Best Practice Life Expectancy: An Extreme Value Approach Anthony - PowerPoint PPT Presentation

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 amedford@health.sdu.dk September 9, 2015

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References Introduction 1 What is Best Practice Life Expectancy? Trends since 1900 Breakpoints Why Extreme Value Theory? 2 Empirical motivation Theoretical motivation The GEV 3 Distribution Function Inference Results 4 Fitted Model Projections Other Inference ARIMA and the GEV 5 Model Residuals Innovations Process Conclusions 6

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 life expectancy observed among nations at a given age. At birth, has been increasing almost linearly - beginning in Scandinavia c. 1840 - at about 3 months per year (Oeppen and Vaupel, 2002). Life expectancy trends may fit better than individual-country trends 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 when they are farther below BPLE and tend to converge towards BPLE (Torri and Vaupel, 2012). It is sensible to consider national mortality trends in a larger international 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 e 0 Female Best Practice e0 Iceland Japan 85 Norway NZ (non−maori) Sweden 80 75 70 e0 65 60 55 50 1900 1920 1940 1960 1980 2000 Year

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References Males e 0 Male Best Practice e0 Australia NZ (non−maori) Denmark Norway 85 Iceland Sweden Japan Switzerland Netherlands 80 75 70 e0 65 60 55 50 1900 1920 1940 1960 1980 2000 Year

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References Females e 65 Female Best Practice e65 24 Canada France Iceland Japan Norway 22 NZ (non−maori) Sweden 20 e65 18 16 14 12 1900 1920 1940 1960 1980 2000 Year

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References Males e 65 Male Best Practice e65 24 Australia Denmark Iceland Japan Norway 22 Switzerland 20 e65 18 16 14 12 1900 1920 1940 1960 1980 2000 Year

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References Breakpoints Females Males 50 55 60 65 70 75 80 85 50 55 60 65 70 75 80 85 e0 e0 1900 1920 1940 1960 1980 2000 1900 1920 1940 1960 1980 2000 Females Males 24 24 22 22 20 20 e65 e65 18 18 16 16 14 14 12 12 1900 1920 1940 1960 1980 2000 1900 1920 1940 1960 1980 2000

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References Empirical motivation Female Best Practice e0 Kernel Density and fitted GEV 1.0 86 Observed kernel density Detrended fitted GEV Trend line 84 0.8 82 0.6 80 Density e0 78 0.4 76 0.2 74 0.0 72 1960 1970 1980 1990 2000 2010 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Year N = 58 Bandwidth = 0.1659 Figure: Left panel: raw and detrended data. Right panel: kernel density and fitted GEV distribution.

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References Theoretical motivation Suppose that X 1 , X 2 , . . . , X n is a sequence of independent, identically distributed random variates all having a common distribution function F ( x ). Let M n = max { X 1 , X 2 , . . . , X n } . The distribution of the maxima, M n , converges (for large n) to the Generalized Extreme Value (GEV) Distribution.

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References The Generalized Extreme Value Distribution � − 1 1 + ξ ( z − u � � ξ � G ( z ) = exp − ) σ u is the location parameter σ is the scale parameter ξ is the shape parameter, which determines the tail behaviour ξ > 0: polynomial tail decay and the Fr´ echet Distribution ξ = 0: exponential tail decay and the Gumbel Distribution ξ < 0: bounded upper finite end point and the Weibull Distribution

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References Inference Quantiles Inverting the GEV distribution function: z p = µ − σ � 1 − {− log (1 − p ) } − ξ � , ξ where p is the tail probability and G ( z p ) = 1 − p Return Levels Simply a different way of thinking about the quantiles. If data are annual the (1 − p )th quantile would be exceeded on average once every 1 / p years.

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References Fitted Model GEV ( u t = 59 . 6 + 0 . 24 t , σ = 1 . 31 , ξ = − 0 . 48) Female Best Practice e0 Median 85 50 Y ear Return Level 75 e0 65 55 1900 1940 1980 Y ear

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References Projections, Females e 0 100 90 e0 80 70 Median 95% Conf Ints 60 1900 1950 2000 2050 Year

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References Projections, Females e 0 100 90 e0 80 70 99th Percentile 95% Conf Ints 60 1900 1950 2000 2050 Year

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 are available. P ( e max P ( e max Year > 90) > 95) 0 0 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 . 1.0 Absolute Differences 0.8 0.6 0.4 0.2 1985 1990 1995 2000 Year Mean Absolute Difference(MAD)= 0.67 years Mean Absolute Percentage Error = 0.8%

Outline Introduction Why Extreme Value Theory? The GEV Results ARIMA and the GEV Conclusions References ARIMA model residuals 2 fitted GEV 0.4 0 0.3 Residuals Density −2 0.2 −4 0.1 −6 0.0 −2 −1 0 1 2 −6 −4 −2 0 2 norm quantiles Residuals Figure: Normality tests for residuals of ARIMA(2,1,1) fitted to female e 0 BPLE. 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 be poorly fitting. GEV is more flexible and is able to capture the shape of different error distributions - not just symmetric. In practice Gaussian often provides a reasonable fit but GEV should be considered as an alternative for the innovations process.

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 preferable Or using the GEV as the innovations process in an ARIMA model EVT can identify in an objective way whether life expectancy is actually at an extreme level rather than just ”high” EVT can be used to obtain probabilities and/ or levels of extreme 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. The Linear Rise in Life Expectancy: History and Prospects , Volume III of Social Insurance Studies . Swedish Social Insurance Agency, Stockholm. Oeppen, J. and J. W. Vaupel (2002). Broken limits to life expectancy. Science 296 (5570), 1029–1031. Torri, T. and J. W. Vaupel (2012). Forecasting life expectancy in an international context. International Journal of Forecasting 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 decline converging toward international trends? Population and Development Review , 593–600.