*4.5. Estimation of CMoN and SMSE by Monte-Carlo Simulations*

Let *K*(*x*ˆ*k*) and *βδ*(*x*ˆ*k*) denote the sample means of the Bates and Watts and direct parameter-effects curvatures calculated from *M* Monte Carlo runs. Subsequently,

$$\mathbb{X}(\mathfrak{x}\_k) := \frac{1}{M} \sum\_{m=1}^{M} \mathbb{X}(\mathfrak{x}\_{k,m}), \quad k = 1, 2, \dots, N\_{\mathbf{x}\_{\mathbf{x}}} \tag{100}$$

$$\overline{\beta}\_{\delta}(\pounds\_{k}) := \frac{1}{M} \sum\_{m=1}^{M} \beta\_{\delta}(\pounds\_{k,m}), \quad k = 1, 2, \dots, N\_{\text{x}}.\tag{101}$$

Correspondingly, we define

$$b\_k := \log\_{10} \text{SMSE}\_k, \quad k = 1, 2, \dots, N\_{\text{x}}.\tag{102}$$

$$\mathfrak{c}\_{k} := \log\_{10} \mathbb{Z}(\mathfrak{k}\_{k}), \quad k = 1, 2, \dots, N\_{\mathbf{x}}.\tag{103}$$

$$d\_k := \log\_{10} \overline{\beta}\_{\delta}(\mathfrak{X}\_k), \quad k = 1, 2, \dots, N\_{\mathfrak{X}}.\tag{104}$$

Define

$$\mathbf{b} := \begin{bmatrix} b\_1 & b\_2 & \dots & b\_{N\_x} \end{bmatrix}',\tag{105}$$

$$\mathbf{c} := \begin{bmatrix} c\_1 & c\_2 & \dots & c\_{N\_x} \end{bmatrix}^\prime \tag{106}$$

$$\mathbf{d} := \begin{bmatrix} d\_1 & d\_2 & \dots & d\_{N\_x} \end{bmatrix}'. \tag{107}$$

Suppose that an affine mapping exists between **b** and **c**. Subsequently,

$$b\_k = \mathbb{A}\_1^K \mathbf{c}\_k + \mathbb{A}\_0^K + \mathbf{c}\_k, \quad k = 1, 2, \dots, N\_{\mathbf{x}\_\prime} \tag{108}$$

where *ek* is a random noise. Afterwards, we can write (108) in the matrix-vector form by

$$\mathbf{b} = \mathbf{H}\_c \mathbf{a} + \mathbf{e},\tag{109}$$

where

$$\mathfrak{a} := \left[ \begin{array}{cc} \mathfrak{a}\_1^K & \mathfrak{a}\_0^K \end{array} \right]',\tag{110}$$

$$\mathbf{e} := \begin{bmatrix} e\_1 & e\_2 & \dots & e\_{N\_x} \end{bmatrix}' \, \, \, \, \tag{111}$$

$$\mathbf{H}\_{\mathbf{c}} := \begin{bmatrix} c\_1 & 1 \\ c\_2 & 1 \\ \dots & \dots \\ c\_{N\_x} & 1 \end{bmatrix}. \tag{112}$$

Given **b** and **H***c*, we can estimate *α* using the linear least squares (LLS).

We can similarly define the affine mapping between other variable pairs. Altogether, we consider the following four:

