Sub-Weibull distributions: generalizing sub-Gaussian and sub-Exponential properties to heavier-tailed distributions

We propose the notion of sub-Weibull distributions, which are characterised by tails lighter than (or equally light as) the right tail of a Weibull distribution. This novel class generalises the sub-Gaussian and sub-Exponential families to potentially heavier-tailed distributions. Sub-Weibull distributions are parameterized by a positive tail index $\theta$ and reduce to sub-Gaussian distributions for $\theta=1/2$ and to sub-Exponential distributions for $\theta=1$. A characterisation of the sub-Weibull property based on moments and on the moment generating function is provided and properties of the class are studied. An estimation procedure for the tail parameter is proposed and is applied to an example stemming from Bayesian deep learning.Comment: 10 pages, 3 figure

An extension of the inductive approach to the lace expansion

We extend the inductive approach to the lace expansion, previously developed to study models with critical dimension 4, to be applicable more generally. In particular, the result of this note has recently been used to prove Gaussian asymptotic behaviour for the Fourier transform of the two-point function for sufficiently spread-out lattice trees in dimensions d>8, and it is potentially also applicable to percolation in dimensions d>6

Modulation of the asymmetry of sea urchin sperm flagellar bending by calmodulin

Sea urchin spermatozoa demembranated with Triton X-100 in the presence of EGTA, termed potentially asymmetric, generate asymmetric bending waves in reactivation solutions containing EGTA. After they are converted to the potentially symmetric condition by extraction with Triton and millimolar Ca++, they generate symmetric bending waves in reactivation solutions containing EGTA. In the presence of EGTA, their asymmetry can be restored by addition of brain calmodulin or the concentrated supernatant obtained from extraction with Triton and millimolar Ca++. These extracts contain calmodulin, as assayed by gel electrophoresis, radioimmunoassay, activation of brain phosphodiesterase, and Ca++-dependent binding of asymmetry-restoring activity to a trifluorophenothiazine-affinity resin. Conversion to the potentially symmetric condition can also be achieved with trifluoperazine substituted for Triton during the exposure to millimolar Ca++, which suggests that the calmodulin-binding activity of Triton is important for this conversion. These observations suggest that the conversion to the potentially symmetric condition is the result of removal of some of the axonemal calmodulin and provide additional evidence for axonemal calmodulin as a mediator of the effect of Ca++ on the asymmetry of flagellar bending

Q-learning: flexible learning about useful utilities

Dynamic treatment regimes are fast becoming an important part of medicine, with the corresponding change in emphasis from treatment of the disease to treatment of the individual patient. Because of the limited number of trials to evaluate personally tailored treatment sequences, inferring optimal treatment regimes from observational data has increased importance. Q-learning is a popular method for estimating the optimal treatment regime, originally in randomized trials but more recently also in observational data. Previous applications of Q-learning have largely been restricted to continuous utility end-points with linear relationships. This paper is the first attempt at both extending the framework to discrete utilities and implementing the modelling of covariates from linear to more flexible modelling using the generalized additive model (GAM) framework. Simulated data results show that the GAM adapted Q-learning typically outperforms Q-learning with linear models and other frequently-used methods based on propensity scores in terms of coverage and bias/MSE. This represents a promising step toward a more fully general Q-learning approach to estimating optimal dynamic treatment regimes