Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References Best Practice Life Expectancy: An Extreme Value Approach Anthony Medford [email protected] September 9, 2015
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Best Practice Life Expectancy: An Extreme ValueApproach
Anthony Medford
September 9, 2015
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
1 IntroductionWhat is Best Practice Life Expectancy?Trends since 1900Breakpoints
2 Why Extreme Value Theory?Empirical motivationTheoretical motivation
3 The GEVDistribution FunctionInference
4 ResultsFitted ModelProjectionsOther Inference
5 ARIMA and the GEVModel ResidualsInnovations Process
6 Conclusions
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Some Facts
Best Practice Life Expectancy (BPLE) is the maximum lifeexpectancy observed among nations at a given age.
At birth, has been increasing almost linearly - beginning inScandinavia c. 1840 - at about 3 months per year (Oeppenand Vaupel, 2002).
Life expectancy trends may fit better than individual-countrytrends in age-standardized (log) death rates (White, 2002).
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Some Facts
Nations experience more rapid life expectancy gains whenthey are farther below BPLE and tend to converge towardsBPLE (Torri and Vaupel, 2012).
It is sensible to consider national mortality trends in a largerinternational context rather than individual projections (Lee,2006; Wilmoth, 1998).
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Females e0
1900 1920 1940 1960 1980 2000
5055
6065
7075
8085
Female Best Practice e0
Year
e0
IcelandJapanNorwayNZ (non−maori)Sweden
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Males e0
1900 1920 1940 1960 1980 2000
5055
6065
7075
8085
Male Best Practice e0
Year
e0
AustraliaDenmarkIcelandJapanNetherlands
NZ (non−maori)NorwaySwedenSwitzerland
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Females e65
1900 1920 1940 1960 1980 2000
1214
1618
2022
24Female Best Practice e65
Year
e65
CanadaFranceIcelandJapanNorwayNZ (non−maori)Sweden
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Males e65
1900 1920 1940 1960 1980 2000
1214
1618
2022
24Male Best Practice e65
Year
e65
AustraliaDenmarkIcelandJapanNorwaySwitzerland
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Breakpoints
1900 1920 1940 1960 1980 2000
5055
6065
7075
8085
Females
e0
1900 1920 1940 1960 1980 2000
5055
6065
7075
8085
Males
e0
1900 1920 1940 1960 1980 2000
1214
1618
2022
24
Females
e65
1900 1920 1940 1960 1980 2000
1214
1618
2022
24
Malese6
5
Figure: Breakpoints in the trend of the highest life expectancies at birthand age 65, males and females separately, from 1900 - 2012.
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Empirical motivation
1960 1970 1980 1990 2000 2010
7274
7678
8082
8486
Year
e0
Female Best Practice e0
ObservedDetrendedTrend line
−1.0 −0.5 0.0 0.5 1.0 1.5 2.00.
00.
20.
40.
60.
81.
0
N = 58 Bandwidth = 0.1659
Den
sity
Kernel Density and fitted GEV
kernel densityfitted GEV
Figure: Left panel: raw and detrended data. Right panel: kernel densityand fitted GEV distribution.
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Theoretical motivation
Suppose that X1,X2, . . . ,Xn is a sequence of independent,identically distributed random variates all having a commondistribution function F (x).
Let Mn = max{X1,X2, . . . ,Xn}.
The distribution of the maxima, Mn, converges (for large n) to theGeneralized Extreme Value (GEV) Distribution.
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
The Generalized Extreme Value Distribution
G (z) = exp{−
[1 + ξ(
z − u
σ)]−1
ξ}
u is the location parameter
σ is the scale parameter
ξ is the shape parameter, which determines the tail behaviourξ > 0: polynomial tail decay and the Frechet Distributionξ = 0: exponential tail decay and the GumbelDistributionξ < 0: bounded upper finite end point and the WeibullDistribution
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Inference
QuantilesInverting the GEV distribution function:
zp = µ− σ
ξ
[1− {−log(1− p)}−ξ
],
where p is the tail probability and G (zp) = 1− p
Return Levels
Simply a different way of thinking about the quantiles.
If data are annual the (1− p)th quantile would be exceededon average once every 1/p years.
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Fitted Model
GEV (ut = 59.6 + 0.24t, σ = 1.31, ξ = −0.48)
1900 1940 1980
5565
7585
Year
e0
Median50 Year Return Level
Female Best Practice e0
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Projections, Females e0
1900 1950 2000 2050
6070
8090
100
Year
e0
Median 95% Conf Ints
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Projections, Females e0
1900 1950 2000 2050
6070
8090
100
Year
e0
99th Percentile95% Conf Ints
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Other Inference
A probability distribution has been fit so the usual tools areavailable.
Year P(emax0 > 90) P(emax
0 > 95)
2020 35% < 0.001%2050 > 99.99% 91%
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
In Sample Comparison
Fit model using data up to 1980.Compare Observed 10 Year Maxima vs 10 Year return Levels .
1985 1990 1995 2000
0.2
0.4
0.6
0.8
1.0
Year
Abs
olut
e D
iffer
ence
s
Mean Absolute Difference(MAD)= 0.67 yearsMean Absolute Percentage Error = 0.8%
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
ARIMA model residuals
−2 −1 0 1 2
−6
−4
−2
02
norm quantiles
Res
idua
ls
Residuals
Den
sity
−6 −4 −2 0 2
0.0
0.1
0.2
0.3
0.4
fitted GEV
Figure: Normality tests for residuals of ARIMA(2,1,1) fitted to female e0BPLE. Left panel: QQ Plot; Right panel: histogram.
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Innovations Process
Assumption of Gaussian errors is often arbitrary and can bepoorly fitting.
GEV is more flexible and is able to capture the shape ofdifferent error distributions - not just symmetric.
In practice Gaussian often provides a reasonable fit but GEVshould be considered as an alternative for the innovationsprocess.
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
Conclusion
Method can be used similarly to the Torri and Vaupel (2012)approach to forecasting life expectancy:
Either through projecting BPLE directly, which is preferableOr using the GEV as the innovations process in an ARIMAmodel
EVT can identify in an objective way whether life expectancyis actually at an extreme level rather than just ”high”
EVT can be used to obtain probabilities and/ or levels ofextreme longevity
Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References
References
Lee, R. (2006). Perspectives on Mortality Forecasting. III. TheLinear Rise in Life Expectancy: History and Prospects, VolumeIII of Social Insurance Studies. Swedish Social Insurance Agency,Stockholm.
Oeppen, J. and J. W. Vaupel (2002). Broken limits to lifeexpectancy. Science 296(5570), 1029–1031.
Torri, T. and J. W. Vaupel (2012). Forecasting life expectancy inan international context. International Journal ofForecasting 28(2), 519–531.
White, K. M. (2002). Longevity advances in high-income countries,1955–96. Population and Development Review 28(1), 59–76.
Wilmoth, J. R. (1998). Is the pace of Japanese mortality declineconverging toward international trends? Population andDevelopment Review , 593–600.