Asymmetric maximum likelihood (AML) is a statistical method used to estimate model parameters in cases where the likelihood function is not symmetrical. This can occur when there is a significant difference in the variance of the data for different values of the model parameters.

One example of when AML may be used is in financial modeling. In the case of a stock price, the likelihood function may be skewed to the right, with a higher variance for higher stock prices compared to lower prices. In this case, using standard maximum likelihood estimation (MLE) would underestimate the true value of the model parameters, leading to biased results. AML can be used to account for this asymmetry and provide more accurate estimates.

Another example is in medical research, where AML may be used to model the efficacy of a new treatment. In this case, the likelihood function may be skewed to the left, with a higher variance for lower efficacy rates compared to higher rates. Using MLE would again lead to biased estimates, and AML can be used to correct for this asymmetry.

The key difference between AML and MLE is that AML allows for different variances for different values of the model parameters. This is achieved by using a weighting function, which adjusts the contribution of each data point to the likelihood function based on its position relative to the model parameters. This weighting function is determined through a process of iterative optimization, in which the model parameters are updated based on the weighted likelihood function until a maximum likelihood estimate is reached.

Overall, AML provides a more flexible and accurate method for estimating model parameters in cases where the likelihood function is not symmetrical. This can lead to more reliable results in applications such as financial modeling and medical research.