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

  • 1
    $\begingroup$ 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

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).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.