Maximum Likelihood Estimator

Under parametric family distributions, there's a much better way of constructing estimators - the maximum likelihood estimator.

The Method of Moments

A fairly simple method of constructing estimators that's not often used now.

Consistency

Introducing consistency, a concept about the convergence of estimators. We start from the convergence of non-random number sequences to convergence in probability, then to consistency of estimators and its properties.

Bias and Variance

The bias, variance and mean squared error of an estimator. The efficiency is used to compare two estimators.

Brief Review Before STAT 6520

A brief review of probability theory and statistics we've learnt so far.

Sampling Distribution and Limit Theorems

We observe a random sample from a probability distribution of interest and want to estimate its properties. The CLT also comes into place.

Functions of Random Variables

Finding the distribution of a real-valued function of multiple random variables. There's the method of distribution functions, transformations and moment generating functions.

Multivariate Probability Distributions

Joint probability distributions of two or more random variables defined on the same sample space. Also covers independence, conditional expectation and total expectation.

Definitions for Discrete Random Variables

The probability mass function, cumulative distribution function, expectation and variance for random variables.

Estimation

In this chapter we introduce the concept of linear models. We use the ordinary least squares estimator to get unbiased estimates of the unknown parameters. $R^2$ is introduced as a measure of the goodness of fit, and the different types of sum of squares in a linear model are briefly discussed.