How are the Normal importance-sampling proposal parameters chosen from the retained $g^*$ samples?

Conditions

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

Reasoning, step by step

1. Compute the sample mean of the retained $g^*$ samples to estimate $\mu$.
2. Compute the sample root second central moment of the retained samples to estimate $\sigma$.

Example

The code computes `mu = np.mean(Xs)` and `sigma = np.sqrt(np.mean((Xs-mu)**2))`, and the narrator describes this as finding maximum likelihood estimates for $\mu$ and $\sigma$ given the samples.

Common misconceptions

• Assuming the parameters are chosen randomly.
• Thinking the standard deviation is calculated using an unbiased sample variance (dividing by $n-1$) rather than the sample second central moment (dividing by $n$).

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Explore next

• What is the final importance-sampling estimator for $P(X>\gamma)$?
• How is the reported confidence interval constructed from the importance-sampling estimates?