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In this paper, we develop statistical inference tools for high-dimensional functional time series. We introduce a new concept of physical dependent processes in the space of square integrable functions, which adopts the idea of basis decomposition of functional data in these spaces, and derive Gaussian and multiplier bootstrap approximations for sums of high-dimensional functional time series. These results have numerous important statistical consequences. Exemplarily, we consider the development of joint simultaneous confidence bands for the mean functions and the construction of tests for the hypotheses that the mean functions in the panel dimension are parallel. The results are illustrated by means of a small simulation study and in the analysis of Canadian temperature data.

Summary Clinical trials often aim to compare two groups of patients for efficacy and/or toxicity depending on covariates such as dose. Examples include the comparison of populations from different geographic regions or age classes or, alternatively, of different treatment groups. Similarity of these groups can be claimed if the difference in average outcome is below a certain margin over the entire covariate range. In this article, we consider the problem of testing for similarity in the case that efficacy and toxicity are measured as binary outcome variables. We develop a new test for the assessment of similarity of two groups for a single binary endpoint. Our approach is based on estimating the maximal deviation between the curves describing the responses of the two groups, followed by a parametric bootstrap test. Further, using a two-dimensional Gumbel-type model we develop methodology to establish similarity for (correlated) binary efficacy–toxicity outcomes. We investigate the operating characteristics of the proposed methodology by means of a simulation study and present a case study as an illustration.

We focus on estimating daily integrated volatility (IV) by realized measures based on intraday returns following a discrete-time stochastic model with a pronounced intraday periodicity (IP). We demonstrate that neglecting the IP-impact on realized estimators may lead to invalid statistical inference concerning IV for a common finite number of intraday returns. For a given IP functional form, we analytically derive robust IP-correction factors for realized measures of IV as well as their asymptotic distributions. We show both in Monte Carlo simulations and empirically that the proposed bias corrections are the robust way to account for IP by computing realized estimators.

Diurnal fluctuations in volatility are a well-documented stylized fact of intraday price data. This warrants an investigation how this intraday periodicity (IP) affects both finite sample as well as asymptotic properties of several popular realized estimators of daily integrated volatility which are based on functionals of a finite number of intraday returns. It turns out that most of the estimators considered in this study exhibit a finite-sample bias due to IP, which can however get negligible when the number of intraday returns diverges to infinity. The appropriate correction factors for this bias are derived based on estimates of the IP. The adequacy of the new corrections is evaluated by means of a Monte Carlo simulation study and an empirical example.

Summary The classical approach to analyze pharmacokinetic (PK) data in bioequivalence studies aiming to compare two different formulations is to perform noncompartmental analysis (NCA) followed by two one-sided tests (TOST). In this regard, the PK parameters area under the curve (AUC) and $C_{\max}$ are obtained for both treatment groups and their geometric mean ratios are considered. According to current guidelines by the U.S. Food and Drug Administration and the European Medicines Agency, the formulations are declared to be sufficiently similar if the $90\%$ confidence interval for these ratios falls between $0.8$ and $1.25 $. As NCA is not a reliable approach in case of sparse designs, a model-based alternative has already been proposed for the estimation of $\rm AUC$ and $C_{\max}$ using nonlinear mixed effects models. Here we propose another, more powerful test than the TOST and demonstrate its superiority through a simulation study both for NCA and model-based approaches. For products with high variability on PK parameters, this method appears to have closer type I errors to the conventionally accepted significance level of $0.05$, suggesting its potential use in situations where conventional bioequivalence analysis is not applicable.

Data based materials science is the new promise to accelerate materials design. Especially in computational materials science, data generation can easily be automatized. Usually, the focus is on processing and evaluating the data to derive rules or to discover new materials, while less attention is being paid on the strategy to generate the data. In this work, we show that by a sequential design of experiment scheme, the process of generating and learning from the data can be combined to discover the relevant sections of the parameter space. Our example is the energy of grain boundaries as a function of their geometric degrees of freedom, calculated via atomistic simulations. The sampling of this grain boundary energy space, or even subspaces of it, represents a challenge due to the presence of deep cusps of the energy, which are located at irregular intervals of the geometric parameters. Existing approaches to sample grain boundary energy subspaces therefore either need a huge amount of datapoints or a~priori knowledge of the positions of these cusps. We combine statistical methods with atomistic simulations and a sequential sampling technique and compare this strategy to a regular sampling technique. We thereby demonstrate that this sequential design is able to sample a subspace with a minimal amount of points while finding unknown cusps automatically.

With an increasing number of novel therapeutic options for lower urinary tract symptoms (LUTS), the spectrum of potential treatment pathways resulting from different combinations of treatment decisions is expanding and evolving. Treatment decisions are frequently made with little or no evidence from randomized controlled trials (RCTs) and thus require evidence from other data sources. Clinical routine data reflect real-world treatment pathways. However, evidence for LUTS from routine data means that heterogeneous pathways need to be simultaneously analyzed for compiling evidence in the absence of RCTs. Statistical multi-state model approaches can provide a powerful framework for achieving this goal. More extensive statistical and methodological efforts in the area of similarity of small data are needed to enable the valid pooling of pathways towards joining evidence. PATIENT SUMMARY: Treatment decisions should rely primarily on evidence from clinical trials. When treatment for which there is limited trial evidence needs to be provided, analysis of results from routine clinical practice can represent valuable complementary evidence, but this requires integration of data from heterogeneous treatment pathways.

For a spatiotemporal process $\{X_j(s,t) | ~s \in S~,~t \in T \}_{j =1, \ldots , n} $, where $S$ denotes the set of spatial locations and $T$ the time domain, we consider the problem of testing for a change in the sequence of mean functions. In contrast to most of the literature we are not interested in arbitrarily small changes, but only in changes with a norm exceeding a given threshold. Asymptotically distribution free tests are proposed, which do not require the estimation of the long-run spatiotemporal covariance structure. In particular we consider a fully functional approach and a test based on the cumulative sum paradigm, investigate the large sample properties of the corresponding test statistics and study their finite sample properties by means of simulation study.

Summary We study the problem of testing equivalence of functional parameters, such as the mean or the variance function, in the two-sample functional data setting. In contrast to previous work where the functional problem is reduced to a multiple testing problem for the equivalence of scalar data by comparing the functions at each point, our approach is based on an estimate of a distance measuring the maximum deviation between the two functional parameters. Equivalence is claimed if the estimate for the maximum deviation does not exceed a given threshold. We propose a bootstrap procedure for obtaining quantiles of the distribution of the test statistic, and we prove consistency of the corresponding test in the large-sample scenario. As the methods proposed here avoid the use of the intersection-union principle, they are less conservative and more powerful than currently available approaches.

This article considers the problem of estimating a change point in the covariance matrix in a sequence of high-dimensional vectors, where the dimension is substantially larger than the sample size. A two-stage approach is proposed to efficiently estimate the location of the change point. The first step consists of a reduction of the dimension to identify elements of the covariance matrices corresponding to significant changes. In a second step, we use the components after dimension reduction to determine the position of the change point. Theoretical properties are developed for both steps, and numerical studies are conducted to support the new methodology. Supplementary materials for this article are available online.

In this paper we study the theoretical properties of the simultaneous multiscale change point estimator (SMUCE) proposed by Frick et al. (2014) in regression models with dependent error processes. Empirical studies show that in this case the change point estimate is inconsistent, but it is not known if alternatives suggested in the literature for correlated data are consistent. We propose a modification of SMUCE scaling the basic statistic by the long run variance of the error process, which is estimated by a difference-type variance estimator calculated from local means from different blocks. For this modification we prove model consistency for physical dependent error processes and illustrate the finite sample performance by means of a simulation study.

Summary We develop methodology for testing relevant hypotheses about functional time series in a tuning-free way. Instead of testing for exact equality, e.g. for the equality of two mean functions from two independent time series, we propose to test the null hypothesis of no relevant deviation. In the two-sample problem this means that an L2-distance between the two mean functions is smaller than a prespecified threshold. For such hypotheses self-normalization, which was introduced in 2010 by Shao, and Shao and Zhang and is commonly used to avoid the estimation of nuisance parameters, is not directly applicable. We develop new self-normalized procedures for testing relevant hypotheses in the one-sample, two-sample and change point problem and investigate their asymptotic properties. Finite sample properties of the tests proposed are illustrated by means of a simulation study and data examples. Our main focus is on functional time series, but extensions to other settings are also briefly discussed.

The recent availability of routine medical data, especially in a university-clinical context, may enable the discovery of typical healthcare pathways, i.e., typical temporal sequences of clinical interventions or hospital readmissions. However, such pathways are heterogeneous in a large provider such as a university hospital, and it is important to identify similar care pathways that can still be considered typical pathways. We understand the pathway as a temporal process with possible transitions from a single initial treatment state to hospital readmission of different types, which constitutes a competing risk setting. In this paper, we propose a multi-state model-based approach to uncover pathway similarity between two groups of individuals. We describe a new bootstrap procedure for testing the similarity of transition intensities from two competing risk models with constant transition intensities. In a large simulation study, we investigate the performance of our similarity approach with respect to different sample sizes and different similarity thresholds. The studies are motivated by an application from urological clinical routine and we show how the results can be transferred to the application example.

We develop methodology for testing hypotheses regarding the slope function in functional linear regression for time series via a reproducing kernel Hilbert space approach. In contrast to most of the literature, which considers tests for the exact nullity of the slope function, we are interested in the null hypothesis that the slope function vanishes only approximately, where deviations are measured with respect to the $L^2$-norm. An asymptotically pivotal test is proposed, which does not require the estimation of nuisance parameters and long-run covariances. The key technical tools to prove the validity of our approach include a uniform Bahadur representation and a weak invariance principle for a sequential process of estimates of the slope function. Both scalar-on-function and function-on-function linear regression are considered and finite-sample methods for implementing our methodology are provided. We also illustrate the potential of our methods by means of a small simulation study and a data example.

We consider the problem of constructing nonparametric undirected graphical models for high-dimensional functional data. Most existing statistical methods in this context assume either a Gaussian distribution on the vertices or linear conditional means. In this article we provide a more flexible model which relaxes the linearity assumption by replacing it by an arbitrary additive form. The use of functional principal components offers an estimation strategy that uses a group lasso penalty to estimate the relevant edges of the graph. We establish statistical guarantees for the resulting estimators, which can be used to prove consistency if the dimension and the number of functional principal components diverge to infinity with the sample size. We also investigate the empirical performance of our method through simulation studies and a real data application.

We consider the moment space M 2 n + 1 d n of moments up to the order 2 n + 1 of d n × d n real matrix measures defined on the interval [ 0 , 1 ] . The asymptotic properties of the Hankel determinant { log det ( M i + j d n ) i , j = 0 , … , ⌊ n t ⌋ } t ∈ [ 0 , 1 ] of a uniformly distributed vector ( M 1 , … , M 2 n + 1 ) t ∼ U ( M 2 n + 1 ) are studied when the dimension n of the moment space and the size of the matrices d n converge to infinity. In particular weak convergence of an appropriately centered and standardized version of this process is established. Mod-Gaussian convergence is shown and several large and moderate deviation principles are derived. Our results are based on some new relations between determinants of subblocks of the Jacobi-beta-ensemble, which are of their own interest and generalize Bartlett decomposition-type results for the Jacobi-beta-ensemble from the literature.

Classical change point analysis aims at (1) detecting abrupt changes in the mean of a possibly non-stationary time series and at (2) identifying regions where the mean exhibits a piecewise constant behavior. In many applications however, it is more reasonable to assume that the mean changes gradually in a smooth way. Those gradual changes may either be non-relevant (i.e., small), or relevant for a specific problem at hand, and the present paper presents statistical methodology to detect the latter. More precisely, we consider the common nonparametric regression model $X_{i} = \mu (i/n) + \varepsilon_{i}$ with possibly non-stationary errors and propose a test for the null hypothesis that the maximum absolute deviation of the regression function $\mu$ from a functional $g (\mu )$ (such as the value $\mu (0)$ or the integral $\int_{0}^{1} \mu (t) dt$) is smaller than a given threshold on a given interval $[x_{0},x_{1}] \subseteq [0,1]$. A test for this type of hypotheses is developed using an appropriate estimator, say $\hat d_{\infty, n}$, for the maximum deviation $ d_{\infty}= \sup_{t \in [x_{0},x_{1}]} |\mu (t) - g( \mu) |$. We derive the limiting distribution of an appropriately standardized version of $\hat d_{\infty,n}$, where the standardization depends on the Lebesgue measure of the set of extremal points of the function $\mu(\cdot)-g(\mu)$. A refined procedure based on an estimate of this set is developed and its consistency is proved. The results are illustrated by means of a simulation study and a data example., Comment: 40 pages

In clinical trials, the comparison of two different populations is a common problem. Nonlinear (parametric) regression models are commonly used to describe the relationship between covariates, such as concentration or dose, and a response variable in the two groups. In some situations, it is reasonable to assume some model parameters to be the same, for instance, the placebo effect or the maximum treatment effect. In this paper, we develop a (parametric) bootstrap test to establish the similarity of two regression curves sharing some common parameters. We show by theoretical arguments and by means of a simulation study that the new test controls its significance level and achieves a reasonable power. Moreover, it is demonstrated that under the assumption of common parameters, a considerably more powerful test can be constructed compared with the test that does not use this assumption. Finally, we illustrate the potential applications of the new methodology by a clinical trial example.

In this article, we propose a new approach for sequential monitoring of a general class of parameters of a d-dimensional time series, which can be estimated by approximately linear functionals of the empirical distribution function. We consider a closed-end method, which is motivated by the likelihood ratio test principle and compare the new method with two alternative procedures. We also incorporate self-normalization such that estimation of the long-run variance is not necessary. We prove that for a large class of testing problems the new detection scheme has asymptotic level α and is consistent. The asymptotic theory is illustrated for the important cases of monitoring a change in the mean, variance, and correlation. By means of a simulation study it is demonstrated that the new test performs better than the currently available procedures for these problems. Finally, the methodology is illustrated by a small data example investigating index prices from the dot-com bubble. Supplementary materials for this article are available online.

A time-domain test for the assumption of second-order stationarity of a functional time series is proposed. The test is based on combining individual cumulative sum tests which are designed to be sensitive to changes in the mean, variance and autocovariance operators, respectively. The combination of their dependent p values relies on a joint-dependent block multiplier bootstrap of the individual test statistics. Conditions under which the proposed combined testing procedure is asymptotically valid under stationarity are provided. A procedure is proposed to automatically choose the block length parameter needed for the construction of the bootstrap. The finite-sample behavior of the proposed test is investigated in Monte Carlo experiments, and an illustration on a real data set is provided.

In this paper we construct optimal designs for frequentist model averaging estimation. We derive the asymptotic distribution of the model averaging estimate with fixed weights in the case where the competing models are non-nested. A Bayesian optimal design minimizes an expectation of the asymptotic mean squared error of the model averaging estimate calculated with respect to a suitable prior distribution. We derive a necessary condition for the optimality of a given design with respect to this new criterion. We demonstrate that Bayesian optimal designs can improve the accuracy of model averaging substantially. Moreover, the derived designs also improve the accuracy of estimation in a model selected by model selection and model averaging estimates with random weights.

We consider the problem of designing experiments for the comparison of two regression curves describing the relation between a predictor and a response in two groups, where the data between and within the group may be dependent. In order to derive effi- cient designs we use results from stochastic analysis to identify the best linear unbiased estimator (BLUE) in a corresponding continuous time model. It is demonstrated that in general simultaneous estimation using the data from both groups yields more precise results than estimation of the parameters separately in the two groups. Using the BLUE from simultaneous estimation, we then construct an efficient linear estimator for finite sample size by minimizing the mean squared error between the optimal solution in the continuous time model and its discrete approximation with respect to the weights (of the linear estimator). Finally, the optimal design points are determined by minimizing the maximal width of a simultaneous confidence band for the difference of the two regression functions. The advantages of the new approach are illustrated by means of a simulation study, where it is shown that the use of the optimal designs yields substantially narrower confidence bands than the application of uniform designs., Discussion Paper / SFB823;2/2021

Optimal portfolio selection problems are determined by the (unknown) parameters of the data generating process. If an investor wants to realise the position suggested by the optimal portfolios, he/she needs to estimate the unknown parameters and to account for the parameter uncertainty in the decision process. Most often, the parameters of interest are the population mean vector and the population covariance matrix of the asset return distribution. In this paper, we characterise the exact sampling distribution of the estimated optimal portfolio weights and their characteristics. This is done by deriving their sampling distribution by its stochastic representation. This approach possesses several advantages, {e.g.} (i) it determines the sampling distribution of the estimated optimal portfolio weights by expressions, which could be used to draw samples from this distribution efficiently; (ii) the application of the derived stochastic representation provides an easy way to obtain the asymptotic approximation of the sampling distribution. The later property is used to show that the high-dimensional asymptotic distribution of optimal portfolio weights is a multivariate normal and to determine its parameters. Moreover, a consistent estimator of optimal portfolio weights and their characteristics is derived under the high-dimensional settings. Via an extensive simulation study, we investigate the finite-sample performance of the derived asymptotic approximation and study its robustness to the violation of the model assumptions used in the derivation of the theoretical results., Comment: 40 pages, 4 figures (this version: accepted version manuscript + corrigendum)

We propose a new sequential monitoring scheme for changes in the parameters of a multivariate time series. In contrast to procedures proposed in the literature which compare an estimator from the training sample with an estimator calculated from the remaining data, we suggest to divide the sample at each time point after the training sample. Estimators from the sample before and after all separation points are then continuously compared calculating a maximum of norms of their differences. For open-end scenarios our approach yields an asymptotic level α procedure, which is consistent under the alternative of a change in the parameter. By means of a simulation study it is demonstrated that the new method outperforms the commonly used procedures with respect to power and the feasibility of our approach is illustrated by analyzing two data examples.

In the common time series model $X_{i,n} = \mu (i/n) + \varepsilon_{i,n}$ with non-stationary errors we consider the problem of detecting a significant deviation of the mean function $\mu$ from a benchmark $g (\mu )$ (such as the initial value $\mu (0)$ or the average trend $\int_{0}^{1} \mu (t) dt$). The problem is motivated by a more realistic modelling of change point analysis, where one is interested in identifying relevant deviations in a smoothly varying sequence of means $ (\mu (i/n))_{i =1,\ldots ,n }$ and cannot assume that the sequence is piecewise constant. A test for this type of hypotheses is developed using an appropriate estimator for the integrated squared deviation of the mean function and the threshold. By a new concept of self-normalization adapted to non-stationary processes an asymptotically pivotal test for the hypothesis of a relevant deviation is constructed. The results are illustrated by means of a simulation study and a data example.

We show that polynomials do not belong to the reproducing kernel Hilbert space of infinitely differentiable translation-invariant kernels whose spectral measures have moments corresponding to a determinate moment problem. Our proof is based on relating this question to the problem of best linear estimation in continuous time one-parameter regression models with a stationary error process defined by the kernel. In particular, we show that the existence of a sequence of estimators with variances converging to 0 implies that the regression function cannot be an element of the reproducing kernel Hilbert space. This question is then related to the determinacy of the Hamburger moment problem for the spectral measure corresponding to the kernel. In the literature it was observed that a nonvanishing constant function does not belong to the reproducing kernel Hilbert space associated with the Gaussian kernel. Our results provide a unifying view of this phenomenon and show that the mentioned result can be extended for arbitrary polynomials and a broad class of translation-invariant kernels.

The Portmanteau test provides the vanilla method for detecting serial correlations in classical univariate time series analysis. The method is extended to the case of observations from a locally stationary functional time series. Asymptotic critical values are obtained by a suitable block multiplier bootstrap procedure. The test is shown to asymptotically hold its level and to be consistent against general alternatives., Keywords: Autocovariance operator, Block multiplier bootstrap, Functional white noise, Time domain test

In traditional pharmacokinetic (PK) bioequivalence analysis, two one-sided tests (TOST) are conducted on the area under the concentration-time curve and the maximal concentration derived using a non-compartmental approach. When rich sampling is unfeasible, a model-based (MB) approach, using nonlinear mixed effect models (NLMEM) is possible. However, MB-TOST using asymptotic standard errors (SE) presents increased type I error when asymptotic conditions do not hold. In this work, we propose three alternative calculations of the SE based on (i) an adaptation to NLMEM of the correction proposed by Gallant, (ii) the a posteriori distribution of the treatment coefficient using the Hamiltonian Monte Carlo algorithm, and (iii) parametric random effects and residual errors bootstrap. We evaluate these approaches by simulations, for two-arms parallel and two-period, two-sequence cross-over design with rich (n = 10) and sparse (n = 3) sampling under the null and the alternative hypotheses, with MB-TOST. All new approaches correct for the inflation of MB-TOST type I error in PK studies with sparse designs. The approach based on the a posteriori distribution appears to be the best compromise between controlled type I errors and computing times. MB-TOST using non-asymptotic SE controls type I error rate better than when using asymptotic SE estimates for bioequivalence on PK studies with sparse sampling.

In this paper we propose statistical inference tools for the covariance operators of functional time series in the two sample and change point problem. In contrast to most of the literature, the focus of our approach is not testing the null hypothesis of exact equality of the covariance operators. Instead, we propose to formulate the null hypotheses in the form that “the distance between the operators is small”, where we measure deviations by the sup-norm. We provide powerful bootstrap tests for these type of hypotheses, investigate their asymptotic properties and study their finite sample properties by means of a simulation study.

Motivated by the need to statistically quantify differences between modern (complex) data-sets which commonly result as high-resolution measurements of stochastic processes varying over a continuum, we propose novel testing procedures to detect relevant differences between the second order dynamics of two functional time series. In order to take the between-function dynamics into account that characterize this type of functional data, a frequency domain approach is taken. Test statistics are developed to compare differences in the spectral density operators and in the primary modes of variation as encoded in the associated eigenelements. Under mild moment conditions, we show convergence of the underlying statistics to Brownian motions and construct pivotal test statistics. The latter is essential because the nuisance parameters can be unwieldy and their robust estimation infeasible, especially if the two functional time series are dependent. In addition to these novel features, the properties of the tests are robust to any choice of frequency band enabling also to compare energy contents at a single frequency. The finite sample performance of the tests are verified through a simulation study and are illustrated with an application to fMRI data.

Functional data analysis is typically conducted within the $L^2$-Hilbert space framework. There is by now a fully developed statistical toolbox allowing for the principled application of the functional data machinery to real-world problems, often based on dimension reduction techniques such as functional principal component analysis. At the same time, there have recently been a number of publications that sidestep dimension reduction steps and focus on a fully functional $L^2$-methodology. This paper goes one step further and develops data analysis methodology for functional time series in the space of all continuous functions. The work is motivated by the fact that objects with rather different shapes may still have a small $L^2$-distance and are therefore identified as similar when using an $L^2$-metric. However, in applications it is often desirable to use metrics reflecting the visualization of the curves in the statistical analysis. The methodological contributions are focused on developing two-sample and change-point tests as well as confidence bands, as these procedures appear do be conducive to the proposed setting. Particular interest is put on relevant differences; that is, on not trying to test for exact equality, but rather for pre-specified deviations under the null hypothesis. The procedures are justified through large-sample theory. To ensure practicability, non-standard bootstrap procedures are developed and investigated addressing particular features that arise in the problem of testing relevant hypotheses. The finite sample properties are explored through a simulation study and an application to annual temperature profiles., updated footnote and acknowledgements

We investigate moment sequences of probability measures on $E\subset\mathbb{R}$ under constraints of certain moments being fixed. This corresponds to studying sections of $n$-th moment spaces, i.e. the spaces of moment sequences of order $n$. By equipping these sections with the uniform or more general probability distributions, we manage to give for large $n$ precise results on the (probabilistic) barycenters of moment space sections and the fluctuations of random moments around these barycenters. The measures associated to the barycenters belong to the Bernstein-Szeg\H{o} class and show strong universal behavior. We prove Gaussian fluctuations and moderate and large deviations principles. Furthermore, we demonstrate how fixing moments by a constraint leads to breaking the connection between random moments and random matrices., Comment: 43 pages

Change point detection in high dimensional data has found considerable interest in recent years. Most of the literature either designs methodology for a retrospective analysis, where the whole sample is already available when the statistical inference begins, or considers online detection schemes controlling the average time until a false alarm. This paper takes a different point of view and develops monitoring schemes for the online scenario, where high dimensional data arrives successively and the goal is to detect changes as fast as possible controlling at the same time the probability of a type I error of a false alarm. We develop a sequential procedure capable of detecting changes in the mean vector of a successively observed high dimensional time series with spatial and temporal dependence. The statistical properties of the method are analyzed in the case where both, thesample size and dimension tend to infinity. In this scenario, it is shown that the new monitoring scheme has asymptotic level alpha under the null hypothesis of no change and is consistent under the alternative of a change in at least one component of the high dimensional mean vector. The approach is based on a new type of monitoring scheme for one-dimensional data which turns out to be often more powerful than the usually used CUSUM and Page-CUSUM methods, and the component-wise statistics are aggregated by the maximum statistic. For the analysis of the asymptotic properties of our monitoring scheme we prove that the range of a Brownian motion on a given interval is in the domain of attraction of the Gumbel distribution, which is a result of independent interest in extreme value theory. The finite sample properties of the new methodology are illustrated by means of a simulation study and in the analysis of a data example.

Residual-based analysis is generally considered a cornerstone of statistical methodology. For a special case of indirect regression, we investigate the residual-based empirical distribution function and provide a uniform expansion of this estimator, which is also shown to be asymptotically most precise. This investigation naturally leads to a completely data-driven technique for selecting a regularization parameter used in our indirect regression function estimator. The resulting methodology is based on a smooth bootstrap of the model residuals. A simulation study demonstrates the effectiveness of our approach., Comment: Keywords: bandwidth selection, indirect regression estimator, inverse problems, regularization, residual-based empirical distribution function, smooth bootstrap

In this note we consider the optimal design problem for estimating the slope of a polynomial regression with no intercept at a given point, say z. In contrast to previous work, which considers symmetric design spaces we investigate the model on the interval $[0, a]$ and characterize those values of $z$, where an explicit solution of the optimal design is possible.

We develop an estimator for the high-dimensional covariance matrix of a locally stationary process with a smoothly varying trend and use this statistic to derive consistent predictors in non-stationary time series. In contrast to the currently available methods for this problem the predictor developed here does not rely on fitting an autoregressive model and does not require a vanishing trend. The finite sample properties of the new methodology are illustrated by means of a simulation study and a data example., Discussion Paper / SFB823;1/2020

The estimation of covariance operators of spatio-temporal data is in many applications only computationally feasible under simplifying assumptions, such as separability of the covariance into strictly temporal and spatial factors.Powerful tests for this assumption have been proposed in the literature. However, as real world systems, such as climate data are notoriously inseparable, validating this assumption by statistical tests, seems inherently questionable. In this paper we present an alternative approach: By virtue of separability measures, we quantify how strongly the data's covariance operator diverges from a separable approximation. Confidence intervals localize these measures with statistical guarantees. This method provides users with a flexible tool, to weigh the computational gains of a separable model against the associated increase in bias. As separable approximations we consider the established methods of partial traces and partial products, and develop weak convergence principles for the corresponding estimators. Moreover, we also prove such results for estimators of optimal, separable approximations, which are arguably of most interest in applications. In particular we present for the first time statistical inference for this object, which has been confined to estimation previously. Besides confidence intervals, our results encompass tests for approximate separability. All methods proposed in this paper are free of nuisance parameters and do neither require computationally expensive resampling procedures nor the estimation of nuisance parameters. A simulation study underlines the advantages of our approach and its applicability is demonstrated by the investigation of German annual temperature data.

In this paper, we present a new method for determining optimal designs for enzyme inhibition kinetic models, which are used to model the influence of the concentration of a substrate and an...

This paper considers the problem of testing if a sequence of means $(\mu_{t})_{t=1,\ldots ,n}$ of a nonstationary time series $(X_{t})_{t=1,\ldots ,n}$ is stable in the sense that the difference of the means $\mu_{1}$ and $\mu_{t}$ between the initial time $t=1$ and any other time is smaller than a given threshold, that is $|\mu_{1}-\mu_{t}|\leq c$ for all $t=1,\ldots ,n$. A test for hypotheses of this type is developed using a bias corrected monotone rearranged local linear estimator and asymptotic normality of the corresponding test statistic is established. As the asymptotic variance depends on the location of the roots of the equation $|\mu_{1}-\mu_{t}|=c$ a new bootstrap procedure is proposed to obtain critical values and its consistency is established. As a consequence we are able to quantitatively describe relevant deviations of a nonstationary sequence from its initial value. The results are illustrated by means of a simulation study and by analyzing data examples.

In drug development, comparability of dissolution profiles of 2 different formulations is usually assessed using the similarity factor f2 . In practice, the drug dissolution profiles are deemed similar if the f2 exceeds 50, which occurs when a 10% maximum difference in the mean percentage of the dissolved drug at each time point between test and reference formulation is obtained. According to the Guideline on the Investigation of Bioequivalence (CPMP/EWP/QWP/1401/98 Rev. 1/ Corr **) use of the f2 is however restricted by a set of validity conditions. If some of these conditions are not satisfied, the f2 is not considered suitable, and alternative statistical methods are needed. In this article, we propose an inferential framework based on the maximum deviation between curves to test the comparability of drug dissolution profiles. The new methodology is applicable regardless whether the validity criteria of the f2 are met or not. Contrary to the f2 , this approach also integrates the variability of the measurements over time and not only their average. To benchmark our method, we performed simulations informed by 3 real case studies provided by the European Medicines Agency and extracted from dossiers submitted to the Centralised Procedure for Marketing Authorisation Application. In the scenarios of the simulation study, the new method controlled its type I error rate when the maximum deviation was greater than the similarity acceptance limit of 10%. The power exceeded 80% for small values of the maximum deviation, while the test was more conservative for intermediate ones. Our results were also very robust to sampling variations. Based on these positive findings, we encourage applicants to consider the new maximum deviation-based method as a valid alternative to the f2 , especially when the validity criteria of the latter are not met.