On derivative based global sensitivity criteria

Consider a mathematical model $f(x)$ defined in the $n$-dimensional unit cube, $x=(x_1,\dots,x_n)$. How to estimate the global sensitivity of $f(x)$ with respect to $x_i$? If $f(x)\in L_2$, global sensitivity indeces provide practical answers to the question. Derivative based criteria are less reliable but sometimes easier for computing.
In this note a new derivative based global sensitivity criterion is compared with the correspondding global sensitivity index. It is proved that in the special case when $f(x)$ is a linear function of $x_i$, the estimates are equal. However the Monte Carlo approximations to the derivative based criterion converge faster.
Thus the derivative based criterion may be useful in situations when the dependence of $f(x)$ on $x_i$ is near to linear. It can also be applied for detecting nonessential variables $x_i$.