Parameter space

The concept of parameter space is found in the theory of statistical inference. In a statistical inference problem, the statistician utilizes a sample to understand from what probability distribution the sample itself has been generated. Attention is usually restricted to a well-defined set of probability distributions that could have generated the sample. When these probability distributions are put into correspondence with a set of real numbers (or real vectors), such set is called the parameter space and its elements are called parameters.

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Definition

A more rigorous definition could be as follows.

Definition Let $\xi$ be a sample (i.e., a vector of observed data). Denote by $\Phi$ the set of all probability distributions that could have generated the sample $\xi$ . Let $\Theta$ be a set of real vectors. Suppose there exists a correspondence that associates a subset of $\Phi$ to each $\theta \in \Theta$ . The set $\Theta$ is called a parameter space for $\Phi$ if and only if [eq2] The members of $\Theta$ are called parameters.

In other words, $\Theta$ is a parameter space for $\Phi$ if and only if all the probability distributions in $\Phi$ are associated to at least one parameter, and all parameters are associated to probability distributions belonging to $\Phi$ .

If the correspondence associates only one probability distribution to each parameter, then we have a parametric model. If there is a one-to-one correspondence between the members of $\Phi$ and $\Theta$ (i.e., only one parameter is associated to each probability distribution), then the parametric model is said to be identified.

More details

A detailed presentation of the concepts of parameter and parameter space can be found in the lecture entitled Statistical inference.

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How to cite

Please cite as:

Taboga, Marco (2021). "Parameter space", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/parameter-space.