In statistics, the data set is viewed as a realization or observation of a @random element defined on a @probability space $(\mathrm{\Omega },\mathcal{F},P)$ related to the @random @experiment.

Given a data set, the @probability measure $P$ is called the population.

A data set or the @random element that produces the data is called a sample from $P.$

The size of the data set is called the sample size.

The population $P$ is known if and only if $P(A)$ is a known value for every event $A\in \mathcal{F}.$

In a statistical problem, the population $P$ is at least partially unknown and we would like to deduce some properties of $P$ based on the available sample.

A statistical model $\mathcal{P}$ is a collection of @probability measures on a common measurable space $(\mathrm{\Omega },\mathcal{F}).$

Equivalently, $\mathcal{P}\subseteq \mathsf{P}(\mathrm{\Omega },\mathcal{F}),$ where $\mathsf{P}(\mathrm{\Omega },\mathcal{F})$ denotes the set of all probability measures on $(\mathrm{\Omega },\mathcal{F}).$

In other words, a statistical model is a set of populations constrained by some set of assumptions about the populations, or, is the assumptions themselves.

Let $\mathcal{P}$ be a statistical model. A parameter is a function $\vartheta :\mathcal{P}\to \mathrm{\Lambda }$ into some set $\mathrm{\Lambda }$. For $P\in \mathcal{P}$, the element $\vartheta (P)$ is the value of the parameter at $P$.

A parameterization of a statistical model $\mathcal{P}$ is a surjective map $\pi :\mathrm{\Theta }\to \mathcal{P}$ written $P_{\theta }=\pi (\theta )$.

A parameterization $\pi$ is identifiable if the @identity map on $\mathrm{\Theta }$ satisfies the condition in Parameters Factor Through the Parameterization, that is, if $\pi$ is injective. Then ${\pi }^{-1}:\mathcal{P}\to \mathrm{\Theta }$ is a parameter, the indexing parameter, and $\mathrm{\Theta }$ is its range.

The parameter space of a parameterization $\pi :\mathrm{\Theta }\to \mathcal{P}$ is its domain $\mathrm{\Theta }.$

A statistical model $\mathcal{P}$ on a measurable space $(\mathrm{\Omega },\mathcal{F})$ is a parametric family if it has a parameterization $\theta \mapsto P_{\theta }$ whose parameter space satisfies $\mathrm{\Theta }\subseteq {\mathbb{R}}^d$ for some fixed positive integer $d$.

Assume that we have a data set that is a realization of a sample $X$ (a @random vector) from an unknown population $P$ on a @probability space.

A @measurable function of $X,$ $T(X),$ is called a statistic if $T(X)$ is a known value whenever $X$ is known, i.e., the function $T$ is a known function (i.e. $T$ doesn't depend on the unknown $P$.)

TODO: define this.

Let $\mathcal{P}$ be a statistical model on $(\mathrm{\Omega },\mathcal{F})$, and let $\vartheta :\mathcal{P}\to \mathrm{\Lambda }$ be a parameter, where $(\mathrm{\Lambda },\mathcal{G})$ is a measurable space. An estimator of $\vartheta$ is a statistic $T:(\mathrm{\Omega },\mathcal{F})\to (\mathrm{\Lambda },\mathcal{G})$. If $x\in \mathrm{\Omega }$ is an observed data set (i.e. one value that a sample $X$ happened to take), the value $T(x)$ is called an estimate of $\vartheta$.

In an estimation problem, the bias of an estimator $T(X)$ of a real-valued parameter $\vartheta$ of the unknown population is defined to be $b_T(P)=\mathbb{E}[T(X)]-\vartheta$ (which is denoted by $b_T(\theta )$ when $P$ is in a parametric family index by $\theta ).$

An estimator is said to be an unbiased estimator for $\vartheta$ if and only if $b_T(P)=0$ for any $P\in \mathcal{P}.$

A function $T(X)$ is said to be a sufficient statistic relative to the family $\{f_{\theta }(x)\}$ if $X$ is conditionally independent of $\theta$ given $T(X)$ for any probability distribution on $\theta ,$ that is, $\theta \to T(X)\to X$ forms a Markov chain.

A statistic $T(X)$ is a minimal sufficient statistic relative to $\{f_{\theta }(x)\}$ if it is a function of every other sufficient statistic $U.$