Functional principal component analysis (FPCA) based on the Karhunen--Lo\`{e}ve decomposition has been successfully applied in many applications, mainly for one sample problems. In this paper we consider common functional principal components for two sample problems. Our research is motivated not only by the theoretical challenge of this data situation, but also by the actual question of dynamics of implied volatility (IV) functions. For different maturities the log-returns of IVs are samples of (smooth) random functions and the methods proposed here study the similarities of their stochastic behavior. First we present a new method for estimation of functional principal components from discrete noisy data. Next we present the two sample inference for FPCA and develop the two sample theory. We propose bootstrap tests for testing the equality of eigenvalues, eigenfunctions, and mean functions of two functional samples, illustrate the test-properties by simulation study and apply the method to the IV analysis.Comment: Published in at http://dx.doi.org/10.1214/07-AOS516 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Implied volatility, Nonparametric regression,

Knowledge growth models, based on primary principles, play a fundamental role in the cognitive sciences. The authors submit an extension of their model (ENKI) from 2005, with the results of the practical testing, which was performed using the method developed for the purpose of model ENKI of solving tasks with immediate feedback. This was applied to the curriculum of parallel configuration of resistors in electrical circuits. There were 73 pupils from six elementary schools in attendance for testing. Analysis based on ENKI indicates that three autonomous units (scopes) were evaluated simultaneously during the assessment. Results showed that 25% of pupils knew the curriculum, 9% of pupils showed no improvement, while 66% of pupils showed an increased success in accordance with the ENKI model (significance level = 0.05). Solving 7.2 typical tasks on average, by a method of immediate feedback resulted in 90% of the pupils mastering the curriculum.

One of the major cost factors in car manufacturing is the painting of body and other parts such as wing or bonnet. Surprisingly, the painting may be even more expensive than the body itself. From this point of view it is clear that car manufacturers need to observe the painting process carefully to avoid any deviations from the desired result. Especially for metallic colors where the shining is based on microscopic aluminium particles, customers tend to be very sensitive towards a difference in the light reflection of different parts of the car.The following study, carried out in close cooperation with a partner from car industry, combines classical tests and nonparametric smoothing techniques to detect trends in the process of car painting. The localized versions motivated by t-test, Mann-Kendall, Cox-Stuart and a change point test are employed in this study. Suitable parameter settings and the properties of the proposed tests are studied by simulations based on resampling methods borrowed from nonparametric smoothing.The aim of the analysis is to find a reliable technical solution which avoids any interaction from a human side.

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