**6. Conclusions**

We considered a polynomial curve in 2D and derived analytic expressions for the ML estimate and associated variance of the independent variable *x* using a vector measurement. The ML estimate is used to evaluate the Jacobian and Hessian of the measurement function appearing in the computation of Bates and Watts and direct parameter-effects curvatures, Beale- s MoN, and Linssen- s MoN. Our numerical results show that the variance of the estimated parameter and the Cramér-Rao lower bound (CRLB) are nearly the same for different powers of *x*. The average normalized estimation error squared (ANEES) lies within the 99% confidence interval, which indicates that the ML based variance is consistent with the estimation error.

We used seven MoNs, including the extrinsic curvature using differential geometry, Beale's MoN (and its least squares variant), Linssen's MoN (and its least squares variant), Bates and Watts parameter-effects curvature, direct parameter-effects curvature, Li's MoN, and the MoN of Straka, Duník, and Simandl. If a MoN has a high value, then the nonlinearity is high. All of the MoNs show ˘ the same type of variation with *x* and the power of of the polynomial. Secondly, as the logarithm of a MoN increases, the logarithm of the MSE also increases linearly for each MoN. This implies that, as a MoN increases, and then the MSE increases. These results are quite surprising, given the fact that these MoNs are derived based on completely different theoretical considerations. The second feature of our analysis is useful in establishing that a MoN in our study can be considered as a candidate metric for quantifying the MSE that represents the complexity of a parameter estimation problem. Our future work will study other practical parameter estimation and non-linear filtering problems.

**Author Contributions:** Formal analysis, M.M. and X.T.; methodology, M.M.; software, M.M. and X.T.; writing, M.M. and X.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The author thanks Sanjeev Arulampalam of Defence Science and Technology Organisation, Edinburgh SA, Australia for useful discussions and insightful comments in improving the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.
