In random-effects meta-analysis, the between-study heterogeneity variance, $\tau ^2$, is often reported but is not easy to interpret. For meta-analyses of differences (such as mean differences, standardized mean differences, or risk differences), the standard deviation (SD), $\tau $, indicates the extent to which studies’ true effects vary about their average. For meta-analyses of (natural) log-transformed measures of effect (such as log risk ratios [RRs]), we explain how the geometric SD, $\exp (\tau )$, is helpful to understand how untransformed measures (such as RRs) vary multiplicatively about their average. We recommend that authors and software developers report $\tau $ for differences and $\exp (\tau )$ for ratios, rather than $\tau ^2$. This will facilitate the interpretation of the magnitude of heterogeneity values, for example, the interpretation of heterogeneity estimates and confidence intervals beyond simple binary statements about the presence or absence of heterogeneity.

Fitting loss distributions in insurance is sometimes a dilemma: either you get a good fit for the small/medium losses or for the very large losses. To be able to get both at the same time, this paper studies generalisations and extensions of the Pareto model that initially look like, for example, the Lognormal distribution but have a Pareto or GPD tail. We design a classification of such spliced distributions, which embraces and generalises various existing approaches. Special attention is paid to the geometry of distribution functions and to intuitive interpretations of the parameters, which can ease parameter inference from scarce data. The developed framework gives also new insights into the old Riebesell (power curve) exposure rating method.

The cross-sectional distribution of government debt is often approximated by a lognormal distribution. This note empirically demonstrates that government debt is more accurately characterized by the double Pareto-lognormal (dPLN) distribution, which features a lognormal body with two Pareto tails. The dPLN assuredly surpasses alternative parametric distributions and passes goodness-of-fit tests. With its analytical tractability, flexibility, and parsimony, coupled with a theoretical foundation, the dPLN may be appealing for different computational and empirical applications.

A variety of distributional assumptions for dissimilarity judgments are considered, with the lognormal distribution being favored for most situations. An implicit equation is discussed for the maximum likelihood estimation of the configuration with or without individual weighting of dimensions. A technique for solving this equation is described and a number of examples offered to indicate its performance in practice. The estimation of a power transformation of dissimilarity is also considered. A number of likelihood ratio hypothesis tests are discussed and a small Monte Carlo experiment described to illustrate the behavior of the test of dimensionality in small samples.

While lognormal distributions have been proposed as useful descriptors of recruitment variability, the very nature of the recruitment distributions is still debated.To account quantitatively for recruitment distributions,I here propose a Weibull exponential model;it derives from a simple and natural hypothesis for uncorrelated recruitment processes of spawning, hatching, growth and survival through the early life stages to the point of vulnerability to the fishery.The quantification of Weibull exponentials is particularly important with regards to extrapolations to low recruits that have not yet been observed.To test the Weibull exponential null-hypothesis, I examine annual time-series of recruitment in major aquatic stocks.The Weibull exponential quite describes the bulk (95%) of the recruitment distributions of widely differing stocks,while the remaining 5% of the largest recruits are occurring with a much larger rate than predicted by the Weibull exponential.Further, I study the inter-event times between unusually high numbers in recruitment time-series data and find that intermittent pulses of strong recruitment follow non-Poisson statistics, which arises from year-to-year persistence of the magnitude of recruitment:large (or small) recruits are more likely to be followed by large (or small) recruits.This recruitment clustering effect is confirmed by the rescaled range analysis method.The empirical results imply that individual survivals on recruitment levels are independent of initial cohort sizes but year-to-year recruiting events exhibit long-term correlations.