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According to Wikipedia, a space in mathematics is:

a set (sometimes called a universe) with some added structure.

In statistical literature, I usually find references to a parameter space in the context of optimization, e.g.:

Occasionally, choice of particular effect size measures imposes constraints on the parameter space of $\mu_\theta$. Moreover, $\sigma^2_\theta$ is constrained to be non-negative in all cases. When the solutions converge to values outside the parameter space, then one should check whether a maximum of the log-likelihood function occurs at the boundaries of the parameter space. (Viechtbauer, 2005, p. 268-269)

I can see that the set of possible parameter values is endowed with an order, but I have failed to find a source that makes explicit what constitutes the structure of a parameter space in such a scenario.

Kuku
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    Parameters of what? The answer varies (tremendously) depending on what is being parameterized. In statistics we often parameterize random variables (those are [stochastic processes](https://stats.stackexchange.com/questions/126791)), distributions (those are [distribution families](https://stats.stackexchange.com/questions/320746/)), [regression models,](https://stats.stackexchange.com/questions/148638) and much more. – whuber May 04 '21 at 15:04
  • A parameter space is a collection of elements, each defining a probability distribution over the set of observations. That seems to me to be the only requirement, hence failing to impose further structure at this level of generality. – Xi'an May 04 '21 at 15:06
  • @Xi'an When you assume the parameterization concerns a family of probability distributions, then at a minimum it inherits the natural properties of *all* distributions, including various topologies, so there is something that can be said generally (although it doesn't amount to much...). – whuber May 04 '21 at 15:07
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    @whuber Added a motivating example, but already this comment suggests that in statistics we are dealing with more than one kind of structure when discussing parameter spaces? – Kuku May 04 '21 at 15:10
  • @whuber I am unsure about this: indeed working backward from the set of probability distribution $dP_\theta$ endows $\Theta$ the set of $\theta$'s with a (projected) structure but this does not mean that $\Theta$ cannot be chosen freely (provided $dP_\theta$ exists). – Xi'an May 04 '21 at 15:14
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    @Xi'an I don't see why not. But I'm not advancing any deep or subtle idea here: it's merely that the set of all probability distributions is endowed with various topologies found useful in certain kinds of statistical analyses and that (trivially) literally any subset, no matter how arbitrary, is automatically endowed with the [subspace topology](https://en.wikipedia.org/wiki/Subspace_topology). Many of these subspaces, as in the example, have additional structure (in much the same way that, say, zero sets of nonsingular differentiable functions on $\mathbb{R}^n$ are differentiable manifolds). – whuber May 04 '21 at 15:19

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