I am trying to characterize the sensitivity of a function $f: R^N\to{}R$ to the perturbations in the input vector $\mathbb{x}=\left[x\_1,\dots{}x\_N\right]$. For that purpose, I evaluate Cramer-Rao bound for Gaussian i.i.d. arguments.


The function has a singularity at the point $\mathbb{x}\_0$ where $f(\mathbb{x}\_0)=0$, in the sense that $\|\nabla{}f\|\sim{}1/f$ as $\mathbb{x}\to\mathbb{x}\_0$


The Cramer-Rao bound then doesn't make sense, because it diverges at $\mathbb{x}\_0$, while the variance of $f$, obviously, remains bounded. What I am looking for, I guess, some type of "concentration of measure"/"deviation inequality"-type sharp bound on the variance of $f$. 


The literature on concentration of measure phenomenon is extensive and deals with fairly advanced topics, whereas I am looking for something rather more basic. If you could point towards some starting point, your help will be appreciated.



Concentration of measure and bounds on variance
Let $X\_i$ and $X'\_i$ be i.i.d random variables. Write $f = f(X\_1, \dots, X\_n),$ and define $$f\_i = f(X\_1, \dots, X'\_i, \dots, X\_n)$$ as the same function with the $i$th input replaced by the independent copy $X'\_i$. 


The [**Efron-Stein inequality**](http://www-stat.wharton.upenn.edu/~steele/Publications/PDF/AEifns.pdf) states that $$\operatorname{Var}(f) \le \tfrac 1 2\sum\_{i=1}^n \ \mathbb E(f - f\_i)^2.$$


Since your random variables are Gaussians, the system satisfies the [**Poincaré inequality**](http://www.stat.berkeley.edu/~sourav/talk_matrix2.pdf) too: $$\operatorname{Var}(F) \le C \ \mathbb E |\nabla f|^2.$$


