A modal age at death approach to forecasting adult mortality

Figures & data

Figure 1 (a) Modal age at death, M, and (b) age-at-death distribution around the mode, D: males in France, 1965–2019 Source: Authors’ analysis of data from HMD (2022).

Figure 2 Observed and forecasted model parameters for the (a) modal age at death, M; (b) time index, ${\kappa }_t$; and (c) age pattern, ${\text{β}}_{x-M}$: males and females in France, 1965–2050

Source: As for Figure 1.

Table 1 Mean forecasting errors across 84 out-of-sample forecasts for life expectancy at age 40 (e₄₀), prediction interval accuracy for life expectancy at age 40, and mean forecasting errors in modal age at death (M), lifespan variation (e^†), and weighted logged death rates (In(m_x)), from the Mode, STAD, CoDA, FDA, and LC models: males and females in 10 West European countries

Males
	CHE	DNK	ESP	FIN	FRA	IRL	NLD	NOR	PRT	SWE	Mean (rank)
$e_{40}$
Mode	0.41	1.57	0.54	0.31	0.42	1.18	1.31	1.08	0.60	0.48	0.79 (1)
STAD	1.42	1.74	0.82	0.42	1.62	1.37	1.31	1.34	0.68	0.73	1.15 (4)
CoDA	0.75	1.74	0.52	0.54	0.45	1.98	1.68	1.44	1.24	0.96	1.13 (3)
FDA	0.92	1.59	1.62	0.59	1.00	1.46	1.80	1.11	1.28	1.05	1.24 (5)
LC	1.17	1.59	0.66	0.60	0.92	1.33	1.38	1.28	0.99	1.27	1.12 (2)
95 per cent prediction interval
Mode	99.2	93.3	99.9	100.0	98.8	98.0	68.6	83.8	99.9	99.0	87.3 (2)
STAD	52.9	55.1	69.9	96.6	56.9	69.1	60.0	57.2	80.0	84.6	68.2 (5)
CoDA	98.4	80.1	97.8	99.9	99.8	86.4	67.4	81.8	98.4	99.0	90.9 (1)
FDA	90.7	67.8	76.7	100.0	82.1	73.6	51.3	81.9	99.7	85.4	80.9 (3)
LC	89.2	39.2	94.7	93.8	96.1	64.7	60.9	49.9	94.7	88.3	77.1 (4)
M
Mode	0.42	2.23	0.71	0.79	0.34	1.96	1.77	1.43	0.90	0.61	1.12 (1)
STAD	1.53	2.93	1.17	1.12	1.59	2.56	2.03	1.95	1.32	1.11	1.73 (2)
CoDA	1.43	3.02	1.10	1.64	0.78	3.13	2.87	2.40	2.09	1.72	2.01 (4)
FDA	1.38	2.85	2.41	1.73	1.63	3.09	3.03	2.10	2.16	1.66	2.20 (5)
LC	1.73	2.80	1.22	1.76	1.51	2.86	2.45	2.07	1.65	1.84	1.99 (3)
$e^{†}$
Mode	0.19	0.17	0.29	0.29	0.23	0.18	0.12	0.19	0.11	0.10	0.19 (4)
STAD	0.38	0.37	0.51	0.36	0.39	0.17	0.11	0.16	0.18	0.16	0.28 (5)
CoDA	0.16	0.32	0.13	0.17	0.14	0.14	0.18	0.17	0.08	0.17	0.17 (2)
FDA	0.16	0.21	0.24	0.18	0.12	0.29	0.17	0.21	0.08	0.14	0.18 (3)
LC	0.19	0.21	0.15	0.22	0.10	0.22	0.08	0.12	0.08	0.19	0.16 (1)
$\ln \left(m_x\right)$
Mode	0.86	1.86	0.82	1.07	0.78	1.75	1.66	1.60	0.98	0.84	1.22 (1)
STAD	2.08	2.15	1.30	1.52	2.02	2.34	1.85	1.85	1.29	1.27	1.77 (5)
CoDA	1.05	1.78	0.78	1.09	0.75	2.27	1.73	1.66	1.46	1.15	1.37 (2)
FDA	1.24	1.85	1.65	1.24	1.13	2.19	1.88	1.62	1.55	1.25	1.56 (4)
LC	1.53	1.82	0.97	1.36	1.14	2.14	1.61	1.72	1.42	1.46	1.52 (3)
Females
	CHE	DNK	ESP	FIN	FRA	IRL	NLD	NOR	PRT	SWE	Mean (rank)
$e_{40}$
Mode	0.53	1.06	0.50	0.40	0.60	0.65	0.37	0.36	0.50	0.29	0.53 (1)
STAD	0.43	1.30	0.78	0.40	0.65	1.49	0.42	1.40	1.22	0.19	0.83 (5)
CoDA	1.11	1.34	0.45	0.68	0.85	1.22	0.53	0.71	1.03	0.28	0.82 (4)
FDA	0.31	1.44	0.38	0.29	0.39	1.02	0.65	1.03	0.56	0.17	0.62 (2)
LC	1.21	1.38	0.33	0.31	0.41	0.89	0.54	0.57	1.67	0.20	0.75 (3)
95 per cent prediction interval
Mode	99.0	91.5	95.2	100.0	92.9	99.8	99.8	100.0	100.0	99.7	97.8 (2)
STAD	99.9	65.0	96.5	99.8	99.3	92.8	91.6	48.1	93.3	99.7	88.6 (5)
CoDA	98.9	99.4	100.0	100.0	100.0	100.0	100.0	100.0	100.0	100.0	99.8 (4)
FDA	98.8	79.4	100.0	100.0	98.2	83.1	93.8	97.8	99.8	100.0	95.1 (1)
LC	98.2	62.2	99.3	99.9	96.5	78.4	93.7	99.3	93.3	100.0	92.1 (3)
M
Mode	0.68	0.50	0.52	0.45	0.52	1.01	0.38	0.22	0.51	0.44	0.52 (1)
STAD	0.46	0.86	0.77	0.46	0.38	2.32	0.54	1.61	1.02	0.36	0.88 (5)
CoDA	0.98	0.81	0.34	0.54	0.39	1.65	0.53	0.78	1.08	0.22	0.73 (4)
FDA	0.25	0.78	0.45	0.47	0.41	1.14	0.64	1.02	0.57	0.19	0.59 (2)
LC	1.05	0.71	0.38	0.42	0.32	1.12	0.49	0.54	1.48	0.21	0.67 (3)
$e^{†}$
Mode	0.27	0.50	0.10	0.15	0.09	0.15	0.22	0.19	0.22	0.15	0.20 (3)
STAD	0.18	0.57	0.16	0.15	0.13	0.11	0.28	0.29	0.23	0.21	0.23 (4)
CoDA	0.27	0.51	0.33	0.19	0.33	0.16	0.15	0.17	0.17	0.08	0.24 (5)
FDA	0.10	0.54	0.18	0.13	0.15	0.12	0.19	0.22	0.10	0.08	0.18 (1)
LC	0.25	0.45	0.21	0.15	0.15	0.12	0.13	0.12	0.21	0.08	0.19 (2)
$\ln \left(m_x\right)$
Mode	1.08	1.31	0.82	1.04	0.87	1.24	0.75	0.91	0.93	0.74	0.96 (1)
STAD	1.16	1.59	1.10	1.41	0.92	2.34	0.94	2.09	1.73	0.88	1.42 (5)
CoDA	1.37	1.46	0.88	1.20	1.10	1.64	0.80	1.09	1.31	0.68	1.15 (3)
FDA	0.91	1.59	0.86	1.16	0.78	1.65	0.96	1.48	1.00	0.73	1.11 (2)
LC	1.48	1.59	0.85	1.30	0.79	1.69	0.92	1.23	1.92	0.82	1.26 (4)

Note: Values in bold indicate the model providing the most accurate forecasts. Mean forecasting errors are across the 84 out-of-sample forecasts for each country. The final column shows the mean errors across all countries combined.

Source: Authors’ analysis of data from HMD (2022).

Figure 3 (a) Life expectancy at age 40, $e_{40}$; (b) modal age at death, M; and (c) lifespan variation, $e^{†}$, observed and forecasted with the Mode and Lee–Carter models: males and females in France, 1965–2050

Note: The shaded areas represent the 95 per cent prediction intervals.

Source: As for Figure 1.

Figure 4 (a) Life expectancy at age 40, $e_{40}$; (b) modal age at death, M; and (c) lifespan variation, $e^{†}$, observed and forecasted with the Mode model: males and females in 10 West European countries, 1965–2050

Note: This figure is best viewed online in colour.

Source: As for Figure 1.

HMD. 2022. Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany), Accessed on 6 September. www.mortality.org.

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