Exercise: Effective sample size and weight diagnostics

The effective sample size (ESS) for a weighted IS estimator with weights $w_1, \dots, w_N$ is

It estimates "how many equivalent unweighted samples" you have.

Tasks

1. Show that $N_{\text{eff}} = N$ exactly when all weights are equal, and $N_{\text{eff}} = 1$ when one weight equals $\sum w_i$ and others are zero (extreme case).

2. Bound on the variance. Prove that the standard error of the IS estimator is bounded approximately by $\sigma_f / \sqrt{N_{\text{eff}}}$ where $\sigma_f$ is the (typical) standard deviation of $f$ on the support of $q$. Use the Cauchy-Schwarz argument.
3. Diagnostic threshold. Practitioners flag IS results as unreliable when $N_{\text{eff}}/N < 0.05$ (less than 5% effective). Why is this a useful threshold?
4. Numerical example. For the deep OTM call $K = 200$ from the previous exercise, compute $N_{\text{eff}}/N$ for $\mu \in \{0, z^*, z^* + 1, z^* + 3\}$ where $z^* \approx 3.22$. At which $\mu$ is the IS most efficient?