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We propose a computationally efficient method to construct nonparametric, heteroscedastic prediction bands for uncertainty quantification, with or without any user-specified predictive model. Our approach provides an alternative to the now-standard conformal prediction for uncertainty quantification, with novel theoretical insights and computational advantages. The data-adaptive prediction band is universally applicable with minimal distributional assumptions, has strong non-asymptotic coverage properties, and is easy to implement using standard convex programs. Our approach can be viewed as a novel variance interpolation with confidence and further leverages techniques from semi-definite programming and sum-of-squares optimization. Theoretical and numerical performances for the proposed approach for uncertainty quantification are analyzed., 21 pages, 4 figures

A portmanteau-type statistic based on sample covariance is proposed to test for the existence of the unit-root of an autoregressive model under a heavy-tailed linear noise, where the noise ɛ t = ∑ j = 0 ∞ d j η t − j and { η t } is a sequence of i.i.d. innovations belonging to the domain of attraction of an α -stable distribution for some α ∈ ( 0 , 2 ) . The proposed procedure is nonparametric and easy to implement. Under certain regular assumptions, the limit distribution of the statistics is shown to be a functional of a standard stable distribution. Further, the good finite sample studies show that the new test outperforms PP, DF and ADF tests in size and power. Applications to the daily world crude oil price and 3-month AA financial commercial paper rate are also given to illustrate the performance of the new test.

In nonparametric models, numerous penalization methods using a nonparametric series estimator have been developed for model selection and estimation. However, a penalization has been poorly understood combined with kernel smoothing. This can be attributed to the intrinsic technical and computational difficulties, which leads to different treatments from the developments of the penalized series estimators. Kernel smoothing is a popular and useful nonparametric estimation method; thus it is desirable to establish theoretical and computational analyses for penalized kernel smoothing. In this paper, we develop a novel penalized kernel quantile regression that has nice theoretical and computational properties in varying coefficient models. We show that the proposed method consistently identifies the partially linear structure of the varying coefficient model even when the number of covariates is allowed to increase with a sample size. We develop an efficient algorithm using the alternating direction method of multipliers with a computational convergence guarantee. We derive the plug-in bandwidth selection using high-dimensional kernel regression theory and the penalty parameter is selected by the proposed Bayesian Information Criterion. Our developments require novel high-dimensional kernel regression and computational analyses. The various simulations and real data analyses conducted in this study demonstrate the effectiveness and numerically verify the proposed method.

Functional data analysis has attracted considerable interest and is facing new challenges, one of which is the increasingly available data in a streaming manner. In this article we develop an online nonparametric method to dynamically update the estimates of mean and covariance functions for functional data. The kernel-type estimates can be decomposed into two sufficient statistics depending on the data-driven bandwidths. We propose to approximate the future optimal bandwidths by a sequence of dynamically changing candidates and combine the corresponding statistics across blocks to form the updated estimation. The proposed online method is easy to compute based on the stored sufficient statistics and the current data block. We derive the asymptotic normality and, more importantly, the relative efficiency lower bounds of the online estimates of mean and covariance functions. This provides insight into the relationship between estimation accuracy and computational cost driven by the length of candidate bandwidth sequence. Simulations and real data examples are provided to support such findings.

An important hypothesis in animal cell biology is that an animal’s acute exercise regimen affects some subcellular structures, for example mitochondrial morphology, in its muscle tissue. This paper investigates that hypothesis using a nonparametric metric-based energy test for comparing mitochondrial populations. It explores two shape spaces—the elastic shape space and Kendall shape space—and five corresponding shape metrics on these spaces. The results overwhelmingly point to the statistical significance of the effect of an acute exercise regimen on the shape of SS-type mitochondria. Although past studies based on specific morphological features derived from mitochondria failed to detect this significance. In this analysis, a potentially significant factor is cell membership and a k-sample generalization of the energy test—the DISCO test shows that the cell effect is indeed significant. The energy test cannot be applied directly due to the hierarchical structure of the distance matrix. We propose a compression method to remove the significant cell effect while testing for the exercise effect. With this compression, only the elastic scaled metric shows statistical significance of the exercise factor in this more complicated scenario. This result is because the elastic-scaled metric is more sensitive to subtle changes in mitochondrial shapes caused by acute exercise.

The new therapeutic agents, such as molecular targeted agents and immuno-oncology therapies, appear more likely to induce multiple toxicities at different grades than dose-limiting toxicities defined in traditional dose-finding trials. In addition, it is often challenging to make adaptive decisions on dose escalation and de-escalation on time because of the fast accrual rate and/or the late-onset toxicity outcomes, causing the potential suspension of the enrollment and the delay of the trials. To address these issues, we propose a time-to-event Bayesian optimal interval design to accelerate the dose-finding process utilizing toxicity grades based on both cumulative and pending toxicity outcomes. The proposed design, named "TITE-gBOIN" design, is a nonparametric and model-assisted design and has the virtues of robustness, simplicity and straightforward to implement in actual oncology dose-finding trials. A simulation study shows that the TITE-gBOIN design has a higher probability of selecting the MTDs correctly and allocating more patients to the MTDs across various realistic settings while reducing the trial duration significantly, therefore can accelerate early-stage dose-finding trials.

This paper investigates statistical reproducibility of the [Formula: see text]-test. We formulate reproducibility as a predictive inference problem and apply the nonparametric predictive inference method. Within our research framework, statistical reproducibility provides inference on the probability that the same test outcome would be reached, if the test were repeated under identical conditions. We present an nonparametric predictive inference algorithm to calculate the reproducibility of the [Formula: see text]-test and then use simulations to explore the reproducibility both under the null and alternative hypotheses. We then apply nonparametric predictive inference reproducibility to a real-life scenario of a preclinical experiment, which involves multiple pairwise comparisons of test groups, where different groups are given a different concentration of a drug. The aim of the experiment is to decide the concentration of the drug which is most effective. In both simulations and the application scenario, we study the relationship between reproducibility and two test statistics, the Cohen’s [Formula: see text] and the [Formula: see text]-value. We also compare the reproducibility of the [Formula: see text]-test with the reproducibility of the Wilcoxon Mann–Whitney test. Finally, we examine reproducibility for the final decision of choosing a particular dose in the multiple pairwise comparisons scenario. This paper presents advances on the topic of test reproducibility with relevance for tests used in pharmaceutical research.

We introduce new shape-constrained classes of distribution functions on R , the bi- s ∗ -concave classes. In parallel to results of Dumbgen et al. (2017) for what they called the class of bi-log-concave distribution functions, we show that every s -concave density f has a bi- s ∗ -concave distribution function F for s ∗ ≤ s ∕ ( s + 1 ) . Confidence bands building on existing nonparametric confidence bands, but accounting for the shape constraint of bi- s ∗ -concavity, are also considered. The new bands extend those developed by Dumbgen et al. (2017) for the constraint of bi-log-concavity. We also make connections between bi- s ∗ -concavity and finiteness of the Csorgő–Revesz constant of F which plays an important role in the theory of quantile processes.

We consider nonparametric estimation of a distribution function when data are collected from two-phase stratified sampling without replacement. We study the inverse probability weighted empirical distribution function and propose a novel computational procedure to construct a confidence band. Two-phase sampling design induces heterogeneity across strata and dependence due to sampling without replacement. Two major statistical challenges from this design are: (1) the standard practice to approximate sampling without replacement by Bernoulli sampling leads to an incorrect coverage probability, and (2) a complicated limiting process of the proposed estimator does not allow one to analytically compute quantiles of the supremum of the limiting process nor to apply existing bootstrap methods to the proposed estimator. To address these issues, we rigorously establish the asymptotic properties of the proposed estimator and develop a simulation-based method to estimate the limiting process. The finite sample performance is evaluated through a simulation study. A Wilms tumor example is provided.

The research in this paper gives a systematic investigation of the asymptotic behaviors of four inverse probability weighting (IPW)-based estimators for conditional average treatment effects, with nonparametrically, semiparametrically, parametrically estimated, and true propensity score, respectively. To this end, we first pay particular attention to semiparametric dimension reduction structure such that we can study the semiparametric-based estimator that can alleviate the curse of dimensionality and greatly avoid model misspecification. We also derive some further properties of the existing estimator with a nonparametrically estimated propensity score. According to their asymptotic variance functions, the studies reveal the general ranking of their asymptotic efficiencies; in which scenarios the asymptotic equivalence can hold; the critical roles of the affiliation of the given covariates in the set of arguments of the propensity score, the bandwidth and kernel function selections. The results show an essential difference from the IPW-based estimator of the unconditional average treatment effects(ATE). The numerical studies indicate that for high-dimensional paradigms, the semiparametric-based estimator performs well in general whereas the nonparametric-based estimator, even sometimes parametric-based estimator, is more susceptible to dimensionality. Some numerical studies are carried out to examine their performance. A real data example is analyzed for illustration.

Epidemic modelling is an essential tool to understand the spread of the novel coronavirus and ultimately assist in disease prevention, policymaking, and resource allocation. In this article, we est...

In general, the change point problem considers inference of a change in distribution for a set of time-ordered observations. This has applications in a large variety of fields, and can also apply to survival data. In survival analysis, most existing methods compare two treatment groups for the entirety of the study period. Some treatments may take a length of time to show effects in subjects. This has been called the time-lag effect in the literature, and in cases where time-lag effect is considerable, such methods may not be appropriate to detect significant differences between two groups. In this paper, we propose a novel non-parametric approach for estimating the point of treatment time-lag effect by using an empirical divergence measure. Theoretical properties of the estimator are studied. The results from the simulated data and the applications to real data examples support our proposed method.

This paper presents a new method called the functional distributional clustering algorithm (FDCA) that seeks to identify spatially contiguous clusters and incorporate changes in temporal patterns across overcrowded networks. This method is motivated by a graph-based network composed of sensors arranged over space where recorded observations for each sensor represent a multi-modal distribution. The proposed method is fully non-parametric and generates clusters within an agglomerative hierarchical clustering approach based on a measure of distance that defines a cumulative distribution function over temporal changes for different locations in space. Traditional hierarchical clustering algorithms that are spatially adapted do not typically accommodate the temporal characteristics of the underlying data. The effectiveness of the FDCA is illustrated using an application to both empirical and simulated data from about 400 sensors in a 2.5 square miles network area in downtown San Francisco, California. The results demonstrate the superior ability of the the FDCA in identifying true clusters compared to functional only and distributional only algorithms and similar performance to a model-based clustering algorithm.

This paper develops a time--varying coefficient spatial autoregressive panel data model with individual fixed effects to capture the nonlinear effects of the regressors, which vary over the time. To effectively estimate the model, we propose a method that incorporates local linear estimation and concentrated quasi-maximum likelihood estimation to obtain consistent estimators for the spatial autoregressive coefficient, variance of error term and nonparametric time-varying coefficient function. The asymptotic properties of these estimators are derived as well, showing regular the standard rate of convergence for the parametric parameters and common standard rate of convergence for the time-varying component, respectively. Monte Carlo simulations are conducted to illustrate the finite sample performance of our proposed method. Meanwhile, we apply our method to study the Chinese labor productivity to identify the spatial influences and the time--varying spillover effects among 185 Chinese cities with comparison to the results on a subregion--East China.

The purpose of this paper is to provide a nonparametric kernel method to estimate nonlinear structural equation models involving the functional effects between the latent variables. This approach is based on the combination of Principal Component Analysis (PCA) and kernel smoothing technique. The results obtained from different simulations on both nonlinear and linear structural models show the great performance of this method. Furthermore, an application on real data using a recovery satisfaction model is presented in this paper. From where, we show the adequacy of our method in capturing the non linearity between some latent variables.

We develop new estimation results for the functional relationship between a regressor and a response which are functions indexed by time or by spatial locations. The regressor is assumed to belong ...

An event encountered for the first time can become a common phenomenon in a population through transmission over time. This study focuses on nonparametric partially sequential test statistics in order to early detect such events. For this purpose, firstly, the partially sequential test statistic given in Mukherjee [1] is considered. Then, with a modification of this statistic, a new nonparametric partially sequential test statistic that works at almost the desired alpha level of significance is revealed. Also, based on Monte-Carlo simulations, the proposed test statistic is shown to be more powerful than Mukherjee's statistic. Lastly, two illustrative numerical examples are also given.

We develop purely nonparametric methods for the analysis of repeated measures designs with missing values. Hypotheses are formulated in terms of purely nonparametric treatment effects. In particular, data can have different shapes even under the null hypothesis and therefore, a solution to the nonparametric Behrens-Fisher problem in repeated measures designs will be presented. Moreover, global testing and multiple contrast test procedures as well as simultaneous confidence intervals for the treatment effects of interest will be developed. All methods can be applied for the analysis of metric, discrete, ordinal, and even binary data in a unified way. Extensive simulation studies indicate a satisfactory control of the nominal type-I error rate, even for small sample sizes and a high amount of missing data (up to 30%). We apply the newly developed methodology to a real data set, demonstrating its application and interpretation.

This article proposes the orthogonal series method for the cumulative distribution function estimation based on a random number of observations Nt, which may be recorded in time (0,t]. We give the ...

A two-stage variational mode decomposition approach is explored in this article to estimate the variance function in a nonparametric heteroscedastic fixed design regression model. This estimate imp...

Highest density regions refer to level sets containing points of relatively high density. Their estimation from a random sample, generated from the underlying density, allows to determine the clusters of the corresponding distribution. This task can be accomplished considering different nonparametric perspectives. From a practical point of view, reconstructing highest density regions can be interpreted as a way of determining hot-spots, a crucial task for understanding COVID-19 space-time evolution. In this work, we compare the behaviour of classical plug-in methods and a recently proposed hybrid algorithm for highest density regions estimation through an extensive simulation study. Both methodologies are applied to analyse a real data set about COVID-19 cases in the United States. [ABSTRACT FROM AUTHOR] Copyright of Journal of Nonparametric Statistics is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Alternative estimation and forecasting procedures for the integer-valued autoregressive process with dependent geometric counting series are investigated. Some nonparametric estimation methods are ...

For widespread nonparametric tests, we point out necessary and sufficient conditions of uniform consistency for nonparametric sets of alternatives. Nonparametric sets of alternatives can be defined both in terms of distribution functions and in terms of density. Such conditions are provided for χ2-tests with an increasing number of cells, for Cramer – von Mises tests, for tests generated by 𝕃2-norms of kernel estimators, and for tests generated by quadratic forms of estimators of Fourier coefficients. Necessary and sufficient conditions on sets of alternatives for the existence of uniformly consistent tests are treated as well.

The benefit–risk balance is a critical information when evaluating a new treatment. The Net Benefit has been proposed as a metric for the benefit–risk assessment, and applied in oncology to simultaneously consider gains in survival and possible side effects of chemotherapies. With complete data, one can construct a U-statistic estimator for the Net Benefit and obtain its asymptotic distribution using standard results of the U-statistic theory. However, real data is often subject to right-censoring, e.g. patient drop-out in clinical trials. It is then possible to estimate the Net Benefit using a modified U-statistic, which involves the survival time. The latter can be seen as a nuisance parameter affecting the asymptotic distribution of the Net Benefit estimator. We present here how existing asymptotic results on U-statistics can be applied to estimate the distribution of the net benefit estimator, and assess their validity in finite samples. The methodology generalizes to other statistics obtained using generalized pairwise comparisons, such as the win ratio. It is implemented in the R package BuyseTest (version 2.3.0 and later) available on Comprehensive R Archive Network.

Rank-based methods are frequently used in the life sciences, and in the empirical sciences in general. Among the best-known examples of nonparametric rank-based tests are the Wilcoxon-Mann-Whitney ...

We investigate the problem of nonparametric estimation of the trend for stochastic differential equations with delay and driven by a fractional Brownian motion through the method of kernel-type est...

Constructions for regression graphs and verification of the statistical model via linear, linearized, and logistic regressions along them have recently been intensively studied. Given the graph, an iterative regression method, using local averaging estimators, is introduced for prediction, based on a complete training sample. The method makes it possible to perform nonparametric regressions recursively, irrespective of the type of the context and response variables. As a consequence, predictions for the multiple response variables of a test sample are performed in the possession of their context variables only. Consistency is proved if the joint distribution is Markov compatible with a DAG or with a regression graph. In the latter case, the prediction goes on from chain component to chain component, with vector valued smoothers, mainly of product kernel types in the implementation. Practical considerations and application to randomly generated and real-world data are also presented.

International audience; The Mann-Whitney (MW) test is one of the most important nonparametric tests used in the comparison of the location parameters of two populations. Unlike the t-test, the MW test can be used when the assumption of normality fails to hold. In this paper, the MW U statistic is used to construct two efficient distribution-free monitoring schemes, namely the mixed generally weighted moving average-cumulative sum (GWMA-CUSUM) MW U scheme (denoted as U-MGC) as well as its reversed version, i.e. the CUSUM-GWMA MW U scheme (denoted as U-MCG). The performances of the proposed schemes are investigated using the average run-length (ARL) and average extra quadratic loss (AEQL) values through extensive simulations. The newly proposed charts are found to be superior in small shifts detection than their competing (existing and others that are briefly introduced here) distribution-free Shewhart, EWMA, CUSUM, mixed EWMA-CUSUM, mixed CUSUM-EWMA and GWMA MW U charts in many situations. A real-life example is used to demonstrate the design and implementation of the new schemes.

A new interpoint distance-based measure is proposed to identify the optimal number of clusters present in a data set. Designed in nonparametric approach, it is independent of the distribution of given data. Interpoint distances between the data members make our cluster validity index applicable to univariate and multivariate data measured on arbitrary scales, or having observations in any dimensional space where the number of study variables can be even larger than the sample size. Our proposed criterion is compatible with any clustering algorithm, and can be used to determine the unknown number of clusters or to assess the quality of the resulting clusters for a data set. Demonstration through synthetic and real-life data establishes its superiority over the well-known clustering accuracy measures of the literature., 30 pages, 3 figures

Summary A nonparametric framework for changepoint detection, based on scan statistics utilizing graphs that represent similarities among observations, is gaining attention owing to its flexibility and good performance for high-dimensional and non-Euclidean data sequences. However, this graph-based framework faces challenges when there are repeated observations in the sequence, which is often the case for discrete data such as network data. In this article we extend the graph-based framework to solve this problem by averaging or taking the union of all possible optimal graphs resulting from repeated observations. We consider both the single-changepoint alternative and the changed-interval alternative, and derive analytical formulas to control the Type I error for the new methods, making them readily applicable to large datasets. The extended methods are illustrated on an application in detecting changes in a sequence of dynamic networks over time. All proposed methods are implemented in an $\texttt{R}$ package $\texttt{gSeg}$ available on CRAN.

In many clinical studies, evaluating the association between longitudinal and survival outcomes is of primary concern. For analyzing data from such studies, joint modeling of longitudinal and survival data becomes an appealing approach. In some applications, there are multiple longitudinal outcomes whose longitudinal pattern is difficult to describe by a parametric form. For such applications, existing research on joint modeling is limited. In this article, we develop a novel joint modeling method to fill the gap. In the new method, a local polynomial mixed-effects model is used for describing the nonparametric longitudinal pattern of the multiple longitudinal outcomes. Two model estimation procedures, that is, the local EM algorithm and the local penalized quasi-likelihood estimation, are explored. Practical guidelines for choosing tuning parameters and for variable selection are provided. The new method is justified by some theoretical arguments and numerical studies.

We study empirical Bayes estimation of the effect sizes of N units from K noisy observations on each unit. We show that it is possible to achieve near-Bayes optimal mean squared error, without any assumptions or knowledge about the effect size distribution or the noise. The noise distribution can be heteroskedastic and vary arbitrarily from unit to unit. Our proposal, which we call Aurora, leverages the replication inherent in the K observations per unit and recasts the effect size estimation problem as a general regression problem. Aurora with linear regression provably matches the performance of a wide array of estimators including the sample mean, the trimmed mean, the sample median, as well as James-Stein shrunk versions thereof. Aurora automates effect size estimation for Internet-scale datasets, as we demonstrate on data from a large technology firm.

Depending on the asymptotical independence of periodograms, exponential tilted (ET) likelihood, as an effective nonparametric statistical method, is developed to deal with time series in this paper...

Motivated by two examples concerning global warming and monthly total import and export by China, we study time series that contain a nonparametric periodic component with an unknown period, a nonp...

Sepehrifar and Yarahmadian (Stat Pap 58:1115–1124, 2017) had developed a non-parametric test for testing exponentiality against decreasing renewal dichotomous Markov noise shock model (DRDMNS) alternatives which when subjected to scrutiny under delivers. Hence, we propose a non-parametric test for testing exponentiality against a class of distributions belonging to DRDMNS models. The asymptotic properties of the test statistic are discussed. An exact null distribution is derived and critical values with different sample sizes are obtained. The proposed test is applied to the censored data also. The results of the Monte Carlo simulations are used to further manifest the quality of the proposed test. Finally, the proposed test is illustrated using two real data sets.

In this paper, nonparametric estimation for a stationary strongly mixing and manifold-valued process $$(X_j)$$ is considered. In this non-Euclidean and not necessarily i.i.d setting, we propose kernel density estimators of the joint probability density function, of the conditional probability density functions and of the conditional expectations of functionals of $$X_j$$ given the past behavior of the process. We prove the strong consistency of these estimators under sufficient conditions, and we illustrate their performance through simulation studies and real data analysis.

In this paper, a new product relative error estimation method for the partially linear multiplicative regression model (PLMM) is proposed. Both parametric and nonparametric parts are estimated by t...

In this paper, we propose a novel nonparametric method based on the adjusted empirical likelihood (AEL) and the transformed empirical likelihood (TEL), which are two modified versions of the empiri...

We propose a nonparametric bivariate varying coefficient generalized linear model to predict a mean response trajectory in the future given an individual's characteristics at present or an earlier time point in a longitudinal study. Given the measurement time of the predictors, the coefficients vary as functions of the future time over which the prediction of the mean response is concerned and illustrate the dynamic association between the future response and the earlier measured predictors. We use a nonparametric approach that takes advantage of features of both the kernel and the spline methods for estimation. The resulting coefficient estimator is asymptotically consistent under mild regularity conditions. We also develop a new bootstrap approach to construct simultaneous confidence bands for statistical inference about the coefficients and the predicted response trajectory based on the coverage rate of bootstrap estimates. We use the Framingham Heart Study to illustrate the methodology. The proposed procedure is applied to predict the probability trajectory of hypertension risk given individuals' health condition in early adulthood and to examine the impact of risk factors in early adulthood on a long-term risk of hypertension over several decades.

We propose wavelet-L1-estimation for non parametric location-scale models from multiple subjects. The advantage of the wavelet is to avoid the restrictive smoothness requirement for the location an...

We introduce a new approach to deciding the number of clusters. The approach is applied to Optimally Tuned Robust Improper Maximum Likelihood Estimation (OTRIMLE; Coretto and Hennig 2016) of a Gaussian mixture model allowing for observations to be classified as "noise", but it can be applied to other clustering methods as well. The quality of a clustering is assessed by a statistic $Q$ that measures how close the within-cluster distributions are to elliptical unimodal distributions that have the only mode in the mean. This nonparametric measure allows for non-Gaussian clusters as long as they have a good quality according to $Q$. The simplicity of a model is assessed by a measure $S$ that prefers a smaller number of clusters unless additional clusters can reduce the estimated noise proportion substantially. The simplest model is then chosen that is adequate for the data in the sense that its observed value of $Q$ is not significantly larger than what is expected for data truly generated from the fitted model, as can be assessed by parametric bootstrap. The approach is compared with model-based clustering using the Bayesian Information Criterion (BIC) and the Integrated Complete Likelihood (ICL) in a simulation study and on two datasets of scientific interest. Keywords: parametric bootstrap; noise component; unimodality; model-based clustering

Competing risks data with discrete lifetime come up in practice. However, only limited literature exists for such data. In this paper, we propose a non-parametric test based on U-statistics for tes...

The expected shortfall is an important risk measure in financial risk management. In this paper, we study the Bahadur-type representation of an improved nonparametric expected shortfall estimator f...

The question whether structural changes in time-resolved images are of statistical significance or merely emerge from random noise is of great relevance in many practical applications such as live cell fluorescence microscopy, where intracellular diffusion processes are investigated. Using bootstrap-methods, we construct nonparametric confidence bands for time-resolved images from fluorescence microscopy and use these to detect and visualize temporal changes between individual frames in imaging of living cells. We model the images frames as two-dimensional fields of Poisson random variables and provide a strong approximation result for independent and standardized but not necessarily identically distributed Poisson random variables. The latter result is used to derive a limit result for the maximal difference between the reconstructed and the true image. This provides the theoretical foundation of our method. We apply regularization techniques to cope with the ill-posedness of the convolution problem induced by the imaging system. Our approach provides a criterion to assess time-resolved small scale structural changes, for example, in the nanometer range. It can also be adopted for use in other imaging systems. Moreover, a data-driven selection method for the regularization parameter based on statistical multiscale methods is discussed.