We analysed the morphological variations through geometric morphometric approaches of the dorsal and ventral views of growing male and female Callinectes danae Smith 1869, based on the hypothesis that swimming crabs present polymorphism during growth. Our research identified six instars for females, seven instars for males and one instar for unsexed individuals (young). The sixth instar of females and the seventh instar of males were composed of adults. We identified 20 landmarks from the dorsal view, and 16 landmarks from the ventral view. Based on canonical analysis and Procrustes distance, we observed the formation of clearly separated instars (both views), indicating a significant change during ontogeny, not only in the pubertal moult. The most prevalent changes occurred in the abdomen, with males thinning and females broadening the abdomen during growth. In the dorsal view, we observed a displacement of anterolateral spines to the posterior region throughout the growth period and that the anterolateral and frontal teeth of juveniles were more ornate than those of adults. The ontogenetic trajectories for males and females have similar origins and follow different directions over the instars, with maximum distance after the pubertal moult (P < 0.05).

This paper presents a practical and simple fully nonparametric multivariate smoothingprocedure that adapts to the underlying smoothness of the true regression function. Ourestimator is easily computed by successive application of existing base smoothers (withoutthe need of selecting an optimal smoothing parameter), such as thin-plate spline or kernelsmoothers. The resulting smoother has better out of sample predictive capabilities thanthe underlying base smoother, or competing structurally constrained models (MARS, GAM) forsmall dimension (3 ≤ d ≤7) and moderate sample size n ≤ 1000. Moreover our estimator is still usefulwhen d > 10and to our knowledge, no other adaptive fully nonparametric regression estimator isavailable without constrained assumption such as additivity for example. On a realexample, the Boston Housing Data, our method reduces the out of sample prediction error by20%. An R package ibr, available at CRAN, implements the proposedmultivariate nonparametric method in R.