Estimator for a binomial distribution How do we define an estimator for data coming from a binomial distribution? For bernoulli I can think of an estimator estimating a parameter p, but for binomial I can't see what parameters to estimate when we have n characterizing the distribution?
Update:
By an estimator I mean a function of the observed data. An estimator is used to estimate the parameters of the distribution generating the data.
 A: Say you have data $k_1, \dots, k_m \sim \text{iid binomial}(n, p)$.
You could easily derive method-of-moment estimators by setting $\bar{k} = \hat{n}\hat{p}$ and $s_k^2 = \hat{n}\hat{p}(1-\hat{p})$ and solving for $\hat{n}$ and $\hat{p}$.
Or you could calculate MLEs (perhaps just numerically), eg using optim in R.
A: I guess what you are looking for is the probability generating function. A derivation of the probability generating function of the binomial distribution can be found under 
http://economictheoryblog.com/2012/10/21/binomial-distribution/
However, having a look at Wikipedia is nowadays always a good idea, although I have to say that the specification of the binomial could be improved.
https://en.wikipedia.org/wiki/Binomial_distribution#Specification
A: Every distribution have some unknown parameter(s). For example in the Bernoulli distribution has one unknown parameter probability of success (p). Likewise in the Binomial distribution has two unknown parameters n and p. It depends on your objective which unknown parameter you want to estimate. you can fix one parameter and estimation other one. For more information see this
A: I  think we could use method of moments estimation to estimate the parameters of the Binomial distribution by the mean and the variance.  

Using the method of moments estimation to estimate The parameters $p$ and $m$.
[{\hat{p}}_n=\frac{\overline{X}-S^2}{\overline{X}}][\hat{m}_n=\frac{\overline{X}^2}{\overline{X}-S^2}]
Proof
 The estimators of the parameters $m$ and $p$ by the Method of Moments are the solutions of the system of equations
$$mp =\bar{X},\quad  mp(1-p) = S^2.$$
Hence our equations for the method of moments are:
[\overline{X}=mp]
[S^2=mp (1-p).]
Simple arithmetic shows:
[S^2 = mp\left(1 - p\right) = \bar{X}\left(1 - p\right)]
[S^2=\bar{X}-\bar{X} p]
[\bar{X}p=\bar{X}-S^2, \mbox{ therefore } \hat{p}=\frac{\bar{X}-S^2}{\bar{X}}.]
Then,
[\bar{X} = mp, \mbox{ that is, }  m \left(\frac{\bar{X}-S^2}{\bar{X}}\right)]
[\bar{X}=m\left(\frac{\bar{X}-S^2}{\bar{X}}\right), \mbox{ or } \hat{m}=\frac{\bar{X}^2}{\bar{X}-S^2}. ]
