The Cramer-Rao bound is well-known: for an unbiased estimator $\hat{\theta}$, $$\text{var}(\hat\theta(X_1,...,X_n)) \geq\frac{1}{n}\mathcal{I}(\theta)^{-1}),$$ where $ X_1,...,X_n\sim P_\theta$ i.i.d., and $\mathcal{I}(\theta)$ is the Fisher information (Fisher info matrix, when $\theta$ is high dim).
But is there any anti-concentration version of this. Heuristically, say usually we can expect $\hat\theta(X_1,...,X_n)$ to be asymptotically a Gaussian (for example when $\hat\theta\frac{X_1+...+X_n}{n}$), so we should expect something like $$P(\hat\theta-\theta>t)\gtrsim\exp(-\frac{nt^2}{2\mathcal{I}}).$$ Since we can truncate $\hat\theta$, e.g. clampped to [-1,1], I guess we should only expect the above inequality holds for small $t$.
The only thing I found is this wiki https://en.wikipedia.org/wiki/Concentration_inequality#Anti-concentration_inequalities, but does not seems to address my question. Is there any references/results related to this?