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The next factor that I considered was the effect the number of stimulus levels had on the monotone estimate. I considered the cases when n = 5,10,20,50. In each of the simulations

that I report in this section I fix the value of r = 5. I also only consider the versions of the

MISE, ISE and cross-validation bandwidths that were optimal for P(x) although mention here that results for the other bandwidths followed a similar pattern. It is clear that the magnitude

of the square bias, variance and MSE decreases as n increases.

Figures 4.3-4.6 show the comparison of square bias variance and MSE for each of the func- tions p2(x)-p5(x) respectively. Recall that I do not consider the MISE optimal bandwidth for

functions p1(x) and p6(x). A general pattern in all of these plots was that as the number of

stimulus levels increases the three methods became more similar.

For functionp2(x) the square bias values are generally higher for lower values ofxwith much higher bias at the boundary. There is little to distinguish between the three models although it

is clear that as the number of stimulus levels increases the bandwidth method performs less well

method remains far higher than the other two methods. This results in higher MSE values for this method.

The bandwidth adjustment method is clearly performs the best in terms of MSE perfor-

mance for estimating p3(x) as can be seen from Figure 4.4. The square bias of the PAVA estimate and the bandwidth estimate are very high at both boundaries compared to the LDNP

estimate but are broadly similar over the rest of the values of x. The variance of the PAVA

estimate is always higher than that of the other two estimates suggesting that this method is not very good for estimatingp3(x).

The bandwidth adjustment method also performs the best in terms of MSE performance

for estimating p4(x) as can be seen from Figure 4.5. This seems to be largely due to the fact that the variance for this estimate is significantly lower than for the other methods with the

bias values being reasonably similar.

Figure 4.6, shows that the bandwidth adjustment method again performs well for estimating

p5(x). The LDNP method seems to perform better than the PAVA method. For this model it

is clear that the methods become more similar as the number of stimulus levels increases. As

the value ofnincreases the bias performance of the LDNP method and PAVA method becomes much better in comparison to the bandwidth method but the variance remains higher. In

general then, as the value ofnincreases there is less of a difference between the three methods.

A comparison of the ISE ratios for the ISE optimal and cross validation bandwidths can be

seen in Figures 4.7-4.12 for functions p1(x)-p6(x) respectively. By comparing the first column

we can see that as the number of stimulus levels increases the PAVA method and the LDNP

method become more and more similar. They also become more tightly bound around 1. This indicates that as the value ofn increases the LDNP estimate and the PAVA estimate approach

the unconstrained estimate. In general the LDNP method performs the best when the ISE

optimal bandwidth is used for the unconstrained regression. The bandwidth method performs poorly in this case. In addition the density estimate for this method becomes much more flat as

the number of stimulus levels increases. This mean that this method is becoming progressively

flatter denstiy estimate indicates a lot of variation in ISE ratio.

By considering the second column of Figures 4.7-4.12 it can be seen what affect increasing

the sample size has on the estimates when a cross-validation estimate has been used for the

unconstrained regression. At low values of n the bandwidth adjustment method performs well in comparison to the others. However as the number of stimulus levels increases the density

estimates for this method become more flat suggesting more variability in this method and

hence poorer performance. The PAVA estimates and the LDNP methods perform comparably to each other although the LDNP method is usually a little bit better. This suggests that

for small values of n the cross-validation bandwidth estimate is too small and hence the best

way to calculate a monotone estimate is to simply increase the bandwidth. As the value of n

increases the cross-validation bandwidth is closer to the optimal one and the bandwidth method

performs less well.

In summary, then, it seems that when a good bandwidth is used (ISE optimal) then the

bandwidth adjustment method does not work well in terms of ISE. The other two methods are similar but the LDNP is usually better. Asnincreases the performance of the methods is more

similar but there are still noticeable differences forn= 50. If a poor bandwidth is chosen then

the best method is to simply increase the bandwidth.