*3.6. Parameter-Effects Curvatures*

The parameter-effects curvature and intrinsic curvature defined by Bates and Watts [21,25,26] are associated with a non-linear parameter estimation problem and are defined at the estimated parameter. We note that in (1), *<sup>h</sup>* : <sup>R</sup> <sup>→</sup> <sup>R</sup>. Since *<sup>h</sup>* is a scalar function, the intrinsic curvature of Bates and Watts *K*N(*x*ˆ) [21] or the direct intrinsic curvature *β*<sup>N</sup> *<sup>δ</sup>* (*x*ˆ) [29] is zero. Thus, only the parameter-effects curvature of Bates and Watts *K*T(*x*ˆ) and the direct parameter-effects curvature *β*<sup>T</sup> *<sup>δ</sup>* (*x*ˆ) are non-zero. Since the intrinsic curvature is zero, for simplicity in notation, we drop the superscript "T" from the parameter-effects curvature and they are given by

$$K(\hat{\mathfrak{x}}) := \frac{||\mathring{\mathbf{H}} \delta^2||}{||\mathring{\mathbf{H}} \delta||^2} = \frac{||\mathring{\mathbf{H}}||}{||\mathring{\mathbf{H}}||^2} \tag{50}$$

$$\beta\_{\delta}(\pounds) := \frac{||\H \delta^2||}{||\H \delta||} = \frac{||\H|| \delta|}{|||\H|||},\tag{51}$$

where

$$\mathbf{H} = \frac{d^2 \mathbf{h}(x)}{dx^2} \Big|\_{x=\pm'} \tag{52}$$

$$
\delta := \mathbf{x} - \mathbf{\hat{x}}.\tag{53}
$$

From (26), we get

$$
\ddot{\mathbf{H}} = a n(n-1) \hat{\mathbf{x}}^{n-2} \mathbf{d}.\tag{54}
$$

Hence, from (27) and (52), we obtain

$$<\langle |\hat{\mathbf{H}}|| = an\mathfrak{X}^{n-1}\sqrt{N}.\tag{55}$$

$$||\mathbf{H}|| = an(n-1)\mathfrak{x}^{n-2}\sqrt{N}.\tag{56}$$

Substitution of results from (55) and (56) in (50) and (51) gives

$$K(\mathfrak{X}) = \frac{n-1}{na\sqrt{N}} \frac{1}{\mathfrak{X}^{n\prime}} \tag{57}$$

$$\beta\_{\delta}(\mathfrak{X}) = \frac{(n-1)|\delta|}{\mathfrak{X}}.\tag{58}$$

We note that the extrinsic curvature in (36) is evaluated at the true *x*, while the parameter-effects curvatures *K*(*x*ˆ) in (50) and *βδ*(*x*ˆ) in (51) are evaluated at the estimate *x*ˆ. Because *x*ˆ is a random variable, *K*(*x*ˆ) and *βδ*(*x*ˆ) are random variables. When we perform Monte Carlo simulations and estimate *x* from measurements, *x*ˆ varies among Monte Carlo runs. Therefore, *K*(*x*ˆ) and the set of all linear *βδ*(*x*ˆ) vary with Monte Carlo runs.
