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Unlike the real line, the real space ℝ d , for d ≥ 2, is not canonically ordered. As a consequence, such fundamental univariate concepts as quantile and distribution functions and their empirical counterparts, involving ranks and signs, do not canonically extend to the multivariate context. Palliating that lack of a canonical ordering has been an open problem for more than half a century, generating an abundant literature and motivating, among others, the development of statistical depth and copula-based methods. We show that, unlike the many definitions proposed in the literature, the measure transportation-based ranks and signs introduced in Chernozhukov, Galichon, Hallin and Henry (Ann. Statist. 45 (2017) 223–256) enjoy all the properties that make univariate ranks a successful tool for semiparametric inference. Related with those ranks, we propose a new center-outward definition of multivariate distribution and quantile functions, along with their empirical counterparts, for which we establish a Glivenko–Cantelli result. Our approach is based on McCann (Duke Math. J. 80 (1995) 309–323) and our results do not require any moment assumptions. The resulting ranks and signs are shown to be strictly distribution-free and essentially maximal ancillary in the sense of Basu (Sankhyā 21 (1959) 247–256) which, in semiparametric models involving noise with unspecified density, can be interpreted as a finite-sample form of semiparametric efficiency. Although constituting a sufficient summary of the sample, empirical center-outward distribution functions are defined at observed values only. A continuous extension to the entire d-dimensional space, yielding smooth empirical quantile contours and sign curves while preserving the essential monotonicity and Glivenko–Cantelli features of the concept, is provided. A numerical study of the resulting empirical quantile contours is conducted.

This work is devoted to the study of rates of convergence of the empirical measures \\mu_{n} = \\frac {1}{n} \\sum_{k=1}^n \\delta_{X_k}, n \\geq 1, over a sample (X_{k})_{k \\geq 1} of independent identically distributed real-valued random variables towards the common distribution \\mu in Kantorovich transport distances W_p. The focus is on finite range bounds on the expected Kantorovich distances \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) or \\big [ \\mathbb{E}(W_{p}^p(\\mu_{n},\\mu )) \\big ]^1/p in terms of moments and analytic conditions on the measure \\mu and its distribution function. The study describes a variety of rates, from the standard one \\frac {1}{\\sqrt n} to slower rates, and both lower and upper-bounds on \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log-concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.

We propose and study properties of maximum likelihood estimators in the class of conditional transformation models. Based on a suitable explicit parameterization of the unconditional or conditional transformation function, we establish a cascade of increasingly complex transformation models that can be estimated, compared and analysed in the maximum likelihood framework. Models for the unconditional or conditional distribution function of any univariate response variable can be set up and estimated in the same theoretical and computational framework simply by choosing an appropriate transformation function and parameterization thereof. The ability to evaluate the distribution function directly allows us to estimate models based on the exact likelihood, especially in the presence of random censoring or truncation. For discrete and continuous responses, we establish the asymptotic normality of the proposed estimators. A reference software implementation of maximum likelihood-based estimation for conditional transformation models that allows the same flexibility as the theory developed here was employed to illustrate the wide range of possible applications.

The increased availability of massive data sets provides a unique opportunity to discover subtle patterns in their distributions, but also imposes overwhelming computational challenges. To fully utilize the information contained in big data, we propose a two-step procedure: (i) estimate conditional quantile functions at different levels in a parallel computing environment; (ii) construct a conditional quantile regression process through projection based on these estimated quantile curves. Our general quantile regression framework covers both linear models with fixed or growing dimension and series approximation models. We prove that the proposed procedure does not sacrifice any statistical inferential accuracy provided that the number of distributed computing units and quantile levels are chosen properly. In particular, a sharp upper bound for the former and a sharp lower bound for the latter are derived to capture the minimal computational cost from a statistical perspective. As an important application, the statistical inference on conditional distribution functions is considered. Moreover, we propose computationally efficient approaches to conducting inference in the distributed estimation setting described above. Those approaches directly utilize the availability of estimators from subsamples and can be carried out at almost no additional computational cost. Simulations confirm our statistical inferential theory.

Drought is projected to become more severe and widespread as global warming continues in the twenty-first century, but hydroclimatic changes and their drivers are not well examined in the latest projections from phase 6 of the Coupled Model Intercomparison Project (CMIP6). Here, precipitation (P), evapotranspiration (E), soil moisture (SM), and runoff (R) from 25 CMIP6 models, together with self-calibrated Palmer drought severity index with Penman—Monteith potential evapotranspiration (scPDSIpm), are analyzed to quantify hydroclimatic and drought changes in the twenty-first century and the underlying causes. Results confirm consistent drying in these hydroclimatic metrics across most of the Americas (including the Amazon), Europe and the Mediterranean region, southern Africa, and Australia, although the drying magnitude differs, with the drying being more severe and widespread in surface SM than in total SM. Global drought frequency based on surface SM and scPDSIpm increases by ~25%–100% (50%–200%) under the SSP2-4.5 (SSP5-8.5) scenario in the twenty-first century together with large increases in drought duration and areas, which result from a decrease in the mean and flattening of the probability distribution functions of SM and scPDSIpm, while the R-based drought changes are relatively small. Changes in both P and E contribute to the SM change, whereas scPDSIpm decreases result from ubiquitous PET increases and P decreases over subtropical areas. The R changes are determined primarily by P changes, while the PET change explains most of the E increase. Intermodel spreads in surface SM and R changes are large, leading to large uncertainties in the drought projections.

The stationary-state distribution function of confined active Brownian particles (ABPs) is analyzed by computer simulations and analytical calculations. We consider a radial harmonic as well as an anharmonic confinement potential. In the simulations, the ABP is propelled with a prescribed velocity along a body-fixed direction, which is changing in a diffusive manner. For the analytical approach, the Cartesian components of the propulsion velocity are assumed to change independently; active Ornstein-Uhlenbeck particle (AOUP). This results in very different velocity distribution functions. The analytical solution of the Fokker-Planck equation for an AOUP in a harmonic potential is presented and a conditional distribution function is provided for the radial particle distribution at a given magnitude of the propulsion velocity. This conditional probability distribution facilitates the description of the coupling of the spatial coordinate and propulsion, which yields activity-induced accumulation of particles. For the anharmonic potential, a probability distribution function is derived within the unified colored noise approximation. The comparison of the simulation results with theoretical predictions yields good agreement for large rotational diffusion coefficients, e.g. due to tumbling, even for large propulsion velocities (Péclet numbers). However, we find significant deviations already for moderate Péclet number, when the rotational diffusion coefficient is on the order of the thermal one.

Sun-induced fluorescence (SIF) in the far-red region provides a new noninvasive measurement approach that has the potential to quantify dynamic changes in light-use efficiency and gross primary production (GPP). However, the mechanistic link between GPP and SIF is not completely understood. We analyzed the structural and functional factors controlling the emission of SIF at 760 nm (F760) in a Mediterranean grassland manipulated with nutrient addition of nitrogen (N), phosphorous (P) or nitrogen–phosphorous (NP). Using the soil–canopy observation of photosynthesis and energy (SCOPE) model, we investigated how nutrient-induced changes in canopy structure (i.e. changes in plant forms abundance that influence leaf inclination distribution function, LIDF) and functional traits (e.g. N content in dry mass of leaves, N%, Chlorophyll a+b concentration (Cab) and maximum carboxylation capacity (V cmax)) affected the observed linear relationship between F760 and GPP. We conclude that the addition of nutrients imposed a change in the abundance of different plant forms and biochemistry of the canopy that controls F760. Changes in canopy structure mainly control the GPP–F760 relationship, with a secondary effect of Cab and V cmax. In order to exploit F760 data to model GPP at the global/regional scale, canopy structural variability, biodiversity and functional traits are important factors that have to be considered.

We obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From these formulas, we deduce exact formulas for the cumulative distribution function of the product of two correlated zero-mean normal random variables.

The 2022 heatwave in China featured record‐shattering high temperatures, raising questions about its origin and possible link to global warming. Here we show that the maximum temperature anomalies over Central China reached 13.1°C in the summer of 2022, which is ∼4.2σ above the 1981–2010 mean with a return period of tens of thousands of years. Our results suggested that the persistent high‐pressure anomaly and associated extreme heatwave likely resulted mainly from internal variability, although anthropogenic warming has increased the probability of such extreme heatwaves. We also estimate that the 2022‐like heatwave becomes six to seven times more likely under persistent high‐pressure conditions when compared to stochastic circulation states. Due to a shift toward warmer mean temperatures and a flattening of the probability distribution function, such rare extreme heatwaves are projected to become much more common at a global warming level of 4°C, occurring once about every 8.5 years. Plain Language Summary China experienced an extensive and long‐lasting heatwave in the summer of 2022, especially in the middle reaches of the Yangtze River. Using state‐of‐the‐art reanalysis and models, here we show that the 2022 heatwave is among the most severe events ever recorded in China with a return period of tens of thousands of years. The atmospheric circulation patterns associated with the 2022 heatwave are not caused by external forcing, but due to internal atmospheric variability characterized by extremely warm anomalies and positive high‐pressure anomalies around central China, which increase the chance of such events by more than six times. A shift of the temperature distribution toward higher mean values and enlarged variability under global warming would increase the chances of new record‐breaking temperatures in the future. Such rare heatwaves in the natural and present climate could become much more common in the middle reaches of the Yangtze River under anthropogenic warming, occurring at least once every eight years at a global warming level of 4°C. Thus, efforts to build social resilience to extreme heatwaves are therefore urgently needed. Key Points Atmospheric circulation patterns associated with the 2022 heatwave likely resulted from internal variability, enhanced by external forcing The 2022‐like heatwave becomes six to seven times more likely under persistent high‐pressure conditions compared to stochastic circulation states Such rare heatwaves in natural and present climate are projected to become more common by 2100, occurring at least once every eight years

In this paper, we explore the utility of resting-state EEG measures as potential biomarkers for the detection and assessment of cognitive decline in mild cognitive impairment (MCI) and Alzheimer’s disease (AD). Neurophysiological biomarkers of AD derived from EEG and FDG-PET, once characterized and validated, would expand the set of existing diagnostic molecular biomarkers of AD pathology with associated biomarkers of disease progression and neural dysfunction. Since symptoms of AD often begin to appear later in life, successful identification of EEG-based biomarkers must account for age-related neurophysiological changes that occur even in healthy individuals. To this end, we collected EEG data from individuals with AD (n = 26), MCI (n = 53), and cognitively normal healthy controls stratified by age into three groups: 18–40 (n = 129), 40–60 (n = 62) and 60–90 (= 55) years old. For each participant, we computed power spectral density at each channel and spectral coherence between pairs of channels. Compared to age matched controls, in the AD group, we found increases in both spectral power and coherence at the slower frequencies (Delta, Theta). A smaller but significant increase in power of slow frequencies was observed for the MCI group, localized to temporal areas. These effects on slow frequency spectral power opposed that of normal aging observed by a decrease in the power of slow frequencies in our control groups. The AD group showed a significant decrease in the spectral power and coherence in the Alpha band consistent with the same effect in normal aging. However, the MCI group did not show any significant change in the Alpha band. Overall, Theta to Alpha ratio (TAR) provided the largest and most significant differences between the AD group and controls. However, differences in the MCI group remained small and localized. We proposed a novel method to quantify these small differences between Theta and Alpha bands’ power using empirically derived distributions of spectral power across the time domain as opposed to averaging power across time. We defined Power Distribution Distance Measure (PDDM) as a distance measure between probability distribution functions (pdf) of Theta and Alpha power. Compared to average TAR, using PDDF enhanced the statistical significance, the effect size, and the spatial distribution of significant effects in the MCI group. We designed classifiers for differentiating individual MCI and AD participants from age-matched controls. The classification performance measured by the area under ROC curve after cross-validation were AUC = 0.85 and AUC = 0.6, for AD and MCI classifiers, respectively. Posterior probability of AD, TAR, and the proposed PDDM measure were all significantly correlated with MMSE score and neuropsychological tests in the AD group.