Let's define the nonnegative random variable $X$ as 
$$ X := |\hat \theta - \theta| \geq 0 .$$ 

[Markov's inequality](https://en.wikipedia.org/wiki/Markov%27s_inequality) states that: if $X$ is a nonnegative random variable and $a > 0$, then 
$$ \operatorname {Pr} (X\geq a)\leq {\frac {\operatorname {E} (X)}{a}}. $$

Therefore: 

**Result 1** 
$$ {\operatorname {E}[ |\hat \theta - \theta| ]} \geq a \cdot \operatorname {Pr} (|\hat \theta - \theta| \geq a). $$ 


A second result uses the [corollary to Markov on the quantile function](https://en.wikipedia.org/wiki/Markov%27s_inequality#Other_corollaries) which states that: for a nonnegative random variable $X$, the [quantile function](https://en.wikipedia.org/wiki/Quantile_function) of $X$ satisfies 
$$ Q_{X}(1-p) \leq {\frac {\operatorname {E} (X)}{p}}, \quad 0 < p < 1. $$
Therefore: 

**Result 2**
$$ {\operatorname {E} [|\hat \theta - \theta|]} \geq p \cdot Q_{|\hat \theta - \theta|}(1-p), \quad 0 < p < 1. $$