Unbiasedness and Consistency of the following Estimator

Question

Let

$T(x)=\frac{\sum_{i=1}^{n}X_{i}+\frac{\sqrt{n}}{2}} {n+\sqrt{n}}$

Where, $X_{i}$'s are from $B(1,\theta)$ or Bernaulli. Then, comment on the Unbiasedness and Consistency of the given estimator.

My approach

I think the given estimator is biased and Consistent. The Estimator is biased which directly follows from structure of estimator. As far as consistency is concerned, the estimator can be written in the following form:

$T(x)=\frac{\overline{X}}{1+\frac{1}{\sqrt{n}}}+\frac{\sqrt{n}}{2(n+\sqrt{n})}=a_n\overline{X}+b_n$

$\overline{X}$ is consistent for $\theta$ and $a_n \rightarrow 1$ and $b_n \rightarrow 0$. Hence, the given estimator is consistent.

But my manual, gives the answer that given estimator is biased and inconsistent. How can this be possible? I hope I used the theorem correctly. Thanks

Answer 1

If $X_i \sim B(1,\theta)$, then $E[X_i] = \dfrac{1}{1+\theta}$. So $\bar{X}$ is consistent for $\dfrac{1}{1+\theta}$, not $\theta$.

EDIT: Assuming $B(1,\theta)$ means Bernoulli($\theta$), then I don't think there's anything wrong with your proof. Are you sure the manual is correct?

EDIT 2: This is a duplicate question.