arXiv subjects

Lutz Duembgen

Publications and source records attributed to Lutz Duembgen.

45 records · Page 3Linked to original sources

arXiv2009

Nemirovski's Inequalities Revisited

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.

math.ST↗

arXiv2009

Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency

We study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least $(\log(n)/n)^{1/3}$ and typically $(\log(n)/n)^{2/5}$, whereas the difference between the empirical and estimated distribution function vanishes with rate $o_{\mathrm{p}}(n^{-1/2})$ under certain regularity assumptions.

math.ST↗

arXiv2009

Extensions of smoothing via taut strings

Suppose that we observe independent random pairs $(X_1,Y_1)$, $(X_2,Y_2)$, >..., $(X_n,Y_n)$. Our goal is to estimate regression functions such as the conditional mean or $β$--quantile of $Y$ given $X$, where $0<β<1$. In order to achieve this we minimize criteria such as, for instance, $$ \sum_{i=1}^n ρ(f(X_i) - Y_i) + λ\cdot \mathop TV\nolimits (f) $$ among all candidate functions $f$. Here $ρ$ is some convex function depending on the particular regression function we have in mind, $\mathop {\rm TV}\nolimits (f)$ stands for the total variation of $f$, and $λ>0$ is some tuning parameter. This framework is extended further to include binary or Poisson regression, and to include localized total variation penalties. The latter are needed to construct estimators adapting to inhomogeneous smoothness of $f$. For the general framework we develop noniterative algorithms for the solution of the minimization problems which are closely related to the taut string algorithm (cf. Davies and Kovac, 2001). Further we establish a connection between the present setting and monotone regression, extending previous work by Mammen and van de Geer (1997). The algorithmic considerations and numerical examples are complemented by two consistency results.

stat.CO↗

arXiv2009

Least Squares and Shrinkage Estimation under Bimonotonicity Constraints

In this paper we describe active set type algorithms for minimization of a smooth function under general order constraints, an important case being functions on the set of bimonotone r-by-s matrices. These algorithms can be used, for instance, to estimate a bimonotone regression function via least squares or (a smooth approximation of) least absolute deviations. Another application is shrinkage estimation in image denoising or, more generally, regression problems with two ordinal factors after representing the data in a suitable basis which is indexed by pairs (i,j) in {1,...,r}x{1,...,s}. Various numerical examples illustrate our methods.

stat.CO↗

arXiv2008

Multiscale inference about a density

We introduce a multiscale test statistic based on local order statistics and spacings that provides simultaneous confidence statements for the existence and location of local increases and decreases of a density or a failure rate. The procedure provides guaranteed finite-sample significance levels, is easy to implement and possesses certain asymptotic optimality and adaptivity properties.

math.ST↗

arXiv2008

P-values for classification

Let $(X,Y)$ be a random variable consisting of an observed feature vector $X\in \mathcal{X}$ and an unobserved class label $Y\in \{1,2,...,L\}$ with unknown joint distribution. In addition, let $\mathcal{D}$ be a training data set consisting of $n$ completely observed independent copies of $(X,Y)$. Usual classification procedures provide point predictors (classifiers) $\widehat{Y}(X,\mathcal{D})$ of $Y$ or estimate the conditional distribution of $Y$ given $X$. In order to quantify the certainty of classifying $X$ we propose to construct for each $θ=1,2,...,L$ a p-value $π_θ(X,\mathcal{D})$ for the null hypothesis that $Y=θ$, treating $Y$ temporarily as a fixed parameter. In other words, the point predictor $\widehat{Y}(X,\mathcal{D})$ is replaced with a prediction region for $Y$ with a certain confidence. We argue that (i) this approach is advantageous over traditional approaches and (ii) any reasonable classifier can be modified to yield nonparametric p-values. We discuss issues such as optimality, single use and multiple use validity, as well as computational and graphical aspects.

math.ST↗

arXiv2008

Explicit bounds for the approximation error in Benford's law

Benford's law states that for many random variables X > 0 its leading digit D = D(X) satisfies approximately the equation P(D = d) = log_{10}(1 + 1/d) for d = 1,2,...,9. This phenomenon follows from another, maybe more intuitive fact, applied to Y := log_{10}(X): For many real random variables Y, the remainder U := Y - floor(Y) is approximately uniformly distributed on [0,1). The present paper provides new explicit bounds for the latter approximation in terms of the total variation of the density of Y or some derivative of it. These bounds are an interesting alternative to traditional Fourier methods which yield mostly qualitative results. As a by-product we obtain explicit bounds for the approximation error in Benford's law.

math.PR↗

arXiv2007

Confidence bands for convex median curves using sign-tests

Suppose that one observes pairs $(x_1,Y_1)$, $(x_2,Y_2)$, ..., $(x_n,Y_n)$, where $x_1\le x_2\le ... \le x_n$ are fixed numbers, and $Y_1,Y_2,...,Y_n$ are independent random variables with unknown distributions. The only assumption is that ${\rm Median}(Y_i)=f(x_i)$ for some unknown convex function $f$. We present a confidence band for this regression function $f$ using suitable multiscale sign-tests. While the exact computation of this band requires $O(n^4)$ steps, good approximations can be obtained in $O(n^2)$ steps. In addition the confidence band is shown to have desirable asymptotic properties as the sample size $n$ tends to infinity.

math.ST↗

arXiv2007

Marshall's lemma for convex density estimation

Marshall's [Nonparametric Techniques in Statistical Inference (1970) 174--176] lemma is an analytical result which implies $\sqrt{n}$--consistency of the distribution function corresponding to the Grenander [Skand. Aktuarietidskr. 39 (1956) 125--153] estimator of a non-decreasing probability density. The present paper derives analogous results for the setting of convex densities on $[0,\infty)$.

math.ST↗