Monte Carlo Sample Size for Estimating Rare Event Probabilities

Summary

The document asks how to choose a Monte Carlo sample size for estimating a Bernoulli mean to a specified absolute precision. It presents the central limit theorem formula for sample size, using a normal quantile, the outcome variance, and the squared error tolerance. Substituting Bernoulli variance, p(1-p), produces smaller sample estimates as the event probability falls across the examples shown.

The apparent paradox comes from the precision measure: the calculation targets the same absolute error for every probability, not the same relative error. An estimate near zero can have a small absolute deviation while still being far off in percentage terms. The post contains no accepted answer or resolution, but its setup highlights why a rare-event simulation may need far more draws when the goal is reliable relative accuracy or a useful chance of observing enough successes. The displayed calculation also relies on a normal approximation, whose adequacy can be limited when the expected number of successes is small.

Key ideas

• The sample-size expression uses the variance of the simulated outcome and the square of the absolute error tolerance.
• For a Bernoulli outcome, variance depends on the success probability as p times one minus p.
• The examples target fixed absolute precision, which does not guarantee fixed relative precision across probabilities.
• Rare events can require many simulations when the goal is to observe enough successes for stable estimation.
• The central limit theorem approximation may be unreliable when the expected success count is small.

Tags

# Monte Carlo convergence sample size


# Monte Carlo convergence sample size












I'm studying Monte Carlo analysis but I find very counter-intuitive the computation of the minimum sample size in order to reach a certain level of precision. As stated in `Montecarlo methods in Financial Engineering`, such sample size comes from the CLT and it can be computed according to

$n = \frac{z^2_{\delta/2} \sigma^2}{\epsilon^2}$

My experiment is the classic bernoulli variable, with $p$ probability of success. Suppose my level of precision is $0.01$, then my $z_{0.01/2} = 2.57$.

I consider three cases.

$p=0.1\%$

$p=1\%$

$p=10\%$

With $\sigma^2 = p(1-p)$, I get

$n_{10\%} = 2.57^2 \cdot 0.1 \cdot 0.9 / 0.01^2 = 5944$

$n_{1\%} = 2.57^2 \cdot 0.01 \cdot 0.99 / 0.01^2 = 654$

$n_{0.1\%} = 2.57^2 \cdot 0.001 \cdot 0.999 / 0.01^2 = 66$

Intuitevely I would think the rarer the event, the greater the number of the simulations needed in order to approach the theoretical mean. What am I getting wrong?

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