Ziv–Zakai Bound

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 $X$ from a noisy observation $Y$. 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 $X$ with the probability density $f_X$ from a noisy observation $Y$, then for any estimator $g$ a simple form of Ziv-Zakai bound is given by

\begin
& \mathbb\bigl IEEE Radar Conference
, year=2010
, pages=678–683
, publisher=IEEE

See also

* Information theory
* Signal processing

References

Estimation theory
Signal processing>X - g(Y), ^2\bigr\ge \frac \int_^ t
\int_^ \bigl(f_X(x) + f_X(x+t)\bigr)\, P_e(x, x+t)\,\mathrmx\,\mathrmt,
\end

where
$P_e(x, x+t)$ is the minimum (Bayes) error probability for the binary hypothesis testing problem between

$\begin \mathcal_0&: Y \mid X = x \\ \mathcal_1&: Y \mid X = x + t \end$

with prior probabilities
$\Pr(\mathcal_0) = \frac$
and
$\Pr(\mathcal_1) = 1 - \Pr(\mathcal_0)$.

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.

* quantum parameter estimation
* time delay estimation
* time of arrival estimation
* direction of arrival estimation
* MIMO radar {{cite conference
, last1=Chiriac
, first1=V. M.
, last2=Haimovich
, first2=A. M.
, title=Ziv–Zakai lower bound on target localization estimation in MIMO radar systems
, book-title=2010 IEEE Radar Conference
, year=2010
, pages=678–683
, publisher=IEEE

See also

* Information theory
* Signal processing

References

Estimation theory
Signal processing