Key Concepts of Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) is a key method in statistical inference for estimating model parameters. By maximizing the likelihood of observed data, MLE provides reliable estimates, especially in large samples, making it a popular choice among statisticians.

1. Definition of Maximum Likelihood Estimation (MLE)

   • MLE is a method for estimating the parameters of a statistical model.
   • It selects the parameter values that maximize the likelihood of observing the given data.
   • MLE is widely used due to its desirable properties in large samples.

2. Likelihood function

   • The likelihood function measures the probability of the observed data given specific parameter values.
   • It is defined as the product of the probability density (or mass) functions for all observed data points.
   • The likelihood function is not a probability distribution itself; it is a function of the parameters.

3. Log-likelihood function

   • The log-likelihood function is the natural logarithm of the likelihood function.
   • It simplifies calculations, especially when dealing with products of probabilities, by converting them into sums.
   • Maximizing the log-likelihood is equivalent to maximizing the likelihood function.

4. Steps to derive MLE

   • Specify the likelihood function based on the statistical model and observed data.
   • Take the natural logarithm to obtain the log-likelihood function.
   • Differentiate the log-likelihood with respect to the parameters and set the derivatives to zero to find critical points.
   • Solve the resulting equations to obtain the MLE estimates.

5. Properties of MLE (consistency, asymptotic normality, efficiency)

   • Consistency: MLE estimates converge in probability to the true parameter values as sample size increases.
   • Asymptotic normality: MLE estimates are approximately normally distributed for large samples.
   • Efficiency: MLE achieves the lowest possible variance among unbiased estimators in large samples.

6. Score function

   • The score function is the gradient (first derivative) of the log-likelihood function with respect to the parameters.
   • It indicates the direction in which the likelihood function increases.
   • Setting the score function to zero helps find the MLE.

7. Fisher Information

   • Fisher Information quantifies the amount of information that an observable random variable carries about an unknown parameter.
   • It is defined as the expected value of the squared score function.
   • Higher Fisher Information indicates more precise estimates of the parameter.

8. Cramér-Rao lower bound

   • The Cramér-Rao lower bound provides a theoretical lower limit on the variance of unbiased estimators.
   • It states that the variance of any unbiased estimator is at least as large as the inverse of the Fisher Information.
   • MLE is asymptotically efficient, meaning it achieves this lower bound in large samples.

9. MLE for common distributions (e.g., Normal, Binomial, Poisson)

   • For the Normal distribution, MLE estimates for mean and variance are the sample mean and sample variance.
   • For the Binomial distribution, MLE for the probability of success is the ratio of successes to total trials.
   • For the Poisson distribution, MLE for the rate parameter is the average count of events in the observed interval.

10. Numerical methods for finding MLE (e.g., Newton-Raphson)

    • The Newton-Raphson method is an iterative technique used to find roots of equations, applicable for MLE.
    • It uses the score function and the Fisher Information to update parameter estimates.
    • Convergence can be rapid, but it requires a good initial guess and may not work if the likelihood function is not well-behaved.