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Problem

Let $X_1,...,X_n$ be a random sample with Bernoulli distribution of parameter $p$. Consider the new parameter $\theta=p(1-p)$. Find the MLE of $\theta$ and show that is an asintotically unbiased estimator but not an unbiased estimator.

So, I know that the maximum likelihood function is $$L(x_1,...,x_n,\theta)=\prod_{i=1}^np^{x_i}(1-p)^{1-x_i}$$

I don't think that an adequate approach to find the point that maximizes this function in terms of $\theta$ is using derivatives, at least I don't see how to derive it in terms of $\theta$. I can't think of another approach so I would really appreciate suggestions or hints and see if I can go from there. Thanks in advance

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    $\begingroup$ Differentiating works pretty well since $\log L=n\bar x\log(p)+n(1-\bar x)\log(1-p)$ and $2p=1+v$ with $v^2=1-4\theta$. Thus, $v\partial_\theta v=-2$, $v\partial_\theta p=-1$ and $\partial_\theta\log L=n\left(\frac{\bar x}p-\frac{1-\bar x}{1-p}\right)\partial_\theta p$. Hence $\partial_\theta\log L=0$ if and only if $\frac{\bar x}p=\frac{1-\bar x}{1-p}$, that is, $p=\bar x$. Thus, $\hat\theta=\bar x(1-\bar x)$. $\endgroup$ – Did Dec 8 '16 at 20:01
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You might find quite interesting this Python implementation of the estimation of the Bernoulli distribution parameter using the maximum likelihood.

It maximumizes the log-likelihood function by looking for the root of the derivative (which is calculated with sympy).

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