Extremes of Many Random Variables

	Simulated	Exact
Mean	14.69	14.69
5th percentile	10.12	10.13
Median	13.56	13.58
Percentile at P (95%)	22.83	22.86
VoseLargest(object, n, P): the percentile in one call		22.86

The business question

10,000 independent values each follow a Pareto distribution with shape 5 and minimum 2: most are close to the minimum (the mean is 2.5), but the tail is long. Sampling all 10,000 in every iteration to find the largest is slow.

ModelRisk's extreme-value functions sample the largest, the smallest, or the top or bottom few directly, at the cost of a single sample, whatever the number of values. The model shows them all and checks them twice: against the exact distribution of a maximum, and against the largest of 1,000 values sampled one by one.

Who it is for: Anyone who needs the extremes of many variables: the largest claim in a portfolio, the highest of many flood levels, the weakest of many components, the slowest of many tasks.
Main outputs: The largest of 10,000: simulated, its percentiles and mean against the exact formula, and its 95th percentile in one call · the six largest and six smallest, in order, and their totals · the smallest and largest together · the largest of 1,000 sampled one by one against VoseLargest.

How the model works

Four sheets after this one, read from left to right. Each one feeds the next.

Inputs sheet
The Pareto's shape and minimum, the number of values (10,000), and the probability at which to read the largest (95%).
Extremes sheet
VoseLargest samples the largest of the 10,000 values. With a third argument it returns a percentile of the largest instead. VoseLargestSet and VoseSmallestSet give the six largest and six smallest, VoseExtremeRange the smallest and largest together. Exact percentiles and the exact mean of the largest are beside them.
Brute force sheet
1,000 cells, each a value from the same Pareto, and their maximum, against VoseLargest of 1,000.
Results sheet
The largest's distribution, simulated and exact; the sets; the brute-force check.

ModelRisk techniques

Pareto object VoseParetoObject
Extremes VoseLargest, VoseSmallest
Extreme sets VoseExtremeRange
Brute-force check
Exact maximum, F(x)ⁿ

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The results in detail

The 6 largest and 6 smallest

Mean of each rank in the simulation.

Rank	Largest set	Smallest set
1	14.64	2.00004
2	11.74	2.00008
3	10.56	2.00012
4	9.86	2.00016
5	9.36	2.00020
6	9.00	2.00024
Total	65.15	12.00084

Brute force: the largest of 1,000

1,000 values sampled one by one against VoseLargest, and exact.

The largest of 1,000, two ways

S-curve (cumulative probability) of One by one, VoseLargest from the saved simulation.

	One by one	VoseLargest	Exact
Mean	9.37	9.26	9.27
Median	8.60	8.57	8.57
95th percentile	14.73	14.39	14.42

How to read these results

Most of the 10,000 values are near the minimum of 2 (the mean is 2.5), but the largest is typically about 13.6 (median) and exceeds 22.9 one time in 20: the extreme of a heavy-tailed variable is far from the typical value.
VoseLargest samples the largest directly at the cost of one sample, whatever n. Sampling the values one by one and taking the maximum gives practically the same distribution (within about 1% here, with 1,000 values) but costs n samples per iteration; VoseLargest also matches the exact formula.
With a third argument U, VoseLargest returns a percentile of the largest instead of a sample: VoseLargest(object, 10,000, 0.95) is 22.86, exactly a ÷ (1 − 0.95^(1/n))^(1/θ). VoseSmallest works the same way.
The smallest values are pressed against the minimum: the six smallest of 10,000 average 2.00004 to 2.00024. VoseLargestSet, VoseSmallestSet and VoseExtremeRange give several extremes of the same set at once, in order.

See it move. Open the workbook, change an input on the Inputs sheet and run the simulation: every table and chart on this page updates in Excel.

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Inputs and assumptions

Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.

The distribution 2 inputs

Every value follows the same Pareto.

Assumption	Unit	Value
Pareto shape θLarger θ: a thinner tail		5.0
Pareto minimum aNo value is below it	value	2.0

How many, and where to read the largest 2 inputs

Assumption	Unit	Value
Number of independent values	values	10000
Probability for the percentile of the largest		95%

Units and conventions

Pareto(θ, a): Shape θ, minimum a: P(X > x) = (a ÷ x)^θ for x ≥ a; mean θa ÷ (θ − 1).
Rank: 1 = the largest (or the smallest) of the set.

Key assumptions and limitations

Independence: The 10,000 values are independent and identically distributed.
Data: The distribution, the 10,000 and the 95% are those of ModelRisk's earlier Pareto Extremes example. The set and range functions follow its Extremes example. Its units ($ billion) are dropped: the values are left unitless.

How to use it

Set the distribution and the number of values on the Inputs sheet (any ModelRisk distribution object works in the extreme functions).
Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
Use VoseLargest or VoseSmallest wherever a model needs the extreme of many identical variables.

ModelRisk functions used

VoseParetoObject: The Pareto distribution as an object, for the extreme functions.
VoseLargest, VoseSmallest: The largest or smallest of n values; with U, its U-th percentile.
VoseLargestSet, VoseSmallestSet: The k largest (descending) or smallest (ascending) of n values.
VoseExtremeRange: The smallest and largest of n values, from the same sample.
VoseSimulate: One value from the object, for the brute-force cells.

Related example models

All example models

An illustrative example: the figures are invented for teaching and do not describe a real company, project or asset. Last updated 9 October 2026.

The largest of 10,000