Summation of rational series twisted by strongly B-multiplicative coefficients

We evaluate in closed form series of the type $\sum u(n) R(n)$, where $(u(n))_n$ is a strongly $B$-multiplicative sequence and $R(n)$ a (well-chosen) rational function. A typical example is: $$\sum_{n \geq 1} (-1)^{s_2(n)} \frac{4n+1}{2n(2n+1)(2n+2)} = -\frac{1}{4}$$ where $s_2(n)$ is the sum of the binary digits of the integer $n$. Furthermore closed formulas for series involving automatic sequences that are not strongly $B$-multiplicative, such as the regular paperfolding and Golay-Shapiro-Rudin sequences, are obtained; for example, for integer $d \geq 0$: $$\sum_{n \geq 0} \frac{v(n)}{(n+1)^{2d+1}} = \frac{\pi^{2d+1} |E_{2d}|}{(2^{2d+2}-2)(2d)!}$$ where $(v(n))_n$ is the $\pm 1$ regular paperfolding sequence and $E_{2d}$ is an Euler number.Comment: Typo in a crossreference corrected in Example 9, page 6. Remark added top of Page 9 about the relation between paperfolding and the Jacobi-Kronecker symbo

A refined Weissman estimator for extreme quantiles

International audienceWeissman extrapolation methodology for estimating extreme quantiles from heavy-tailed distributions is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail-index. The common practice is to select the same intermediate sequence for both estimators. In this work, we show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined Weissman estimator. The asymptotic normality of the latter estimator is established and a data-driven method is introduced for the practical selection of the intermediate sequences. Our approach is compared to Weissman estimator and to six bias reduced estimators of extreme quantiles on a large scale simulation study. It appears that the refined Weissman estimator outperforms its competitors in a wide variety of situations, especially in the challenging high bias cases. Finally, an illustration on an actuarial real data set is provided

Breakthrough infections due to SARS-CoV-2 Delta variant: relation to humoral and cellular vaccine responses

IntroductionCOVID-19 vaccines are expected to provide effective protection. However, emerging strains can cause breakthrough infection in vaccinated individuals. The immune response of vaccinated individuals who have experienced breakthrough infection is still poorly understood.MethodsHere, we studied the humoral and cellular immune responses of fully vaccinated individuals who subsequently experienced breakthrough infection due to the Delta variant of SARS-CoV-2 and correlated them with the severity of the disease.ResultsIn this study, an effective humoral response alone was not sufficient to induce effective immune protection against severe breakthrough infection, which also required effective cell-mediated immunity to SARS-CoV-2. Patients who did not require oxygen had significantly higher specific (p=0.021) and nonspecific (p=0.004) cellular responses to SARS-CoV-2 at the onset of infection than those who progressed to a severe form.DiscussionKnowing both humoral and cellular immune response could allow to adapt preventive strategy, by better selecting patients who would benefit from additional vaccine boosters.Trial registration numbershttps://clinicaltrials.gov, identifier NCT04355351; https://clinicaltrials.gov, identifier NCT04429594

A refined Weissman estimator for extreme quantiles

International audienceWeissman's extrapolation methodology for estimating extreme quantiles from heavy-tailed distributions is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail-index. The common practice is to select the same intermediate sequence for both estimators. We show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined Weissman estimator. The asymptotic normality of the latter estimator is established and a data-driven method is introduced for the practical selection of the intermediate sequences. Our approach is compared to Weissman estimator and to six bias-reduced estimators of extreme quantiles in a large-scale simulation study. It appears that the refined Weissman estimator outperforms its competitors in a wide variety of situations, especially in challenging high-bias cases. Finally, an illustration of an actuarial real data set is provided

A refined Weissman estimator for extreme quantiles

International audienceWeissman extrapolation methodology for estimating extreme quantiles from heavy-tailed distributions is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail-index. The common practice is to select the same intermediate sequence for both estimators. In this work, we show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined Weissman estimator. The asymptotic normality of the latter estimator is established and a data-driven method is introduced for the practical selection of the intermediate sequences. Our approach is compared to Weissman estimator and to six bias reduced estimators of extreme quantiles on a large scale simulation study. It appears that the refined Weissman estimator outperforms its competitors in a wide variety of situations, especially in the challenging high bias cases. Finally, an illustration on an actuarial real data set is provided

A refined Weissman estimator for extreme quantiles

International audienc

A refined Weissman estimator for extreme quantiles

Weissman extrapolation methodology for estimating extreme quantiles from heavy-tailed distributions is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail-index. The common practice is to select the same intermediate sequence for both estimators. In this work, we show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined Weissman estimator. The asymptotic normality of the latter estimator is established and a data-driven method is introduced for the practical selection of the intermediate sequences. Our approach is compared to Weissman estimator and to six bias reduced estimators of extreme quantiles on a large scale simulation study. It appears that the refined Weissman estimator outperforms its competitors in a wide variety of situations, especially in the challenging high bias cases. Finally, an illustration on an actuarial real data set is provided

A refined Weissman estimator for extreme quantiles

International audienceWeissman extrapolation methodology for estimating extreme quantiles from heavy-tailed distributions is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail-index. The common practice is to select the same intermediate sequence for both estimators. In this work, we show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined Weissman estimator. The asymptotic normality of the latter estimator is established and a data-driven method is introduced for the practical selection of the intermediate sequences. Our approach is compared to Weissman estimator and to six bias reduced estimators of extreme quantiles on a large scale simulation study. It appears that the refined Weissman estimator outperforms its competitors in a wide variety of situations, especially in the challenging high bias cases. Finally, an illustration on an actuarial real data set is provided