I have a discrete data distribution: (1,1,2,2,1,1). I wish to estimate its Fisher information. How should I proceed? FI is normally inversely proportional to the variance of the data. The variance of the data comes out to be 0.267.
How should I compute FI according to the following equation? $$ I(\theta) = 4 \sum_{i=1}^{n}{{(\sqrt{p_{i+1}}-\sqrt{p_i})}^2} $$
Is your (1,1,2,2,1,1) vector giving observed counts over 6 distinct outcomes, or is it sequence of observations over 2 outcomes?
Either way, you can treat empirical distribution as a multinomial distribution with 1 draw, use observed counts to estimate parameters of that distribution, compute Fisher Information matrix using recipe here -- https://stats.stackexchange.com/a/578715/511