also assume that h ≤ pT. Then recursive substitution reveals that ˆ
where µ(pT, h) is the true value of the plug-in estimator. From the previous argument about the convergence rate of ˜gi¡φ(pˆ T)¢
, we see that the second term is Op(1/√
T ) and hence will have no contribution in the limit. We can then apply the same reasoning as in the proof of Proposition 3.1, but with ˜y(pT) = (0, . . . , 0, yT, . . . , yT +h−pT)t of length pT, which proves the validity of Extension
3.6.1. 2
A.3 CRM classification efficiencies
Proof of Proposition 5.2: Under the conditions listed in the proposition, and at the canonical model distribution, we have that
RL,Hm(a, b) = 1 restric-tion that σ(b) = σ. Because L(·) is a decreasing funcrestric-tion, it follows immediately
Appendix A. Proofs and computations 113
that RL,Hm(a, b) decreases as b1 increases. The Cauchy-Schwarz inequality yields b1∆ ≤ kbk2∆ = σ∆,
and this inequality becomes an equality if b = σe1. Hence, the minimal value of the expected risk for a and σ fixed becomes
RL,Hm(a, σ) = 1 which has to be minimised with respect to σ and a.
Setting the first order derivative with respect to a to zero implies that the optimal σ and a have to satisfy
E
Since θ = 0, we conclude that (5.7) holds, from which Fisher consistency follows.
2 to a and b, evaluated in ˜BL(H), are equal to zero. This holds in particular for Hε = (1 − ε)Hm+ ε∆x˜. With ˜bε= ˜BL(Hε), we have that holds for all ε. From this it follows that
d ψ(ε, ˜bε)
114 A.3. CRM classification efficiencies
or in other words, that
∂˜bε The second factor is easily seen to be
∂ψ¡
For the first factor, observe that
∂ψ¡
Because we work in the canonical model defined in Section 5.3, it immediately follows from symmetry arguments that
K = for the first-order influence functions as stated in proposition 5.3. 2
Proof of Corollary 5.4: We have
x1lim→−∞IF¡
Because L is a decreasing, convex function, we find that L0 is a negative, increas-ing function, where it is defined. Hence, it holds that
x1lim→−∞L0¡
yCL(Hm)(θ + ∆x1)¢
< 0,
Appendix A. Proofs and computations 115
An analogous reasoning holds for the first order influence functions on the slope parameters. Thus, the first order influence functions on the parameters are un-bounded, and by extension, the second order influence function on the error rate
is unbounded as well. 2
Proof of Proposition 5.5: To determine whether Fisher consistency holds for any θ, the values of c and d minimising
EHm£ exp¡
− Y (cθ + d + c∆X1)¢¤
must satisfy d = 0, and c = CL(Hm).
We rewrite the expected risk as EHm£ The first order conditions with respect to c and d give
0 = ∂RL,Hm(c, d)
116 A.3. CRM classification efficiencies 1/2. Hence, AdaBoost is Fisher consistent for all ∆ and θ, which concludes the
proof. 2
Proof of Proposition 5.6: For computing the asymptotic loss, we first need to find an expression for the asymptotic variances of the parameters, see equation (5.9). The general form of these asymptotic parameters can be found in equation (5.10), and the unknown A0, A1, and A2 are obtained using equation (5.8).
First, observe that we have L00(u) = exp(−u) = L(u). Using this result, we find that
Using the above expression with n = 1, we obtain that A1 = 0. In general, for Z ∼ N (0, 1) and any real c it holds that E[exp(cZ)] = exp(c2/2), E[exp(cZ)Z] =
Appendix A. Proofs and computations 117
Inserting the above quantities in (5.9), together with CL(Hm) = 12, yields the
result in (5.12), proving the proposition. 2
118 A.3. CRM classification efficiencies
Proof of Proposition 5.7: Because we assume that θ = 0, the expression in (5.9) simplifies to respectively the density and the distribution function of the standard normal distribution, and δ(·) is the Dirac-delta function. Using (5.8), and noting that L00(u) = δ(u − 1), we evaluate the expressions for A0, A1, and A2 as follows. where we used the notation
T = 1 − CL(Hm)∆2/2
Appendix A. Proofs and computations 119
For the evaluation of the asymptotic losses ASV(A) and ASV(B2) we further use that for Z ∼ N (0, 1) and any real c, E[I{Z ≤ c}] = Φ(c), E[I{Z ≤ c}Z] = −φ(c),
which proves the proposition. 2
List of Figures
2.1 Boxplots of the log(MSE) and log(MAE) of the 500 observations to pre-dict in the test sample for the sampling scheme with ntrain = 50 and q = 5. The MSE and MAE have been simulated for estimators of a model selected by the criteria AIC, BIC, FICMSE, FICMAE, or FICER, as well as for the model averaged versions of the estimators (indicated by the prefix “a”). . . . 22
2.2 Boxplots of the Error Rates of the 500 observations to predict in the test sample. These Error Rates have been simulated for estimators of a model selected by the criteria AIC, BIC, FICMSE, FICMAE, or FICER, as well as for the model averaged versions of the estimators (indicated by the prefix “a”). In the top panel (a) ntrain= 50, and q = 5 variables, in (b) ntrain= 50, and q = 9 variables, and in panel (c) ntrain= 200, and q = 5 variables. . . . 23
121
122 List of Figures
3.1 3D-surface plot for the ratios of mean squared errors for the 2-step ahead prediction of the series {xt}, comparing model order selection using the series {xt} with model order selection using the series {yt}, and where prediction is according to the direct method.
An ARMA(1,1)-process generated both series {xt} and {yt}. The autoregression parameter φ can be found on the phi axis, and the moving average parameter η is indicated on the eta axis. The surface shows the ratios of MSEs where the selection criterion used in both cases is the FIC. Where this surface lies above 1, signified by the grey-shaded facets, the two-series case had a smaller MSE than the one-series case. . . 45
3.2 3D-surface plot for the ratios of mean squared errors for the 2-step ahead prediction of the series {xt}, with model order selection us-ing the series {xt}, comparing prediction with the plug-in method and with the direct method. An ARMA(1,1)-process generated the series {xt}. The autoregression parameter φ can be found on the phi axis, and the moving average parameter η is indicated on the eta axis. The surface shows the ratios of MSEs where the se-lection criterion used in both cases is the FIC. Where this surface lies above 1, signified by the grey-shaded facets, the direct method for prediction resulted in a lower MSE than the plug-in method. . 51
3.3 3D-surface plot for the ratios of mean squared errors for the esti-mation of the impulse response function of the series {xt} at lag 2, with model order selection using the same series. An ARMA(1,1)-process generated the series {xt}. The autoregression parameter φ can be found on the phi axis, and the moving average param-eter η is indicated on the eta axis. The surface shows the ratios of MSEs where the AIC is compared with the FIC. Where this surface lies above 1, signified by the grey-shaded facets, the FIC selected models which results in a lower MSE than the AIC. . . 53
List of Figures 123
4.1 Values of KRIC and SVMICa in a simulation experiment, showing high correlation (0.975). . . 70 4.2 Generalization error rates for 100 simulation experiments, for n =
100, p = 25 (a) linear kernel, ranking with kwk2, (b) linear kernel, ranking with Fisher score, (c) quadratic kernel, ranking with kwk2, and for (d) n = 25, 100 variables, linear kernel and ranking with kwk2. . . 74 5.1 Asymptotic loss for Fisher’s linear discriminant rule (solid),
Ad-aBoost (dashed), logistic regression (dotted) and support vector machines (dash-dotted) with p = 2. (a) θ = 0; (b) θ = 1; (c)
∆ = 1. . . 99 5.2 Asymptotic relative classification efficiencies for AdaBoost (solid),
logistic regression (dashed), and SVM (dotted) for p = 2. (a) θ = 0; (b) θ = 1; (c) ∆ = 1. . . . 100 5.3 ARCEs for: AdaBoost (solid), logistic regression (dashed), and in
(a), SVM (dotted), for p = ∞. (a) θ = 0; (b) θ = 1; (c) ∆ = 1. . . 101
List of Tables
2.1 Average values, together with their standard errors (SE), of the log(MSE), log(MAE) and Error Rates over the 500 observations to predict in the test sample for the sampling scheme with ntrain = 50 and q = 5. The MSE, MAE, and Error rates have been simulated for estimators of a model selected by the criteria AIC, FICMSE, FICMAE, and FICER, as well as for the model averaged versions of the estimators (indicated by the prefix “a”). . . . 21
2.2 Error rates for the WESDR data, obtained via cross-validation. The models are selected using AIC, BIC FICMSE, FICMAE FICER and also results for the model-averaged estimates are reported. . . . 26
2.3 Model selection methods FICMSE and FICER are applied to each sub-ject within a group of the WESDR data. The table shows the selection percentages of the four most frequently selected variables per group. For completeness, the last 2 rows show the first four variables considered for inclusion by AIC and BIC, and whether they have been selected (“yes”) or not (“no”). . . . 28
125
126 List of Tables
3.1 Ratios of mean squared errors for the 2-step ahead prediction of the series {xt}, with model order selection using the same series, and prediction according to the direct method. An ARMA(1,1)-process generated the series {xt}. The autoregression parameter φ can be found in the leftmost column, and the moving average parameter η is indicated in the top row. The upper table shows the rMSE(·, FIC, AIC), the lower table shows the rMSE(·, FIC, BIC), as defined in (3.10). . . 42 3.2 Comparison of models selected by the information criteria FIC,
AIC, and BIC. A further comparison is made with a model selected based on the MSE of a hold-out sample. The table contains the estimated mean squared errors (×10−3) for each prediction horizon h, with the average value of the selected order within parenthesis.
Furthermore, t-values (p-values) of the Diebold-Mariano test for pairwise differences in MSE are presented. Results are given in (a) for the US Liquor sales data, and in (b) for the life insurance data. 47 4.1 Simulated average generalization error rate (%) for the six methods
using two different kernels. For each method, the number on the left resulted from ranking by variable influence on kwk2, and the number on the right in each column is from ranking by the Fisher scores Sj. . . 73 4.2 Simulated frequencies of selected models, with variable ranking
done by influence on kwk2. Here ‘C’ denotes correct selection, ‘U’
is underfitting, ‘O’ is overfitting, and ‘R’ for all other situations. . 76 4.3 As Table 1, but now for two populations with different variances . 77 4.4 As Table 2, but now for two populations with different variances . 78 4.5 Generalization error rates (%) for variable selection applied to four
data sets. Two variable ranking schemes and three types of kernel are used for each of the criteria. . . 80 5.1 Commonly used loss functions for general risk minimisation . . . . 87
List of Tables 127
5.2 Estimated values for a and b for SVM in the normal model, for several values of ∆ and θ. . . . 95
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