Functional data analysis (FDA) (Ramsay et al. (2009);Ramsay and Silverman (2005)) is a part of modern multivariate statistics that analyses data providing information about curves, surfaces or anything else varying over a certain continuum. In economics and empirical finance we often have to deal with time series of functional data, where we cannot easily decide, whether they are to be considered as homogeneous or heterogeneous. At present a discussion on adequate tests of homogenity for functional data is carried (see e.g. Flores et al. (2015)). We propose a novel statistic for detetecting a structural change in functional time series based on a local Wilcoxon statistic induced by a local depth function proposed in Paindaveine and Van Bever (2013).

Markov chain analysis was applied to studies of cyclic sedimentation in the Coal Complex of the Bełchatów mining field (part of the Bełchatów lignite deposit). The majority of ambiguous results of statistical testing that were caused by weak, statistically undetectable advantage of either cyclicity over environmental barriers or vice versa, could be explained if only the above-mentioned advantages appeared in the neighbourhood. Therefore, in order to enhance the credibility of statistical tests, a new approach is proposed here in that matrices of observed transition numbers from different boreholes should be added to increase statistical reliability if they originated in a homogeneous area. A second new approach, which consists of revealing statistically undetectable cyclicity of lithofacies alternations, is proposed as well. All data were derived from the mining data base in which differentiation between lithology and sedimentary environments was rather weak. For this reason, the methodological proposals are much more important than details of the sedimentation model in the present paper. Nevertheless, they did reveal some interesting phenomena which may prove important in the reconstruction of peat/lignite environmental conditions. First of all, the presence of cyclicity in the sedimentation model, i.e., cyclic alternation of channel and overbank deposits, represents a fluvial environment. It was also confirmed that the lacustrine subenvironment was cut off from a supply of clastic material by various types of mire barriers. Additionally, our analysis revealed new facts: (i) these barriers also existed between lakes in which either carbonate or clay sedimentation predominated; (ii) there was no barrier between rivers and lakes in which clay sedimentation predominated; (iii) barriers were less efficient in alluvial fan areas but were perfectly tight in regions of phytogenic or carbonate sedimentation; (iv) groundwater, rather than surface flow, was the main source of CaCO3 in lakes in which carbonate sedimentation predominated; (v) a lack of cyclic alternation between abandoned channels and pools with clayey sedimentation; (vi) strong evidence for autocyclic alternation of phytogenic subenvironments and lakes in which carbonate sedimentation predominated was found in almost all areas studied.

Methods of functional outliers detection in functional setting have been discussed, i.e. shape outliers and magnitude outliers. Outliergram has been discussed, a tool for functional shape outliers detection. Robust adjusted functional boxplot has been discussed as well, a tool for functional magnitude outliers detection. „The elements of functional outliers analysis have been applied to air pollution data for Katowice and Kraków.”

The subject of this article is to present the beta - regression model, where we assume that one parameter in the model is described as a combination of algebraically independent continuous functions. The proposed beta model is useful when the dependent variable is continuous and restricted to the bounded interval. The parameters are obtained by maximum likelihood estimation. We prove that estimators are consistent and asymptotically normal

Shang and Hyndman (2017) proposed a grouped functional time series forecasting approach as a combination of individual forecasts obtained using generalized least squares method. We modify their methodology using generalized exponential smoothing technique for the most disaggregated functional time series in order to obtain more robust predictor. We discuss some properties of our proposals basing on results obtained via simulation studies and analysis of real data related to a prediction of a demand for electricity in Australia in 2016.