The Priestley-Chao Estimator - Bias, Variance and Mean-Square Error

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1 The Priestley-Chao Estimator - Bias, Variance and Mean-Square Error Bias, variance and mse properties In the previous section we saw that the eact mean and variance of the Pristley-Chao estimator ˆm() when estimating the regression function for an (0, 1), the interval over which m is defined, are given by the epressions: bias[ ˆm()] = 1 nh n ( i K i=1 h ) m( i ) m() V [ ˆm()] = σ2 n 2 h 2 n ( ) i 2 K i=1 h Then mse[ ˆm()] = [bias[ ˆm()]] 2 + V [ ˆm()]. To see these in practice we can calculate the functions bias( ˆm()), V [ ˆm()] and mse[ ˆm()] for (0, 1) for the simulated regression data and plot the results for the four different values of the smoothing parameter h that we previously used. The results are shown in figures 1-3. Eact bias of Priestley Chao estimates using the simulated data bias Figure 1: Eact bias of Pristley-Chao estimates of the simulated regression data. h = red line; h = blue line; h = green line; h = orange line. A biweight (quartic) kernel was used in each case. The plot of the eact biases shows that, overall, the bias curves become flatter as the smoothing parameter h gets smaller. The bias is largest in 1

2 intervals corresponding to the main peak and trough of the true curve. For the main peak the bias is negative whereas for the main trough the bias is positive. Eact variance of Priestley Chao estimates using the simulated data variance Figure 2: Eact variance of Pristley-Chao estimates of the simulated regression data. h = red line; h = blue line; h = green line; h = orange line. A biweight (quartic) kernel was used in each case. The eact variances of the regression estimates are all constant in the middle of the range of -values and then fall away as gets closer to zero and to one. The variances are ordered in terms of smoothing parameter value with the lowest variance corresponding to h = 0.4 and the highest to h = The variance function for h = 0.03 falls away quickest at the etreme -values and that for h = 0.4 the slowest. 2

3 Eact mse of Priestley Chao estimates using the simulated data mse Figure 3: Eact mean squared error of Pristley-Chao estimates of the simulated regression data. h = red line; h = blue line; h = green line; h = orange line. A biweight (quartic) kernel was used in each case. The eact mse s for the four smoothing parameters considered are ordered by size of h here. The lowest mse function is when h = 0.03 and the highest when h = 0.4. The differences between them are most marked in the main peak and trough. Based on performance in terms of mse, the estimate using h = 0.03 can be regarded as being the best of the four here. Later in the last section we saw that, asymtotically, the bias and variance of ˆm() are given by: bias[ ˆm()] h2 2 m(2) ()σ 2 K, n 0, h V [ ˆm()] δσ2 h 1 1 K(u) 2 du as n The mean squared error (mse) of ˆm at the point is mse[ ˆm()] = [bias[ ˆm()]] 2 + V [ ˆm()] h4 4 m(2) () 2 σk 4 + δσ2 K(u) 2 du h 3

4 These functions are plotted in figures 4-6, again using the four different levels of smoothing. Asymptotic bias of Priestley Chao estimates using the simulated data abias Figure 4: Asymptotic bias of Pristley-Chao estimates of the simulated regression data. h = red line; h = blue line; h = green line; h = orange line. A biweight (quartic) kernel was used in each case. The asymptotic bias depends on the second derivative of the regression function m which, for the running simulated data eample, is given by m (2) () = 216sin(2π 3 )cos(2π 3 ) 2 π sin(2π 3 ) 3 π sin(2π 3 ) 2 cos(2π 3 )π The asymptotic bias functions in figure 4 are again ordered by the size of the smoothing parameter used. That for h = 0.03 is the lowest whilst that for h = 0.4 shows the largest peaks and troughs. For any h, the intervals when the bias has largest magnitude correspond to the main peak and trough of the true regression curve. The asymptotic bias curves are generally not as detailed as the true bias curves in this eample. 4

5 Asymptotic variance of Priestley Chao estimates using the simulated data avar Figure 5: Asymptotic variance of Pristley-Chao estimates of the simulated regression data. h = red line; h = blue line; h = green line; h = orange line. A biweight (quartic) kernel was used in each case. The asymptotic variances here are each constant functions of. That for h = 0.03 is the highest while that for h = 0.4 is the lowest. 5

6 Asymptotic mse of Priestley Chao estimates using the simulated data amse Figure 6: Asymptotic mean squared error of Pristley-Chao estimates of the simulated regression data. h = red line; h = blue line; h = green line; h = orange line. A biweight (quartic) kernel was used in each case. Again, the asymptotic mse s are ordered by the values of the four smoothing parameters used with that for h = 0.03 being the lowest while that for h = 0.4 is the highest. Out of the four values of h considered here, the asymptotic results again indicate that using h = 0.03 leads to the best estimate in terms of asymptotic mse. To see how close the asymptotic approimations are to the true performance measures the eact and asymptotic values were plotted on the same diagrams, just for the case when h = The asymptotics assume that as n h 0 so that the best approimations in our case should be when h = The results are shown in figures 7-9. That for the bias shows very good agreement in the interval (0, 0.65) but less good in (0.65, 1) where the changes in curvature of the true regression curve are greatest. The eact and asymptotic variances show etremely good agreement. Comparing the eact and asymptotic mse s, we can see that the asymptotic curve here has a less detailed shape and overestimates the true values when the curvature in m is greatest. It would be of interest to repeat this eercise with a much larger sample size n to see whether then the eact and asymptotic results are more similar overall. 6

7 Eact and Asymptotic Bias for a P C Estimator of the Simulated Data (Quartic kernel, h=0.03) bias Figure 7: Eact and asymptotic bias of Pristley-Chao estimates of the simulated regression data. Eact bias - solid line; asymptotic bias - dashed line. A biweight (quartic) kernel with h = 0.03 was used in each case. Eact and Asymptotic Variance for a P C Estimator of the Simulated Data (Quartic kernel, h=0.03) variance Figure 8: Eact and asymptotic variance of Pristley-Chao estimates of the simulated regression data. Eact variance - solid line; asymptotic variance - dashed line. A biweight (quartic) kernel with h = 0.03 was used in each case. 7

8 Eact and Asymptotic MSE for a P C Estimator of the Simulated Data (Quartic kernel, h=0.03) mse Figure 9: Eact and asymptotic mean squared error of Pristley-Chao estimates of the simulated regression data. Eact mse - solid line; asymptotic mse - dashed line. A biweight (quartic) kernel with h = 0.03 was used in each case. 8

The Priestley-Chao Estimator

The Priestley-Chao Estimator Te Priestley-Cao Estimator In tis section we will consider te Pristley-Cao estimator of te unknown regression function. It is assumed tat we ave a sample of observations (Y i, x i ), i = 1,..., n wic are