Talk:Minimum-distance estimation

Population Distribution

This population distribution $F(x;\theta )$, what is its domain and range? I know it says that $\theta \in \Theta \subset \mathbb {R} ^{n}$ with $n\geq 1$, but where does x live, and where does $\displaystyle F(x;\theta )$ live? It might look nice if it were to be written explicitly, like: $\displaystyle F:X\times \Theta \to Y$ where the spaces X and Y need to be given.

 Declan Davis   (talk)  19:17, 24 September 2008 (UTC)

Goodness, I'm just a lowly actuary, when you say spaces I think of the QWERTY keyboard . Seriously, I'm not 100% certain as to the space of $\displaystyle x$. Drossos & Phillippou did not explicitly state it, as they did for $\displaystyle \theta$. Although they do discuss ${\mathcal {F}}$ as a class of distribution functions and define $d(\cdot ,\cdot )$ as being defined on ${\mathcal {F}}\times {\mathcal {F}}$.

Kim & Lee (1999) describe the distance without referring to the population of $\displaystyle x$ at all, talking solely about the space of $\displaystyle \theta$.

Anderson & Darling (1952) define $\displaystyle -\infty <x<\infty$. -- Avi (talk) 19:38, 24 September 2008 (UTC)

From my foray through the literature, I do not see why the samples need to be one-dimensional, although they almost always are, so $\mathbb {R} ^{n}$ seems reasonable for the domain and range of $\displaystyle x$ as well. $\displaystyle F(x;\theta )$ has to live on the closed interval between 0 and 1, of course, as it is a distribution function. I do hope someone more erudite in this area than I drops by, though. -- Avi (talk) 00:41, 25 September 2008 (UTC)

I'm not an expert, but I'll try to explain it. $\{F_{\theta }|\theta \in \Theta \}$ is a statistical model. The set $\Theta$ is a parameter space and theoretically it could be any non-empty set. In practice parameters are real numbers and $\Theta \subset \mathbb {R} ^{n}$ where n is the number of parameters. Each function $F_{\theta }$ is probability distribution. The range of $F(.;\theta )$ is closed interval [0,1]. If the random samples $X_{1},...,X_{n}$ are in the sample space X, then domain of each $F(.;\theta )$ is X (or the set of all measurable subsets of X). Usually X is either the set of real number $\mathbb {R}$ or the set of integers $\mathbb {Z}$ (or subset of either). Tlepp (talk) 08:33, 26 September 2008 (UTC)

Thank you, Tlepp. While I agree that the vast majority of the time, the random variables under consideration are real numbers or integers, isn't it possible to be sampling ordered pairs or vectors, in which case $\displaystyle X$ actually belongs to $\mathbb {R} ^{n}$? -- Avi (talk) 15:26, 26 September 2008 (UTC)