Branching processes with immigration in atypical random environment

Motivated by a seminal paper of Kesten et al. (Ann. Probab., 3(1), 1–31, 1975) we consider a branching process with a conditional geometric offspring distribution with i.i.d. random environmental parameters An, n ≥ 1 and with one immigrant in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution F of $\xi _{n}:=\log ((1-A_{n})/A_{n})$ ξ n : = log ( ( 1 − A n ) / A n ) is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n th generation which becomes even heavier with increase of n. More precisely, we prove that, for all n, the distribution tail $\mathbb {P}(Z_{n} \ge m)$ ℙ ( Z n ≥ m ) of the n th population size Zn is asymptotically equivalent to $n\overline F(\log m)$ n F ¯ ( log m ) as m grows. In this way we generalise Bhattacharya and Palmowski (Stat. Probab. Lett., 154, 108550, 2019) who proved this result in the case n = 1 for regularly varying environment F with parameter α > 1. Further, for a subcritical branching process with subexponentially distributed ξn, we provide the asymptotics for the distribution tail $\mathbb {P}(Z_{n}>m)$ ℙ ( Z n > m ) which are valid uniformly for all n, and also for the stationary tail distribution. Then we establish the “principle of a single atypical environment” which says that the main cause for the number of particles to be large is the presence of a single very small environmental parameter Ak.

Firm size and economic concentration: An analysis from a lognormal expansion

This paper studies the distribution of the firm size for the Colombian economy showing evidence against the Gibrat’s law, which assumes a stable lognormal distribution. On the contrary, we propose a lognormal expansion that captures deviations from the lognormal distribution with additional terms that allow a better fit at the upper distribution tail, which is overestimated according to the lognormal distribution. As a consequence, concentration indexes should be addressed consistently with the lognormal expansion. Through a dynamic panel data approach, we also show that firm growth is persistent and highly dependent on firm characteristics, including size, age, and leverage −these results neglect Gibrat’s law for the Colombian case.

Improved Statistical Method for Quality Control of Hydrographic Observations

Realistic ocean state prediction and its validation rely on the availability of high quality in situ observations. To detect data errors, adequate quality check procedures must be designed. This paper presents procedures that take advantage of the ever-growing observation databases that provide climatological knowledge of the ocean variability in the neighborhood of an observation location. Local validity intervals are used to estimate binarily whether the observed values are considered as good or erroneous. Whereas a classical approach estimates validity bounds from first- and second-order moments of the climatological parameter distribution, that is, mean and variance, this work proposes to infer them directly from minimum and maximum observed values. Such an approach avoids any assumption of the parameter distribution such as unimodality, symmetry around the mean, peakedness, or homogeneous distribution tail height relative to distribution peak. To reach adequate statistical robustness, an extensive manual quality control of the reference dataset is critical. Once the data have been quality checked, the local minima and maxima reference fields are derived and the method is compared with the classical mean/variance-based approach. Performance is assessed in terms of statistics of good and bad detections. It is shown that the present size of the reference datasets allows the parameter estimates to reach a satisfactory robustness level to always make the method more efficient than the classical one. As expected, insufficient robustness persists in areas with an especially low number of samples and high variability.

Evaluating CMIP6 Model Fidelity at Simulating Non-Gaussian Temperature Distribution Tails

<p>Under global warming, changes in extreme temperatures will manifest in more complex ways in locations where temperature distribution tails deviate from Gaussian. For example, uniform warming applied to a temperature distribution with a shorter-than-Gaussian warm tail would lead to greater exceedances in warm-side temperature extremes compared with a Gaussian distribution. Confidence in projections of future temperature extremes and associated impacts under global warming therefore relies on the ability of global climate models (GCMs) to realistically simulate observed temperature distribution tail behavior. This presentation examines the ability of the latest state-of-the-art ensemble of GCMs from the Coupled Model Intercomparison Project phase six (CMIP6) to capture historical global surface temperature distribution tail shape in hemispheric winter and summer seasons. Comparisons between the multi-model ensemble mean and a reanalysis product reveal strong agreement on coherent spatial patterns of longer- and shorter-than-Gaussian tails for the cold and warm sides of the temperature distribution, suggesting that CMIP6 is broadly capturing tail behavior for plausible physical and dynamical reasons. Most individual GCMs are also reasonably skilled at capturing historical tail shape on a global scale, but a division of the domain into sub-regions reveals considerable model and spatial variability. To explore potential mechanisms driving these differences, a back trajectory analysis examining patterns in the origin of air masses on days experiencing extreme temperatures is also discussed.</p>

Inferences on parametric estimation of distribution tails

We propose a general method of parameter estimation of a distribution tail that does not depend on the fulfillment of the conditions of Gnedenko theorem. We prove the consistency of the proposed estimator and its asymptotic normality under the stronger conditions imposed on the parametric family of distribution tails. Additionally, the adaptation of the proposed method to Weibull and log-Weibull tail indices estimation is provided.