Draft:Ziv-Zakai bound

The Ziv–Zakai bound (named after Jacob Ziv and Moshe Zakai ^[1]) is used in theory of estimations to provide a lower bound on possible-probable error involving some random parameter ${\displaystyle X}$ from a noisy observation ${\displaystyle Y}$. 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 ^[2] subsequent of the first version which was published 1969. ^[1]

Simple Form of the Bound

Suppose we want to estimate a random variable ${\displaystyle X}$ with the probability density ${\displaystyle f_{X}}$ from a noisy observation ${\displaystyle Y}$, then for any estimator ${\displaystyle g}$ a simple form of Ziv-Zakai bound is given by^[1]

${\displaystyle {\begin{aligned}&\mathbb {E} {\bigl [}|X-g(Y)|^{2}{\bigr ]}\geq {\frac {1}{2}}\int _{0}^{\infty }t\int _{-\infty }^{\infty }{\bigl (}f_{X}(x)+f_{X}(x+t){\bigr )}\,P_{e}(x,x+t)\,\mathrm {d} x\,\mathrm {d} t,\end{aligned}}}$

where ${\displaystyle P_{e}(x,x+t)}$ is the minimum (Bayes) error probability for the binary hypothesis testing problem between

${\displaystyle {\begin{aligned}{\mathcal {H}}_{0}&:Y\mid X=x\\{\mathcal {H}}_{1}&:Y\mid X=x+t\end{aligned}}}$

with prior probabilities ${\displaystyle \Pr({\mathcal {H}}_{0})={\frac {f_{X}(x)}{f_{X}(x)+f_{X}(x+t)}}}$ and ${\displaystyle \Pr({\mathcal {H}}_{1})=1-\Pr({\mathcal {H}}_{0})}$.

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 ^[3]
• time delay estimation ^[4]
• time of arrival estimation ^[5]
• direction of arrival estimation ^[6]
• MIMO radar ^[7]

See also

• Information theory
• Signal processing
• Weiss–Weinstein bound

References

1. Ziv, J.; Zakai, M. (1969). "Some lower bounds on signal parameter estimation". IEEE Transactions on Information Theory. 15 (3): 386–391. doi:10.1109/TIT.1969.1054301.
2. Bell, K.; Steinberg, Y.; Ephraim, Y.; Van Trees, H. (1997). "Extended Ziv–Zakai lower bound for vector parameter estimation". IEEE Transactions on Information Theory. 43 (2): 624–637. doi:10.1109/18.556118.
3. Tsang, M. (June 2012). "Ziv–Zakai error bounds for quantum parameter estimation". Physical Review Letters. 108 (23): 230401. arXiv:1111.3568. Bibcode:2012PhRvL.108w0401T. doi:10.1103/PhysRevLett.108.230401. PMID 23003924. Retrieved 2025-02-16.
4. Mishra, K. V.; Eldar, Y. C. (2017). "Performance of time delay estimation in a cognitive radar". 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE. pp. 3141–3145.
5. Driusso, M.; Comisso, M.; Babich, F.; Marshall, C. (2015). "Performance analysis of time of arrival estimation on OFDM signals". IEEE Signal Processing Letters. 22 (7): 983–987. Bibcode:2015ISPL...22..983D. doi:10.1109/LSP.2014.2378994. hdl:11368/2830716.
6. Wen, S.; Zhang, Z.; Zhou, C.; Shi, Z. (2024). "Ziv–Zakai bound for DOA estimation with gain–phase error". ICASSP 2024 – 2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE. pp. 8681–8685.
7. Chiriac, V. M.; Haimovich, A. M. (2010). "Ziv–Zakai lower bound on target localization estimation in MIMO radar systems". 2010 IEEE Radar Conference. IEEE. pp. 678–683.

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Categories:

• Estimation theory
• Signal processing
• Mathematical inequalities