This lecture deals with maximum likelihood estimation of the parameters of the normal distribution. Before reading this lecture, you might want to revise the lecture entitled Maximum likelihood, which presents the basics of maximum likelihood estimation.

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 regularity conditions needed for the consistency and asymptotic normality of maximum likelihood estimators are assumed to be satisfied.

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 is

Proof

By taking the natural logarithm of the likelihood function, we get

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.

The vector is asymptotically normal with asymptotic mean equal to and 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

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