Extreme value theory provides methods to analyze the most extreme parts of data. We used the generalized extreme value (GEV) distribution to predict the human lifespan in the world and Japan. The diagnostic plots, which assessed the accuracy of the GEV model, were fitted to the human lifespan, validating the model. The human lifespan in the world and Japan had shape parameters of ?0.1623 and ?0.2949 and had an upper limit. The calculated upper limit in the world was 128.7 years. The world’s oldest record holder, Jeanne Calment’s age of 122.45 years, was close to the 260-year return level and was far from the calculated upper limit. The calculated upper limit in Japan was 120.4 years. Japan’s oldest record holder, Kane Tanaka’s age of 119 years in 2022, was the 500-year return level and was close to the upper limit. In the world, achieving the calculated limit was difficult, but the human lifespan will soon reach the upper limit in Japan.
Both for and against a limit on the human lifespan were argued [
EVT has emerged as one of the most important statistical disciplines in applied science. Extreme value techniques are also widely used in other disciplines, such as financial market risk assessment and telecommunications traffic prediction [
In our study, we predict the human lifespan in the world and Japan using the extreme value theory.
We used the human lifespan in the world and Japan from 1953 to 2018 [
The extreme value theory concerns the phenomena of extreme data. We used the block maxima method. A method for modeling the extremes of a stationary time series is block maxima, in which consecutive observations are grouped into non-overlapping blocks of length n, generating a series of m block maxima, Mn, 1, …, Mn, m, to which the Generalized Extreme Value (GEV) distribution can be fitted for some large value of n. The usual approach considers blocks of a given time length, thus yielding maxima at regular intervals [
was considered as a year, i.e., annual maxima values were used. Although the block maxima method is suitable for analyzing maximum value data, it has the disadvantage of being easily affected by one realization value and having a large variance of the estimator.
When data are taken to be the maxima (or minima) over certain blocks of time (such as annual maximum precipitation), it is appropriate to use the GEV distribution:
G ( z ) = { exp { − [ 1 + ξ ( z − μ σ ) ] − 1 / ξ } , ξ ≠ 0 , exp { − exp [ − ( z − μ σ ) ] } , ξ = 0 , (1)
where z are extreme values from blocks, μ a location parameter, σ a scale parameter, and ξ a shape parameter. G(z) is defined for all z such that (1 + ξ(z − μ)/σ) > 0 for ξ ≠ 0, and all z for ξ = 0. Three families of GEV distributions are defined depending on the value of ξ. We get the Fréchet distribution with a heavy tail for ξ > 0, the Gumbel distribution with a lighter tail for ξ = 0, and the Weibull distribution with a finite tail for ξ < 0.
If a GEV distribution is fitted to observations, it becomes possible to estimate the probability of an event that has not yet been observed. Estimates of extreme quantiles of the annual maximum distribution are obtained by inverting Equation (1):
z p = { μ − σ ξ [ 1 − { − log ( 1 − p ) } − ξ ] , ξ ≠ 0 , μ − σ log { − log ( 1 − p ) } , ξ = 0 , (2)
where G(zp) = 1 − p. zp is the return level associated with the return period 1/p, since zp is expected to be exceeded on average once every 1/p year with a reasonable degree of accuracy. More accurately, zp is exceeded by the annual maximum in any particular year with probability p [
Modeling was performed using the evd package in R for the GEV calculations. We also tried a non-stationary model in the GEV, but it did not work.
The human lifespan in the world from 1953 to 2018 is shown in
Estimates of the maximum return levels for the return periods of 10, 20, 50, 100, and 500 years along with their respective 95% CI are shown in
The diagnostic plots for assessing the accuracy of the GEV model fitted to the human lifespan in the world are shown in
| μ | σ | ξ | |
|---|---|---|---|
| Parameter estimate | 113.3 | 2.504 | –0.1623 |
| Standard error | 0.4292 | 0.2964 | 0.08698 |
| 95% CI | (112.4, 114.1) | (1.923, 3.084) | (−0.3298, 0.005280) |
| Return period (years) | 10 | 20 | 50 | 100 | 500 |
|---|---|---|---|---|---|
| Return level (years) | 118.0 | 119.2 | 120.5 | 121.4 | 123.1 |
| Standard error | 0.6483 | 0.7894 | 1.046 | 1.283 | 1.917 |
| 95% CI | (116.7, 119.3) | (117.6, 120.7) | (118.5, 122.6) | (118.9, 123.9) | (119.3, 126.8) |
of points follows a near-linear behavior. The corresponding density estimate is consistent with the data. In the return level curve, the estimated curve is nonlinear with a downward convex shape due to the negative ξ. Consequently, all diagnostic plots supported the fitted GEV model.
The human lifespan in Japan from 1953 to 2018 is shown in
Estimates of the maximum return levels for the return periods of 10, 20, 50, 100, and 500 years along with their respective 95% CI are shown in
The diagnostic plots for assessing the accuracy of the GEV model fitted to the human lifespan in Japan are shown in
The human lifespan in the world and Japan from 1953 to 2018 is shown in
The return levels at each return period for the human lifespan in the world and Japan are shown in
The return levels at each return period for the human lifespan in the world and Japan in a log-log plot are shown in
| μ | σ | ξ | |
|---|---|---|---|
| Parameter estimate | 111.6 | 2.576 | –0.2949 |
| Standard error | 0.5310 | 0.4090 | 0.1735 |
| 95% CI | (110.6, 112.7) | (1.774, 3.377) | (−0.6350, 0.04516) |
| Return period (years) | 10 | 20 | 50 | 100 | 500 |
|---|---|---|---|---|---|
| Return level (years) | 115.9 | 116.7 | 117.6 | 118.1 | 119.0 |
| Standard error | 0.5771 | 0.7162 | 1.030 | 1.307 | 1.949 |
| 95% CI | (114.7, 117.0) | (115.3, 118.1) | (115.6, 119.6) | (115.6, 120.7) | (115.2, 122.8) |
| World | Japan | |
|---|---|---|
| a | 1.014 × 10−2 | 6.102 × 10−3 |
| b | 115.6 | 114.5 |
| r | 0.9935 | 0.9817 |
law. A power low, known as a scaling law, is a relation of the type y = bxa, where y and x are variables of interest, a is called the power exponent, and b is a constant. This indicates a correlation between the return level and the return period. The inclination of the world (1.014 × 10−2) is larger than that of Japan (6.102 × 10−3). The human lifespan in the world will grow more in the future.
The calculated upper limit in the world was 128.7 years with a 95% CI (114.2, 143.2). The world’s oldest record holder, Jeanne Calment’s age of 122.45 years, was close to the 260-year return level and was far from the calculated upper limit. Based on a tendency for survival probability to be maximized in modern human survival curves, they developed an estimation method for maximum human lifespan and obtained about 126 years for Swedish females for 1950-2005 [
The calculated upper limit in Japan was 120.4 years with a 95% CI (112.5, 128.3). Japan’s oldest record holder, Kane Tanaka’s age of 119 years in 2022, was the 500-year return level and was close to the upper limit. Hence, in the world, achieving the calculated limit was difficult. However, in Japan, the human lifespan will soon reach the upper limit.
Extreme value theory can provide methods to analyze the most extreme parts of data. We used the generalized extreme value (GEV) distribution to predict the human lifespan in the world and Japan. The results are summarized as follows:
1) The diagnostic plots, which assessed the accuracy of the GEV model, were fitted to the human lifespan, validating the model.
2) The human lifespan in the world and Japan had shape parameters of −0.1623 and −0.2949 and had an upper limit.
3) The calculated upper limit in the world was 128.7 years. The world’s oldest record holder, Jeanne Calment’s age of 122.45 years, was close to the 260-year return level and was far from the calculated upper limit.
4) The calculated upper limit in Japan was 120.4 years. Japan’s oldest record holder, Kane Tanaka’s age of 119 years in 2022, was the 500-year return level and was close to the upper limit. In the world, achieving the calculated limit was difficult, but the human lifespan will soon reach the upper limit in Japan.
5) The relationship between the return level of the human lifespan and the return period follows a power law.
The author declares no conflicts of interest regarding the publication of this paper.
Maruyama, F. (2022) An Analysis of the Maximum Lifespan in the World and Japan. Journal of Biosciences and Medicines, 10, 254-262. https://doi.org/10.4236/jbm.2022.104021