Multivariate Distributions
Multivariate distributions describe the joint probabilistic behavior of several random variables and provide the foundation on which multivariate inference is built.
Definition
A multivariate distribution is a probability law for a random vector that specifies the joint distribution of its components, including their marginal behavior and their dependence.
Scope
This area covers the central probability models of multivariate statistics: the multivariate normal distribution and its properties, the Wishart distribution that governs sample covariance matrices, and copula models that separate marginal behavior from the dependence structure. It addresses joint, marginal, and conditional distributions, moments, and the role of these distributions in estimation and hypothesis testing.
Sub-topics
Core questions
- How is the joint behavior of several random variables specified and characterized?
- What sampling distributions arise from multivariate normal data?
- How can dependence be modeled separately from marginal distributions?
- Which distributional assumptions justify standard multivariate procedures?
Key theories
- Multivariate normal as a foundation
- The multivariate normal distribution is closed under linear transformation, marginalization, and conditioning, and its mean vector and covariance matrix fully specify it, making it the central model for multivariate inference.
- Separation of margins and dependence
- By Sklar's theorem any joint distribution can be decomposed into its marginal distributions and a copula encoding dependence, allowing dependence to be modeled independently of the margins.
Clinical relevance
Multivariate distributions underpin the assumptions and sampling theory of nearly every multivariate method, and copula models in particular are used to model dependence in finance, hydrology, and risk analysis.
History
The multivariate normal distribution and the Wishart sampling distribution of covariance matrices were established in the early twentieth century and systematized in the classical theory of multivariate analysis. Copula theory, formalized through Sklar's theorem in 1959, later provided a flexible framework for dependence modeling.
Key figures
- T. W. Anderson
- John Wishart
- Abe Sklar
Related topics
Seminal works
- anderson2003
- mardia1979
- muirhead1982
Frequently asked questions
- Why is the multivariate normal distribution so central?
- It arises as a limiting distribution through multivariate central-limit behavior, is mathematically tractable, and underlies the sampling theory for means, covariances, and many test statistics in multivariate analysis.
- What does a copula add beyond the marginal distributions?
- A copula captures the dependence structure linking the variables, allowing arbitrary marginal distributions to be combined with a chosen pattern of dependence.