Asymmetric Maximum Likelihood (AML)
- Estimates model parameters when the likelihood function is skewed or not symmetrical.
- Allows different variances for different parameter values via a weighting function.
- Used to reduce bias that standard maximum likelihood estimation (MLE) can introduce in such asymmetric cases.
Definition
Section titled “Definition”Asymmetric maximum likelihood (AML) is a statistical method used to estimate model parameters in cases where the likelihood function is not symmetrical.
Explanation
Section titled “Explanation”Asymmetry in the likelihood function arises when the variance of the data differs for different values of the model parameters. In such situations, standard maximum likelihood estimation (MLE) can produce biased parameter estimates because it assumes a symmetric contribution from data points. AML addresses this by allowing different variances for different parameter values. It does this through a weighting function that adjusts each data point’s contribution to the likelihood according to its position relative to the model parameters. The weighting function is determined by iterative optimization: model parameters are updated based on the weighted likelihood until a maximum (weighted) likelihood estimate is reached.
Examples
Section titled “Examples”Financial modeling — stock prices
Section titled “Financial modeling — stock prices”For 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 MLE would underestimate the true value of the model parameters, leading to biased results. AML can account for this asymmetry and provide more accurate estimates.
Medical research — treatment efficacy
Section titled “Medical research — treatment efficacy”When modeling the efficacy of a new treatment, the likelihood function may be skewed to the left, with a higher variance for lower efficacy rates compared to higher rates. Using MLE in this situation can again lead to biased estimates; AML can be used to correct for that asymmetry.
Use cases
Section titled “Use cases”- Financial modeling (example: skewed stock price likelihoods)
- Medical research (example: skewed treatment-efficacy likelihoods)
Notes or pitfalls
Section titled “Notes or pitfalls”- Applying standard MLE when the likelihood is asymmetric can underestimate parameter values and produce biased results.
- AML relies on a weighting function to allow different variances across parameter values; this weighting function is found through iterative optimization, updating model parameters until a weighted maximum likelihood estimate is reached.
Related terms
Section titled “Related terms”- Maximum likelihood estimation (MLE)
- Likelihood function
- Weighting function