The mean function is a central object of inquiry in the analysis of functional data. Typical questions related to the mean function include quantifying estimation uncertainty, testing parametric models, and making comparisons between populations. To make probabilistic statements about the mean function over its entire domain, rather than at a single location, it is necessary to infer all of its values simultaneously. Pointwise inference is not appropriate for this task and indeed produces anticonservative results, i.e., the coverage level of confidence regions is too low and the significance level of hypothesis tests too high. In contrast, simultaneous confidence bands (SCB) provide a flexible framework for conducting simultaneous inference on the mean function and other functional parameters. They also offer powerful visualization tools for communicating analytic results to interdisciplinary audiences. The construction of SCB in the context of functional data requires specific theory and methods. In particular, it is not addressed by the nonparametric regression literature. Although software is available to perform individual steps of an SCB procedure, resources that provide end-to-end computations are scarce. Applications of SCB to one- and two-sample inferences are illustrated here with the R package SCBmeanfd. *WIREs Comput Stat* 2017, 9:e1397. doi: 10.1002/wics.1397

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Simultaneous confidence bands of level 95% for the average DNA copy number profile of patients with colorectal tumors. Chromosomal regions with significant gain/loss in DNA copy number are marked with black rectangles above the x-axis. These regions may contain cancer-causing genes.

The Poisson distribution has been widely studied and used for modeling univariate count-valued data. However, multivariate generalizations of the Poisson distribution that permit dependencies have been far less popular. Yet, real-world, high-dimensional, count-valued data found in word counts, genomics, and crime statistics, for example, exhibit rich dependencies and motivate the need for multivariate distributions that can appropriately model this data. We review multivariate distributions derived from the univariate Poisson, categorizing these models into three main classes: (1) where the marginal distributions are Poisson, (2) where the joint distribution is a mixture of independent multivariate Poisson distributions, and (3) where the node-conditional distributions are derived from the Poisson. We discuss the development of multiple instances of these classes and compare the models in terms of interpretability and theory. Then, we empirically compare multiple models from each class on three real-world datasets that have varying data characteristics from different domains, namely traffic accident data, biological next generation sequencing data, and text data. These empirical experiments develop intuition about the comparative advantages and disadvantages of each class of multivariate distribution that was derived from the Poisson. Finally, we suggest new research directions as explored in the subsequent Discussion section. *WIREs Comput Stat* 2017, 9:e1398. doi: 10.1002/wics.1398

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This paper reviews and empirically compares multivariate models derived from the Poisson distribution which can be categorized into three model classes based on primary modeling assumptions.

Unit roots are nonstationary autoregressive (AR) or autoregressive moving average (ARMA) time series processes which may include an intercept and/or a trend. These processes are used often in economics and finance, but can also be found in other scientific fields. Unit root tests address the null hypothesis of a unit root, and an alternative hypothesis of a stationary (or trend stationary) time series. Critical values for unit root tests are typically derived via simulation of limiting distributions expressed as functionals of Brownian motions. The critical values for the Dickey Fuller unit root test with a constant and linear trend are derived via simulation in the R language. Simulation studies are presented showing that linear regressions with unit root processes often produce spurious results. Additional simulation studies are reviewed providing statistical evidence that near-unit roots can often result in spurious cointegration relationships. Various unit root tests are presented, including ones that allow for structural breaks in intercept and/or trend. Threshold unit root tests are introduced. Simulation studies are used to compare the unit root tests under various scenarios. The case where the analyzed time series may have stationary and nonstationary segments is also considered. *WIREs Comput Stat* 2017, 9:e1396. doi: 10.1002/wics.1396

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Mean and standard deviation of ADF URT p-values for Gaussian white noise(I(0)) and random walks(I(1)) with 150 time steps.