1. J.-M. Robine and M. Allard in Validation of Exceptional Longevity B. Jeune and J. W. Vaupel Eds. (Odense Univ. Press Odense Denmark 1999) pp.145–172.
2. Wilmoth J. R., Lundström H., Eur. J. Popul. 12, 63 (1996).
3. The least-squares regression equation for the trend lines in Fig. 1 is as follows: age = 101.5369 + 0.0444 (year – 1861) + 0.0667 (year – 1969) I year>1969 – 1.741 I male where age is the maximum age at death recorded for a calendar year (in A.D.) I year>1969 is an indicator variable that equals one after 1969 and zero otherwise and I male equals one for males and zero for females. Thus the slope of the trend lines is 0.0444 per annum before 1969 and 0.0444 + 0.0667 = 0.1111 afterwards. The year 1969 was chosen as the turning point for the slope because this choice maximizes goodness-of-fit (in R 2 ). This model provides a significantly better description of the data than a comparable one-slope model [ F (1 274) = 17.94; P < 0.0001] whereas a four-slope model (different trends for men and women both before and after 1969) is only marginally better than the model shown here [ F (2 272) = 2.52; P = 0.0825].
4. Hill M., Preston S. H., Rosenwaike I., Demography 37, 175 (2000).
5. The maximum age at death can be thought of as an extreme value of a statistical distribution (24 25). Suppose that S ( x ) is the probability of survival from birth to age x for an individual chosen at random from a cohort of N births. The probability that the maximum age at death for this cohort lies above age x is given by S N ( x ) = 1 – [1 – S( x )] N . Accordingly the maximum age at death is itself a random variable with a probability distribution and this distribution is determined by N and the S ( x ) function or alternatively by N and the probability distribution of ages at death given by the function f(x)=–dS(x)dx.