General recipe for finding unbiased or consistent estimator? I am wondering whether there is a general recipe for finding unbiased and consistent estimators of some non-random quantity.
For concreteness, I will discuss only discrete probability distributions with a finite support, but I would appreciate answers also for the continuous case.
Suppose that there is a discrete distribution over $M$ values:
$p_1=P(x_1)\\p_2=P(x_2)\\...\\p_M=P(x_M)$
Denote $x=(x_1,x_2,...,x_M)$ and $p=(p_1,p_2,...,p_M)$.  The $x_i$ values are all different. The $p_i$ values are all non-negative and add up to $1$.
Any quantity that we may want to estimate can be written as $g(x,p)$ for some function $g$.
I am assuming that we know the function $g$ exactly, and that we know the value of $M$, but we do not know the values of $x$ and $p$.
So, given the function $g$ and given $M$:

*

*Is there a general method that tells us whether an unbiased estimator for $g(x,p)$ exists?

*Is there a general method that can find an unbiased estimator for $g(x,p)$, whenever it exists?

*Is there a general method that tells us whether a consistent estimator for $g(x,p)$ exists?

*Is there a general method that can find a consistent estimator for $g(x,p)$, whenever it exists?

(It may be the case that, given the function $g$ and given $M$, an unbiased or consistent estimator can be found only provided that $x$ and $p$ belong to some subset of values. In this case, I also wonder about whether there are methods to find this set).

For the sake of completeness, let me provide some definitions:

*

*An estimator is a function $f(y_1,y_2,...)$ that takes as input a finite sequence of numbers $y_1,y_2,...\in \mathbb{R}$, and returns a value from $\mathbb{R}$. ($f$ does not depend on $p$ or $x$ defined above)


*The estimator is said to be an unbiased estimator of $g(x,p)$ (defined above) (for a specific $N$), if for all $x$ and $p$, the expected value of  $f(Y_1,Y_2,...,Y_N)$ equals $g(x,p)$, where $Y_1,Y_2,...,Y_N$ are independent random variables whose distribution is defined by $x$ and $p$ (see above).


*(informal) The estimator is said to be a consistent estimator of $g(x,p)$ (defined above), if for all $x$ and $p$, in the limit of large $N$, the expected value of $f(Y_1,Y_2,...,Y_N)$ has limit $g(x,p)$, where $Y_1,Y_2,...,Y_N$ are independent random variables whose distribution is defined by $x$ and $p$ (see above).
 A: Let $D$ be the distribution determined by $x$ and $p$, and let $\phi$ be the mean of $(g(x, p)_0, g(x, p)_1, ..., g(x, p)_N)$, where $g(x, p)_i$ are i.i.d. random variables distributed as $g(D)$.
The answer to your first question is given in Halmos (1946): An unbiased estimator for $\phi$ exists if and only if $\phi$ is a homogeneous function over $D$ of degree at most $N$. In particular, if $M$ is 2 and $x$ is {0, 1} (so that the distribution is Bernoulli), then the only functions for which an unbiased estimator exists are polynomials of degree at most $N$ (Wästlund 1999, Goyal and Sigman 2012; see also this answer).
However, if $N$ is unlimited and $M$ is 2, then $g$ has an unbiased estimator in some subset of (0, 1) if and only if $g$ is continuous and is identically 0, identically 1, or polynomially bounded away from 0 and 1 in that subset (Keane and O'Brien 1994, Łatuszyński et al. 2009/2011).
I am not aware of any results that show whether a consistent estimator exists for a function if and only if an unbiased estimator exists for that function.
For more general distributions, Jacob and Thiery (2015) investigates the following question: Given a sequence of unbiased estimators, can we turn that sequence into an unbiased estimator of $g$ that is nonnegative almost surely?  Without knowledge on what values $x$ can take on, the answer is no.  However, if upper and lower bounds on $x$ are known, for example, this is possible for certain functions $g$; see the paper for details.  See also Glynn (2016).
REFERENCES:

*

*Wästlund, J., "Functions arising by coin flipping", 1999.

*Goyal, V. and Sigman, K., 2012. On simulating a class of Bernstein polynomials. ACM Transactions on Modeling and Computer Simulation (TOMACS), 22(2), pp.1-5.

*Keane, M. S., and O'Brien, G. L., "A Bernoulli factory", ACM Transactions on Modeling and Computer Simulation 4(2), 1994.

*Jacob, P.E., Thiery, A.H., "On nonnegative unbiased estimators", Ann. Statist., Volume 43, Number 2 (2015), 769-784.

*Glynn, P.W., "Exact simulation vs exact estimation", Proceedings of the 2016 Winter Simulation Conference, 2016.

*Halmos, P.R., "The theory of unbiased estimation", Annals of Mathematical Statistics 17(1), 1946.

