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The spatial and temporal patterns of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) cases and COVID-19 deaths in the United States are poorly understood. We show that variations in the cumulative reported cases and deaths by county, state, and date exemplify Taylor’s law of fluctuation scaling. Specifically, on day 1 of each month from April 2020 through June 2021, each state’s variance (across its counties) of cases is nearly proportional to its squared mean of cases. COVID-19 deaths behave similarly. The lower 99% of counts of cases and deaths across all counties are approximately lognormally distributed. Unexpectedly, the largest 1% of counts are approximately Pareto distributed, with a tail index that implies a finite mean and an infinite variance. We explain why the counts across the entire distribution conform to Taylor’s law with exponent two using models and mathematics. The finding of infinite variance has practical consequences. Local jurisdictions (counties, states, and countries) that are planning for prevention and care of largely unvaccinated populations should anticipate the rare but extremely high counts of cases and deaths that occur in distributions with infinite variance. Jurisdictions should prepare collaborative responses across boundaries, because extremely high local counts of cases and deaths may vary beyond the resources of any local jurisdiction. 

A novel methodology is proposed for clustering multivariate time series data using energy distance defined in Székely and Rizzo (2013). Specifically, a dissimilarity matrix is formed using the energy distance statistic to measure the separation between the finite‐dimensional distributions for the component time series. Once the pairwise dissimilarity matrix is calculated, a hierarchical clustering method is then applied to obtain the dendrogram. This procedure is completely nonparametric as the dissimilarities between stationary distributions are directly calculated without making any model assumptions. In order to justify this procedure, asymptotic properties of the energy distance estimates are derived for general stationary and ergodic time series. The method is illustrated in a simulation study for various component time series that are either linear or nonlinear. Finally, the methodology is applied to two examples; one involves the GDP of selected countries and the other is the population size of various states in the U.S.A. in the years 1900–1999.

 

Pillai & Meng (Pillai & Meng 2016 Ann. Stat. 44 , 2089–2097; p. 2091) speculated that ‘the dependence among [random variables, rvs] can be overwhelmed by the heaviness of their marginal tails ·· ·’. We give examples of statistical models that support this speculation. While under natural conditions the sample correlation of regularly varying (RV) rvs converges to a generally random limit, this limit is zero when the rvs are the reciprocals of powers greater than one of arbitrarily (but imperfectly) positively or negatively correlated normals. Surprisingly, the sample correlation of these RV rvs multiplied by the sample size has a limiting distribution on the negative half-line. We show that the asymptotic scaling of Taylor’s Law (a power-law variance function) for RV rvs is, up to a constant, the same for independent and identically distributed observations as for reciprocals of powers greater than one of arbitrarily (but imperfectly) positively correlated normals, whether those powers are the same or different. The correlations and heterogeneity do not affect the asymptotic scaling. We analyse the sample kurtosis of heavy-tailed data similarly. We show that the least-squares estimator of the slope in a linear model with heavy-tailed predictor and noise unexpectedly converges much faster than when they have finite variances. 

We estimate the parameter of a stationary time series process by minimizing the integrated weighted mean squared error between the empirical and simulated characteristic function, when the true characteristic functions cannot be explicitly computed. Motivated by Indirect Inference, we use a Monte Carlo approximation of the characteristic function based on i.i.d. simulated blocks. As a classical variance reduction technique, we propose the use of control variates for reducing the variance of this Monte Carlo approximation. These two approximations yield two new estimators that are applicable to a large class of time series processes. We show consistency and asymptotic normality of the parameter estimators under strong mixing, moment conditions, and smoothness of the simulated blocks with respect to its parameter. In a simulation study we show the good performance of these new simulation based estimators, and the superiority of the control variates based estimator for Poisson driven time series of counts.

 

Studies of spatial point patterns (SPPs) are often used to examine the role that density‐dependence (DD) and environmental filtering (EF) play in community assembly and species coexistence in forest communities. However, SPP analyses often struggle to distinguish the opposing effects that DD and EF may have on the distribution of tree species.

We tested percolation threshold analysis on simulated tree communities as a method to distinguish the importance of thinning from DD EF on SPPs. We then compared the performance of percolation threshold analysis results and a Gibbs point process model in detecting environmental associations as well as clustering patterns or overdispersion. Finally, we applied percolation threshold analysis and the Gibbs point process model to observed SPPs of 12 dominant tree species in a Puerto Rican forest to detect evidence of DD and EF.

Percolation threshold analysis using simulated SPPs detected a decrease in clustering due to DD and an increase in clustering from EF. In contrast, the Gibbs point process model clearly detected the effects of EF but only identified DD thinning in two of the four types of simulated SPPs. Percolation threshold analysis on the 12 observed tree species' SPPs found that the SPPs for two species were consistent with thinning from DD processes only, four species had SPPs consistent with EF only and SPP for five reflected a combination of both processes. Gibbs models of observed SPPs of living trees detected significant environmental associations for 11 species and clustering consistent with DD processes for seven species.

Percolation threshold analysis is a robust method for detecting community assembly processes in simulated SPPs. By applying percolation threshold analysis to natural communities, we found that tree SPPs were consistent with thinning from both DD and EF. Percolation threshold analysis was better suited to detect DD thinning than Gibbs models for clustered simulated communities. Percolation threshold analysis improves our understanding of forest community assembly processes by quantifying the relative importance of DD and EF in forest communities.