Asymptotically unbiased estimator using MLE

Question

I am learning Maximum likelihood estimators for a inference class. And this is a problem I came across.

Let $X_1,X_2,X_3,\ldots, X_n$ be a random sample with p.m.f $$p(X)=\theta(1-\theta)^x; x=0,1,2,\ldots\quad \mathrm{and}\quad 0<\theta<1$$.
As maximum Likelihood estimator I obtained $\hat{\theta}=\frac{1}{\bar{x}+1}$.

Now the question asks to obtain a unbiased estimator for $\theta$ when $n$ is large and considering it's distribution for large sample approximate 95% confidence interval for $\theta$.

What I am unable to do is to find a unbiased estimator. Then if $\hat\theta $ is unbiased for $\theta$ using the property that $\hat{\theta}\sim \mathrm{N}\left(\theta,{1\over I_x(\theta)}\right)$. I can construct a confidence interval.

But how can I find a unbiased estimator for $\theta$?
I thought since MLE's are not in general unbiased but are asymptotically unbiased to compute $\mathrm{E}\left(\frac{1}{\bar{x}+1}\right)$.

Here $\sum X_i$ follows a negative binomial distribution right?

Then $\mathrm{E}\left(\frac{1}{\bar{x}+1}\right) =\mathrm{E}\left(\frac{n}{T+n}\right)$ where T=$\sum X_i$.

Then I come up with $\sum n\left(\frac{1}{t+n}\right)\binom{n+t-1}{t} \theta^n(1-\theta)^t $. I am unable to show that this is asymptotically unbiased.

Please help me to find a unbiased estimator for $\theta$ when $n$ is large.

Answer 1

One unbiased estimator of $\theta$ is $I[X_1=0]$. The joint pmf is $\theta^n(1-\theta)^t$, where $t=\sum_i x_i$ is a sufficient statistic for $\theta$. By the Rao-Blackwell theorem, a better estimator, also unbiased, is $$\begin{aligned} \mathbb{E}\Big[ I[X_1=0] \Big| \sum_i X_i = t \Big] &= \sum_{k_{2:n}\geq0}I\big[{\textstyle\sum} k_i=t\big] \Big/ \sum_{k_{1:n}\geq0}I\big[{\textstyle\sum} k_i=t\big] \\&= \binom{t+n-2}{n-2} \Big/ \binom{t+n-1}{n-1} \\&= \frac{n-1}{t+n-1}, \qquad (n\geq2). \end{aligned}$$

Note also that $T=\sum_i X_i$ is a complete sufficient statistic, so by the Lehmann-Scheffe theorem, $\frac{n-1}{T+n-1}$ is the uniformly minimum variance unbiased estimator for $\theta$. When $n=1$, this is just $I[T=0]$.