I been reading the following text on randomized estimators, I cant manage to understand how the randomisation is incoparated into the randomized estimator.
How does the random mechanism fit in, where does it appear in the definition of the random estimator?
The answer posted is a good answer, I been updating the question since it was poorly formulated and that might be why the answer seams wierd in relation to the question.
Consider the following example.
You have a coin whose probability of landing heads when tossed is $\theta$. You know the precise and true value of $\theta$.
You give this coin to Experimenter 1, but you do not tell him $\theta$. He conducts an experiment consisting of tossing the coin $n$ times, and recording the results ($0$ for tails and $1$ for heads) as a vector $\boldsymbol x \in \{0,1\}^n$. This vector is the sample, and it is "original" in the sense that it comprises the entirety of the information gained from the experiment; namely $$\boldsymbol x = (x_1, x_2, \ldots, x_n), \quad x_i \in \{0,1\}$$ not only tells Experimenter 1 how many heads were obtained, but also which tosses were heads and which were tails.
Now suppose Experimenter 1 now tells Experimenter 2 the value $$T(\boldsymbol x) = \sum_{i=1}^n x_i.$$ Experimenter 2 does not flip the coin; he too is unaware of the true value of $\theta$, and he does not know which tosses were heads and which were tails from Experimenter 1's experiment. All he is told is the number of heads and the sample size $n$.
Does Experimenter $2$ have the same ability to make inferences on $\theta$ that Experimenter $1$ does? If $T(\boldsymbol x)$ is a sufficient statistic for $\theta$, the answer to this question is yes.
With respect to the "randomization" notion, what the author is trying to get at here is that what matters in regard to the estimation of $\theta$ is not the order in which the heads were obtained, but merely the number of heads observed. Therefore, Experimenter 2 can use a randomization device to simulate outcomes from the original sampling distribution that satisfy $T(\boldsymbol x) = y$. Concretely, if the coin was tossed $n = 17$ times and Experimenter 2 was told $T(\boldsymbol x) = 8$, then Experimenter 2 can create a simulation of a sample that has $8$ heads and $9$ tails; e.g., $$\boldsymbol x = (1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1).$$ Does this mean that this sample was what Experimenter 1 actually observed? Of course not. Experimenter 1 could have observed some other sample, but it doesn't matter. That $T(\boldsymbol x)$ is sufficient for $\theta$ is why it doesn't matter.
Now, it is important to remember that $T(\boldsymbol x)$ may be sufficient for some parameter $\theta$, but that does not mean it estimates $\theta$. Sufficiency pertains to data reduction, not estimation per se. However, estimators that have "good" properties (such as those that do not unnecessarily throw away information) are usually based on sufficient statistics.