Draft:Ziv-Zakai bound
{{Short description|A theoretical bound used in estimation theory}}
{{Draft topics|mathematics}}
{{AfC topic|other}}
{{AfC submission|||ts=20250223184611|u=BobyKrasav|ns=2}}
The Ziv–Zakai bound (named after Jacob Ziv and Moshe Zakai {{cite journal
|last1=Ziv
|first1=J.
|last2=Zakai
|first2=M.
|title=Some lower bounds on signal parameter estimation
|journal=IEEE Transactions on Information Theory
|volume=15
|issue=3
|pages=386–391
|year=1969
|doi=10.1109/TIT.1969.1054301
}}) is used in theory of estimations to provide a lower bound on possible-probable error involving some random parameter from a noisy observation . 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 {{cite journal
|last1=Bell
|first1=K.
|last2=Steinberg
|first2=Y.
|last3=Ephraim
|first3=Y.
|last4=Van Trees
|first4=H.
|title=Extended Ziv–Zakai lower bound for vector parameter estimation
|journal=IEEE Transactions on Information Theory
|volume=43
|issue=2
|pages=624–637
|year=1997
|doi=10.1109/18.556118
}} subsequent of the first version which was published 1969.
Simple Form of the Bound
Suppose we want to estimate a random variable with the probability density from a noisy observation , then for any estimator a simple form of Ziv-Zakai bound is given by
\begin{aligned}
& \mathbb{E}\bigl[|X - g(Y)|^2\bigr] \ge \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
is the minimum (Bayes) error probability for the binary hypothesis testing problem between
\begin{aligned}
\mathcal{H}_0&: Y \mid X = x \\
\mathcal{H}_1&: Y \mid X = x + t
\end{aligned}
with prior probabilities
and
.
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 {{cite journal
|last=Tsang
|first=M.
|title=Ziv–Zakai error bounds for quantum parameter estimation
|journal=Physical Review Letters
|volume=108
|page=230401
|date=June 2012
|issue=23
|url=https://link.aps.org/doi/10.1103/PhysRevLett.108.230401
|doi=10.1103/PhysRevLett.108.230401
|pmid=23003924
|arxiv=1111.3568
|bibcode=2012PhRvL.108w0401T
|access-date=2025-02-16
}}
- time delay estimation {{cite conference
|last1=Mishra
|first1=K. V.
|last2=Eldar
|first2=Y. C.
|title=Performance of time delay estimation in a cognitive radar
|book-title=2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
|year=2017
|pages=3141–3145
|publisher=IEEE
}}
- time of arrival estimation {{cite journal
|last1=Driusso
|first1=M.
|last2=Comisso
|first2=M.
|last3=Babich
|first3=F.
|last4=Marshall
|first4=C.
|title=Performance analysis of time of arrival estimation on OFDM signals
|journal=IEEE Signal Processing Letters
|volume=22
|issue=7
|pages=983–987
|year=2015
|doi=10.1109/LSP.2014.2378994
|bibcode=2015ISPL...22..983D
|hdl=11368/2830716
}}
- direction of arrival estimation {{cite conference
|last1=Wen
|first1=S.
|last2=Zhang
|first2=Z.
|last3=Zhou
|first3=C.
|last4=Shi
|first4=Z.
|title=Ziv–Zakai bound for DOA estimation with gain–phase error
|book-title=ICASSP 2024 – 2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
|year=2024
|pages=8681–8685
|publisher=IEEE
}}
- 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
}}