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 ifTherefore, 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 ifThus, 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 vectoris asymptotically normal with asymptotic mean equal toand asymptotic covariance matrix equal to

Proof

The first entry of the score vector isThe second entry of the score vector isIn order to compute the Hessian we need to compute all second order partial derivatives. We haveandFinally, which, as you might want to check, is also equal to the other cross-partial derivative . Therefore, the Hessian isBy the information equality, we have thatAs 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

How to cite

Please cite as:

Taboga, Marco (2021). "Normal distribution - Maximum Likelihood Estimation", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-statistics/normal-distribution-maximum-likelihood.

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