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What is Maximum Likelihood Estimation?
Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a statistical model by maximizing the likelihood function. It is a widely used approach in quantitative finance and other fields of statistics, including parameter estimation in asset pricing models, option pricing models, risk management models, and econometric models. It allows for data-driven estimation of model parameters, enabling statistical inference, hypothesis testing, and predictive analysis.
Within Maximum Likelihood Estimation, the likelihood function represents the probability of observing the given data as a function of the model parameters. It quantifies how well the model, with specific parameter values, explains the observed data. The goal of MLE is to find the parameter values that maximize the likelihood function. MLE aims to estimate the values of unknown parameters in a statistical model. By maximizing the likelihood function, MLE selects the parameter values that make the observed data most probable. The estimated parameter values are referred to as the maximum likelihood estimates.
In practice, it is common to work with the log-likelihood function instead of the likelihood function. Taking the logarithm of the likelihood function simplifies the mathematical calculations, as the log transforms the product of probabilities into a sum of logarithms.
Maximizing the likelihood (or log-likelihood) function typically involves solving an optimization problem. Various optimization techniques, such as gradient descent, Newton's method, or the Expectation-Maximization algorithm, can be used to find the maximum likelihood estimates.
Under certain regularity conditions, the maximum likelihood estimates possess desirable statistical properties, such as consistency, asymptotic normality, and efficiency. Consistency means that as the sample size increases, the estimates converge to the true parameter values. Asymptotic normality implies that the estimates follow an approximately normal distribution as the sample size increases.
It's important to note that the success of MLE relies on several assumptions, such as the correct specification of the statistical model, independence of observations, and absence of measurement errors. Careful consideration and diagnostic checks are necessary to ensure the validity of the MLE results.
MLE is covered in more detail in module 4 of the CQF program.