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Parameter space

by , PhD

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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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 [eq1] 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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