# estimating a parametric function

I am working on this problem and am stumped. Can anyone take a look at it?

$X_1, \ldots ,X_n$ are distributed Bernoulli$(p)$ where $p$ is unknown. Consider the parametric function $\tau(p)=(p+(1-p)e^3)^2$.

1.) Find a suitable unbiased estimator of $\tau(p)$.

2.) Since the complete sufficient statistic is $\sum_{i=1}^n X_i$, use the Lehman-Scheffe theorem and evaluate $E_p[T|U=u]$.

There were two ways I looked at it. Immediately apparent is that the MGF of Bernoulli looks a lot like it, taken at $t=3$. But the problem there is that $p$ and $(1-p)$ are switched around. So I wasn't sure where to go from there.

The other way I looked at it is distributing it out, and I got $p^2+2p(1-p)e^3+(1-p)^2e^6$

I could find an unbiased estimator of this, but that conditional expectation looks nasty...could anyone start me out? I really have no clue about this problem.

Since the probabilities are flipped you can look instead at $1 - X_i$. An unbiased estimator then could be

$$T = e^{3(1 - X_1) + 3(1 - X_2)}$$

since we have

\begin{align} \text{E}(T) &= \text{E}(e^{3(1 - X_1) + 3(1 - X_2)}) \\ &= \text{E}(e^{3(1 - X_1)})^2 \\ &= (p + (1 - p) e^3)^2. \end{align}

Finding the conditional expectation is the challenging part but it's not impossible. We just use the fact that conditional on $S = \sum_{i=1}^{n} X_i$ all $\binom{n}{S}$ combinations are equally likely, and consider how they affect $X_1$ and $X_2$.

There are three cases to consider: when both $X_1$ and $X_2$ are zero, when both are one, and when one is zero and the other one. There are $\binom{n-2}{S}$ combinations corresponding to $X_1 = 0$ and $X_2 = 0$, $\binom{n-2}{S-2}$ corresponding to $X_1 = 1$ and $X_2 = 1$, and $2 \binom{n - 2}{S - 1}$ consistent with $X_1 = 1$ and $X_2 = 0$ or $X_1 = 0$ and $X_2 = 1$. Collecting everything together we get

\begin{align} \text{E}(T \mid S) &= \frac{\binom{n-2}{S} e^6 + \binom{n-2}{S-2} + 2 \binom{n - 2}{S - 1} e^3}{\binom{n}{S}} \end{align}

where we set $\binom{n - 2}{k} = 0$ if $k < 0$.