A polynomial spline estimator is proposed for the mean function of dense functional data together with a simultaneous confidence band which is asymptotically correct. In addition, the spline estimator and its accompanying confidence band enjoy oracle efficiency in the sense that they are asymptotically the same as if all random trajectories are observed entirely and without errors. The confidence band is also extended to the difference of mean functions of two populations of functional data. Simulation experiments provide strong evidence that corroborates the asymptotic theory while computing is efficient. The confidence band procedure is illustrated by analyzing the near infrared spectroscopy data.

Core Ideas Meta‐analysis showed that poultry litter’s influence on crop productivity is comparable to that of inorganic fertilizer.Poultry litter’s effectiveness on crop yield is influenced by soil properties, tillage, application practice, and crop species.More positive effects were found in acidic soil compared with neutral or alkaline, in loam soil compared with sand or clay, under conservation tillage compared with conventional, by subsurface banded poultry litter compared with broadcast or incorporation through tillage.The full benefits of using poultry litter was achieved from long‐term studies, with litter improving crop yield compared with inorganic fertilizer. Research has shown that poultry litter (PL) can be used as a nutrient source for crop production. However, yield responses often varied when compared with inorganic fertilizer (IF) depending on soil type, management conditions, and PL application practices. Therefore, we reviewed the literature and conducted a meta‐analytic assessment to summarize the effects of PL vs. IF on yield response under different agricultural practices. A total of 866 observations from 90 studies were evaluated to determine how soil properties, tillage, application practices, crop species, and repeated applications influenced yield. Poultry litter significantly increased yield in loam, sandy loam, and silty‐clay loam soils, whereas yields were significantly greater with IF in sand and silty‐clay soils; no differences were observed between PL and IF with clay loams or silt loams. Under conventional tillage, IF’s effect on yield was positive, albeit not significant, whereas PL had a significant positive effect under strip‐till or no‐till. Poultry litter produced slightly lower yield when surface incorporated, but higher yield with subsurface band application when compared with IF. Poultry litter had significantly higher yield with cotton (Gossypium hirsutum L.), corn (Zea mays L.), soybean [Glycine max (L.) Merr.], and peanut (Arachis hypogaea L.), significantly lower with bermudagrass [Cynodon dactylon (L.) Pers] than IF, and no effects on tall fescue (Festuca arundinacea Schreb.), corn silage, rice (Oryza sativa L.), and wheat (Triticum aestivum L.). Overall, PL was comparable to IF. However, the greatest benefits of PL on yield when compared to IF tended to occur following repeated (three or more) annual applications.

Abstract:We consider nonparametric estimation of the covariance function for dense functional data using computationally efficient tensor product B-splines. We develop both local and global asymptotic distributions for the proposed estimator, and show that our estimator is as efficient as an "oracle" estimator where the true mean function is known. Simultaneous confidence envelopes are developed based on asymptotic theory to quantify the variability in the covariance estimator and to make global inferences on the true covariance. Monte Carlo simulation experiments provide strong evidence that corroborates the asymptotic theory. Examples of near infrared spectroscopy data and speech recognition data are provided to illustrate the proposed method.

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A central topic in functional data analysis is how to design an optimal decision rule, based on training samples, to classify a data function. We exploit the optimal classification problem when data functions are Gaussian processes. Sharp convergence rates for minimax excess misclassification risk are derived in both settings that data functions are fully observed and discretely observed. We explore two easily implementable classifiers based on discriminant analysis and deep neural network, respectively, which are both proven to achieve optimality in Gaussian settings. Our deep neural network classifier is new in literature which demonstrates outstanding performance even when data functions are non-Gaussian. In case of discretely observed data, we discover a novel critical sampling frequency that governs the sharp convergence rates. The proposed classifiers perform favorably in finite-sample applications, as we demonstrate through comparisons with other functional classifiers in simulations and one real data application.