Unbised estimator of Binomial Distribution

[leon1127]

[SOLVED] unbised estimator of Binomial Distribution

I have no idea how to find such an estimator
Suppose$$X_1, ..., X_n \sim Bern(p)$$
find an unbiased estimator of $$p^m, for m < n$$
Induction on m was a nasty mess that should not be expected. The power of m causes some problem when I try to go from the definition of expectation. Any one have some hint?

 

[EnumaElish]

Statistic u is unbiased if E = p^m. Suppose m = 2. To find u, you count all sequences that have two successes in a row, then express them as a ratio of the total number of trials. Would that satisfy E = p^2?

 

[leon1127]

I approached in a similar way.
It is clear that $$Y = X_1...X_m$$ is unbiased for p^m. I was asked to find an unbiased estimator that is a function of sample total. Thus I construct a R.V which
$$Z(T) = 1 for T = 26; 0 elsewhere$$
I might have to normalise Z so that it is the same as Y. However I don't know how to sum it.

 

[EnumaElish]

Where did 26 come from?

If you look at my previous post you will see that it is a function of the sample total.

 

[leon1127]

it is not 26, it should be m...

 

[EnumaElish]

Very well, my previous post applies.

 

[leon1127]

k gotcha thx