Draft:Ziv-Zakai Bound

The Ziv–Zakai bound is a theoretical bound used in estimation theory which provides a lower bound on the estimation error of estimation some random parameter $X$ from a noisy observation $Y$. The bound work by connecting probability of the excess error to the hypothesis testing. The Ziv–Zakai bound was first introduced by Jaco Ziv and Moshe Zakai in 1969. The bound is considered to be tighter than Cramer-Rao bound albeit more involved. Several modern version of the bound have been subsequently introduced.

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

$\begin{array}{cc}& 𝔼[|X-g(Y){|}^2]\ge \frac{1}{2}\underset{0}{\overset{\mathrm{\infty }}{\int }}t\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(f_X(x)+f_X(x+t))P_e(x,x+t)\mathrm{d}x\mathrm{d}t,\end{array}$

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

$\begin{array}{cc}{\mathscr{H}}_0& :Y\mid X=x\\ {\mathscr{H}}_1& :Y\mid X=x+t\end{array}$

with prior probabilities $\mathrm{Pr}({\mathscr{H}}_0)=\frac{f_X(x)}{f_X(x)+f_X(x+t)}$ and $\mathrm{Pr}({\mathscr{H}}_1)=1-\mathrm{Pr}({\mathscr{H}}_0)$.

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

Related bounds or concepts

• Cramér–Rao bound
• Weiss–Weinstein bound

See also

• Estimation theory
• Signal processing
• Information theory