The derivation of the Cramér–Rao lower bound in [Kay](https://books.google.com/books?id=W7BWngEACAAJ "Fundamentals of statistical signal processing: Estimation theory") uses the weighted Cauchy–Schwarz inequality: 
$$
\left[ \int w(\mathbf{x})g(\mathbf{x})h(\mathbf{x})d\mathbf{x} \right]^2 \leq \int w(\mathbf{x})g^2(\mathbf{x})d\mathbf{x} \int w(\mathbf{x}) h^2(\mathbf{x}) d\mathbf{x}
$$

where $g$ and $h$ are arbitrary scalar functions, and $w(\mathbf{x}) \geq 0$ for all $\mathbf{x}$. 

Instead, we can use Hölder's more general inequality:
$$
\left| \int w(\mathbf{x})g(\mathbf{x})h(\mathbf{x})d\mathbf{x} \right| \leq \left( \int w(\mathbf{x})\left|g(\mathbf{x})\right|^pd\mathbf{x}\right)^{\frac{1}{p}} \left(\int w(\mathbf{x}) \left|h(\mathbf{x}) \right|^q d\mathbf{x} \right)^\frac{1}{q}
$$
where $\frac{1}{p}+\frac{1}{q} = 1$. Cauchy's inequality is the special case $p = q = 2$.

If the estimator is unbiased:
$$
E[\hat{\theta}] = \theta 
$$
or
$$
\int \hat{\theta} \: p(\mathbf{x};\theta) \: d\mathbf{x} = \theta
$$

differentiating with respect to $\theta$ and using $\dfrac{\partial p(\mathbf{x}; \theta)}{\partial \theta} = \dfrac{\partial \ln p(\mathbf{x};\theta)}{\partial \theta} p(\mathbf{x};\theta) $ yields:

$$
\int \hat{\theta} \: \dfrac{\partial \ln p(\mathbf{x}; \theta)}{\partial \theta} p(\mathbf{x}; \theta) \: d\mathbf{x} = 1.
$$

$\hat{\theta}$ in this expression can be replaced with $(\hat{\theta} - \theta)$ because the CRLB assumes the regularity condition $\displaystyle E\left[\frac{\partial \ln p(\mathbf{x}; \theta)}{\partial \theta}\right] = 0$, so 
$
\displaystyle \int \theta \: \dfrac{\partial \ln p(\mathbf{x}; \theta)}{\partial \theta} p(\mathbf{x}; \theta) \: d\mathbf{x} = \theta \: E\left[\dfrac{\partial \ln p(\mathbf{x}; \theta)}{\partial \theta}\right] = 0
$ and then we have:

$$
\int (\hat{\theta} - \theta) \: \dfrac{\partial \ln p(\mathbf{x}; \theta)}{\partial \theta} p(\mathbf{x}; \theta) \: d\mathbf{x} = 1.
$$

Using Hölder's inequality with $w(\mathbf{x}) = p(\mathbf{x};\theta)$, $g(\mathbf{x}) = \hat{\theta} - \theta$, and $h(\mathbf{x}) = \dfrac{\partial \ln p(\mathbf{x};\theta}{\partial \theta}$

$$
1 \leq \left(\int \left|(\hat{\theta} - \theta)\right|^p p(\mathbf{x}; \theta) \: d\mathbf{x} \right)^\frac{1}{p} \left(\int \left| \frac{\partial \ln p(\mathbf{x}; \theta)}{\partial \theta} \right|^q p(\mathbf{x}; \theta) d\mathbf{x} \right)^\frac{1}{q}.
$$

In the limit $p \rightarrow 1$, $q \rightarrow \infty$
$$
\lim_{p \to 1}  \left(\int \left|(\hat{\theta} - \theta)\right|^p p(\mathbf{x}; \theta) \: d\mathbf{x} \right)^\frac{1}{p} = E\left[ \left|(\hat{\theta} -\theta)\right| \right]
$$

$$
\lim_{q \to \infty} \left(\int \left| \frac{\partial \ln p(\mathbf{x}; \theta)}{\partial \theta} \right|^q p(\mathbf{x}; \theta) d\mathbf{x} \right)^\frac{1}{q} = \sup \left| \frac{\partial \ln p(\mathbf{x}; \theta)}{\partial \theta} \right|.
$$

Rearranging:
$$
E\left[ \left|(\hat{\theta} -\theta)\right| \right] \geq \frac{1}{\sup\left| \frac{\partial \ln p(\mathbf{x}; \theta)}{\partial \theta} \right|}.
$$

The expectation in the denominator is no longer the Fisher Information $I(\theta)$, but the supremum of $\left| \frac{\partial \ln p(\mathbf{x}; \theta)}{\partial \theta} \right|$.