The Ziv–Zakai bound (named after Jacob Ziv and Moshe Zakai) is used in theory of estimations to provide a lower bound on possible-probable error involving some random parameter formula_1 from a noisy observation formula_2. The bound work by connecting probability of the excess error to the hypothesis testing. The bound is considered to be tighter than Cramér–Rao bound albeit more involved. Several modern version of the bound have been introduced subsequent of the first version which was published 1969.
Simple Form of the Bound.
Suppose we want to estimate a random variable formula_3 with the probability density formula_4 from a noisy observation formula_2, then for any estimator formula_6 a simple form of Ziv-Zakai bound is given by
formula_7
where 
formula_8 is the minimum (Bayes) error probability for the binary hypothesis testing problem between
formula_9
with prior probabilities 
formula_10 
and 
formula_11.
Applications.
The Ziv-Zakai bound has several appealing advantages. Unlike the
other bounds, in fact, the Ziv-Zakai bound only requires one regularity
condition, that is, the parameter under estimation needs to
have a probability density function; this is one of the
key advantages of the Ziv-Zakai bound . Hence, the Ziv-Zakai bound has a broader
applicability than, for instance, the "Cramér-Rao bound", which
requires several smoothness assumptions on the probability density function of the
estimand.