Estimating the Parameters of the Univariate and Multivariate Normal Distribution

Many multivariate statistical analysis procedures are based on an assumption that the responses have a multivariate Normal distribution. Multivariate Normal distributions have a number of parameters, and research questions are framed in terms of those parameters. Answering these research questions is not as simple as observing the values of the parameters or making mathematical comparisons among them, because the true values of these parameters are unknown in any real-world situation. Rather, statistical inference must be used in answering these questions. This involves estimating the various parameters of the Multivariate Normal distribution from data, testing hypotheses about the parameters, and constructing confidence intervals for individual parameters and confidence regions for sets of parameters.

In this lecture video we are going to use the maximum likelihood principle to derive estimators for Normal distributions. We’re going to do this for the univariate Normal distribution and for the multivariate Normal distribution. We’ll begin by deriving estimators for the two parameters in the univariate Normal distribution, the mean and the variance. We’ll start here because from our previous statistics courses we’re already familiar with the univariate Normal distribution and the estimators of its parameters. Another reason we start here is because it’s mathematically easier to present notationally, to discuss, and understand. Once we’ve become familiar with the process for deriving the estimators for the univariate case, we’ll take what we’ve learned and proceed by analogy, so to speak, into a discussion of the derivation process for the multivariate case.

Here’s an outline of what we will be looking at in this lecture video:

Derive Estimators of the Parameters of the Univariate Normal Distribution
The Likelihood Function
Maximum Likelihood: The Big Idea
Derivation
Evaluation and adjustment where necessary
Derive Estimators of the Parameters of the Multivariate Normal Distribution
The Likelihood Function
Maximum Likelihood: The Big Idea
Derivation (setup and discussion)
Evaluation and adjustment where necessary

Next lecture in this series:
First lecture in this series:

0:00 Intro Music
0:21 Introduction
2:15 Outline
4:38 Deriving the parameter estimates of the univariate Normal distribution
4:52 Constructing the likelihood function for the univariate Normal distribution
11:33 The idea behind maximum likelihood estimation
14:25 Using calculus to derive the maximum likelihood estimators
17:43 Evaluation and bias adjustment of the maximum likelihood estimators
20:11 Deriving the parameter estimates of the multivariate Normal distribution
20:24 Constructing the likelihood function for the multivariate Normal distribution
23:12 The idea behind maximum likelihood estimation
23:49 Using calculus to derive the maximum likelihood estimators
27:01 Evaluation and bias adjustment of the maximum likelihood estimators
29:01 Calculation and construction of the estimators of the mean vector and covariance matrix
32:35 Summary
34:05 Outro

[benoitpeacemaker]

fantastic, please how can I contact you?