Calculate the Gumbel \(p\)-value for a scan statistic.

Given an observed scan statistic \(\lambda^*\) and a vector of replicate scan statistics \(\lambda_i\), \(i=1,\ldots,R\), fit a Gumbel distribution to the replicates and calculate a \(p\)-value for the observed statistic based on the fitted distribution. $$ \frac{1 + \sum_{i=1}^R \mathrm{I}(\lambda_i > \lambda^*)}{1 + R} $$ The function is vectorized, so multiple \(p\)-values can be calculated if several scan statistics (e.g. statistics from secondary clusters) are supplied.

Usage

gumbel_pvalue(observed, replicates, method = "ML", ...)

Arguments

observed

A scalar containing the observed value of the scan statistic, or a vector of observed values from secondary clusters.

replicates

A vector of Monte Carlo replicates of the scan statistic.

method

Either "ML", for maximum likelihood, or "MoM", for method of moments.

...

Additional arguments passed to ismev::gum.fit, which may include arguments passed along further to optim.

Value

The \(p\)-value or \(p\)-values corresponding to the observed scan statistic(s).