Lecture

Likelihood Ratio Tests: Optimality and Extensions

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In course

MATH-442: Statistical theory

This course gives a mostly rigourous treatment of some statistical methods outside the context of standard likelihood theory. ...

Description

This lecture covers Likelihood Ratio Tests (LRT) and their optimality in hypothesis testing. It discusses the breakdown of optimality theory in higher dimensions and general cases, leading to the combination of Neyman-Pearson paradigm with Maximum Likelihood. The lecture explores the definition of the Likelihood Ratio statistic, its application in simple vs simple settings, and the construction of reasonable tests. It also delves into the asymptotic distribution of the Likelihood Ratio, including Wilks' Theorem and its extensions to general cases. The content includes examples, proofs, and discussions on the significance level, asymptotic properties, and the relationship between Confidence Intervals and Hypothesis Tests.

Instructor

Yoav Zemel

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics, Statistical hypothesis testing

Related concepts (32)

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.

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.

Statistics

Statistics (from German: Statistik, () "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal".

Statistical theory

The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistical inference, and the actions and deductions that satisfy the basic principles stated for these different approaches. Within a given approach, statistical theory gives ways of comparing statistical procedures; it can find a best possible procedure within a given context for given statistical problems, or can provide guidance on the choice between alternative procedures.

Statistic

A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypothesis. The average (or mean) of sample values is a statistic. The term statistic is used both for the function and for the value of the function on a given sample. When a statistic is being used for a specific purpose, it may be referred to by a name indicating its purpose.

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Related concepts (32)

Likelihood-ratio test