How can I show that the following function is convex in which Z is a random variable?
$$\rho(Z)=\frac{2}{3}\mathrm{argmin}_{t}\{t+10\mathbf{E}[Z-t]_{+}\}+\frac{1}{3}\mathrm{argmin}_{t}\{t+5\mathbf{E}[Z-t]_{+}\}$$
I know that each argmin function taken seperatly is actually not convex. I am confused as to how to prove that the entire function is convex. The argmin is what is causing me difficulty.
First, we need to explain the notation $[\cdot]_+$. It is $$ [z]_+ = \max\{z,0\}. $$
If your functions was, instead, defined as $$ r(Z)=\tfrac{2}{3}\inf_{t}\{t+10\mathbb{E}[Z-t]_{+}\}+\tfrac{1}{3}\mathrm{argmin}_{t}\{t+5 \mathbb{E}[Z-t]_{+}\}, $$ it would make sense to ask whether it is convex because $r$ is not single-valued. In fact, this would be the weighted sum of two expected shortfall functionals and it is known to be convex for all $Z\in\mathcal{L}_p(\Omega, \mathcal{F}, \mathrm{P};\mathbb{R})$ with $p\in [1,\infty)$.
I tend to believe that you mean $\inf$ instead of $\arg\min$ since your expression is reminiscent of weighted sums of expected shortfall functionals which are very common in the theory of risk measures.
Your function $\rho$ is typically not single-valued. If we write it as $$\rho(Z) = \tfrac{2}{3} \rho_1(Z) + \tfrac{1}{3}\rho_2(Z),$$ where $$\rho_1(Z) = \arg\min_{t}\{t+10\mathbb{E}[Z-t]_{+}\},$$ and $$\rho_2(Z) = \arg\min_{t}\{t+5\mathbb{E}[Z-t]_{+}\}.$$ Then assume that the cumulative distribution function of $Z$ is some $H_Z$. Then $\rho_1(Z)$ is equal to the interval $$ \rho_1(Z) = [t_1^\star(Z), t_1^{\star\star}(Z)], $$ with $$ t_1^\star(Z) = \inf_{t}\{z: H_Z(z) \geq 1-\tfrac{1}{10}\} , $$ and $$ t_1^{\star\star}(Z) = \sup_{t}\{z: H_Z(z) \leq 1-\tfrac{1}{10}\} $$ And the same holds for $\rho_2(Z) = [t_2^\star(Z), t_2^{\star\star}(Z)]$.
In order for $\rho_1$ and $\rho_2$ to be single-valued we need $t_{i}^\star=t_i^{\star\star}$ and for the to happen $H_Z$ must be continuous (but, I believe that this would be a rather restrictive assumption). Then, $$ \rho_1(Z) = p_1 + 10 \mathbb{E}[Z-p_1]_+, $$ where $p_1 = H_Z^{-1}(1-\tfrac{1}{10})$ and similarly for $\rho_2$. In that case, yes, your function $\rho$ would be convex.