Polynomial based impact

As a next step we investigate the confidence intervals based on the shrinkage like ap-proach of Section 1.7.2 for polynomial regression. We chose to use polynomials of degree 3. Table 15 shows that the confidence intervals resulting from the shrinkage like ap-proach perform very good in all scenarios where the mean impact is greater than zero.

However, the coverage probability of 0.837 in the case where the mean impact is zero is very low and far from tolerable. Hence, other methods for the computation of confidence

Model ιX(Y) Cover P ower

Y = 0.3X+ǫ 0.3 0.988 0.731

Y = sin((X+ 1)³₂π) +ǫ 0.6745 1.000 0.820 Y = sin(5X) +ǫ 0.707 1.000 0.169 Y = sin(12(X+0.2))

(X+0.2) +ǫ 1.2698 1.000 0.541

Y =ǫ 0 0.837 0.163

Heteroscedasticity 0 0.693 0.307

Table 15: Simulation results for the shrinkage approach confidence intervals for the cubic polynomial based mean impact. Given are the coverage probability of the interval for the (unrestricted) mean impact and the probability of exclusion of zero (power).

intervals are needed.

Since the confidence intervals based on the shrinkage like approach to not perform very good, we want to investigate the performance of the intervals based on the polynomial mean impact. We now compute studentized bootstrap intervals. We used the functional delta method variance estimate which is further introduced in Section A.3.2. The results

Model ι_X(Y) Coverage P ower

Y = 0.3X+ǫ 0.3 0.949 0.610

Y = sin((X+ 1)³₂π) +ǫ 0.6745 1.000 0.732

Y = sin(5X) +ǫ 0.707 1.000 0.066

Y = sin(12(X+0.2))

(X+0.2) +ǫ 1.2698 1.000 0.180

Y =ǫ 0 0.936 0.064

Heteroscedasticity 0 0.855 0.145

Table 16: Simulation results for studentized bootstrap intervals for the cubic polynomial based mean impact. Given are the coverage probability of the interval for the (unrestricted) mean impact and the probability of exclusion of zero (power).

of Table 16 indicate, that the coverage probability of the studentized intervals is main-tained in scenarios 2-4. In the case of a linear model and when there is no relationship betweenX and Y the coverage probability lies slightly below 0.95. However, this small deviations from the nominal confidence level is explainable by simulation error.

In Section 1.7.5 it was also mentioned that it is possible to compute bootstrap in-tervals based directly on the estimate ˆι^lin_X(Y) instead of using its square (as we did in

the calculations for Table 16), when we can preclude that the mean impact is zero. To this end we computed studentized bootstrap intervals, again using the functional delta method estimate of the variance, and pre performed different test for the hypothesis that the polynomial based impact is zero. The first test we used is the test from Section 1.7.1.

Since this test is based on the results of White (1980b) we denote the coverage prob-ability and power of the confidence intervals arising from pre performing this test by Cover_white and P ower_white. We also investigated whether or not the use of the global F-test (H₀ : ξ₁ = ξ₂ = ξ₃ = 0 where ξ₀, ..., ξ₃ are the coefficients from the projection of Y onto span(1, X, X², X³)) from the linear regression (without robust variance esti-mate) as pre-performed test delivers better results (coverage probability and power of this procedure are denoted by Cover_F and P ower_F in Table 17). The performance of the procedure where both, the F-test and the test from Section 1.7.1 are performed prior to the calculation of confidence intervals was also investigated. The results of Table 17 Model ι_X(Y) Cover_{no test} P ower_{no test} Cover_white P ower_white

Y = 0.3X+ǫ 0.3 0.931 0.927 0.931 0.689

Y = sin((X+ 1)³₂π) +ǫ 0.6745 1.000 0.950 1.000 0.855

Y = sin(5X) +ǫ 0.707 1.000 0.315 1.000 0.045

Y = sin(12(X+0.2))

(X+0.2) +ǫ 1.2698 1.000 0.592 1.000 0.334

Y =ǫ 0 0.688 0.312 0.960 0.040

Heteroscedasticity 0 0.543 0.457 0.592 0.408

Model ι_X(Y) Cover_F P ower_F Cover_both P ower_both

Y = 0.3X+ǫ 0.3 0.931 0.836 0.931 0.685

Y = sin((X+ 1)³₂π) +ǫ 0.6745 1.000 0.862 1.000 0.841

Y = sin(5X) +ǫ 0.707 1.000 0.167 1.000 0.043

Y = sin(12(X+0.2))

(X+0.2) +ǫ 1.2698 1.000 0.391 1.000 0.320

Y =ǫ 0 0.821 0.179 0.960 0.040

Heteroscedasticity 0 0.671 0.329 0.679 0.321

Table 17: Simulation results for studentized bootstrap intervals for the cubic polynomial based mean impact (not using transformations). Given are the coverage prob-ability of the interval for the (unrestricted) mean impact and the probprob-ability of exclusion of zero (power), when pre-performing different test for the impact being zero.

show that the studentized bootstrap intervals based on the polynomial mean impact, not using the transformation performs very bad in the case, where the mean impact is

zero. This is due to the fact, that in this case the smooth function model does not hold, and we do not have any theoretical justification for the use of bootstrap methods in this case. This issue is resolved by the use of the test from Section 1.7.1. However, when using this test, we still have slight undercoverage in the case of a linear relationship between X and Y. The use of the F-test does not give any benefit. The improvement of the coverage probability to 0.821 in the zero impact case is not sufficient. Therefore, the best choice seems to be to use the test from Section 1.7.1 prior to the calculation of the confidence intervals.

When comparing the results of the procedure where we compute the studentized boot-strap intervals via transformation of the estimated polynomial mean impact (Table 16) with the results of the procedure where we did not use the transformation but pre performed a test for the mean impact being zero (Table 17), we can see that the lat-ter procedure yields a higher coverage probability as well as a higher power in most scenarios. In the presence of heteroscedasticity both procedures do not perform very good, although we should mention that the transformation procedure beats the non-transformation procedure in this case. However, this scenario is not covered by the theory derived in this thesis.

The computation of studentized bootstrap confidence intervals can be very time consum-ing, especially when using the functional delta method variance estimate. This is why it might be preferable to calculate the less cpu-intensive basic bootstrap intervals. We performed the same simulations as for Tables 16 and 17 but with basic bootstrap inter-vals instead of studentized interinter-vals. We can see from Table 18 that the basic bootstrap

Model ι_X(Y) Coverage P ower

Y = 0.3X+ǫ 0.3 0.988 0.010

Y = sin((X+ 1)³₂π) +ǫ 0.6745 1.000 0.358

Y = sin(5X) +ǫ 0.707 1.000 0.001

Y = sin(12(X+0.2))

(X+0.2) +ǫ 1.2698 1.000 0.010

Y =ǫ 0 0.997 0.003

Heteroscedasticity 0 0.994 0.006

Table 18: Simulation results for basic bootstrap intervals for the cubic polynomial based mean impact. Given are the coverage probability of the interval for the (un-restricted) mean impact and the probability of exclusion of zero (power).

intervals using the transformation of the estimated mean impact are very conservative

and have close to no power.

Model ιX(Y) Coverno test P owerno test Cover_white P ower_white

Y = 0.3X+ǫ 0.3 0.931 0.940 0.931 0.700

Y = sin((X+ 1)³₂π) +ǫ 0.6745 1.000 0.973 1.000 0.857

Y = sin(5X) +ǫ 0.707 1.000 0.329 1.000 0.045

Y = sin(12(X+0.2))

(X+0.2) +ǫ 1.2698 1.000 0.614 1.000 0.361

Y =ǫ 0 0.554 0.446 0.573 0.427

Heteroscedasticity 0 0.457 0.180 0.200 0.673

Model ι_X(Y) Cover_f P ower_f Cover_both P ower_both

Y = 0.3X+ǫ 0.3 0.931 0.835 0.931 0.685

Y = sin((X+ 1)³₂π) +ǫ 0.6745 1.000 0.864 1.000 0.842

Y = sin(5X) +ǫ 0.707 1.000 0.172 1.000 0.043

Y = sin(12(X+0.2))

(X+0.2) +ǫ 1.2698 1.000 0.411 1.000 0.338

Y =ǫ 0 0.818 0.182 0.959 0.041

Heteroscedasticity 0 0.656 0.344 0.662 0.338

Table 19: Simulation results for basic bootstrap intervals for the cubic polynomial based mean impact (not using transformations). Given are the coverage probabil-ity of the interval for the (unrestricted) mean impact and the probabilprobabil-ity of exclusion of zero (power), when pre-performing different test for the impact being zero.

Moving to the intervals where we do not use the transformation of the estimated impact (Table 19) increases the power at the cost of resulting severe undercoverage in the case where the mean impact is zero. In this case pre-performing the test from Section 1.7.1 does not resolve the issue. However, using this test and the F-test leads to confidence intervals that maintain the coverage probability and have a power which is comparable to the power of the studentized intervals using no transformation but the test from Section 1.7.1.

Im Dokument Non-Linear Mean Impact Analysis (Seite 116-120)