Empirical likelihood is a statistical method that is used to estimate the likelihood of a given set of observations. It is often used when the underlying distribution of the data is unknown or when the standard methods of likelihood estimation are not applicable.

One example of empirical likelihood is in the analysis of survey data. Suppose that a researcher is interested in estimating the proportion of individuals in a population who have a certain characteristic, such as having a college degree. In a survey, the researcher collects data on the education level of a random sample of individuals from the population.

To estimate the proportion of individuals with a college degree using empirical likelihood, the researcher first calculates the observed proportion of individuals with a college degree in the sample. This is the empirical likelihood estimate of the population proportion.

The researcher then uses this estimate to construct a confidence interval around the estimate. This confidence interval represents the range of values that the population proportion is likely to fall within, given the observed data.

Another example of empirical likelihood is in the analysis of regression models. Suppose that a researcher is interested in estimating the relationship between two variables, such as income and education level. In a study, the researcher collects data on the income and education level of a sample of individuals from the population.

To estimate the relationship between income and education level using empirical likelihood, the researcher first fits a regression model to the data. This model estimates the relationship between the two variables, based on the observed data.

The researcher then uses this model to construct a confidence interval around the estimated relationship. This confidence interval represents the range of values that the relationship between the two variables is likely to fall within, given the observed data.

Overall, empirical likelihood is a useful statistical method for estimating the likelihood of a given set of observations, particularly when the underlying distribution of the data is unknown or when the standard methods of likelihood estimation are not applicable. It allows researchers to construct confidence intervals around their estimates, providing a measure of the uncertainty associated with the estimates.