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

Samples, statistical models, estimation, statistical decisions

Examples of mean estimation and properties of common mean estimators

Estimates and estimators of a parameter and criteria to evaluate them

Point estimation of the variance

Examples of variance estimation and properties of common variance estimators

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

Confidence intervals, confidence coefficients, how to evaluate them

Set estimation of the variance

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

Testing hypotheses about the mean

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

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

Testing hypotheses about the variance

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

Introduction to extremum estimators, including ML, GMM, NLS

MLE - Covariance matrix estimation

How to estimate the covariance matrix of a maximum likelihood estimator

Maximum likelihood estimators and their asymptotic properties

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

A test of hypothesis involving both restricted and unrestricted ML estimates

Introduction to conditional models, regression and classification

Properties of the OLS estimator

Asymptotic properties of the OLS estimators of regression coefficients

Introduction to linear regression models: notation, assumptions, inference.

R squared of a linear regression

A measure of how well a linear regression fits the data

The Normal Linear Regression Model

A regression model in which errors are conditionally normal

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

Linear regression - Hypothesis testing

How to test hypotheses about coefficients estimated by OLS

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

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

How to diagnose (and solve) problems with MCMC samples

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

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

Introduction to Bayesian inference: prior, likelihood, posterior distributions

Normal distribution - Bayesian estimation

Bayesian inference about the parameters of a normal distribution

The book

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

Featured pages

- Conditional probability
- Normal distribution
- Independent events
- Central Limit Theorem
- Combinations
- Mean square convergence

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Main sections

- Mathematical tools
- Fundamentals of probability
- Probability distributions
- Asymptotic theory
- Fundamentals of statistics
- Glossary

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Glossary entries

- Alternative hypothesis
- Type II error
- Loss function
- Probability mass function
- IID sequence
- Precision matrix

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