$v_{jack}$ is always unbiased for any estimator $\hat{\theta}$ regardless of linearity.

$v_{jack}$ is consistent for smooth, differentiable statistics as $n \to \infty$.

$v_{jack}$ tends to zero as $n \to \infty$ for all statistics.

$v_{jack}$ is only valid when the sampling fraction $f = n/N$ is exactly 0.5.