*4.3. MSE and Direct Parameter-Effects Curvature*

The expression for the direct parameter-effects curvature *βδ*(*x*ˆ) [29,30] is given by (58). Similar to the previous section, we define

$$L\_{\beta}(\mathbf{x}) := \log\_{10} \left( E\{\beta\_{\delta}(\mathbf{f})\} \right). \tag{87}$$

Now, taking the expected value of *β*, we have

$$E\{\boldsymbol{\beta}(\hat{\boldsymbol{x}})\} \approx \frac{(n-1)}{\mathbf{x}} E\{ |\boldsymbol{\delta}| \} = \frac{(n-1)}{\mathbf{x}} E\{ |\hat{\boldsymbol{x}} - \boldsymbol{x}| \}. \tag{88}$$

The RHS of (88) can be simplified by assuming that *x*ˆ is unbiased and that it achieves the CRLB. Additionally, we approximate this error to be Gaussian and the variance of *x*ˆ is given in (29). Then,

$$E\{|\pounds-x|\} = \sqrt{\text{CRLB}\_x} \sqrt{\frac{2}{\pi}}.\tag{89}$$

Substituting (89) into (88) and using (32) for CRLB*x* we have

$$\mathbb{E}[\beta(\mathbf{t})] \approx \frac{(n-1)}{na} \sigma \sqrt{\frac{2}{N\pi}} \mathbf{x}^{-n}.\tag{90}$$

Thus,

$$L\_{\beta}(\mathbf{x}) \approx \log\_{10} \left[ \frac{(n-1)\sigma\sqrt{\frac{2}{N\pi}}}{na} \right] - n\log\_{10} \mathbf{x}.\tag{91}$$

From (91) and (81) we can write the affine mapping

$$L\_{\rm CRLB}(\mathbf{x}) = a\_1^{\beta} L\_{\beta}(\mathbf{x}) + a\_{0^{\prime}}^{\beta} \tag{92}$$

where

$$\begin{aligned} \alpha\_1^6 &= \frac{2(n-1)}{n},\\ \alpha\_0^6 &= \log\_{10}\left(\frac{\sigma^2}{Nn^2a^2}\right) - \frac{2n-2}{n}\log\_{10}\left[\frac{(n-1)\sigma\sqrt{2}}{na\sqrt{N\pi}}\right]. \end{aligned} \tag{93}$$

We also observe that *α<sup>β</sup>* <sup>1</sup> is positive and, hence, *Lβ*(*x*) and *L*CRLB(*x*) have the same sign of the non-zero slopes. As a result, *βδ*(*x*ˆ) and CRLB have the same sign of the non-zero slopes.
