StatlectThe Digital Textbook

# 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.

## Definition

A more rigorous definition could be as follows.

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

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

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 and (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.