Suppose we wish to estimate the probability $\theta \in(0,1)$ that a potentially biased coin lands heads up when tossed. After $n$ independent tosses, we observe $X$ heads.

(a) Write down the maximum likelihood estimator $\hat{\theta}$ of $\theta$.

(b) Find the mean squared error $f(\theta)$ of $\hat{\theta}$ as a function of $\theta$. Compute $\sup _{\theta \in(0,1)} f(\theta)$.

(c) Suppose a uniform prior is placed on $\theta$. Find the Bayes estimator of $\theta$ under squared error loss $L(\theta, a)=(\theta-a)^{2}$.

(d) Now find the Bayes estimator $\tilde{\theta}$ under the $\operatorname{loss} L(\theta, a)=\theta^{\alpha-1}(1-\theta)^{\beta-1}(\theta-a)^{2}$, where $\alpha, \beta \geqslant 1$. Show that

$$\tilde{\theta}=w \hat{\theta}+(1-w) \theta_{0},$$

where $w$ and $\theta_{0}$ depend on $n, \alpha$ and $\beta$.

(e) Determine the mean squared error $g_{w, \theta_{0}}(\theta)$ of $\tilde{\theta}$ as defined by $(*)$.

(f) For what range of values of $w$ do we have $\sup _{\theta \in(0,1)} g_{w, 1 / 2}(\theta) \leqslant \sup _{\theta \in(0,1)} f(\theta)$ ?

[Hint: The mean of a Beta $(a, b)$ distribution is $a /(a+b)$ and its density $p(u)$ at $u \in[0,1]$ is $c_{a, b} u^{a-1}(1-u)^{b-1}$, where $c_{a, b}$ is a normalising constant.]