Problem 1
Assume the linear model:
$$ Y = X \beta + \epsilon$$
where $X’X=I$ and $\epsilon \sim N(0,\sigma^2I)$
Find the numerical solution for the elastic net in the form:
$$\hat{\beta_{en}} = argmin_b \frac{1}{2} |Y-Xb|2^2 + \lambda\biggl(\frac{1}{2}(1-\alpha)|b|2^2 + \alpha\sum\limits{i=1}^p|b_i| \biggr)$$
$$\hat{\beta{en}} = argmin_b \frac{1}{2} |Y-Xb|_2^2 + \lambda\biggl(\frac{1}{2}(1-\alpha)|b|2^2 + \alpha\sum\limits{i=1}^p|b_i| \biggr)$$
- What would be the value of the elastic net estimator with $\lambda = 1$ and $\alpha = 0.5$ if $\hat{\beta}_{OLS} = 3?$
- How does the number of discoveries depend on the parameter $\alpha$
- Provide the numerical value for the expected number of false discoveries when $n = p = 1000$, $p_0 = 950$, $\sigma = 1$, and $\lambda = 2$ , and the power of detection of $X_1$ when $\beta_1 = 3$
