Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Statistical Hypothesis Testing

In course

MATH-232: Probability and statistics (for IC)

Cours d'introduction à la théorie des probabilités et aux statistiques. ...

Description

This lecture covers the comparison of two simple hypotheses, Neyman-Pearson lemma, rejection regions, optimal tests, probabilities, likelihood, maximum likelihood estimation, observed and expected information, and the construction of confidence intervals. It also discusses the likelihood ratio statistic, limit distribution of the maximum likelihood estimator, and the elements of a statistical hypothesis test. The instructor explains the importance of p-values, power of tests, and decision-making based on statistical evidence.

Instructors (2)

Raphaël Jean Berthier

Emmanuel Abbé

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics, Statistical hypothesis testing

Related concepts (32)

Statistical hypothesis testing

A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. While hypothesis testing was popularized early in the 20th century, early forms were used in the 1700s. The first use is credited to John Arbuthnot (1710), followed by Pierre-Simon Laplace (1770s), in analyzing the human sex ratio at birth; see .

Maximum likelihood estimation

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

Likelihood function

In statistical inference, the likelihood function quantifies the plausibility of parameter values characterizing a statistical model in light of observed data. Its most typical usage is to compare possible parameter values (under a fixed set of observations and a particular model), where higher values of likelihood are preferred because they correspond to more probable parameter values.

Likelihood-ratio test

In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models, specifically one found by maximization over the entire parameter space and another found after imposing some constraint, based on the ratio of their likelihoods. If the constraint (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero.

Statistical significance

In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by , is the probability of the study rejecting the null hypothesis, given that the null hypothesis is true; and the p-value of a result, , is the probability of obtaining a result at least as extreme, given that the null hypothesis is true. The result is statistically significant, by the standards of the study, when .

Related lectures (49)

Probability and Statistics

Delves into probability, statistics, paradoxes, and random variables, showcasing their real-world applications and properties.

Estimation and Confidence Intervals

Explores bias, variance, and confidence intervals in parameter estimation using examples and distributions.

Probabilities and Statistics

Covers fundamental concepts in probabilities and statistics, including linear regression, exploratory statistics, and the analysis of probabilities.

Confidence Intervals and Hypothesis Tests

Covers confidence intervals, hypothesis tests, standard errors, statistical models, likelihood, Bayesian inference, ROC curve, Pearson statistic, goodness of fit tests, and power of tests.

Statistical Significance: Maximum Likelihood Estimation and Confidence Intervals