Table 14 Validation theorem conditions using the Allsvenskan dataset.

Non-robust estimator

Theorem 1

-

\(\sum _{j=1}^{r}\frac{1}{\omega _{j}} - \frac{\omega _j^2{(\omega }_j+d)^2\psi _{jj}}{(\omega _j+k)^2(\omega _j+1)^2}\)

35.307

Satisfied

Theorem 2

k=\({\hat{k}}_1\)

\(\sum _{j=1}^{r}\frac{\omega _j}{(\omega _j+k)^2} - \frac{\omega _j^2{(\omega }_j+d)^2\psi _{jj}}{(\omega _j+k)^2(\omega _j+1)^2}\)

21.202

Satisfied

k=\({\hat{k}}_2\)

\(\sum _{j=1}^{r}\frac{\omega _j}{(\omega _j+k)^2} - \frac{\omega _j^2{(\omega }_j+d)^2\psi _{jj}}{(\omega _j+k)^2(\omega _j+1)^2}\)

13.999

Satisfied

Theorem 3

k=\({\hat{k}}_1\), d=\({\hat{d}}\)

\(\sum _{j=1}^r \frac{\omega _j}{(\omega _j+k(d+1))^2} - \frac{\omega _j^2{(\omega }_j+d)^2\psi _{jj}}{(\omega _j+k)^2(\omega _j+1)^2}\)

13.999

Satisfied

k=\({\hat{k}}_2\), d=\({\hat{d}}\)

\(\sum _{j=1}^r \frac{\omega _j}{(\omega _j+k(d+1))^2} - \frac{\omega _j^2{(\omega }_j+d)^2\psi _{jj}}{(\omega _j+k)^2(\omega _j+1)^2}\)

12.783

Satisfied

Theorem 4

k=\({\hat{k}}_1\), d=\({\hat{d}}\)

\(\sum _{j=1}^r \frac{\omega _j{(\omega }_j+d)^2}{(\omega _j+k)^2(\omega _j+1)^2} - \frac{\omega _j^2{(\omega }_j+d)^2\psi _{jj}}{(\omega _j+k)^2(\omega _j+1)^2}\)

16.492

Satisfied

k=\({\hat{k}}_2\), d=\({\hat{d}}\)

\(\sum _{j=1}^r \frac{\omega _j{(\omega }_j+d)^2}{(\omega _j+k)^2(\omega _j+1)^2} - \frac{\omega _j^2{(\omega }_j+d)^2\psi _{jj}}{(\omega _j+k)^2(\omega _j+1)^2}\)

15.413

Satisfied

Robust estimator

Theorem 5

-

\(\sum _{j=1}^{r}\psi _{jj} - \frac{\omega _j^2{(\omega }_j+d)^2\psi _{jj}}{(\omega _j+k)^2(\omega _j+1)^2}\)

14.439

Satisfied

Theorem 6

k=\({\hat{k}}_1\)

\(\sum _{j=1}^{r}\frac{\omega _j^2\psi _{jj}}{(\omega _j+k)^2} - \frac{\omega _j^2{(\omega }_j+d)^2\psi _{jj}}{(\omega _j+k)^2(\omega _j+1)^2}\)

12.489

Satisfied

k=\({\hat{k}}_2\)

\(\sum _{j=1}^{r}\frac{\omega _j^2\psi _{jj}}{(\omega _j+k)^2} - \frac{\omega _j^2{(\omega }_j+d)^2\psi _{jj}}{(\omega _j+k)^2(\omega _j+1)^2}\)

11.365

Satisfied

Theorem 7

k=\({\hat{k}}_1\), d=\({\hat{d}}\)

\(\sum _{j=1}^r \frac{\omega _j^2\psi _{jj}}{(\omega _j+k(d+1))^2} - \frac{\omega _j^2{(\omega }_j+d)^2\psi _{jj}}{(\omega _j+k)^2(\omega _j+1)^2}\)

11.365

Satisfied

k=\({\hat{k}}_2\), d=\({\hat{d}}\)

\(\sum _{j=1}^r \frac{\omega _j^2\psi _{jj}}{(\omega _j+k(d+1))^2} - \frac{\omega _j^2{(\omega }_j+d)^2\psi _{jj}}{(\omega _j+k)^2(\omega _j+1)^2}\)

11.566

Satisfied