Concept Architecture
Concept
Theoretically, Extreme Value Analysis is a branch of statistical inference concerned with modelling the probability and magnitude of rare or extreme observations occurring in the tails of a probability distribution. It is founded on Extreme Value Theory, which provides asymptotic models for the behaviour of maxima, minima and threshold exceedances. In health economics, extreme value analysis is used to investigate unusually high healthcare costs, catastrophic expenditure, rare adverse events and other low-frequency outcomes that may substantially influence economic evaluations.
Mathematically, extreme value analysis models either block maxima using the Generalised Extreme Value (GEV) distribution or observations exceeding a high threshold using the Generalised Pareto Distribution (GPD). Model parameters are typically estimated by maximum likelihood estimation, allowing estimation of tail probabilities, return levels and the likelihood of future extreme events.
In practice, analysts first identify an appropriate threshold or block structure before fitting an extreme value model and evaluating model adequacy using diagnostic plots and goodness-of-fit measures. Within health economics, these models are applied to quantify financial risk, evaluate catastrophic healthcare costs, estimate rare-event frequencies and support scenario analyses involving extreme but plausible outcomes.
Purpose
Used to quantify the probability and magnitude of rare or extreme outcomes, enabling robust assessment of catastrophic costs, infrequent adverse events and tail risks within health economic evaluations.
Mathematical Formulae
Primary Formula
Generalised Extreme Value distribution:
F(x) = exp{?[1 + ?((x ? ?)/�)]^(?1/?)}
where:
- ? = location parameter
- � > 0 = scale parameter
- ? = shape parameter
Supporting Formulae
Generalised Pareto Distribution:
F(x) = 1 ? [1 + ?(x/�)]^(?1/?)
for x > threshold.
Log-likelihood:
ln(L) = ?ln(f(x? | ?, �, ?))
Return level:
z? = ? + (�/?)[{?ln(1 ? p)}^(??) ? 1]
Related Mathematical Methods
- Extreme Value Theory
- Generalised Extreme Value Distribution
- Generalised Pareto Distribution
- Maximum Likelihood Estimation
- Peak-Over-Threshold Analysis
- Block Maxima Analysis
Example
A health insurer analyses annual inpatient costs to estimate the probability of catastrophic claims.
The largest annual claims are modelled using a Generalised Extreme Value distribution with estimated parameters:
? = �82,000
� = �18,500
? = 0.12
The fitted model estimates that the probability of an annual claim exceeding �150,000 is approximately 1.8%, allowing the insurer to incorporate extreme-cost scenarios into its economic risk assessment.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MAX | =MAX(B2:B1001) | Identify block maxima for extreme value modelling |
| LARGE | =LARGE(B2:B1001,1) | Extract extreme observations |
| PERCENTILE.INC | =PERCENTILE.INC(B2:B1001,0.95) | Select a threshold for peak-over-threshold analysis |
| LN | =LN(B2) | Calculate log-likelihood components |
| Solver | Maximise the log-likelihood by varying ?, � and ? | Estimate GEV or GPD parameters |
VBA (Optional)
A VBA procedure can automate identification of extreme observations, estimate extreme value model parameters using Solver and generate return-level summaries for health economic risk analyses.
Sources
- Coles S. An Introduction to Statistical Modelling of Extreme Values.
- Embrechts P, Kl�ppelberg C, Mikosch T. Modelling Extremal Events for Insurance and Finance.
- de Haan L, Ferreira A. Extreme Value Theory: An Introduction.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- ISPOR Good Practice Reports.
Related Concepts (2)
Library
Tools & Resources
1
SAVI — Sheffield Accelerated Value of Information — Mark Strong, Jeremy Oakley & Penny Breeze (University of Sheffield), Web application ed., 2024 (University of Sheffield)
A free, open-access web calculator that computes value-of-information measures (EVPI, partial EVPI/EVPPI and EVSI) directly from a model’s probabilistic sensitivity analysis output — no need to re-run the model. Also reports payer strategy-specific and uncertainty burden.
Web Tool (R Shiny)View source →
Frequently Asked Questions (6)
What is extreme value analysis?
A statistical approach characterising the behaviour of the most extreme, rather than typical, values a variable can take, based on extreme value distribution theory.
Source: Gumbel 1958
Why can ordinary statistics mislead about extreme events?
Ordinary statistical methods describe the typical behaviour of a variable, its average and usual spread, and say little about the rare, extreme values in the far tail of its distribution. Yet it is often those extremes, such as an unusually severe outbreak or a very large cost, that matter most for planning. Extreme value analysis focuses on the tail directly, using distributions derived to describe maxima and minima, so that the frequency of rare events can be estimated rather than assumed. It fills the gap ordinary analysis leaves. Coles (2001) sets out the theory.
Source: Coles 2001
What distributions arise in extreme value analysis?
Extreme value analysis rests on the result that the maxima or minima of samples tend, under broad conditions, to follow one of a limited family of extreme value distributions, encompassed by the generalised extreme value distribution with its Gumbel, Frechet, and Weibull forms. Which form applies depends on the tail behaviour of the underlying variable. These distributions describe the extremes rather than the whole, so fitting them to observed maxima or exceedances allows the probabilities of extreme values to be estimated.
Source: Gumbel 1958
Why is extreme value analysis used?
Extreme value analysis is used when the interest lies in rare, extreme outcomes rather than typical values, since ordinary distributions fitted to the bulk of the data can describe the tails poorly, and extreme events may carry disproportionate consequences. By modelling the extremes directly with the appropriate limiting distributions, the approach estimates the probability and size of rare values more reliably. It is applied in fields concerned with rare large events, and where understanding worst-case or best-case magnitudes of a variable matters.
Source: Gumbel 1958
How does extreme value analysis differ from ordinary statistical analysis?
Extreme value analysis focuses on the tails of a distribution, the largest or smallest values, and uses distributions suited to extremes, whereas ordinary statistical analysis usually describes central tendency and the bulk of the data with distributions fitted to typical values. A model that fits the centre well may misrepresent the tails, so extreme value analysis treats the extremes specifically. This difference in focus, extremes versus typical values, means the two use different distributions and estimation targeted at their respective parts of the range.
Source: Gumbel 1958
What are the limitations of extreme value analysis?
Extreme value analysis relies on limited data about rare events, since by definition extremes are scarce, so estimates of very rare values carry wide uncertainty and depend heavily on the assumed tail distribution. Extrapolating beyond the observed range to rarer events is uncertain, and the choice of how to define extremes, such as block maxima or exceedances over a threshold, affects results. These limitations mean extreme value estimates are treated with caution, with their sensitivity to assumptions and data scarcity acknowledged.
Source: Gumbel 1958
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 27 Oct 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-EM-UA-020
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