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  • 1.
    Leonenko, Nikolaj
    et al.
    School of Mathematics, Cardi® University, Senghennydd Road, Cardi® CF24 4AG, UK.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Statistical inference for the epsilon-entropy and the quadratic Renyi entropy2010In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 101, no 9, p. 1981-1994Article in journal (Refereed)
    Abstract [en]

    Entropy and its various generalizations are widely used in mathematical statistics, communication theory, physical and computer sciences for characterizing the amount of information in a probability distribution. We consider estimators of the quadratic Rényi entropy and some related characteristics of discrete and continuous probability distributions based on the number of coincident (or-close) vector observations in the corresponding independent and identically distributed sample. We show some asymptotic properties of these estimators (e.g., consistency and asymptotic normality). These estimators can be used in various problems in mathematical statistics and computer science (e.g., distribution identi¯cation problems, average case analysis for random databases, approximate pattern matching in bioinformatics, cryptography).

  • 2.
    Pini, Alessia
    et al.
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Spreafico, Lorenzo
    Vantini, Simone
    Vietti, Alessandro
    Multi-aspect local inference for functional data: Analysis of ultrasound tongue profiles2019In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 170, p. 162-185Article in journal (Refereed)
    Abstract [en]

    Motivated by the analysis of a dataset of ultrasound tongue profiles, we present multi-aspect interval-wise testing (IWT), i.e., a local nonparametric inferential technique for functional data embedded in Sobolev spaces. Multi-aspect IWT is a nonparametric procedure that tests differences between groups of functional data, jointly taking into account the curves and their derivatives. Multi-aspect IWT provides adjusted multi-aspect p-value functions that can be used to select intervals of the domain that are imputable for the rejection of a null hypothesis. As a result, it can impute the rejection of a functional null hypothesis to specific intervals of the domain and to specific orders of differentiation. We show that the multi-aspect p-value functions are provided with a control of the family wise error rate and that they are consistent. We apply multi-aspect IWT to the analysis of a dataset of tongue profiles recorded for a study on Tyrolean, a German dialect spoken in South Tyrol. We test differences between five different ways of articulating the uvular /r/: vocalized /r/, approximant, fricative, tap, and trill. Multi-aspect IWT-based comparisons result in an informative and detailed representation of the regions of the tongue where a significant difference occurs. 

  • 3.
    Pini, Alessia
    et al.
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics. MOX - Dept. of Mathematics, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 Milano, Italy; Department of Statistical Sciences, Università Cattolica del Sacro Cuore, Largo A. Gemelli 1, 20123 Milano, Italy.
    Stamm, Aymeric
    Vantini, Simone
    Hotelling's T-2 in separable Hilbert spaces2018In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 167, p. 284-305Article in journal (Refereed)
    Abstract [en]

    We address the problem of finite-sample null hypothesis significance testing on the mean element of a random variable that takes value in a generic separable Hilbert space. For this purpose, we propose a (re)definition of Hotelling's T-2 that naturally expands to any separable Hilbert space that we further embed within a permutation inferential approach. In detail, we present a unified framework for making inference on the mean element of Hilbert populations based on Hotelling's T-2 statistic, using a permutation-based testing procedure of which we prove finite-sample exactness and consistency; we showcase the explicit form of Hotelling's T-2 statistic in the case of some famous spaces used in functional data analysis (i.e., Sobolev and Bayes spaces); we demonstrate, by means of simulations, that Hotelling's T-2 exhibits the best performances in terms of statistical power for detecting mean differences between Gaussian populations, compared to other state-of-the-art statistics, in most simulated scenarios; we propose a case study that demonstrate the importance of the space into which one decides to embed the data; we provide an implementation of the proposed tools in the R package fdahotelling available at https://github.com/astamm/fdahotelling. 

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