On the Computational Methods in Non-linear Design of Experiments

Abstract

In this paper the non-linear problem is discussed, for point and interval computational estimation. For the interval estimation an adjusted formulation is discussed due to Beale’s measure of non-linearity. The non-linear experimental design problem is regarded when the errors of observations are assumed i.i.d. and normally distributed as usually. The sequential approach is adopted. The average-per-observation information matrix is adopted to the developed theoretical approach. Different applications are discussed and we provide evidence that the sequential approach might be the panacea for solving a non-linear optimal experimental design problem.

Appendix

1. Local Selection Criteria for NLM

• C1: Mallow’s C _p-statistic $C_p(\theta )=(\frac {RSS}{s^2})-n+2p$

• C2: Akaike’s criterion $AC(\theta )=nln(\frac {RSS}{n}+\frac {n(n+p)}{n-p-2}$

• C3: Weighted $WRSS(\theta )=\sum _{i}[\frac {(y_i-\hat {y}_i)^2}{1-W_i}]$

• C4: p—Weighted $WRSS_p(\theta )=\frac {WRSS(\theta )}{s^2}-n+2p$

• C5: B-criterion $B_{max}=\frac {RSS}{s^2}-(n-p)+3s^2B$

with W _i as defined in (9).

2. Proof of Proposition 5

Indeed from (19) for the true 𝜗, 𝜗 _t and the estimate $\hat {\vartheta }$ it holds for the corresponding sum of squares:

$$\displaystyle \begin{aligned} SS(\vartheta_t)-SS(\hat{\vartheta}){=}(\hat{\vartheta}-\vartheta_t)F'F(\hat{\vartheta}-\vartheta){=}(\hat{\vartheta}-\vartheta_t)'M^{-1}(\hat{\vartheta}-\vartheta)\leq Bps^2F(a;p,n-p) \end{aligned} $$

(29)

from ([9], pg.81) with B as in (26).

But B is as in [23] and we extend it to (28). We approximate B ₀ by the value of the intrinsic curvature of the model γ _max. This curvature provides a measure to “how good is the target-plane approximation,” in other words “how far from linear” is the model. But there is an upper bound of γ _max, see ([28], pg.135), namely

$$\displaystyle \begin{aligned} B_0\cong \gamma_{max} < \frac{1}{2}(F(a;p,n-p))^{-\frac{1}{2}}=\gamma_0. \end{aligned} $$

(30)

So, with p = 1, it is $F(a;1;n-p)=t_{n-p}^2$, i.e. $\gamma _0=\frac {1}{2}t(a;n-1)$, that is, $B=1+\frac {n}{n+1}B_0<1+\frac {n}{n-1}\frac {1}{2}t(a;n-1)\approx 2.2:=B_k$.

When p = 2, recall (26) and (30):

$$\displaystyle \begin{aligned} B=1+\frac{n(p+2)}{(n-p)p}B_0\cong 1+\frac{4n}{2(n-2)}\gamma_{max}<1+\frac{n}{n-2}\frac{1}{\sqrt{F}}=B_k. \end{aligned}$$

With p = 3 similar calculations provide evidence that $B_k=1+0.41\frac {n}{n-3}$ approximating $\sqrt {F}$ by a rough approximation 2 with n > 9 at a = 0.05.

Also with p ≥ 4 considering (29) and (30) the last part of (28) is defined, q.e.d.