accompanied by a higher variance. However, looking at the total root mean squared er- rors, we see that the new estimators outperform the naive two-step estimators in almost all cases. This is also true for the non-normal models 7 and 8, so the warped maximum likelihood estimators are robust to mild departures from normality. 

## 6 Example: beetle growth data 

In this section we study mass growth curves of flour beetles fr om birth to pupation, from Irwin and Carter (2013). A total of 122 insects are considered. This is a subset of a larger dataset that includes both siblings and half-siblings, but in order to apply the one-way ANOVA model, which assumes independence between groups, we consider only the half- siblings. (The full data set can be modeled as a nested two-way ANOVA, with the mother factor nested within the father factor.) The insects were sired by 29 different fathers, which will constitute the grouping variable. The number of insects per father varies between 2 and 5, with a median of 4. 

Part of the raw data is shown in Figure 1(a); for better visualization we only plotted half of the sample curves. The mass measures were taken about every 3 days early in the growth curve, and up to once per day late in the growth curve. However, only 18 of the 122 larvae were measured for mass for the first time on the day t hey hatched; 76 were measured for mass for the first time on the second day, 22 on thethird day, 5 on the fourth day, and one was not measured for mass until the seventh day. Therefore, the starting points of the curves are unequal. The endpoints are also irregular, because larvae reached pupation at different points between days 16 and 25. However, while the unequal starting points are due to missing data, the unequal endpoints are due to a well-defined biological landmark which is reached at different times. Therefore we rescaled the time grids so that all trajectories end at the median pupation day 19, but we did not align the starting points 

