Suppose $X\_1,\ldots,X\_n\sim \text{i.i.d.} \operatorname N(\mu,\sigma^2).$ Consider these functions of the foregoing random variables:


* Any weighted average of $X\_1,\ldots,X\_n,$ with constant (i.e. non-random) weights not depending on $\mu$ or $\sigma^2.$
* $\operatorname{median}\{X\_1,\ldots,X\_n\}.$ (If $n$ is even, take this to be the mean of the two middle observations.)
* $(\max\{X\_1,\ldots,X\_n\} + \min\{X\_1,\ldots,X\_n\})/2$
* $\operatorname{mean} \{X\_i : i\in I\},$ where $I\subset\{1,\ldots,n\}$ is randomly chosen from among all $2^n-1$ non-empty subsets, in a way that is independent of the values of $X\_1,\ldots,X\_n.$
* $\operatorname{mean} \{X\_i : i\in I\},$ where $I\subset\{1,\ldots,n\}$ is randomly chosen from among all $2^n$ subsets, in a way that does depend on $X\_1,\ldots,X\_n$ but for which this mean nonetheless has the properties stated below. (The third item in this list is a special case of this.)
* $\operatorname{median}\{X\_i : i\in I \}$ with $I$ as in either of the last two situations above.


All of these functions of $X\_1,\ldots,X\_n$ have the following properties:


* The value of each can be found without knowing the values of $\mu$ or $\sigma^2,$ i.e. each of these is a "statistic."
* Each has expected value $\mu$ regardless of the values of $\mu$ and $\sigma^2,$ i.e. each is an unbiased estimator of $\mu.$


**My question is** whether there is a way to classify all such functions. Can we describe some set of functions without mentioning anything about $\{\operatorname N(\mu,\sigma^2) : \mu\in\mathbb R, \sigma^2\in[0,+\infty)\}$ or any other family of probability distributions and show that those and only those have the properties stated in the last two bullet points above?



Classification of all unbiased estimators of the mean of a normal population
$\newcommand{\R}{\mathbb R}\newcommand{\si}{\sigma} $At least in your setting, statistics (also called estimators) are random variables of the form $T(X)$, where $T$ is a Borel-measurable function from $\R^n$ to $\R$ and $X:=(X\_1,\ldots,X\_n)$.


An estimator $T(X)$ of $\mu$ is called unbiased (for $\mu$) if
\begin{equation}
 E\_{\mu,\si^2}T(X):=\int\_{\R^n}dx\,T(x)f\_{\mu,\si^2}(x)=\mu
\end{equation}
for all $(\mu,\si^2)\in\R\times(0,\infty)$, where $f\_{\mu,\si^2}$ is the (joint) density of $X$, so that
\begin{equation}
 f\_{\mu,\si^2}(x)=(2\pi\si^2)^{-n/2}\exp-\sum\_1^n\frac{(x\_i-\mu)^2}{2\si^2}
\end{equation}
for $x:=(x\_1,\ldots,x\_n)\in\R^n$.


E.g.,
\begin{equation}
 \bar X:=\frac1n\,\sum\_1^n X\_i
\end{equation}
is unbiased (for $\mu$).


So, $T(X)$ is unbiased for $\mu$ iff $E\_{\mu,\si^2}(T(X)-\bar X)=0$ for all $(\mu,\si^2)\in\R\times(0,\infty)$ -- that is, iff the function $\R^n\ni x=(x\_1,\ldots,x\_n)\mapsto\bar x:=T(x)-\frac1n\,\sum\_1^n x\_i\in\R$ is orthogonal in $L^2(\R)$ to all the normal density functions $f\_{\mu,\si^2}$, for all $(\mu,\si^2)\in\R\times(0,\infty)$.


This rather trivial characterization of unbiased estimators of $\mu$ for $N(\mu,\si^2)$ is apparently the least non-tautological characterization possible.


Indeed, the most comprehensive study of unbiased estimators is apparently the two-volume [book by Voinov and Nikulin](https://books.google.com/books?id=xgH1CAAAQBAJ&printsec=copyright#v=onepage&q&f=false). There is no non-tautological characterization of all unbiased estimators of any function of the parameter of **any** parametric family of distributions there.


In particular, in the long table of unbiased estimators of functions of the parameter $(\mu,\si^2)$ of the family of normal distributions, the only unbiased estimator of $\mu$ listed there is $\bar X$.


Indeed, all the other unbiased estimator of $\mu$ for the normal family do not really matter. Indeed, by the completeness, $\bar X$ is almost surely (a.s.) the only unbiased estimator of $\mu$ that is a function of a minimal sufficient statistic. So, by the [Blackwell--Rao](https://en.wikipedia.org/wiki/Rao%E2%80%93Blackwell_theorem)--[Lehmann–Scheffé](https://en.wikipedia.org/wiki/Rao%E2%80%93Blackwell_theorem#Completeness_and_Lehmann%E2%80%93Scheff%C3%A9_minimum_variance) theory, $\bar X$ is the (a.s. unique) uniformly minimum variance unbiased estimator of $\mu$; actually, $\bar X$ is the best unbiased estimator of $\mu$ with respect to any convex [loss function](https://en.wikipedia.org/wiki/Loss_function).


