**5. Numerical Examples**

*5.1. Parameter Estimation in the Kalman Observation System*

Let us consider the linear Gaussian discrete-time (Kalman) observation system:

$$\begin{cases} X\_t = aX\_{t-1} + bV\_{t\prime} & t = \overline{1\_\prime}\overline{T}\_{\prime} \\ Y\_t = cX\_t + fW\_{t\prime} & t = \overline{0,T}\_{\prime} \end{cases} \tag{42}$$

where:


Our goal is to calculate the proposed minimax estimates of the uncertain vector *γ* and analyze their performance depending on the specific form of the loss function (6). To vary the loss function we can either specify the estimated test signal *h*(·) or determine different Euclidean weighted norms. We choose the second approach and define the following norm -·*<sup>ξ</sup>X*,*ξγ* for the compound vector: *Z* col(**X***T*, *γ*):

$$\|\|Z\|\|\_{\mathfrak{F}\_{X'\mathfrak{F}\_{Y}}} \stackrel{\triangle}{=} \sqrt{\mathfrak{f}\_X^2 \sum\_{t=1}^T X\_t^2 + \mathfrak{f}\_{\gamma}^2 (a^2 + b^2)}$$

and the corresponding loss function takes the form:

$$J\_{\mathbb{Z}\chi,\mathbb{Z}\_{\mathcal{T}}}(\mathbb{Z}\_{\prime}F|\mathbf{Y}\_{T}) \stackrel{\scriptstyle \Delta}{=} \mathbb{E}\_{F}\left\{||Z-\mathbb{Z}(\mathbf{Y}\_{T})\|\_{\mathbb{Z}\_{\mathcal{X}},\mathbb{Z}\_{\mathcal{T}}}^{2}|\mathbf{Y}\_{T}\right\}.\tag{43}$$

In the case *ξγ* = 1 and *ξ<sup>X</sup>* = 0 we obtain "the traditional" case of the mean-square loss conditional function *J*0,1(*Z*, *F*|**Y***T*) = E*<sup>F</sup>* ? *γ* − *γ*(**Y***T*)-<sup>2</sup>|**Y***<sup>T</sup>* @ , and the estimation quality of *γ*(·) is determined directly through the loss function. Using *ξγ* = 0 and *ξ<sup>X</sup>* = 1 we transform the loss function into *J*1,0(*Z*, *F*|**Y***T*) = E*<sup>F</sup>* ? -*X* − *X*(**Y***T*)-<sup>2</sup>|**Y***<sup>T</sup>* @ , and the estimation of *γ* appears indirectly via the estimation of the state trajectory **X***T*.

The minimax estimation is calculated by the numerical procedure introduced in Section 4.1 with the uniform mesh C*ha*,*hb* of the uncertainty set C; *ha* and *hb* are corresponding mesh steps along each coordinate.

We calculate the minimax estimate with the likelihood conformity constraint of the form:

$$\frac{\mathfrak{L}(\mathbf{Y}\_{T},F) - \min\_{(a,b)\in \mathcal{C}\_{h\_{b},h\_{b}}} \mathcal{L}(\mathbf{Y}\_{T}|(a,b))}{\max\_{(a,b)\in \mathcal{C}\_{h\_{b},h\_{b}}} \mathcal{L}(\mathbf{Y}\_{T}|(a,b)) - \min\_{(a,b)\in \mathcal{C}\_{h\_{b},h\_{b}}} \mathcal{L}(\mathbf{Y}\_{T}|(a,b))} \geqslant r\_{r}$$

where *r* ∈ (0, 1) is a confidence ratio.

We compare the proposed minimax estimate with some known alternatives.

The calculations have been executed with the following parameter values: C = [−0.1; 0.1] × [0.1; 1], *a* = −0.1, *b* = 0.1, *P*<sup>0</sup> = 0.5, *c* = 1, *f* = 0.5, *T* = 1000, *ha* = 0.01, *hb* = 0.045. The choice of the parameters can be explained by the following facts. First, the point (−0.1; 0.1) of actual parameter values belongs to the domain of the LFD for both loss functions *J*0,1 and *J*1,0. This means the appearance of just the LFD for both cases. Second, in spite of sufficient observation length, the signal-to-noise ratio is rather small, which prevents high performance of the asymptotic estimation methods.

Figure <sup>1</sup> presents the evolution of the minimax estimates *a*0,1(*r*) and *a*1,0(*r*) of a drift coefficient depending on the confidence ratio *r* ∈ (0, 1). The minimax estimates are compared with;

• The estimate *aMS*(*YT*) calculated by the moment/substitution method [12]:

$$\overline{\pi}^{\text{MS}} = \sum\_{t=1}^{T} y\_{t-1} y\_t \Big/ \left(\sum\_{t=1}^{T} y\_t^2 - T f^2\right), \quad \overline{\mathbb{B}}^{\text{MS}} = \sqrt{\frac{1}{c^2} \left(1 - (\overline{\pi}^{\text{MS}})^2\right) \left(\sum\_{t=1}^{T} y\_t^2 - T f^2\right)};$$


Figure <sup>2</sup> contains a similar comparison of the diffusion coefficient estimates *b*0,1(*r*) and *b*1,0(*r*).

**Figure 1.** Estimation of the drift coefficient *a*.

**Figure 2.** Estimation of the diffusion coefficient *b*.

The results of this experiment allow us to make the following conclusions.

