While reading this book

Mathematical Statistics: Basic Ideas and Selected Topics of Kjell A. Doksum and Peter J. Bickel

I came across these sentences that I'm trying to understand as they're fundamental

Dual to the notion of a parametrization, a map from some $\Theta$ to $\mathcal{P}$, is that of a parameter, formally a map, $\mathcal{V}$, from $\mathcal{P}$ to another space $\mathcal{N}$. A parameter is a feature $\mathcal{V}(P)$ of the distribution of X.

I know what parametrization means but I don't see why we would need a second map to go from the distribution space of our model to another space, $\mathcal{N}$.

Why this $\mathcal{N}$ ? is it a set of natural numbers as $\theta$ is strictly positive?

Is it just to show that it is possible to go from knowing/assuming what a model is to guessing the real param or the set of the parameter by using the map $\mathcal{V}$ ?

Any input on the subject would be more than welcome

  • $\begingroup$ There's nothing here about guessing or estimation. "Features" are not necessarily parameters. For instance, one feature of the space of Normal distributions is the sign of their means, which is an element of $\mathcal{N}=\{-1,0,1\}.$ This map is not invertible; you cannot use the set $\{-1,0,1\}$ in a meaningful way as part of a (continuous) parameterization of the Normal distributions. $\endgroup$ – whuber Sep 25 '19 at 16:01

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