# Fundamentals of mathematical statistics

Learn the mathematical foundations of statistics, through a series of rigorous but accessible lectures on the most frequently utilized statistical concepts.

## The foundations

Samples, statistical models, estimation, statistical decisions

## Point estimation

Examples of mean estimation and mathematical properties of common mean estimators

Estimates and estimators of a parameter and criteria to evaluate them

Examples of variance estimation and mathematical properties of common variance estimators

## Interval estimation

Examples of confidence intervals for the mean, with detailed derivations of their properties

Confidence intervals, confidence coefficients, how to evaluate them

Examples of confidence intervals for the variance, with detailed derivations of their properties

## Hypothesis testing

Examples of hypothesis tests about the mean, with detailed derivations of their properties

Null and alternative hypothesis, types of errors, size and power

Examples of hypothesis tests about the variance, with detailed derivations of their properties

## Estimation methods

Introduction to estimators used in mathematical statistics, including ML, GMM, NLS

## Maximum likelihood estimation

How to estimate the covariance matrix of a maximum likelihood estimator

The fundamentals of the theory of maximum likelihood estimation

How to carry out tests of hypothesis in a maximum likelihood framework

How to solve numerically the maximum likelihood optimization problem

A test of hypothesis involving only restricted ML estimates

A test of hypothesis involving only unrestricted ML estimates

Criteria used to select the best model among a set of candidate models estimated by ML

A test of hypothesis involving both restricted and unrestricted ML estimates

Recursive algorithm used for ML estimation of latent-variable models

Parametric families that are particularly important in maximum likelihood estimation

## Conditional models

The fundamentals of conditional models, regression and classification

## Linear regression

Asymptotic properties of the OLS estimators of regression coefficients

Introduction to the mathematics of linear regression models: notation, assumptions, inference.

A measure of how well a linear regression fits the data

A regression model in which errors are conditionally normal

The OLS estimator is the best among those that are linear and unbiased

How to test hypotheses about coefficients estimated by OLS

Linear regression where all the variables are centered and divided by their standard deviation

How to estimate the regression coefficients efficiently when the errors are heteroskedastic or correlated

A biased estimator of linear regression coefficients whose MSE can be lower than that of OLS

If regressors are highly correlated, then OLS coefficient estimates have high variance

How to separately estimate the regression coefficients of two groups of regressors

Variables used in regression models to encode categorical features

Use our calculator to run your regressions effortlessly and without coding

## Classification models

Binary classification model in which the logistic function is used to transform inputs

Conditional models in which the output variable has a discrete distribution

Binary model in which the cdf of a standard normal distribution is used to transform inputs

## Topics in stochastic processes and time series

Definition of autocorrelation, autocorrelation function (ACF), sample ACF, ACF plots.

Sequences of random vectors whose future does not depend on the past conditional on the present

## Markov Chain Monte Carlo (MCMC) methods

How to diagnose (and solve) problems with MCMC samples

Monte Carlo methods based on sequences of dependent draws from a distribution

MCMC algorithm based on acceptance/rejection of draws from a proposal distribution

## Bayesian statistics

Bayesian models in which the parameters of the prior are assigned a hyper-prior

The fundamentals of Bayesian inference: prior, likelihood, posterior distributions

Bayesian inference about the parameters of a normal linear regression model

Bayesian inference about the parameters of a normal distribution

A simple and intuitive way of comparing two different models or hypotheses

When prior and posterior distribution belong to the same parametric family

An "objective" prior that has little influence on the posterior distribution

A scale used to translate the value of the Bayes factor into a qualitative judgement on the evidence

The books

Most of the learning materials found on this website are now available in a traditional textbook format.

Glossary entries
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