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  • 1. Ahmad, M. Rauf
    et al.
    Pavlenko, Tatjana
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    A U-classifier for high-dimensional data under non-normality2018In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 167, p. 269-283Article in journal (Refereed)
    Abstract [en]

    A classifier for two or more samples is proposed when the data are high-dimensional and the distributions may be non-normal. The classifier is constructed as a linear combination of two easily computable and interpretable components, the U-component and the P-component. The U-component is a linear combination of U-statistics of bilinear forms of pairwise distinct vectors from independent samples. The P-component, the discriminant score, is a function of the projection of the U-component on the observation to be classified. Together, the two components constitute an inherently bias-adjusted classifier valid for high-dimensional data. The classifier is linear but its linearity does not rest on the assumption of homoscedasticity. Properties of the classifier and its normal limit are given under mild conditions. Misclassification errors and asymptotic properties of their empirical counterparts are discussed. Simulation results are used to show the accuracy of the proposed classifier for small or moderate sample sizes and large dimensions. Applications involving real data sets are also included. 

  • 2.
    Olsson, Jimmy
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Pavlenko, Tatjana
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Rios, Felix
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Bayesian structure learning in graphical models using sequential Monte CarloManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper we present a family of algorithms, the junction tree expanders, for expanding junction trees in the sense that the number of nodes in the underlying decomposable graph is increased by one. The family of junction tree expanders is equipped with a number of theoretical results including a characterization stating that every junction tree and consequently every de- composable graph can be constructed by iteratively using a junction tree expander. Further, an important feature of a stochastic implementation of a junction tree expander is the Markovian property inherent to the tree propagation dynamics. Using this property, a sequential Monte Carlo algorithm for approximating a probability distribution defined on the space of decompos- able graphs is developed with the junction tree expander as a proposal kernel. Specifically, we apply the sequential Monte Carlo algorithm for structure learning in decomposable Gaussian graphical models where the target distribution is a junction tree posterior distribution. In this setting, posterior parametric inference on the underlying decomposable graph is a direct by- product of the suggested methodology; working with the G-Wishart family of conjugate priors, we derive a closed form expression for the Bayesian estimator of the precision matrix of Gaus- sian graphical models Markov with respect to a decomposable graph. Performance accuracy of the graph and parameter estimators are illustrated through a collection of numerical examples demonstrating the feasibility of the suggested approach in high-dimensional domains. 

  • 3. Stepanova, N.
    et al.
    Pavlenko, Tatjana
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Goodness-of-fit tests based on sup-functionals of weighted empirical processes2018In: Theory of Probability and its Applications, ISSN 0040-585X, E-ISSN 1095-7219, Vol. 63, no 2, p. 292-317Article in journal (Refereed)
    Abstract [en]

    A large class of goodness-of-fit test statistics based on sup-functionals of weighted empirical processes is proposed and studied. The weight functions employed are the Erdős–Feller– Kolmogorov–Petrovski upper-class functions of a Brownian bridge. Based on the result of M. Csörgő, S. Csörgő, L. Horváth, and D. Mason on this type of test statistics, we provide the asymptotic null distribution theory for the class of tests and present an algorithm for tabulating the limit distribution functions under the null hypothesis. A new family of nonparametric confidence bands is constructed for the true distribution function and is found to perform very well. The results obtained, involving a new result on the convergence in distribution of the higher criticism statistic, as introduced by D. Donoho and J. Jin, demonstrate the advantage of our approach over a common approach that utilizes a family of regularly varying weight functions. Furthermore, we show that, in various subtle problems of detecting sparse heterogeneous mixtures, the proposed test statistics achieve the detection boundary found by Yu. I. Ingster and, when distinguishing between the null and alternative hypotheses, perform optimally adaptively to unknown sparsity and size of the non-null effects. 

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