What would you like to understand?

The parameters of the Normal proposal distribution are chosen using maximum likelihood estimation. The mean $\mu$ is set to the sample mean of the retained $g^*$ samples, and the standard deviation $\sigma$ is set to the square root of the sample second central moment of those samples.

Conditions: The proposal family is restricted to Normal distributions.; The data used for fitting are the retained samples satisfying $X>\gamma$.

The parameters of the Normal proposal distribution are chosen using maximum likelihood estimation. The mean $\mu$ is set to the sample mean of the retained $g^*$ samples, and the standard deviation $\sigma$ is set to the square root of the sample second central moment of those samples.

Conditions: The proposal family is restricted to Normal distributions.; The data used for fitting are the retained samples satisfying $X>\gamma$.

The zero-variance density $g^*$ is the conditional distribution of the random variable $X$ given that the rare event $X>\gamma$ occurs. It is the ideal importance-sampling proposal that would make the estimator have zero variance.

Conditions: The target quantity is the rare-event probability $P(X>\gamma)$.; $g^*$ is supported only on the region $x>\gamma$.

The zero-variance density $g^*$ is the conditional distribution of the random variable $X$ given that the rare event $X>\gamma$ occurs. It is the ideal importance-sampling proposal that would make the estimator have zero variance.

Conditions: The target quantity is the rare-event probability $P(X>\gamma)$.; $g^*$ is supported only on the region $x>\gamma$.

Rejection sampling is used because $g^*$ is exactly the conditional distribution of $X$ given $X>\gamma$. By drawing many samples from the original standard normal distribution and keeping only those that exceed the threshold $\gamma$, the retained samples are distributed according to $g^*$.

Conditions: Direct sampling from the original distribution $X \sim \mathrm{Normal}(0,1)$ is possible.; The acceptance rule is $X_i > \gamma$.

Rejection sampling is used because $g^*$ is exactly the conditional distribution of $X$ given $X>\gamma$. By drawing many samples from the original standard normal distribution and keeping only those that exceed the threshold $\gamma$, the retained samples are distributed according to $g^*$.

Conditions: Direct sampling from the original distribution $X \sim \mathrm{Normal}(0,1)$ is possible.; The acceptance rule is $X_i > \gamma$.

The final estimator is the sample mean of the likelihood ratios multiplied by the rare-event indicator. It is computed by drawing samples from the fitted Normal proposal, calculating the ratio of the standard normal density to the proposal density, multiplying by the indicator that the sample exceeds $\gamma$, and averaging the results.

Conditions: Samples are drawn from the fitted Normal proposal, not the original distribution.; The event indicator is $\mathbf{1}_{\{Z_j>\gamma\}}$.

The final estimator is the sample mean of the likelihood ratios multiplied by the rare-event indicator. It is computed by drawing samples from the fitted Normal proposal, calculating the ratio of the standard normal density to the proposal density, multiplying by the indicator that the sample exceeds $\gamma$, and averaging the results.

Conditions: Samples are drawn from the fitted Normal proposal, not the original distribution.; The event indicator is $\mathbf{1}_{\{Z_j>\gamma\}}$.