I'm having a hard time trying to understand some basic statistical definitions presented in the section II.9 of the book "Mathematical Statistics" written by Wiebe R. Pestman.
Below I'll write the excerpt of the book that I'm trying to understand.
Let $\Theta$ be an arbitrary set and let $f:\mathbb{R}\times\Theta \to [0,+\infty )$ be a function. For every fixed $\theta\in\Theta$ denote the function $x\mapsto f(x,\theta)$ by $f(\bullet ,\theta)$. If for all $\theta\in\Theta$ this function presents a (possibly discrete) probability density, then one talks about a family of probability densities with parameter $\theta\in\Theta$. [...]
Now, once and for all, let $\big(f(\bullet,\Theta)\big)_{\theta\in\Theta}$ be any family of probability densities. It is assumed that the population from which the samples are drawn has a probability density $f(\bullet,\theta)$, where $\theta\in\Theta$. [...]
Definition II.9.1. A function $\kappa:\Theta\to\mathbb{R}$ is said to be a characteristic of the population. [...]
Definition II.9.2: Let $X_1,\cdots,X_n$ be a sample from a population with probability density $f(\bullet,\theta)$, where $\theta\in\Theta$. Suppose $\kappa$ is a characteristic of the population. A statistic $T=g(X_1,\cdots,X_n)$, the outcome of which is used as an estimate of $\kappa(\theta)$, is said to be an estimator of $\kappa$ (based on the sample $X_1,\cdots,X_n$).
Note that the expectation value of a statistic $T=g(X_1,\cdots,X_n)$ will, if existing, in general depend on the parameter $\theta$. [...]. If in some context one has to count with this, one could write $\mathbb{E}_\theta(T)$ instead of $\mathbb{E}(T)$.
Definition II.9.3: A statistic $T$ is said to be an unbiased estimator of $\kappa$ if $\mathbb{E}_\theta(T)=\kappa(\theta)$ for all $\theta \in\Theta$.
My question is (EDITED): What does the definition II.9.3 mean? Does it mean that given any $\theta\in \Theta$, any $n\in\mathbb{N}$ and any sample $X_1,\cdots,X_n$ with distribution $f(\bullet,\theta)$ there's a Borel measurable function $g:\mathbb{R}^n\to\mathbb{R}$ such that $T=g(X_1,\cdots,X_n)$ and $\mathbb{E}_\theta (T)=\kappa(\theta)$?
I tried to interpret the content of that excerpt with the following definitions.
Definition 1: We say that $(\Omega,\Sigma,\mathbb{P},f_\theta :\theta\in\Theta)$ is a statistical model if $(\Omega,\Sigma,\mathbb{P})$ is probability space, $\Theta$ is a non-empty set and $f_\theta:\mathbb{R}\to\mathbb{R}$ is a density function for all $\theta\in\Theta$.
Definition 2: Let $(\Omega,\Sigma,\mathbb{P},f_\theta :\theta\in\Theta)$ be a statistical model. We say that random variables $X_1,\cdots,X_n:\Omega\to\mathbb{R}$ is a $\theta$-sample if $\theta\in\Theta$, $X_1,\cdots,X_n$ are iid with respect to $(\Omega,\Sigma,\mathbb{P})$ and have common distribution $f_\theta$.
Definition 3: Let $(\Omega,\Sigma,\mathbb{P},f_\theta :\theta\in\Theta)$ be a statistical model and $X_1,\cdots,X_n$ be a $\theta$-sample. We say that $T:\Omega\to \mathbb{R}$ is a statistic (with respect to that sample) if there's a Borel measurable function $g:\mathbb{R}^n\to\mathbb{R}$ such that $T=g(X_1,\cdots,X_n)$.
Definition 4 (EDITED): Let $(\Omega,\Sigma,\mathbb{P},f_\theta :\theta\in\Theta)$ be a statistical model and $\kappa:\Omega\to\mathbb{R}$ be any function. We say that a Borel function $g:\mathbb{R}^n\to \mathbb{R}$ is unbiased estimator of $\kappa$ if for any $\theta\in \Theta$ and any $\theta$-sample $X_1,\cdots,X_n$ we have $\mathbb{E}[g(X_1,\cdots,X_n)]=\kappa (\theta)$
Another question: Are these definitions really a good interpretation of that excerpt? Also, are they in line with standard statistical definitions?
Thank you for your attention!
As to your first question about $II.9.3$, I would say you have two too many "any"s there. More on that later. Definition $4$ is defining "estimator", but the condition that $\mathbb{E}(T)=\kappa(\theta)$ is a requirement of an unbiased estimator. That condition isn't a part of the definition of "estimator." In general, one has to say what something is an unbiased estimator of. By that, I mean that you can have an unbiased estimator of $\theta$, or of $\theta^2$, etc. Something that is an unbiased estimator of $\theta$ will generally not be an unbiased estimator of $\theta^2$, and vice versa. Quite often you will hear "unbiased" without reference to what the unbiased estimator is an unbiased estimator of, but it's formally a part of it. The omission should be in the case when what is being estimated is understood.
In general, you can think of a statistic as a (Borel measurable) function of a random sample, as you have: $T=g(X_1, \ldots, X_n)$. A key point here is that it does not explicitly depend on the underlying $\theta$.
If you use this statistic to estimate $\kappa(\theta)$, then it's an estimator of $\kappa(\theta)$. This isn't a very "mathy" definition, in that it's not a property of $T$ which determines whether or not it's an estimator of $\kappa(\theta)$. It's how you use it which determines whether or not it's an estimator. If you use it to estimate $\kappa(\theta)$, then it's an estimator of $\kappa(\theta)$. The fact that this definition does not reference any properties of $T$, and only how one uses $T$, struck me as a little odd, because it seemed like there must be more to it than that.
The bias of an estimator is the difference between its expectation and the value it's estimating: $$\text{Bias}(T)=\mathbb{E}_\theta(T)-\kappa(\theta).$$ This could be different for different $\theta$. If $\text{Bias}(T)=0$ for all $\theta$, then $T$ is an unbiased estimator.
To get back to $II.9.3$, it shouldn't say "for any $T$". You're defining what it means for a specific estimator $T$ to be unbiased. "We say the estimator $T$ is unbiased ..." or "We say $T$ is an unbiased estimator if...", so all of this is happening for a specific, fixed $T$, and the statement shouldn't include "for any $T$".
Also, formally speaking, different values of $n$ will generally be different statistics. If $T=g(X_1,X_2)$, then you're working in a specific case where $n=2$. So the definition should not include "for any $n\in\mathbb{N}$". However, this formality is sometimes overlooked. For example, if we're dealing with random variables which have a normal distribution with mean $\theta$ and variance $1$, $$S_2=\frac{1}{2}\Bigl[X_1+X_2\Bigr]$$ is an estimator of $\theta$ (and an unbiased estimator). The estimator $$S_3=\frac{1}{3}\Bigl[X_1+X_2+X_3\Bigr]$$ is a different unbiased estimator. For $S_2$, you'd be dealing with samples $(X_1, X_2)$ that specifically have $n=2$. So when answering the question of whether $S_2$ is unbiased, you wouldn't be dealing with multiple $n$, but only $n=2$.