  We consider unbiased estimation of a sparse nonrandom vector corrupted by
additive white Gaussian noise. We show that while there are infinitely many
unbiased estimators for this problem, none of them has uniformly minimum
variance. Therefore, we focus on locally minimum variance unbiased (LMVU)
estimators. We derive simple closed-form lower and upper bounds on the variance
of LMVU estimators or, equivalently, on the Barankin bound (BB). Our bounds
allow an estimation of the threshold region separating the low-SNR and high-SNR
regimes, and they indicate the asymptotic behavior of the BB at high SNR. We
also develop numerical lower and upper bounds which are tighter than the
closed-form bounds and thus characterize the BB more accurately. Numerical
studies compare our characterization of the BB with established biased
estimation schemes, and demonstrate that while unbiased estimators perform
poorly at low SNR, they may perform better than biased estimators at high SNR.
An interesting conclusion of our analysis is that the high-SNR behavior of the
BB depends solely on the value of the smallest nonzero component of the sparse
vector, and that this type of dependence is also exhibited by the performance
of certain practical estimators.
