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Does the human lifespan have an impenetrable biological upper limit which ultimately will stop further increase in life lengths? This question is important for understanding aging, and for society, and has led to intense controversies. Demographic data for humans has been interpreted as showing existence of a limit, or even as an indication of a decreasing limit, but also as evidence that a limit does not exist. This paper studies what can be inferred from data about human mortality at extreme age. We find that in western countries and Japan and after age 110 the risk of dying is constant and is about 47% per year. Hence data does not support that there is a finite upper limit to the human lifespan. Still, given the present stage of biotechnology, it is unlikely that during the next 25 years anyone will live longer than 128 years in these countries. Data, remarkably, shows no difference in mortality after age 110 between sexes, between ages, or between different lifestyles or genetic backgrounds. These results, and the analysis methods developed in this paper, can help testing biological theories of aging and aid confirmation of success of efforts to find a cure for aging.A discussion of this paper will follow in a later issue of Extremes Electronic supplementary material The online version of this article (https://doi

Define Y(t) = max 0≤s≤1 W(t+s)−W(t), where W(·) is a standard Wiener process. We study the maximum of Y up to time T: M T = max 0≤t≤T Y(t) and determine an asymptotic expression for P (M T > u) when u → ∞. Further we establish the limiting Gumbel distribution of M T when T → ∞ and present the corresponding normalization sequence.

We use a combination of extreme value statistics, survival analysis and computer-intensive methods to analyse the mortality of Italian and French semi-supercentenarians. After accounting for the effects of the sampling frame, extreme-value modelling leads to the conclusion that constant force of mortality beyond 108 years describes the data well and there is no evidence of differences between countries and cohorts. These findings are consistent with use of a Gompertz model and with previous analysis of the International Database on Longevity and suggest that any physical upper bound for the human lifespan is so large that it is unlikely to be approached. Power calculations make it implausible that there is an upper bound below 130 years. There is no evidence of differences in survival between women and men after age 108 in the Italian data and the International Database on Longevity, but survival is lower for men in the French data.

Let (ξ i , i ≥ 1) be a sequence of independent standard normal random variables and let S k = k i=1 ξ i be the corresponding random walk. We studydetermine asymptotic expressions for P M (N) T > u when u, N and T → ∞ in a synchronized way. There are three types of relations between u and N that give different asymptotic behavior. For these three cases we establish the limiting Gumbel distribution of M (N) T when T, N → ∞ and present corresponding normalization sequences.

This paper develops tail estimation methods to handle false positives in multiple testing problems where testing is done at extreme significance levels and with low degrees of freedom, and where the true null distribution may differ from the theoretical one. We show that the number of false positives, conditional on the total number of positives, approximately has a binomial distribution, and find estimators of its parameter. We also develop methods for estimation of the true null distribution, and techniques to compare it with the theoretical one. Analysis is based on a simple polynomial model for very small p-values. Asymptotics which motivate the model, properties of the estimators, and model checking tools are provided. The methods are applied to two large genomic studies and an fMRI brain scan experiment.

There is sustained and widespread interest in understanding the limit, if there is any, to the human life span. Apart from its intrinsic and biological interest, changes in survival in old age have implications for the sustainability of social security systems. A central question is whether the endpoint of the underlying lifetime distribution is finite. Recent analyses of data on the oldest human lifetimes have led to competing claims about survival and to some controversy, due in part to incorrect statistical analysis. This article discusses the particularities of such data, outlines correct ways of handling them, and presents suitable models and methods for their analysis. We provide a critical assessment of some earlier work and illustrate the ideas through reanalysis of semisupercentenarian lifetime data. Our analysis suggests that remaining life length after age 109 is exponentially distributed and that any upper limit lies well beyond the highest lifetime yet reliably recorded. Lower limits to 95% confidence intervals for the human life span are about 130 years, and point estimates typically indicate no upper limit at all. Expected final online publication date for the Annual Review of Statistics and Its Application, Volume 9 is March 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

What can be learned from data about human survival at extreme age? In this rejoinder we give our views on some of the issues raised in the discussion of our paper Rootzén and Zholud (Extremes 20(4), 713-728, 2017). Keywords Extreme human life lengths • Supercentenarians • Jeanne Calment • Truncation • Censoring • Limit for human lifespan • Age-biased sampling • Generalized Pareto distribution AMS 2000 Subject Classifications 62P10 • 62N01 Introduction We thank the discussants 1 for very stimulating, thought-provoking, and educational comments. We were impressed by the title of Davison's contribution,

Let T be the Student one-or two-sample t-, F -, or Welch statistic. Now release the underlying assumptions of normality, independence and identical distribution and consider a more general case where one only assumes that the vector of data has a continuous joint density. We determine asymptotic expressions for P(T > u) as u → ∞ for this case. The approximations are particularly accurate for small sample sizes and may be used, for example, in the analysis of High-Throughput Screening experiments, where the number of replicates can be as low as two to five and often extreme significance levels are used. We give numerous examples and complement our results by an investigation of the convergence speed -both theoretically, by deriving exact bounds for absolute and relative errors, and by means of a simulation study.

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