Informative selection mechanisms for extreme value analyses

# Informative selection mechanisms for extreme value analyses

**EVA 2021**

.light[Léo Belzile, HEC Montréal]
<br>
Based on joint work with Anthony Davison, Jutta Gampe, Holger Rootzén, Dmitrii Zholud
]

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# Vargas catastrophe

Daily precipitation (in mm) at Maiquetía airport

- Yearly maximum for 1951-1960
- Daily measurements for 1961-1999

1999 catastrophe: cumulative rainfall of 820mm over 3 days (backcast)

- largest record: ~410mm mid-December.
- measurements stop shortly after.
- Accounting (or not) for this observation has a huge impact on inference.

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# Inferential strategy

- Poisson point process formulation: combine yearly maximum with exceedances and stopping rule

- View the largest record as a
 .section-title-1[**stopping rule** ] : set to `$s$`=282mm, the largest two-day sum previously reported in early 1950s

- Observations before trigger are *right-truncated*, the largest record is *left-truncated* at `$s$`.

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# Results

<div class="figure" style="text-align: center">
<img src="EVA2021-Belzile_files/figure-html/maiquetia-1.png" alt="Profile likelihood for standard analysis and conditioning on stopping rule (left) and P-P plot of model fitted above 27mm." width="82%" />
<p class="caption">Profile likelihood for standard analysis and conditioning on stopping rule (left) and P-P plot of model fitted above 27mm.</p>
</div>

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class: title title-1 title-1-inv

# How rare was this event?

Q: was the event much larger than one could expect?

Target: median of 50-years maximum

Consider one-sided likelihood ratio test

- `$H_0$` : median 50-year maximum > 410.4mm,

- `$p$`-values 9.5% (unconditional) versus 1.9% (conditional)

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# Estimating human lifespan

The study of longevity is full of pitfalls...

The quality of the data is particularly important.

.box-6.medium.sp-after-half[International Database on Longevity (IDL)]

- .section-title-2[ **validated supercentenarian (110+) **] from 13 countries

- plus (partly validated) .section-title-2[** semi-supercentenarian (105-109)**] for 9 countries

- Age-ascertainement bias-free

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class: title title-2

# Sampling mechanisms

Data are obtained by .section-title-2[**casting a net**] on the population of potential (semi)-supercentenarians.

- for IDL, (only) supercentenarians in a country **who died** between dates `$c_1$` and `$c_2$`.

- records for the candidates are then individually validated.

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.box-6.medium.sp-after-half[Lexis diagrams]

<div class="figure" style="text-align: center">
<img src="EVA2021-Belzile_files/figure-html/lexisdiag-1.png" alt="Lexis diagrams showing the selection mechanism." width="82%" />
<p class="caption">Lexis diagrams showing the selection mechanism.</p>
</div>

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.box-6.medium.sp-after-half[More complex schemes!]

Semisupercentenarians (105-109) who died in window `$(d_1, d_2)\neq (c_1, c_2)$`

<div class="figure" style="text-align: center">
<img src="EVA2021-Belzile_files/figure-html/lexisdiag2-1.png" alt="Lexis diagrams for IDL data with semisupercentenarian and supercentenarians" width="85%" />
<p class="caption">Lexis diagrams for IDL data with semisupercentenarian and supercentenarians</p>
</div>

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# Why does it matter?

- .small[Ignoring truncation leads here to underestimation of the survival probability]
- .small[Because of the population increase and reduction in mortality at lower age, larger impact for later cohorts]

<div class="figure" style="text-align: center">
<img src="EVA2021-Belzile_files/figure-html/artefacts_truncation-1.png" alt="Impact of truncation on quantile-quantile plots (left) and maximum age by birth year (right)." width="65%" />
<p class="caption">Impact of truncation on quantile-quantile plots (left) and maximum age by birth year (right).</p>
</div>

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# Complications ahead

Accounting for interval truncation.

- e.g., the plotting position for `$x$`-axis of Q-Q plot for observation `$y_i$` is
$$
F_0^{-1}\left[F_0(a_i) +\left\\{ F_0(b_i)-F_0(a_i) \right\\} \frac{F_n(y_i) - F_n(a_i)}{F_n(b_i)-F_n(a_i)}\right]
$$
where
- `$F_0$` is the postulated (i.e., fitted) parametric distribution,
- `$F_0^{-1}$` is the corresponding quantile function,
- `$F_n$` is the NPMLE of the distribution function (Turnbull, 1976).

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# Truncation can be hidden

- .section-title-2[ Extinct cohort ]: Birth cohorts for which no death has been reported for X consecutive years.
- counts cross-tabulated by years of birth, age (rounded down to nearest year) and gender

<div class="figure" style="text-align: center">
<img src="figures/Sibuya_tabular.png" alt="Annual Vital Statistics Report of Japan (Hanamaya &amp; Sibuya, 2014)." width="55%" />
<p class="caption">Annual Vital Statistics Report of Japan (Hanamaya & Sibuya, 2014).</p>
</div>

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class: title title-2
# Impacts of interval censoring

Japanese (unvalidated) data are interval-censored and right-truncated

<div class="figure" style="text-align: center">
<img src="EVA2021-Belzile_files/figure-html/japanese-1.png" alt="Profile likelihood ratio confidence intervals by threshold (left) and sampling distribution with(out) rounding (right)." width="65%" />
<p class="caption">Profile likelihood ratio confidence intervals by threshold (left) and sampling distribution with(out) rounding (right).</p>
</div>

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name: outline
class: title title-2

# Lack of (threshold) stability

- .small[The hazard doesn't stabilize until 106 years or so.]

<div class="figure" style="text-align: center">
<img src="EVA2021-Belzile_files/figure-html/thstab-1.png" alt="Threshold stability plots for France and Italy (left), and Netherlands (right)." width="60%" />
<p class="caption">Threshold stability plots for France and Italy (left), and Netherlands (right).</p>
</div>
Take home message: one cannot use low thresholds for extrapolation.

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# Is the lifetime distribution bounded?

Mathematically speaking, is `$t_F=\sup\{t: F(t)<1\} = \infty$`?

<div class="figure" style="text-align: center">
<img src="EVA2021-Belzile_files/figure-html/profile_lifetime-1.png" alt="Profile likelihood for endpoint for various countries and three thresholds." width="80%" />
<p class="caption">Profile likelihood for endpoint for various countries and three thresholds.</p>
</div>

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# You have no power in here!

The power of a likelihood ratio test for detecting a finite endpoint (obtained by simulating records with a generalized Pareto distribution with lifespan `$t_F$`) is high: based on France/Italy/IDL data (2016 version),

- 125 years: combined power of 97%;
- 130 years: combined power of 83%;
- 135 years: combined power of 66%.

Suggests that the human lifespan lies well beyond any lifetime yet observed
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# Supercentenarians [don't] live forever...

- Estimated exponential distribution above 110 years has mean 0.5 (0.46, 0.53)

- Surviving until 130 years conditional on surviving until 110 years
  - is equivalent to obtaining 20 heads in a row
  - a less than one-in-a-million chance...

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# References

- Léo R. Belzile, Anthony C. Davison, Jutta Gampe, Holger Rootzén and Dmitrii Zholud (2022). Is there a cap on longevity? A statistical review., Annual Reviews of Statistics and its Applications.
- Léo R. Belzile and Anthony C. Davison (2020). Improved inference for risk measures of univariate extremes (2021+), arXiv:2007.10780 preprint.
- Léo R. Belzile, Anthony C. Davison, Holger Rootzén and Dmitrii Zholud (2021+). Human mortality at extreme age., arXiv:2001.04507 preprint.