Empirical Likelihood

Ordinary maximum likelihood needs you to pick a distribution — normal, exponential, whatever — and then finds the parameters that make the observed data most probable under that assumed shape. Empirical likelihood keeps the same spirit of "how probable is this data" but refuses to assume any shape at all. Instead, it puts an unknown probability weight on each observed data point itself and asks: what weights make the sample most likely to have occurred, subject to some constraint you care about (like "the mean equals a candidate value")? The data effectively supplies its own distribution.

Think of it like a weighted vote among the data points you already have, rather than fitting a smooth curve through them. If you want to test whether the true mean return of a strategy could plausibly be zero, empirical likelihood finds the weights on the observed daily returns that are as close to equal (as close to "no information added") as possible while still forcing the weighted average to equal zero, and measures how much that forcing costs in likelihood. A candidate value that requires very lopsided weights to satisfy is rejected; one reachable with nearly equal weights is not.

The resulting test statistic behaves like a chi-squared statistic asymptotically (this is Owen's theorem), so confidence intervals can be built in a familiar way, but without ever assuming returns are normal, symmetric, or belong to any named family — useful for financial returns, which are typically fat-tailed and skewed, where a standard t-test's normality assumption is shaky.