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 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

Confidence interval for the mean

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

Confidence intervals, confidence coefficients, how to evaluate them

Confidence interval for 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 estimators used in mathematical statistics, including ML, GMM, NLS

MLE - Covariance matrix 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

Exponential family of distributions

Parametric families that are particularly important in maximum likelihood estimation

The fundamentals of conditional models, regression and classification

Properties of the OLS estimator

Asymptotic properties of the OLS estimators of regression coefficients

Introduction to the mathematics of 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

Standardized linear regression

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

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

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

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

Normal distribution - Bayesian estimation

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

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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
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About

Glossary entries

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

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