Ulrich Stadtmüller

Generalized functional linear models

We propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function. A linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function, and the expected value of the response is related to this linear predictor via a link function. If, in addition, a variance function is specified, this leads to a functional estimating equation which corresponds to maximizing a functional quasi-likelihood. This general approach includes the special cases of the functional linear model, as well as functional Poisson regression and functional binomial regression. The latter leads to procedures for classification and discrimination of stochastic processes and functional data. We also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data, using a semiparametric quasi-likelihood procedure. An essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated Karhunen-Loeve expansion. We develop asymptotic inference for the proposed class of generalized regression models. In the proposed asymptotic approach, the truncation parameter increases with sample size, and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics. We establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable L 2 metric and is defined through a generalized covariance operator. As a consequence, we obtain asymptotic tests and simultaneous confidence bands for the parameter function that determines the model. The proposed estimation, inference and classification procedures and variants with unknown link and variance functions are investigated in a simulation study. We find that the practical selection of the number of components works well with the AIC criterion, and this finding is supported by theoretical considerations. We include an application to the classification of medflies regarding their remaining longevity status, based on the observed initial egg-laying curve for each of 534 female medflies.We propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function. A linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function, and the expected value of the response is related to this linear predictor via a link function. If in addition a variance function is specified, this leads to a functional estimating equation which corresponds to maximizing a functional quasi-likelihood. This general approach includes the special cases of the functional linear model, as well as functional Poisson regression and functional binomial regression. The latter leads to procedures for classification and discrimination of stochastic processes and functional data. We also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data, using a semiparametric quasi-likelihood procedure. An essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated Karhunen-Loève expansion. We develop asymptotic inference for the proposed class of generalized regression models. In the proposed asymptotic approach, the truncation parameter increases with sample size, and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics. We establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable L2 metric and is defined through a generalized covariance operator. As a consequence, we obtain asymptotic tests and simultaneous confidence bands for the parameter function that determines the model. The proposed estimation, inference and classification procedures and variants with unknown link and variance functions are investigated in a simulation study. We find that the practical selection of the number of components works well with the AIC criterion, and this finding is supported by theoretical considerations. We include an application to the classification of medflies regarding their remaining longevity status, based on the observed initial egg-laying curve for each of 534 female medflies.

Critical systematic review of the level of evidence for routine use of probiotics for reduction of mortality and prevention of necrotizing enterocolitis and sepsis in preterm infants

BACKGROUND & AIMS Probiotics have been suggested to prevent severe necrotizing enterocolitis (NEC) and decrease mortality in preterm infants. The aim of this paper was to systematically analyze the level of evidence (LoE) of published controlled randomized trials (RCTs) on probiotics in preterm infants. METHODS Literature searches were made up to November 2010. LoE of recommendations based on single trials or meta-analyses were scored following the Oxford Center for Evidence based Medicine approach (1a - meta-analyses of 1b LoE studies; 1b - well designed RCT; 2a - meta-analyses which include 2b LoE studies; 2b - lesser quality RCT). RESULTS Fifteen trials were included (Two 1b LoE trials and thirteen 2b LoE trials). Methodological assessment revealed considerable heterogeneity. Some probiotics may be beneficial in relation to reduction of severe NEC (2b LoE) and reduction of mortality (2b LoE). Probiotics do not accelerate feeding advancement (1b and 2b LoE). There was no convincing benefit with regard to prevention of sepsis (1b and 2b LoE). CONCLUSION There is insufficient evidence to recommend routine probiotics. However, there is encouraging data (2b LoE) which justifies the further investigation regarding the efficacy and safety of specific probiotics in circumstances of high local incidence of severe NEC.

Ruin probabilities in the presence of heavy tails and interest rates

Abstract We study the infinite time ruin probability for the classical Cramer-Lundberg model, where the company also receives interest on its reserve. We consider the large claims case, where the claim size distribution F has a regularly varying tail. Hence our results apply for instance to Pareto, loggamma, certain Benktander and stable claim size distributions. We prove that for a positive force of interest δ the ruin probability ψδ (u) ∼ κδ (1 - F(u)) as the initial risk reserve u→∞. This is quantitatively different from the non-interest model, where ψ(u) ∼ κ (1 – F(y)) dy.

On almost sure max-limit theorems

We prove an almost sure version of a maximum limit theorem using logarithmic means and show that essentially only logarithmic means work as it is the case for almost sure central limit theorems.

Multivariate boundary kernels and a continuous least squares principle

Whereas there are many references on univariate boundary kernels, the construction of boundary kernels for multivariate density and curve estimation has not been investigated in detail. The use of multivariate boundary kernels ensures global consistency of multivariate kernel estimates as measured by the integrated mean‐squared error or sup‐norm deviation for functions with compact support. We develop a class of boundary kernels which work for any support, regardless of the complexity of its boundary. Our construction yields a boundary kernel for each point in the boundary region where the function is to be estimated. These boundary kernels provide a natural continuation of non‐negative kernels used in the interior onto the boundary. They are obtained as solutions of the same kernel‐generating variational problem which also produces the kernel function used in the interior as its solution. We discuss the numerical implementation of the proposed boundary kernels and their relationship to locally weighted least squares. Along the way we establish a continuous least squares principle and a continuous analogue of the Gauss–Markov theorem.

Recovering band-limited signals under noise

We consider the problem of recovering a band-limited signal f(t) from noisy data yk=f(k/spl tau/)+/spl Gt/epsilon//sub k/, where /spl tau/ is the sampling rate. Starting from the truncated Whittaker-Shannon cardinal expansion with or without sampling windows (both cases yield inconsistent estimates of f(t)) we propose estimators that are convergent to f(t) in the pointwise and uniform sense. The basic idea is to cut down high frequencies in the data and to use suitable oversampling /spl tau//spl les//spl pi///spl Omega/, /spl Omega/ being the bandwidth (maximum frequency) of f(t). The simplest estimator we propose is given by f/spl circ//sub n/(t)=/spl tau/ /spl Sigma//|t-k/spl tau/|/spl les/n/spl tau/ yksin(/spl Omega/(t-k/spl tau/))//spl pi/(t-k/spl tau/),|t|/spl les/n/spl tau/. Generalizations of f/spl circ//sub n/ including sampling windows are also examined. The main aim is to examine the mean squared error (MSE) properties of such estimators in order to determine the optimal choice of the sampling rate /spl tau/ yielding the fastest possible rate of convergence. The best rate for the MSE we obtain is O(In(n)/n).

On the uniform modulus of continuity of certain discrete approximation operators

Abstract For a certain class of discrete approximation operators B n f defined on an interval I and including, e.g., the Bernstein polynomials, we prove that for all f ϵ C ( I ), the ordinary moduli of continuity of B n f and f satisfy ω ( B n f ; h ) ⩽ cω ( f ; h ), n = 1,2,…, 0 h c > 0. A similar result is shown to hold for a different modulus of continuity which is suitable for functions of polynomial growth on unbounded intervals. Some special operators are discussed in this connection.

Functional Variance Processes

We introduce the notion of a functional variance process to quantify variation in functional data. The functional data are modeled as samples of smooth random trajectories observed under additive noise. The noise is assumed to be composed of white noise and a smooth random process—the functional variance process—which gives rise to smooth random trajectories of variance. The functional variance process is a tool for analyzing stochastic time trends in noise variance. As a smooth random process, it can be characterized by the eigenfunctions and eigenvalues of its autocovariance operator. We develop methods to estimate these characteristics from the data, applying concepts from functional data analysis to the residuals obtained after an initial smoothing step. Asymptotic justifications for the proposed estimates are provided. The proposed functional variance process extends the concept of a variance function, an established tool in nonparametric and semiparametric regression analysis, to the case of functional data. We demonstrate that functional variance processes offer a novel data analysis technique that leads to relevant findings in applications, ranging from a seismic discrimination problem to the analysis of noisy reproductive trajectories in evolutionary biology.

Spatial Smoothing of Geographically Aggregated Data, with Application to the Construction of Incidence Maps

Abstract We address the commonly encountered situation in spatial statistics where data such as counts of incidences of a certain disease are available only in geographically aggregated form. We develop fairly general models and propose a modified version of the locally weighted least squares method to recover the unknown smooth spatial function that is assumed to generate the observations. In the special case of count data, the target function is the intensity function, conditional on the total number of observations. Our method avoids the arbitrariness of selecting a point within each geographic area at which the measurement for the whole area is supposed to be located. We derive basic asymptotic properties, and apply our methods to acquired immune deficiency syndrome (AIDS) incidence data in San Francisco for 1980–1992, where counts are available aggregated over zip code areas.

Smoothing histograms by means of lattice-and continuous distributions

SummaryUsing lattice distributions or an auxiliary density function each satisfying certain moment conditions a general type of estimator for a one dimensional density functionf is developed. This estimator can be looked at as a smoothed histogram. As a measure of quality the exact order of magnitude for the mean squared error is established (pointwise and uniformly) in terms of the size of an iid sample drawn fromf and depending on a design parameter. The methods in deriving the asymptotic behaviour of the mean squared error are based on Edgeworth expansions for the auxiliary distributions.