). DOI:10.1214/13-AOS1089 · 2.18 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Let $P$ be a probability distribution on $q$-dimensional space. The so-called
Diaconis-Freedman effect means that for a fixed dimension $d << q$, most
$d$-dimensional projections of $P$ look like a scale mixture of spherically
symmetric Gaussian distributions. The present paper provides necessary and
sufficient conditions for this phenomenon in a suitable asymptotic framework
with increasing dimension $q$. It turns out, that the conditions formulated by
Diaconis and Freedman (1984) are not only sufficient but necessary as well.
Moreover, letting $\hat{P}$ be the empirical distribution of $n$ independent
random vectors with distribution $P$, we investigate the behavior of the
empirical process $\sqrt{n}(\hat{P} - P)$ under random projections, conditional
on $\hat{P}$.
[Show abstract][Hide abstract] ABSTRACT: This paper introduces and analyzes a stochastic search method for parameter
estimation in linear regression models in the spirit of Beran and Millar
(1987). The idea is to generate a random finite subset of a parameter space
which will automatically contain points which are very close to an unknown true
parameter. The motivation for this procedure comes from recent work of
Duembgen, Samworth and Schuhmacher (2011) on regression models with log-concave
error distributions.
[Show abstract][Hide abstract] ABSTRACT: We review various inequalities for Mills' ratio (1 - \Phi)/\phi, where \phi and \Phi denote the standard Gaussian density and distribution function, respectively. Elementary considerations involving finite continued fractions lead to a general approximation scheme which implies and refines several known bounds.
[Show abstract][Hide abstract] ABSTRACT: This note proves Hellinger-consistency for the non-parametric maximum likelihood estimator of a log-concave probability density on .
[Show abstract][Hide abstract] ABSTRACT: We study the approximation of arbitrary distributions $P$ on $d$-dimensional
space by distributions with log-concave density. Approximation means minimizing
a Kullback--Leibler-type functional. We show that such an approximation exists
if and only if $P$ has finite first moments and is not supported by some
hyperplane. Furthermore we show that this approximation depends continuously on
$P$ with respect to Mallows distance $D_1(\cdot,\cdot)$. This result implies
consistency of the maximum likelihood estimator of a log-concave density under
fairly general conditions. It also allows us to prove existence and consistency
of estimators in regression models with a response $Y=\mu(X)+\epsilon$, where
$X$ and $\epsilon$ are independent, $\mu(\cdot)$ belongs to a certain class of
regression functions while $\epsilon$ is a random error with log-concave
density and mean zero.
The Annals of Statistics 02/2010; 39(2). DOI:10.1214/10-AOS853 · 2.18 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
The American Mathematical Monthly 02/2010; 117(2):138-160. DOI:10.4169/000298910X476059 · 0.25 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In diesem Kapitel betrachten wir das einfache Modell unabhängiger, identisch verteilter Zufallsvariablen X1, X2, …, Xn mit Wertebereich X und unbekannter Verteilung P; siehe Abschnitt 3.2.
[Show abstract][Hide abstract] ABSTRACT: Anhand von Beispiel 1.2 illustrieren wir nun ein wichtiges statistisches Verfahren, nämlich Fishers exakten Test, und erläutern relevante Grundbegriffe des Testens. Die zugrundeliegenden allgemeinen Konzepte und Beweise werden dann in
späteren Abschnitten präsentiert.
[Show abstract][Hide abstract] ABSTRACT: Erfahrungsgemäß kann man viele (empirische) Verteilungsfunktionen durch Normalverteilungsfunktionen approximieren. Dies legt
nahe, sich statistische Verfahren für diese Verteilungsfamilie zu überlegen. Historisch gesehen waren dies sogar die ersten
statistischen Verfahren. In diesem Kapitel betrachten wir eine Stichprobe X = (Xi )i = 1n X = (X_i )_{i = 1}^n von stochastisch unabhängigen, nach N (μ,σ2) verteilten Zufallsvariablen. Dabei sind µ ∈ ℝ und σ > 0 Parameter, von denen mindestens einer unbekannt ist und durch ein
Konfidenzintervall eingegrenzt werden soll.
[Show abstract][Hide abstract] ABSTRACT: In diesem Kapitel beschäftigen wir uns mit vektorwertigen Beobachtungen. Im Gegensatz zu früher ist es hier wichtig, zwischen
Zeilen- und Spaltenvektoren zu unterscheiden. Vektoren in ℝk betrachten wir stets als Spaltenvektoren bzw. als (k × 1)–Matrizen.