Modeling Expected Shortfall Using Tail Entropy

. 2019 Dec 7;21(12):1204. doi: 10.3390/e21121204

Algorithm 1. Estimation of the entropy of a probability density function.

Estimate the probability density function, obtaining values ${\hat{f}}_{n} (X_{i})$ for $i = 0, \dots n - 1$ ;
Sample from the probability density function using the sampled function $S_{n} ({\hat{f}}_{n}) (i) = {\hat{f}}_{n} (X_{i})$ for $i = 0, \dots n - 1$ ;
Define a quantum $q > 0$ ; then $Q_{q} S_{n} ({\hat{f}}_{n}) (j) = (i + 1 / 2) q$ , if ${\hat{f}}_{n} (X_{j}) \in [i q, (i + 1) q)$ ;
Compute the probabilities $p_{n} (i) = \frac{c_{n} (i)}{\sum_{j} c_{n} (j)} = \frac{c_{n} (i)}{n} = \frac{c a r d {{\hat{f}}_{n} (X_{j}) \in [i q, (i + 1) q)}}{n}$ ;
Estimate the entropy of the probability density function, i.e., $H_{q} ({\hat{f}}_{n}) = - \sum_{i} p_{n} (i) \log_{2} p_{n} (i)$