 StatLect

# Normal distribution - Maximum Likelihood Estimation

This lecture deals with maximum likelihood estimation of the parameters of the normal distribution.

Before continuing, you might want to revise the basics of maximum likelihood estimation (MLE). ## Assumptions

Our sample is made up of the first terms of an IID sequence of normal random variables having mean and variance .

The probability density function of a generic term of the sequence is The mean and the variance are the two parameters that need to be estimated. ## The likelihood function

The likelihood function is Proof

Given the assumption that the observations from the sample are IID, the likelihood function can be written as ## The log-likelihood function

The log-likelihood function is Proof

By taking the natural logarithm of the likelihood function, we get ## The maximum likelihood estimators

The maximum likelihood estimators of the mean and the variance are Proof

We need to solve the following maximization problem The first order conditions for a maximum are The partial derivative of the log-likelihood with respect to the mean is which is equal to zero only if Therefore, the first of the two first-order conditions implies The partial derivative of the log-likelihood with respect to the variance is which, if we rule out , is equal to zero only if Thus, the system of first order conditions is solved by Thus, the estimator is equal to the sample mean and the estimator is equal to the unadjusted sample variance.

## Asymptotic variance

The vector is asymptotically normal with asymptotic mean equal to and asymptotic covariance matrix equal to Proof

The first entry of the score vector is The second entry of the score vector is In order to compute the Hessian we need to compute all second order partial derivatives. We have and Finally, which, as you might want to check, is also equal to the other cross-partial derivative . Therefore, the Hessian is By the information equality, we have that As a consequence, the asymptotic covariance matrix is In other words, the distribution of the vector can be approximated by a multivariate normal distribution with mean and covariance matrix ## Other examples

StatLect has several pages that contain detailed derivations of MLEs. Learn how to find the estimators of the parameters of the following distributions and models.

TypeSolution
Exponential distributionUnivariate distributionAnalytical
Poisson distributionUnivariate distributionAnalytical
T distributionUnivariate distributionNumerical
Multivariate normal distributionMultivariate distributionAnalytical
Gaussian mixtureMixture of distributionsNumerical (EM)
Normal linear regression modelRegression modelAnalytical
Logistic classification modelClassification modelNumerical
Probit classification modelClassification modelNumerical