Jenny Häggström

Xavier De Luna
email: xavier.deluna@stat.umu.se

Estimating prediction error: Cross-validation vs accumulated prediction error

Keywords: Local polynomial regression, Non-parametric regression, Out-ofsample validation, Smoothing parameter Local polynomial regression, Non-parametric regression, Out-of-sample validation, Smoothing parameter

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This paper is concerned with the validation of general prediction rules for a variable of interest y.

We consider situations where a prediction rule μn for y is obtained using a sample of size n, y = (y 1 , y 2 , . . . , y n ) ′ . Typical examples of prediction rules are regression and classification applications.

As an illustration of the former, Figure 1 (left panel) displays 221 observations from a light detection and ranging (LIDAR) experiment, where the horizontal axis is the distance, x, travelled by laseremitted light before it is reflected back to its source and the vertical axis is the logarithm of the ratio, y, of received light from two laser sources, one of which had a frequency equal to the resonance frequency of mercury. LIDAR experiments are used to detect such chemical compounds in the atmosphere; see Ruppert et al. (2003, Sec. 2.7) for more details. Together with the data two regression curves are displayed (plain and dashed line), which are prediction rules of y given x.

As [START_REF] Efron | The estimation of prediction error: Covariance penalties and cross-validation[END_REF] pointed out there are two main schools to approach the problem of assessing the performance of different prediction rules for a given variable y: out-of-sample methods and covariance penalty methods. The latter class of methods is model based, i.e. parametric statistical models for the data generating mechanism are assumed, while the former is non-parametric and allow for the comparison of prediction rules obtained with different inferential frameworks and/or different modelling strategies. Although covariance penalty methods may be more efficient when the assumptions made hold (e.g., [START_REF] Efron | The estimation of prediction error: Covariance penalties and cross-validation[END_REF], we consider in this paper out-of-sample validation which is more widely applicable. The evaluation of prediction rules is concerned with the question: "what is the generalizability of a given prediction rule?" A common approach to validation arises from a translation of this question to, here quoting [START_REF] Efron | The estimation of prediction error: Covariance penalties and cross-validation[END_REF]: (Question 1) "We wonder how well [a prediction rule] will predict a future dataset [of same size n] independently generated by the same mechanism that produced y." Under certain conditions (see Section 2), this statement justifies the use of a cross-validation (CV, [START_REF] Stone | Cross-validatory choice and assessment of statistical predictions (with discussion)[END_REF]

criterion 1 n n i=1 L (y i , μn-1 i ), ( 1
)
where L is a loss function and μn-1 i is the prediction rule μn obtained with a sample where observation y i has been omitted.

Another approach to measuring the generalizability of a prediction rule arises from asking: (Question 2) How good is a prediction rule at predicting n observations arriving sequentially? As described in Section 2, this question justifies the use of the accumulated prediction error criterion (APE) (due 2 to [START_REF] Dawid | Present position and potential developments: Some personal views: Statistical theory: the prequential approach (with discussion)[END_REF] and [START_REF] Rissanen | Order of estimation by accumulated prediction errors[END_REF])

1 n -m n i=m+1 L (y i , μi-1 i ), (2)
where μi-1 i is the prediction rule μn obtained with the sub-sample y i-1 = (y 1 , y 2 , . . . , y i-1 ) ′ and m is the size of the sub-sample on which the first prediction rule is obtained.

CV is widely used, particularly in non-parametric applications, and the study of its properties has received much attention in the literature. The application of APE methods is less spread, except for time series situations (e.g., [START_REF] Sjöstedt | Resampling m-dependent random variables with applications to forecasting[END_REF][START_REF] De Luna | Predictive spatio-temporal models for spatially sparse environmental data[END_REF][START_REF] Wagenmakers | Accumulative prediction error and the selection of time series models[END_REF], where the time ordering of the observed units is used in accumulating prediction errors in APE. However, APE is also applicable in situations where units are exchangeable (no natural ordering exists) as was advocated by [START_REF] Dawid | Present position and potential developments: Some personal views: Statistical theory: the prequential approach (with discussion)[END_REF] and later on by de [START_REF] De Luna | Choosing a model selection strategy[END_REF], where APE is shown to be consistent in discriminating between model selection strategies.

In this paper we focus on the APE criterion and use the framework of [START_REF] Efron | Estimating the error rate of a prediction rule: Improvement on cross-validation[END_REF] to make comparisons with CV. This allows us to shed some new light on the properties of these criteria.

In particular, the APE criterion is an unbiased estimate of a measure of prediction error which is different from the one used to justify CV. The use of CV and APE to select the smoothing parameter in nonparametric regression is further studied and we show, for instance, that CV and APE yield different choices of smoothing parameters, where the difference does not vanish as the sample size increases.

The paper is organized as follows. The next section presents the two different measures of prediction errors which can be used to justify the use of CV and APE respectively. Section 3 focuses on linear prediction rules in regression situations, where out-of-sample methods are typically used to select a smoothing parameter. We review some asymptotics on the use of CV for this purpose and note that few results are available for APE. We then give finite sample expressions for the measures of prediction errors estimated by CV and APE. These results allows us in Section 4 to present numerical experiments to study and compare CV and APE. The paper is concluded in Section 5.

Measures of prediction error: new data versus sequential principle

Validation methods can be justified as estimators of measures of prediction errors. In this respect, a measure justified by Question 1 mentioned above is (e.g., [START_REF] Efron | The estimation of prediction error: Covariance penalties and cross-validation[END_REF]) where y 0 1 , y 0 2 , . . . , y 0 n is a new independent sample generated by the same mechanism as y. For instance, with a linear prediction rule (see Section 3.1 below) and squared error loss, L (y, µ) = (yµ) 2 , it can be shown [START_REF] Hastie | Generalized Additive Models[END_REF]) that the CV criterion (1) yields an approximately unbiased estimate of Err new , E(

Err new = 1 n n i=1 E{L (y 0 i , μn i )|y}, (3)
1 n n i=1 (y i -μn-1 i ) 2 ) ≈ E(Err new ) .
A measure of prediction error based on Question 2 is:

Err seq = 1 n -m n i=m+1 E{L (y i , μi-1 i )|y i-1 }. (4)
An advantage of this prediction rule is that an unbiased estimator of E(Err seq ) is readily available in wide generality by using the APE criterion (2): by the law of iterated expectations where S(z) is an n z × n matrix with n z the dimension of the design vector z at which predictions are made. For z = x, μn x contains fitted values (i.e. prediction of the response at the observed design points). When parametric linear models are fitted with ordinary least squares we have S(x) = X(X ′ X) -1 X ′ , which is called the hat matrix, and where X = (1 x) with 1 an n-length vector of 1:s.

E{ 1 n-m n i=m+1 L (y i , μi-1 i )} = E(Err seq
Examples of nonparametric linear prediction rules include kernel smoothers, cubic smoothing splines and local polynomial regression (see e.g., Schimek, 2000, and references therein). We focus on the latter in this paper to illustrate the use of CV and APE. Local polynomial regression with weights assigned by the tricube kernel, called loess in [START_REF] Cleveland | Locally weighted regression: An approach to regression analysis by local fitting[END_REF], consists of fitting a polynomial of degree p at a design point z using only the part of the data that is deemed to be sufficiently close to the target. The fit, at z, is

μn z (h) = e ′ 1 (Z ′ W h,z Z) -1 Z ′ W h,z y
where e 1 = (1, 0, . . . , 0) ′ , a (p + 1)-length vector,

Z =    1 (x 1 -z) . . . (x 1 -z) p . . . . . . . . . 1 (x n -z) . . . (x n -z) p    and W h,z = diag(K (x 1 -z)/b 1 (h) /b 1 (h), . . . , K (x n -z)/b n (h) /b n (h)).
The tricube kernel, K(•), is defined as

K(u) = 70 81 (1 -|u| 3 ) 3 , if |u| < 1 0, if |u| ≥ 1 ,
and, assuming no ties, b j (h) = the (hn):th nearest (in Euclidean distance) to z among the x j :s for h). This implies that if h is held constant for i = 1, . . . , n a constant number of observations are used for the local fits while b j (h), the size of the local neighborhood, varies with the data points x j .

x j = z, h ∈ [1/n, 1].
An illustrative example is provided in Figure 1. On the left hand panel, the LIDAR data described in the introduction is plotted together with two fitted curves with loess (p = 1) in R (R Development Core Team, 2008), one obtained with h = 0.23 (dashed) and one with h = 0.31. The difference between the two fit is most obvious by looking at the corresponding fitted first derivatives (right hand panel). We can see that a larger value for h yields a less wiggly behaviour. Out-of-sample validation is often used to select a relevant value for the smoothing parameter. The values used in Figures 1 correspond to the choices made with CV and APE (m = 50). The larger value for h, and hence the smoother curve, is obtained with APE. Note that the derivative is of main interest in this application since its proportional to mercury concentration at distance x, see [START_REF] Ruppert | Local polynomial variance function estimation[END_REF].

Choice of smoothing parameter

While CV is routinely used to choose a relevant smoothing parameter, the use of APE is rare. As a consequence there are many theoretical results available on the properties of CV as a method to select smoothing parameters, while these are scarce for APE. We make a review of these results (selective for CV and up to our knowledge exhaustive for APE).

Reviewing some asymptotics on the use of CV and APE

In the literature concerning smoothing parameter selection, the optimal smoothing parameter is often defined (using the squared error loss) either as ĥ0 = arg min

h 1 n n i=1 (µ i -μn i (h)) 2 or h 0 = arg min h 1 n n i=1 E[(µ i -μn i (h)) 2 ],
where µ i = E(y i ), which is allowed to vary at different design points x i . Note that throughout the design x is considered as non-random. Since Err new (h) can be expressed as

σ 2 + 1 n n i=1 (µ i -μn i (h)) 2 ,
where σ 2 = V ar(y i ), ĥ0 and h 0 are the minimizers of Err new (h) and E(Err new (h)), respectively. terms of the rate of convergence of the resulting ĥ to one of the optimal smoothing parameters defined above. The relative asymptotic success of CV depends on if one considers ĥ as an estimator of ĥ0 or h 0 . From [START_REF] Hall | Empirical functionals and efficient smoothing parameter selection[END_REF], in a setting where the prediction rule is a kernel estimator and the smoothing parameter representing a local neighborhood of constant size, we have

that h CV -h 0 h 0 = O p (n -1/10 ),
where h CV is the CV smoothing parameter, i.e. chosen by minimizing (1). This is a relatively slow rate of convergence, since there exist plug-in selectors for which the relative error is of O p (n -1/2 ) (albeit requiring strong smoothness assumptions on f [START_REF] Hall | Empirical functionals and efficient smoothing parameter selection[END_REF]). In the case of approximating ĥ0 CV does perform somewhat better. The relative error is still of O p (n -1/10 ) but in this situation h CV is optimal in the sense that there is no empirical smoothing parameter for which the relative error can be reduced below n -1/10 ; see [START_REF] Hall | Empirical functionals and efficient smoothing parameter selection[END_REF]. Similar results for selection of a constant size neighborhood and selection of a smoothing parameter of the k nearest neighbor type (i. e equivalent to h in Section 3.1) in local linear regression can be found in [START_REF] Li | Cross-validated local linear nonparametric regression[END_REF] and [START_REF] Ouyang | Cross-validation and non-parametric k-nearest-neighbour estimation[END_REF], respectively.

For APE there are no results qualitatively comparable to those reviewed above. [START_REF] Modha | Prequential and cross-validated regression estimation[END_REF] gave, however, rates of convergence for the integrated mean-squared errors in estimating non-parametrically the regression functions, showing that CV and APE achieved the same rates. Finally, de [START_REF] De Luna | Choosing a model selection strategy[END_REF] showed a consistency result for APE holding under weak assumptions. Loosely, their results say that APE will eventually (as the sample size grows) choose the prediction strategy that has lowest E(Err seq ). This result is limited to the comparison of a finite number of strategies and, therefore, it does not apply to the selection of a smoothing parameter. On the other hand, APE could consistently choose between a collection of prediction strategies defined by using different linear smoothers associated with different bandwidth selection criteria.

Finite sample properties

Explicit finite sample expressions for prediction errors can be obtained for linear prediction rules allowing for direct comparison, without the need to use asymptotic approximations. Thus, assuming independently distributed observations, a fixed design vector x and using the squared error loss, we 7

(h) = 1 n n i=1 E{(y 0 i -μn i ) 2 |y} = 1 n n i=1 E y 0 i - n j=1 {S h (x)} ij y j 2 y = 1 n n i=1 E y 0 i -µ i + µ i - n j=1 {S h (x)} ij y j 2 y = σ 2 + 1 n n i=1 µ i - n j=1 {S h (x)} ij y j 2 , E(Err new (h)) = σ 2 + 1 n n i=1 E µ i - n j=1 {S h (x)} ij y j 2 = σ 2 + 1 n n i=1 µ 2 i -2µ i n j=1 {S h (x)} ij E(y j ) + V ar({S h (x)} ij y j ) (5) + E n j=1 {S h (x)} ij y j 2 = σ 2 + σ 2 n n i=1 n j=1 {S h (x)} 2 ij + 1 n n i=1 µ i - n j=1 {S h (x)} ij µ j 2 (6) (7)
where {S h (x)} ij is the (ij):th entry in the n × n smoothing matrix. Analogous derivations yield

Err seq (h) = σ 2 + 1 n -m n i=m+1 µ i - i-1 j=1 {S i-1 h (x i )} ij y j 2 , E(Err seq (h)) = σ 2 + σ 2 n -m n i=m+1 i-1 j=1 {S i-1 h (x i )} 2 ij (8) + 1 n -m n i=m+1 µ i - i-1 j=1 {S i-1 h (x i )} ij µ j 2 , (9)
where {S i-1 h (x i )} ij is the (ij):th entry in the i × (i -1) smoothing matrix based on the design for sub-sample i = 1, 2, . . . , i -1 with

x i = (x 1 , x 2 , . . . , x i ).
When the data generating mechanism is known (in particular µ i and σ 2 ) these prediction errors can be computed for different values of h and compared. Such numerical illustrations are provided in the next section giving some insights on how these two measures of prediction errors differ.

Numerical experiments

We have several objectives in this section. First we want to illustrate numerically that the minimization of E(Err new ) and E(Err seq ) may yield different optimal smoothing parameters, where

Design of the experiments

The following data generating mechanisms are considered:

y k,i = f k (x i ) + ǫ k,i , k = 1, 2, 3, i = 1, . . . , n, n = 100, 200, 500, 1000 where f 1 (x i ) = 0.3 + 0.55x i , f 2 (x i ) = sin(x i ) + sin(2x i ) and f 3 (x i ) = 0.3 + 0.55x i + 2x 2 i , either x i ∼ U(0, 2π) or x i ∼ N (π, 1), x i is generated only once since it is considered fixed. ǫ k,i ∼ N (0,Var(f k (x i ))
), thereby ensuring that the signal-to-noise ratio equals one. For x i ∼ U(0, 2π), and local quadratic regression (loess with p = 2) estimators. For (2), ( 4) and ( 9) we fix m = 50. In order to investigate if the behavior of APE becomes more similar to CV if m is allowed to increase with n, as suggested by [START_REF] Ing | Accumulated prediction errors, information criteria and optimal forecasting for autoregressive time series[END_REF], we also let m = n/2 for n = 200, 1000, f 1 , f 2 , p = 1, 2 with x i ∼ U. We call this criterion APE δ . Moreover, an arbitrary ordering of the observations is used to accumulate prediction errors. Using a non-random ordering should be avoided when observations are exchangeable since, for instance, accumulating prediction errors for observations ordered with increasing x value would imply that the predictions made are always outside the support of the fitted regression function. We also include in the study the popular model based covariance penalty criteria AIC [START_REF] Akaike | A new look at the statistical model identification[END_REF],

Var(f 1 (x i )) = (0.3025/3)π 2 ≈ 0.9952, Var(f 2 (x i )) = 1 and Var(f 3 (x i )) = (0.3025/3)π 2 + (4.
1 n n i=1 (y i -μn i (h)) 2 + 2 n σ2 n i=1 {S h (x)}
ii , and BIC [START_REF] Schwarz | Estimating the dimension of a model[END_REF],

1 n n i=1 (y i -μn i (h)) 2 + log(n) n σ2 n i=1 {S h (x)} ii for comparison. σ2 = P n i=1 (yi-μ n i (h * )) 2 n- P n i=1 (2{S h * (x)}ii- P n j=1 {S 2 h * (x)}ij
) with h * = 0.13. When looking for optimal h values we restrict attention to the span [0.1, 1]. Figure 4 displays data simulated with the design described.

The design of this study is chosen to be able to distinguish two types of situations: cases where the true regression function f (x) is a polynomial function of order smaller or equal to the order p used in loess (correct specification case), and cases where f (x) is not a polynomial function or it is a polynomial function of order larger than p (misspecification case).

Results

Results are based on 1000 replicates. For all the simulated situations we show results on the minimal values obtained for the different measures of prediction errors and their estimates (Tables 1-2 and We first comment on the minima obtained for E(Err new ) and E(Err seq ) and their respective smoothing parameters. We can note that for designs with polynomial regression functions, the minima of E(Err new ) and E(Err seq ) obtained by using a value of p such that we are in the correct specification case are smaller, for n large enough, than the minima obtained with lower value of p (such that we are in the misspecification case, Table 2 f 3 ) and the minima obtained with higher values of p (Table 1, f 1 ). Estimates of the minimum of E(Err new ) and E(Err seq ) obtained with CV and APE seem to be comparable with a slight systematic advantage for CV. This difference may be due to minima of Err seq having a larger variability than the minima of Err new .

We focus now on Table 3-4 summarized in Figures 45, 8-9. A striking result in Table 3 and4 with the sample size in all misspecification situations. This shows that there is an intrinsic difference between the two measures of prediction errors. In the correct specification cases we observe that the selected values of h approach one as the sample size grows, while they tend to zero in the misspecification cases. It can be noted that APE yields h values closer to the ideal value one in the former situations. On the other hand, CV is slightly less biased (as an estimator of h minimizer of E(Err new )) than APE (as an estimator of h minimizer of E(Err seq )). However, APE yields h values which are less variable than those obtained with CV and this results in lower MSE for APE, in both types of situations, when samples are large enough.

The discrepancy, in selected values of h, between APE δ and CV decreases with n. This is in accordance with the theoretical results by [START_REF] Ing | Accumulated prediction errors, information criteria and optimal forecasting for autoregressive time series[END_REF] mentioned in Section 4.1.

Finally, while AIC behaves similarly to CV (note that they have been shown to be asymptotically equivalent in parametric situations, [START_REF] Stone | Asymptotics for and against cross-validation[END_REF]; Shao (1997)), BIC yields systematically higher values than APE. This is to be put in contrast with results showing that BIC and APE are asymptotically equivalent for linear time series models [START_REF] Ing | Accumulated prediction errors, information criteria and optimal forecasting for autoregressive time series[END_REF]. Thus, while AIC may be a reasonable substitute for CV in the situations studied, BIC cannot be used if the interest lies in minimizing E(Err seq ).

Discussion

Cross-validation and accumulated prediction error are two methods used in order to validate and compare different prediction rules without making parametric assumptions. In this paper, we highlight that these two methods are estimating two different measures of prediction error, answering thereby different questions.

In order to compare CV and APE we use the framework developed by [START_REF] Efron | Estimating the error rate of a prediction rule: Improvement on cross-validation[END_REF] to study CV and give its counterpart for APE. This allows us to show that when the prediction rules are linear smoothers, and the above methods are used to select the amount of smoothing (smoothing parameter) then the prediction errors estimated by CV and APE lead to different optimal smoothing parameters. Moreover, the latter difference does not vanish as the sample increases. This result is important since it shows that the choice between CV and APE does matter.

Our comparative results are exact for finite samples and do not rely on asymptotic approximations. Moreover, the non-equivalence result of CV and APE in our non-parametric context is in 1), ( 2), ( 3) and ( 4) for the minimizing h:s. First row n = 200, second row n = 1000. First two columns µ i = f 1 (x i ), last two columns µ i = f 2 (x i ). MSE for ( 1) and ( 2 3 in the appendix. Values of h minimizing (1), ( 2), AIC and BIC. First row n = 200, second row n = 1000. First two columns µ i = f 1 (x i ), last two columns µ i = f 2 (x i ). MSE for ( 1) and ( 2 2. Values of ( 1), ( 2), ( 3) and ( 4) for the minimizing h:s . First row n = 200, second row n = 1000. 1) and ( 2 1) and ( 2 1. Values of ( 1), ( 2), ( 3) and ( 4) for the minimizing h:s. First row n = 100, second row n = 500. First two columns µ i = f 1 (x i ), last two columns µ i = f 2 (x i ). MSE for ( 1) and ( 2 2), ( 3) and ( 4) for the minimizing h:s. First row n = 100, second row n = 500.

First two columns µ

i = f 3 (x i ) with x i ∼ U, last two columns µ i = f 3 (x i ) with x i ∼ N . MSE for (
µ i = f 3 (x i ) with x i ∼ U, last two columns µ i = f 3 (x i ) with x i ∼ N . MSE for (
First two columns µ i = f 3 (x i ) with x i ∼ U, last two columns µ i = f 3 (x i ) with x i ∼ N . MSE for ( 1) and ( 2 3. Values of h minimizing (1), ( 2), AIC and BIC. First row n = 100, second row n = 500. First two columns µ i = f 1 (x i ), last two columns µ i = f 2 (x i ). MSE for ( 1) and ( 2 1) and ( 2 1), ( 2), ( 3), ( 4) and values of ( 6) and ( 9), for the minimizing h:s, for f 1 , and f 2 with x i ∼ U. Sample standard deviation in parenthesis and bias (difference between the mean of the minima of ( 1) or ( 2) and the minimum of ( 6) or ( 9)) in brackets. 2), AIC, BIC, ( 6)and ( 9) for f 1 and f 2 with x i ∼ U. Sample standard deviation in parenthesis and bias (difference between the mean of the minimizing h:s and h minimizing ( 6) or ( 9)) in brackets.

i = f 3 (x i ) with x i ∼ U, last two columns µ i = f 3 (x i ) with x i ∼ N . MSE for (

Figure 1 :

 1 Figure 1: Left hand panel: LIDAR data with two loess fit obtained with h = 0.23 (plain line, CV choice) and with h = 0.31 (dashed line, APE choice). Right hand panel: Fitted first derivatives corresponding to the curve fitted on the left hand panel.

  not vanish with increasing sample sizes, thereby showing that both measures of prediction errors are not measuring the same thing. Secondly, we want to study the finite sample properties of smoothing parameters obtained by estimating E(Err new ) and E(Err seq ) with CV and APE respectively.

Figure 2 :

 2 Figure 2: Regression functions and 100 generated observations: f 1 and f 2 from left to right in the upper panel, f 3 with x i ∼ U and f 3 with x i ∼ N from left to right in the lower panel.

Page

  4/3)π 3 + (256/45)π 4 ≈ 600.6206. For x i ∼ N (π, 1), Var(f 3 (x i )) = (8.3025/3) + 4.4π + 16π 2 ≈ 180.0392. 9We consider local linear regression (loess with p = 1 in R, (R Development Core Team, 2008))

Figures 2

 2 [6][7] in the appendix), as well as for the argument of these minima, i.e. the h values selected (Tables3-4 and Figures 4-5, 8-9; tables and Figure8-9 in the appendix).

Page

  is the fact that the difference between the values for h minimizing E(Err new ) and E(Err seq ) increases 10

Page

  accordance with previous results obtained with parametric prediction rules; see de Luna and Skouras 11

Figure 4 :

 4 Figure 3: Boxplots for results from Table1in the appendix. Values of (1), (2), (3) and (4) for the minimizing h:s. First row n = 200, second row n = 1000. First two columns µ i = f 1 (x i ), last two columns µ i = f 2 (x i ). MSE for (1) and (2) are given on top. The dashed and dotted lines indicates E(Err new ) and E(Err seq ) for the minimizing smoothing parameter, respectively.

Figure 5 :

 5 Figure 4: Boxplots for results from Table3in the appendix. Values of h minimizing (1), (2), AIC and BIC. First row n = 200, second row n = 1000. First two columns µ i = f 1 (x i ), last two columns µ i = f 2 (x i ). MSE for (1) and (2) are given on top. The dashed and dotted lines indicates the smoothing parameter minimizing E(Err new ) and E(Err seq ), respectively.

Figure 6 :

 6 Figure 5: Boxplots for results from Table2. Values of (1), (2), (3) and (4) for the minimizing h:s . First row n = 200, second row n = 1000.First two columns µi = f 3 (x i ) with x i ∼ U, last two columns µ i = f 3 (x i ) with x i ∼ N . MSE for (1) and (2) are given on top. The dashed and dotted lines indicates E(Err new ) and E(Err seq ) for the minimizing smoothing parameter, respectively.

Figure 7 :

 7 Figure 6: Boxplots for results from Table 4. Values of h minimizing (1), (2), AIC and BIC. First row n = 200, second row n = 1000. First two columnsµ i = f 3 (x i ) with x i ∼ U, last two columns µ i = f 3 (x i ) with x i ∼ N . MSE for (1) and (2) are given on top. The dashed and dotted lines indicates the smoothing parameter minimizing E(Err new ) and E(Err seq ), respectively.

  Figure 7: Boxplots for results from Table1. Values of (1), (2), (3) and (4) for the minimizing h:s. First row n = 100, second row n = 500. First two columns µ i = f 1 (x i ), last two columns µ i = f 2 (x i ). MSE for (1) and (2) are given on top. The dashed and dotted lines indicates E(Err new ) and E(Err seq ) for the minimizing smoothing parameter, respectively.

Figure 9 :

 9 Figure 8: Boxplots for results from Table2. Values of (1), (2), (3) and (4) for the minimizing h:s. First row n = 100, second row n = 500.First two columns µi = f 3 (x i ) with x i ∼ U, last two columns µ i = f 3 (x i ) with x i ∼ N . MSE for (1) and (2) are given on top. The dashed and dotted lines indicates E(Err new ) and E(Err seq ) for the minimizing smoothing parameter, respectively.

Figure 10 :

 10 Figure 9: Boxplots for results from Table3. Values of h minimizing (1), (2), AIC and BIC. First row n = 100, second row n = 500. First two columns µ i = f 1 (x i ), last two columns µ i = f 2 (x i ). MSE for (1) and (2) are given on top. The dashed and dotted lines indicates the smoothing parameter minimizing E(Err new ) and E(Err seq ), respectively.

  Figure 10: Boxplots for results from Table4. Values of h minimizing (1), (2), AIC and BIC. First row n = 100, second row n = 500. First two columns µ i = f 3 (x i ) with x i ∼ U, last two columns µ i = f 3 (x i ) with x i ∼ N . MSE for (1) and (2) are given on top. The dashed and dotted lines indicates the smoothing parameter minimizing E(Err new ) and E(Err seq ) , respectively.

ca Communications in Statistics -Simulation and Computation

	3 CV and APE applied to nonparametric regression
	3.1 Linear prediction rules
	We consider situations where y is observed at fixed design points x = (x 1 , x 2 , . . . , x n ) ′ . Then,
	prediction rules are typically constructed by regressing y on x, for example with linear predictors of
	the form
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4

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	is global, while if h=0.2 then the 0.2n observations closest to z are used to produce μn z
	F o r
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	The parameter h is typically called smoothing parameter and in this case is the proportion of
	observations being used to produce the local fit. Thus, if h=1 all observations are used and the fit
	5

ca Communications in Statistics -Simulation and Computation

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	have
	Err new
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	y

URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.mcmaster.

  Err CV Err APE Err APEδ Err new Err seq Boxplots for results from Table1in the appendix. Values of (

	Observed error for selected h	0.6 0.8 1.0 1.2 1.4	0.01 0.01	p=1	Observed error for selected h	0.6 0.8 1.0 1.2 1.4	0.01 0.01	p=2	Observed error for selected h	0.6 1.0 1.4	0.01 0.02	p=1	Observed error for selected h	0.6 1.0 1.4	0.01 0.02	p=2
							Err CV Err APE Err APEδ Err new Err seq			Err CV Err APE Err APEδ Err new Err seq			Err CV Err APE Err APEδ Err new Err seq
	Observed error for selected h	0.8 0.9 1.0 1.1 1.2	0.002 0.002	p=1	Observed error for selected h	0.8 0.9 1.0 1.1 1.2	0.002 0.002	p=2	Observed error for selected h	0.8 0.9 1.0 1.1 1.2	0.002 0.002	p=1	Observed error for selected h	0.9 1.0 1.1 1.2	0.002 0.002	p=2
			Err CV Err APE Err APEδ Err new Err seq			Err CV Err APE Err APEδ Err new Err seq			Err CV Err APE Err APEδ Err new Err seq			Err CV Err APE Err APEδ Err new Err seq
	Figure 3:

Table 1 :

 1 Mean values of (

Table 2 :

 2 Similar to Table 1 but for f 3 with x i ∼ U and x i ∼ N .

	n	d Err CV	d Err AP E	Err new	Err seq	E(Err new ) E(Err seq )

Table 3 :

 3 Mean of the h:s minimizing (1), (

Table 4 :

 4 Similar to Table but 3 but for f 3 with x i ∼ U and x i ∼ N .

	n	CV	APE	AIC	BIC	E(Err new ) E(Err seq )
	f 1 , x ∼ U , p = 1
	100	0.807	0.756	0.810	0.955	0.971	0.994
		(0.228)	(0.221)	(0.229)	(0.086)
		[-0.164] [-0.237]
	200	0.833	0.839	0.834	0.974	0.995	0.996
		(0.224)	(0.187)	(0.222)	(0.075)
		[-0.162] [-0.156]
	500	0.828	0.869	0.830	0.988	0.996	0.996
		(0.240)	(0.175)	(0.237)	(0.045)
		[-0.168] [-0.127]
	1000	0.816	0.903	0.817	0.991	0.996	0.994
		(0.244)	(0.147)	(0.244)	(0.035)
		[-0.180] [-0.090]
	f 1 , x ∼ U , p = 2
	100	0.836	0.771	0.839	0.970	0.971	0.996
		(0.200)	(0.199)	(0.209)	(0.063)
		[-0.135] [-0.225]
	200	0.860	0.861	0.858	0.982	0.994	0.996
		(0.195)	(0.160)	(0.197)	(0.049)
		[-0.134] [-0.134]
	500	0.856	0.897	0.854	0.993	0.994	0.994
		(0.209)	(0.141)	(0.210)	(0.022)
		[-0.138] [-0.097]
	1000	0.856	0.924	0.855	0.995	0.994	0.994
		(0.209)	(0.116)	(0.210)	(0.010)
		[-0.138] [-0.070]
	f 2 , x ∼ U , p = 1
	100	0.267	0.280	0.266	0.358	0.261	0.281
		(0.053)	(0.085)	(0.055)	(0.113)
		[0.006]	[-0.001]
	200	0.224	0.255	0.225	0.300	0.228	0.257
		(0.037)	(0.041)	(0.037)	(0.027)
		[-0.004] [-0.001]
	500	0.179	0.218	0.179	0.252	0.176	0.216
		(0.028)	(0.026)	(0.028)	(0.017)
		[0.003]	[0.002]
	1000	0.154	0.200	0.157	0.223	0.153	0.197
		(0.022)	(0.020)	(0.022)	(0.011)
		[0.001]	[0.003]
	f 2 , x ∼ U , p = 2
	100	0.438	0.449	0.435	0.533	0.439	0.460
		(0.069)	(0.090)	(0.071)	(0.070)
		[-0.001] [-0.010]
	200	0.423	0.447	0.422	0.494	0.430	0.462
		(0.055)	(0.056)	(0.056)	(0.030)
		[-0.008] [-0.015]
	500	0.380	0.422	0.380	0.455	0.393	0.423
		(0.051)	(0.040)	(0.051)	(0.021)
		[-0.013] [-0.001]
	1000	0.364	0.403	0.364	0.438	0.363	0.404
		(0.041)	(0.032)	(0.041)	(0.015)
		[0.001]	[-0.001]

URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.mcmaster.caCommunications in Statistics -Simulation and Computation

Communications in Statistics -Simulation and Computation

Acknowledgments

We acknowledge the financial support of the Swedish Research Council (grant 70246501, the Ageing and Living Condition Program and the Swedish Initiative for Microdata Research in the Medical and Social Sciences). The simulation results obtained were run on facilities made available by the High Performance Computing Center North at Umeå University. 12


Cornelis J Van Der Haven

GERMANIC MYTHS AND GENDER CONSTRUCTIONS IN GERMAN AND DUTCH THEATRE TEXTS 1660-1780

Keywords:

Germanic liberation myths play an important role in the development of a national consciousness in both the Dutch Republic and the German countries. This article discusses a selection of German and Dutch theatre texts from the seventeenth and eighteenth century about the uprisings and battles of Germanic tribes against the Romans. In existing studies to the Hermannsschlacht-motif in German literature, some scholars conclude that in the course of the eighteenth century male protagonists (especially Arminius/Hermann) were occupying more and more space at the expense of the female characters. By analysing theatre texts on Germanic myths from a long-term international perspective, I will test the correctness of this hypothesis. In the critical reading of key works like Klopstock"s bardic drama piece Hermanns Schlacht (1769), I will especially pay attention to the presupposed masculine power fantasies stemming from a more aggressive form of patriotism that was emerging after 1750. Did these fantasies, as they are represented in theatre texts on early national myths, indeed push off female protagonists from the domain of military and political power? What does it mean that these texts were written in a period in which enlightened visions on the fatherland made that the very notion of patriotic heroism was changing radically? How then should we determine the main field of power in these plays, given the fact that male and female characters could use different power instruments in their attempts to act like "good patriots"?

against Rome was important as a source for reflections on a distinct enemy to whom feelings of cultural superiority could be related quite easily.

Literary adaptations of Germanic liberation myths 2 thus confronted the reader extensively with negative stereotypes of the Romanssuch as weakness, servitude, and effeminacywhereas the Germanic warriors were depicted as strong, proud, independent, and manly. Especially the main protagonists of two important uprisings -Hermann (also called Arminius), prince of the Cherusci tribe, and Claudius Civilis, leader of the Batavian tribecould be seen as personifications of that masculine self-image of the patriotic warrior. The use of feminine stereotypes, on the other hand, served to project masculine fantasies of heroic acting in favour of the fatherland; they illustrated the unmanly and cowardly character of the warrior"s enemy, as well as the helpless situation of the suffering fatherland. Writing on the Arminius-myth in German literature in the mid-nineties, Hans Peter Herrmann showed that during the second half of the eighteenth century the role of female protagonist, especially of Thusnelda, Hermann"s companion, would be progressively marginalised in favour of her masculine counterpart. Whereas in the earlier Hermann-plays women played a modest role in political and military actions, their role in later textsespecially Friedrich Klopstock"s Hermanns Schlacht of 1769, to which I will return later would be reduced to being the domestic support of male protagonists before and after the battle. Herrmann interprets this removal of women from the public domain in the light of masculine power fantasies stemming from a more aggressive form of patriotism that was emerging in the German countries after 1750. 3 A problematic notion in Herrmann"s analysis is the underlying notion of passivity and activity in relation to the role of female protagonists in Klopstock"s drama text. Behaving like male protagonists, which means taking part in political and military actions, is interpreted as active and independent; while their acting in other (i.e. private and domestic) domains would imply that women are playing a passive and subordinate role. This narrative of a public-private division in early modern literature and society that separates the genders has been criticised or at least nuanced and complicated in modern gender studies. 4 The question therefore I would like to raise here is whether this is the right dichotomy to work with for interpreting the heroism of male as well as female protagonists, especially given the fact that this was a period in which enlightened visions on the fatherland made that the very notion of patriotic heroism was changing radically. In a critical response to Hans Peter Herrmann, Elystan Griffiths for instance concludes in relation to the Hermann-play of Schlegel, that especially the participation of female protagonists in war acts should be understood as a by-product of the author"s "quasi-democratic" enlightened intentions, whereas the "gender imbalance in battle" in Klopstock"s Hermanns Schlacht, who presents military and political power as a privilege of male producers of culture (bards and poets), would be a result of the author"s limited understanding of the notion of democratic participation. 5 Both Griffiths and Herrmann conclude that Klopstock in his play thus sets apart private-domestic forms of patriotic acts from professional-public ones. Female agency then was primarily connected with the first domain, which would make their acting less important and less heroic then that of their male counterparts. But since, especially for the late 18 th century, civic emancipation often resulted in efforts to instil civic-private interpretations of patriotism into notions of political and military heroism, a strict separation of gender-roles 2 For an overview of literary texts based on the Arminius and Batavian myths, see Kuehnemund (1966) and Duits (2004) respectively. 3 Herrmann (1996), 172-173. 4 See for instance conclusions of recent studies to gender in early modern English society like: [START_REF] Rose | Gender and Heroism in Early Modern English Literature[END_REF]), 115-116 and Shoemaker (1998), 313-317. 5 Griffiths (2008), 135. related to both the public and private domain seems to be less obvious than scholars like Griffiths and Herrmann want to suggest.

THE MALE PRIVILEGE OF POLITICAL SPEECH

Since the years between 1740 and 1770 are the heyday of Germanic mythology in German literature, it seems to be obvious that many literary scholars who write about early patriotism focus on that very period. But we should certainly not overlook the long previous history of the reception of Germanic myths in German and Dutch literature, especially during the seventeenth century. As Hans-Martin Blitz, one of Herrmann"s students, shows in his dissertation Aus Liebe zum Vaterland (Hamburg 2000), the figure of Arminius or Hermann played already from the early seventeenth century onwards an important role in literary representations of the fatherland. In Lohenstein"s novel Arminius und Thusnelda (Leipzig 1689) for instance, the author uses the motive of the raped German princess Walpurgis to describe the violated fatherland and to legitimate the bloodthirsty revenge fantasies of the masculine hero. 6 Written in the tradition of the French heroic-gallant novel, Lohenstein"s novel pays a lot of attention to the character of Thusnelda, Arminius" wife. As we can see at the title page (plate 1), it is a woman (Thusnelda) who very clearly here symbolises the sufferings of the German countries, which are liberated by Arminius and his servants from Roman terror. This image of the suffering woman, often in the role of a raped and wounded virgin, is an image we meet again and again also in Dutch allegorical representations of the defence of the fatherland during and after the Eighty Years" War. The virgin represents the suffering fatherlandthe city, the region or the Netherlands as a wholea fatherland that cannot free itself and needs good patriots to be liberated. With her passive and chaste character, the virgin is of course the ideal object for manly warriors to fight for. Freeing her from the aggression of her oppressors clearly has an erotic connotation, especially since the verb vrijen could mean in Dutch both making love as well as to liberate. 7 The role of women as the symbolic representation of the suffering fatherland is not necessarily a marginal one. In Joost van den Vondel"s play De Batavische Gebroeders (Amsterdam 1662), for example, the long Chorus of Batavian Women dominates the play"s first act with extensive lamentations on the sufferings of the country. Speaking in dialogue with the other protagonists, the women express the hope that men will come in action and free them from Roman tyranny. But the two Batavians the women are addressing with their lamentations, the princes Julius and Burgerhart, are not in the position to organise resistance against the dominant Roman army. This masculine powerlessness of the two brothers is presented by Vondel as unnatural and unbatavian, whereas the helplessness of the Batavian women is natural and self-evident. The princes do not recommend the women to resist but to resign themselves to "patient suffering". 8 Because women are thus considered as apolitical characters, their attempts to negotiate with the Romans are interpreted by their Roman oppressors as a trick of men: Dat schijnt geen vrouwenwerck, maer eer een mans bedrijf.

De man staet achter, en vermomt zich met zijn wijf. 9 [This seems not to be the work of women, but rather the business of men.

The man is behind this, and disguises himself with his wife.] It is Fronto, the Roman centurion, who interprets the lamentations of the Batavian women as manly, because it seems to be aimed at creating resistance. It must have been a "man"s head" which stimulated these women to speak out their lamentations in public 10 , and by doing so, making out of emotions a public performative act aimed at changing politics. Women alone, he believes, would not be able to give such a political direction to their vengeful feelings. The Batavian women however defend their position by pointing to the origins of their lamentations, which would stem from natural passions arising from the mother"s heart and not from any political intentions. 11 In the Dutch Arminius-play by Pieter Bernagie (Amsterdam 1686), we meet again Arminius" wife Thusnelda as the personification of the suffering fatherland. Her lamentations cover at least half the play and until the end she remains dependent on the help of the male protagonists to save her from the hands of her jealous admirer Flavius, who kidnapped her by order of the Romans. Inspired by the tragédie amoureuse, and especially the famous French Arminius-plays of Madelaine de Scudéry (Paris 1643) and Jean-Galbert Campistron (Paris 1685) 12 , Bernagie presents the struggle of Arminius in the context of a love intrigue. It is Thusnelda who inspires Arminius with her protestations of love for both him and the fatherland to defend that fatherland against the enemy, while she accuses her admirer Flavius of unpatriotic behaviour, because he supports the Roman occupation of Germany. As a reference to that occupation, Thusnelda"s kidnapping has a strong metaphoric dimension. Flavius" betrayal of his original admirer Radwich, because of his love for Thusnelda, likewise has a political counterpart. Flavius is not only a traitor to his admirer but also, together with his father, to the Germanic troops, while secretly cooperating with the Romans. Since the love intrigue thus functions as metaphoric reference to the main plot, it does not really create an independent power structure that might enable Thusnelda to use different moral instruments from the male protagonist. What makes Thusnelda a true patriot in the tragedy of Bernagie is the fact that she is able to suppress her instable female emotions and put in action the male qualities of reason and courage. 13 She is not vengeful but reasonable, and it is not by using her female qualities but by adopting the male rhetoric of reason that she is able to convince the Roman General Germanicus to set her free.

SUFFERING WOMEN AND FIGHTING MEN

With the mentioned French Arminius-plays and a couple of Arminius-operas written during the early eighteenth century, Thusnelda became a central figure of theatre texts on the Arminius-myth. 14 The dramatic kidnapping of Thusnelda must have inspired the Dutch author Joannes le Francq van Berkhey to pay special attention to comparable adventures as experienced by the hero"s bride in the Batavian liberation myth. In his play called Claudius Civilis (Amsterdam 1764) Berkhey describes the battle of the Batavian leader against the Romans, around the year 70. Like the other protagonists, the character of Civilia is derived from Tacitus, who 10 Vondel (1690), 115: "Geloof hier heult / Een manshooft me, die "t spel aenschout met koele zinnen, / En broet en koestert het verraet, en houdt zich binnen." 11 Vondel (1690), 104: "Och "t is de natuurdrift, diep in "t moeders hart geprent." 12 Forchert (1975), 49-50 and[START_REF] Kösters | Freiheit der Germanen. Eine französische Karriere[END_REF], 172-174. 13 Bernagie (1686), 37 & 40. 14 In 1749 a libretto entitled Thusnelde appeared in Leipzig (text by Johann Adolf Scheibe), implicating that Thusnelda took over the role of main protagonist from Arminius. See Barbon & Plachta (1995), 281-285.

parenthetically mentions in his Histories the existence of Civilis" bride. 15 As Berkhey explains in his preface, his decision to highlight the sufferings of Civilia and her son, bringing them in action on the battlefield (as vengeful prisoners of the enemy) was made in order to activate a more natural love for the fatherland on the side of the male protagonist. Civilis" love for his family, his son and wife, makes that he acts like a good patriot, whereas his enemy is the aggressive, heerschzugtige (imperious) Roman governor Cerialis, who is inspired by verwyfde laagheden (womanish turpitudes)impulsive and sexual passions, as Berkhey explains in the preface to his play. 16 Although at first sight Civilis" fighting spirit seems to be obvious, static, and unrestricted, the references to his compassion with Civilia and her child are intended to give his readiness to fight a more natural character, but not in the form of bloode verliefdheden (silly love affairs) of the tragédie amoureuse, as the author assures his readers in the preface. 17 It is Civilia who shows how feelings of revenge and patriotism in the hearts of men could be activated, and how sufferings can be transformed into active resistance. Together with Civilis" sister Jula and her son Bato, who all get kidnapped by the enemy, she thus activates a more natural understanding of the love of the fatherland on the part of the hero. By adding three characters to the Batavian liberation myth who all have family bonds with the hero, Berkhey is able to depict Civilis as a free combatant, whose willingness to fight is not mixed up with the ambition for power, as in the case of Cerialis. True patriots do not go to the battlefield for "gold, crowns, and treasures", but for their Huisgezinnen (families) and the Batavian soil, as Civilis explains to his people after the battle against the Roman army. 18 The domestic narrative twist therefore makes Civilis" civic patriotism manly, real, and strong; whereas the Romans and their German alleys are depicted as womanly, depraved, and weak. This femininity (verwyfdheid) and weakness of the depraved enemy also has an ethnic connotation in Berkhey"s play. Being interested in anthropological surveys of the history of the Batavian people, Berkhey relates the depraved nature of the Romans" German alleys to their cultural and ethnical relationship with the Roman people. The inhabitants of Cologne, who organised the kidnapping of Civilia and her companions, are unable to show solidarity to the Batavians because the Germanic women of Cologne had married with Roman soldiers, creating a blood relationship between the two ethnical groups. 19 In his anthropological study to the natural history of Holland (Natuurlijke historie van Holland, Amsterdam 1769-1811), Berkhey qualifies such relationships as disasters of mixed ethnicity (geslagtvermengende rampen). 20 The chief of Cologne, Agrippinus, thus explains to Civilis in the play that because of the mixed identity of the city"s inhabitants, it will be impossible to join the Batavians in a fight against an enemy (the Romans) they really consider as their own people. Civilis in return condemns the slavish soul of the Cologne inhabitants who are disloyal to the Germanic tribes, using a popular Dutch patriotic image, that of the imprisoned lion. 21 Living in a mixed urban environment, the lion will become tame, denaturalised and domesticated, whereas in the woods and open fields he would recover his natural freedom and willingness to fight. 22 In Berkhey"s tragedy, then, the female protagonist is no longer solely a passive symbol of the suffering fatherland, like in the examples discussed for the seventeenth century. The connection between the suffering of the fatherland and the suffering of the family means that, as the agent of a more natural love for the fatherland, Civilia is an essential character for Berkhey"s enlightened interpretation of the Batavian myths. Both in Berkhey"s play and in Johann Elias Schlegel"s tragedy Hermann (Leipzig 1743), the female heroine"s commitment to the struggle of the male protagonists also results in willingness to actively support their husbands during the battle. This appearance of women in the rear of the army also derives from Tacitus" Histories. The presence of children en women during the battle of Vetera/Xanten, was one of the motives in Dutch seventeenthcentury art on the Batavian liberation myth (see plate 2). 23 But is a mistake, in my opinion, to understand this support of the fatherland by women during military operations as an activation of feminine power, since their inspiration to fight is presented here as an ultimate adaption to and identification with classic masculine heroism.

The moral authority and power of women is based on the contrary on their position as political and military outsiders. This position enables them to judge autonomously between true and false patriots. Thusnelda explains in Schlegel"s play that good patriots act independently and are inspired by a natural love for both the fatherland and the family, whereas the false hero is [How happy will I be to come to the army with you! 23 Like the etching in Vaenius (1612) and two sketches for decorations of the Amsterdam city hall by Ferdinand Bol (ca. 1659) and Jurgen Owens (ca. 1662) with the same subject, see Van de Waal (1952), II, plate 78-80. See also Morford (2001), 71 and figure 11. 24 Schlegel (1963), 149. 25 Schlegel (1963), 150 How happy will I be to see you in lead of it! O! that I would give you your victory you rightfully deserve, Just to share in your glory forever:

Your courage would also raise me, and your victory would also be mine.]

Although the intention to fight means here that the heroine aims to become absorbed by masculine heroism, the author uses this image to promote the activation of a träge (slow, lazy)an adjective Schlegel uses again and again in relation to the German companions of the Romanfatherland soothed to sleep by a despotic politics.

The fatherland should do more then lament about its sufferings and has to actively liberate itself. Patriotism thus is related strongly to the enlightened bellicist discourse of the eighteenth century, in which civic self-defence should not depend on paid armies but on the readiness of all citizens to fight as good patriots. 26 Yet this enlightened interpretation of patriotism does not lead to a radical change in gender roles; the image of a passive and slow fatherland continuously carries feminine connotations. The lazy fatherland is represented by characters who are unable to choose between Rome and Germany, like Flavius, Hermann"s brother. His slow understanding of patriotism is described in Schlegel"s tragedy as characterising a weibisch (womanish) fatherland, a fatherland unable to fight and defend itself. Also Claudius Civilis in Berkhey"s play taunts the tribes who (like for instance the Gallic tribes) do not want to fight against the Romans as verwyfd (womanish). 27 The intention of female protagonists to fight, or in any case their fascination with an armed struggle, as Schlegel and Berkhey describe it, should not therefore be understood primarily as an example of crossing the borders of gender-determined behaviour. As a form of female admiration for male military heroism, it still fits into the old hierarchic order of the patriotic community, in which men fight and kill and women care for the wounded soldiers and raise new young patriots. 28 Many ancient historiographers refer to that role of women on the battlefield, like Tacitus, who characterises the shrieking Germanic women with their wailing babes on the battlefield as "each man"s most sacred witnesses and greatest eulogists". 29 In the plays of Berkhey and Schlegel however, their presence on the battlefield does in the first place have a strong symbolic character, referring to the activation of citizens as armed patriots in general. The plays are not about fighting as such, but about the will to fight, symbolised by the activation of powers that are actually presumed not to use arms. Other characters who were in earlier texts characterised by passive sufferings, are likewise transformed into patriots who want to take up arms and become active in the defence of the fatherland. In Claudius Civilis, Bato, the son of the protagonist, for instance tries to convince his father of his readiness to fight, or at least to observe the fight, and thus become absorbed in his father"s courageousness too: Myn Vader! sta my toe, dat ik met vuurige oogen, Al kan ik in "t gevegt u nog geen bystand biên, Ten minsten u van ver mag dapper stryden zien. 30 [My Father! allow me at least to observe you with fiery eyes, 26 On the relation between patriotism and bellicism during the eighteenth century, see: Leonhard (2008), 181-218. On the semantics of German patriotism in the late eighteenth century, see: Fuchs (1991). 27 Francq van Berkhey (1763), 30. 28 See also the remarks of Blitz (here in relation to Schönaich"s Hermann-epic of 1751) in: Blitz (2000), 128-129. 29 Tacitus (1999), 80 (7.2). See also Bruder (1973), 128-151 and Amstadt (1994), 81-84. 30 Francq van Berkhey (1763), 10.

Even though I cannot offer you assistance during the battle, Let me observe from a distance your courageous fighting.] Since women are, together with the elderly and children, the classic symbols of the suffering fatherland, their transformation into subjects with an intention to fight, clearly has to inspire the nominal warriors Civilis and Hermann. Mother and child represent a generative and circular concept of love for the fatherland in the tradition of patriotism as the enlightened ideal of good education. They remind the male protagonist of his duty both to fight and to raise new young patriots, educating them in the love for a fatherland which should be served both at home and on the battlefield.

PATRIOTIC INSPIRATION AND POLITICAL POWER

The transformation of women as suffering objects for which manly warriors risk their life into characters with an own intention to fight can be observed more strongly in the Claudius Civilis-play written by Willem Haverkorn in 1779. 31 The main male hero here is not Civilis but Brinio, leader of the Germanic tribe of the Cananefates, who admires Civilis" courageous niece Adelheide. Adelheide"s second admirer is her old boyfriend Briganticus, the unstable and irresolute Batavian, who serves the Roman army. On the one hand, Brinio describes how the fatherland suffers from Roman tyranny, using the image of the elderly and women who are incompetent to use arms and to defend themselves against the Romans.32 On the other hand, he mentions the heroine Adelheide, who, together with a battalion of armed Batavian women, attacked the false patriots under Briganticus. 33 The courageous women created such a confusion and respect with their armed attack that Briganticus decided to fly from the battlefield, together with the rest of the army. With her courageous behaviour, Adelheide represents the subject who by nature is incompetent to take up arms, but disposes of such an incredibly strong patriotic nature, that she nevertheless takes them up to confront the enemy. Here, the heroine not only dissolves in her adoration for manly warriors by accompanying them in battle, like Thusnelda in Schlegel"s Hermann-tragedy, but consciously crosses the natural borders of her supposed feminine passivity, transforming her initial weakness into an instrument of patriotic strength. In doing so, she represents the strength that patriotism can exert on its subjects. Her courage implicitly accuses the weak and degenerated male patriots who do not even dare to defend themselves and who decide to fly from the battlefield.

Haverkorn"s tragedy shows how the love for the fatherland, as a patriotic passion defended by reasonable, virtuous, and manly warriors, can be so strong, that the patriotic subject crosses the borders of nature. In Klopstock"s play Hermanns Schlacht. Ein Bardiet für die Schaubühne (Hamburg/Bremen 1769), the first part of his trilogy on the life of Hermann the Cherusker34 , reason becomes replaced by a form of patriotic motivation which depends more and more on inspiration and enthusiasm. The tragedy does not concentrate on the Schlacht, the battle itself, but on the origins of patriotic passions. The real heroes in this play, which is called a Bardiet, a song of bards, are not the fighters but the ancient Germanic bards, presented by Klopstock as an ancient caste of poets and singers who inspire the fighting Germans with their performances of songs and rituals.

The bards are standing at the top of a cliff, whereas the Germanic soldiers remain invisible characters during the battle at the feet of that cliff. By focusing on the power poetry can exert on military acts, Klopstock has Hermann and his fellow-fighters share their heroism with the bards and druids, which is explained to the bards by Hermann"s aged father, who states that not only his son"s weapons, but also the songs of the bards are a crucial factor in deciding the fight ("Eure Gesänge stärken des Streitenden Arm.") [START_REF] Klopstock | On the relation between the bardic songs and the political intentions of the author[END_REF] The image of a loving and caring woman is an important source of inspiration for the fighting men in Klopstock"s Bardiet. The first bardic song is about mothers and brides who are nursing the soldiers" wounds, an image inspired by some remarks in Tacitus" Germania. [START_REF] Tacitus | to their mothers and wives they take their wounds, and the women are not afraid to count and examine the blows; they bring to the warriors both food and exhortations[END_REF] This song, which praises the women who lick the blood from the wounds of Germanic warriors 37 , certainly has maternal as well as sexual connotations. Bleeding rather than fighting is the keyword of the play in referring to the patriotic quality of men who do not shun sufferings and self-sacrifice. Only by showing their readiness to bleed can men be good patriots. Segest for instance, the companion of the Romans, is only allowed to fight together with the Germans after showing his readiness to shed his own blood, stating that "Spätes Blut ist auch Blut" (also late blood is blood). 38 It is not Hermann or Siegmar, but Brenno, the chief bard, who decides after this confession that Segest is allowed to fight, but not after he has listened to their songs that will inflame the spirit of war in his heart. An important difference with the idea of patriotic motivation in the earlier tragedies is that this need for inspiration of the indecisive warrior is no longer presented as a weakness. On the contrary, Segest"s change of attitude is presented as a positive example and even his Trägheit is not a problem, since every patriot has his doubts and has to overcome these doubts by patriotic inspiration before joining the battlefield.

As Elystan Griffiths states in his article on the Hermannschlacht-motive in German literature, women are presented in the songs of the bards as helpless characters in need of male protection. 39 But it is important to conclude also that in other songs, Thusnelda and her mother Bercennis are presented as fellow-combatants during the battle inspired by the image of the tender and warlike Diana, the goddess of the hunt. 40 However, this is not what empowers the female protagonists in the play. While in the tragedies of Schlegel and Berkhey women do in the end adopt the masculine rhetoric of patriotism or indeed take part in the fighting itself (Haverkorn), in Klopstock"s play women and bards own instruments of their own to support the fatherland: the instrument of inspiration. The source of inspiration is patriotic speech itself, which can be so powerful that it can transform slow patriots as well as elderly and children into active and vital combatants. This transformation is described as a kind of ecstasy caused directly by the patriotic songs. One of the Opferknaben (young assistants of the druids during the sacrifices) for instance cannot hold himself back from the battlefield because of the great appeal of the songs. 41 The boy"s father praises his wife for having born such a proud young patriot and thanks Wodan for calling his son to the battlefield. The child indeed goes to the battlefield, fights and returns to the rock of the bards deadly wounded.

Unlike the elderly and children, women are not to be seen on the battlefield in the play of Klopstock. At first sight, the author realises a clear separation between gender roles in correlation to the public and private domain, reducing the role of Thusnelda and other women to that of taking care for the soldiers: whetting their spears, nursing their wounds, decorating their shields, and cooking their meals. Nevertheless, we should hesitate to consider the Bardiet solely as manifesting the marginalisation of women within patriotic discourse. Central to Klopstock"s play is an enlightened holistic interpretation of the fatherland in which all citizens should become active and understand their own duties as good patriots. A civic discourse on war and fatherland had to be explored before the domains of private and public life could be described both separately and in relation to each other. Although, in the case of Klopstock, the distinction of roles leads to a reduction of female power in the military field and to the fact that women lose their capacity of interfering directly in the political and military domain, they do become more important in the new domain of patriotic inspiration, a domain to which military agency as such has become subordinated Although Klopstock was a critic of offensive war politics, his Hermann-play clearly can be read as propagating an aggressive non-rational discourse on patriotism in which revenge towards the foreign enemy is represented as a justified motivation for war. 42 While in the earlier tragedies discussed here feminine heroines try to present themselves as non-vengeful, mirroring manly patriotic reason, Klopstock presents their revenge in the end as a positive emotion, because it can activate a more warlike form of patriotism in the hearts of men. This creates a new power structure, in which especially women are playing a crucial role, as in the case of Hermann"s mother who pleads to avenge their people, even after the battle against the Romans has come to an end. She asks her son to kill especially those Romans Die unsre Knaben erwürgt, die unsre Jungfrauen gezwungen haben, gegen ihr eignes Leben zu wüten, die Ihn getödtet haben! Die sollen nicht um seine Leiche her in dem Dampfe des Todtenfeuers liegen? 43 [Who strangled our boys, who forced our maids to rage against their own lives, who killed them! Should they not just lie here around his dead body in the fumes of death fires?] Though the argumentative structure of these lamentations shows some resemblance to the Chorus of Batavian Women in Vondel"s tragedy, the effect of these lamentations on the hero is very different. The call for revenge cannot be neglected, nor can it be suppressed by reason anymore, and Hermann therefore has to capitulate to his mother"s aggressive patriotism. Hermann initially rebukes his mother for her vengeful feelings, meanwhile trying to control his own feelings of revenge both by reason and by showing his compassion with the enemies, but in the end he is not able to neglect her rhetoric of aggressive revenge. Remaining silent for some moments after her vengeful outburst, Hermann is persuaded by the emotions of his mother, and looking at the dead body of his father, he swears to kill the new legions of Roman soldiers expected to arrive soon. Vengeful patriotism which defeats political reason thus is especially remembered, advocated, and activated by the formerly passive forces: women and the elderly. These forces cause the male hero to dispel the discourse on fatherland and patriotism that is solely based on reason, as is illustrated by the outburst of revenge on the part of the hero: 42 See also the remarks of Von Essen (1998), 102-126. 43 Klopstock (1769), 133.

… Du sollst gerochen werden, mein Vater! … ja du sollst gerochen an den neuen Legionen werden! an allen ihren Tribunen! und Legaten! und Feldherrn! …. Ha, an ihren Tribunen gerochen mit Todesrache! …. 44 [… You will be avenged, my Father! … yes you will be avenged on the new legions! and on all their tribunes! and legates! and generals! … Hah, on their tribunes avenged with the vengeance of death!] This emphatic form of patriotism culminating in patriotic vengeful ecstasy on the part of male warriors who are inspired by the songs of bards and the image of fighting and caring women is absent from the Dutch plays I discussed earlier. Although Berkhey and Haverkorn certainly had political intentions with writing their tragedies, at least in relation to early bellicism and the civic re-appropriation of discourse on war and fatherland in general, they did not succeed in transforming the discourse on patriotism as such. Standing in the long tradition of Dutch neo-classicism, the literary style of both Berkhey and Haverkorn remains stiff and distant, which seems to preclude a more emphatic patriotic rhetoric. Not only is the representation of the Batavian women restricted to their role of merging into the manly heroism of the old Batavian liberation myth, but patriotic inspiration also remains an abstract phenomenon, motivating the heroic patriot"s acts but yet an invisible force for the reader or spectator. Neither author was able to write a tragedy which highlighted patriotic enthusiasm and inspiration itself. Even while defending revolutionary political concepts, the patriotic discourse in Dutch literature of around 1770 still was a conservative one, falling back on the idea of political and cultural crisis and decline and on the necessity to restore the fatherland to its former glory. 45 A majority of the Dutch revolutionary Batavian patriots began to see quite lately that it would be necessary to develop a more visionary political concept for the reorganisation of the Dutch Republic and to find a new political, cultural, and literary language to revitalise the Dutch discourse on patriotism. Klopstock, however, did succeed in constituting a play which took shape in a performative (theatrical) speech-act on stage, praising an imaginative German nation by foregrounding the origins of patriotic discourse itself. With his Hermann-play Klopstock stages a competing discourse to the older national discourses based on the idea of a German nation, like the early modern Reichspatriotismus. Although he chose to dedicate his Bardiet to the emperor Joseph II, Klopstock knew very well that it would not be the old political construction of the German Empire which could realise his patriotic ideals. Klopstock believed in the importance of culture and science in the process of political modernisation. The ancient songs of the bards in Klopstock"s play could be seen as representing such a pre-modern state of the cultural constructed by the cotemporaneous bards, i.e.

Klopstock and the circle of German writers and intellectuals defending the idea of German literature as originating patriotic inspiration and enthusiasm. But Klopstock"s Hermanns Schlacht is more then a defence of German cultural patriotism, it is, as Hans Peter Herrmann rightly states in his article, a self-constitution of masculinity and especially of the right of male warriors to defend the country and kill the enemy. 46 In this discursive space of patriotism, the role of women indeed is reduced to an object of masculine possession, constituting a justification to fight and kill. Still, we should not stick to that single observation, underestimating the symbolical value of the activation of women as a source of inspiration for male patriots, without relating their role to that of the singing bards. Although one could 44 Klopstock (1769), 135. 45 [START_REF] Kloek | Blauwdrukken voor een samenleving[END_REF]), 215-217. 46 Herrmann (1996), 172. say that in Klopstock"s play the role of women is reduced once again to that of the seventeenth-century wailing woman, we must bear in mind that these lamentations are now transformed into emphatic songs, activating patriotic pleas for revenge. Both to constitute Hermann and the other Germanic warriors as free patriotic combatants and to legitimate a more emphatic aggressive patriotism, it was necessary to stage independent women and bards as the vengeful commentators of a patriotic war providing the nominal fighters with the inspiration they need. This way of staging inspired patriotism is blocked in the plays by Berkhey and Haverkorn, where the direct participation of women in the armed battle against the Romans does not create an emphatic intensification of patriotic discourse, because it results in submitting femininity to classic forms of masculine heroism. Klopstock literally pushes women back to the side-line of the battlefield, representing them as the uninvolved commentators on the battle. But it is by reducing their role to that of inspiring spectators, that he draws them towards the centre of a patriotic discourse which is not interested in highlighting fighting as such but in the will to fight as well as the inspiration needed to let male warriors kill and bleed for the fatherland as free patriots.

  […] ein fauler Held, den nur die Liebe zieht.Den edler Thaten Reiz nicht von sich selbst beweget, Und der im Schlafe liegt bis ihn ein Blick erreget.24 [a lazy Hero, who only is attracted by love Who is not moved by the impact of noble deeds And who remains in sleep until he is raised by what his eyes see.] Female authority here is not based on reasonable political speech, like in Berganie"s Arminius-play, but on the reward of the true patriot with love and admiration; whereas the unpatriotic enemy is confronted with aversion and sexual rejection.While Thusnelda shows her appreciation of Hermann"s active and courageous understanding of love for the fatherland by accompanying her man to the battlefield, this instrument of reward is transformed into admiration and total submission incited by the female desire of becoming absorbed in the euphoria of masculine heroism during the fight: Wie froh will ich mit dir bis zum dem Heere gehn! Wie froh will ich dich sehn an seiner Spitze stehn! O! daß mich dir dein Sieg zu eigen wiederbrächte, Nur, daß ich deinen Ruhm auf ewig theilen möchte: Dein Muth erhüb auch mich, und dein Sieg wär auch mein.25

[START_REF] Blitz | Aus Liebe zum Vaterland[END_REF], 82-83.

Meijer Drees (1999), 124-126.

[START_REF] Vondel | Batavische Gebroeders of Onderdrukte Vrijheit[END_REF],

98: "Geduldigh lijden zal u allerminst berouwen." 9 Vondel (1690), 100.

On the reception of the play, see[START_REF] Teitler | De opstand der 'Batavieren[END_REF], 62-63. Summary of the play in: DeLeeuwe (2003), 28-30.

[START_REF] Haverkorn | Claudius Civilis; Treurspel[END_REF], 3: "Wyl ze onbekwaam zyn tot het voeren van "t geweer."

[START_REF] Haverkorn | Claudius Civilis; Treurspel[END_REF], 5-6.

The other two plays of the trilogy are: Hermann und die Fürsten (Hamburg 1784) and Hermanns Tod (Hamburg 1787).