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For many bird species, growth is negative or equivocal during development. Traditional, parametric growth curves assume growth follows a sigmoidal form with prescribed inflection points and is positive until asymptotic size. Accordingly, these curves will not accurately capture the variable, sometimes considerable, fluctuations in avian growth over the course of the trajectory. We evaluated the fit of three traditional growth curves (logistic, Gompertz, and von Bertalanffy) and a nonparametric spline estimator to simulated growth data of six different specified forms over a range of sample sizes. For all sample sizes, the spline best fit the simulated model that exhibited negative growth during a portion of the trajectory. The Gompertz curve was the most flexible for fitting simulated models that were strictly sigmoidal in form, yet the fit of the spline was comparable to that of the Gompertz curve as sample size increased. Importantly, confidence intervals for all of the fitted, traditional growth curves were wholly inaccurate, negating the apparent robustness of the Gompertz curve, while confidence intervals of the spline were acceptable. We further evaluated the fit of traditional growth curves and the spline to a large data set of wood thrush Hylocichla mustelina mass and wing chord observations. The spline fit the wood thrush data better than the traditional growth curves, produced estimates that did not differ from known observations, and described negative growth rates at relevant life history stages that were not detected by the growth curves. The common rationale for using parametric growth curves, which compress growth information into a few parameters, is to predict an expected size or growth rate at some age or to compare estimated growth with other published estimates. The suitability of these traditional growth curves may be compromised by several factors, however, including variability in the true growth trajectory. Nonparametric methods, such as the spline, provide a precise description of empirical growth yet do not produce such parameter estimates. Selection of a growth descriptor is best determined by the question being asked but may be constrained by inherent patterns in the growth data.

We study compact operator equations with noisy data in Hilbert space. Instead of assuming that the error in the data converges strongly to 0, we only assume weak convergence. Under the usual source conditions, we derive optimal convergence rates for convexly constrained Phillips-Tikhonov regularization. We also discuss a discrepancy principle and prove its asymptotic behavior. As an example, we discuss compact integral equations in L 2 (0, 1) with data perturbed by white noise, as well as the discrete case.

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