I have given $(x_i)_{i=1}^n$ $d$-dimensional iid. random variables with $x_i\sim\mathcal N(0,I_d)$ and $y_i=|\langle x_i,\theta\rangle|$ with $\theta\in\mathbb R^d$.
First I have to assume that $\|\theta\|_2=1$ and define $\hat\theta$ as the maximal eigenvector of the matrix $\frac1n\sum_{i=1}^n y_i^2 x_i x_i^T$.
Now I have to prove that $\hat\theta$ is a consistent estimator of $\theta$.
My ideas:
That means I have to show that $\lim\limits_{n\to\infty} P(\|\hat\theta-\theta\|_2\geq\varepsilon)=0\quad \forall\theta\in\mathbb R^d$.
I guess the first thing to do is to find a explicit representation of $\hat\theta$ depending on $n$. Maybe the min-max theorem could be helpfull.
In the second part I have to provide an estimator for arbitrary $\theta\in\mathbb R^d$ based on the $\hat\theta$ from above and show its consistency.
For this case there was also a hint to construct a random matrix $Z$ for a pair $(x,y)$ such that $E[Z] =\sqrt\frac2\pi(\theta^*\otimes\theta^*+I_d)$.
The Exercise can similarly be found on page 257 in Wainwright, Exercise 8.7. I hope someone can help me. Greetings