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  • 1.
    Ahmad, M. Rauf
    et al.
    Uppsala University, Sweden; Swedish University of Agriculture Science, Sweden; University of Munich, Germany.
    von Rosen, Dietrich
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Swedish University of Agriculture Science, Sweden.
    Tests for high-dimensional covariance matrices using the theory of U-statistics2015In: Journal of Statistical Computation and Simulation, ISSN 0094-9655, E-ISSN 1563-5163, Vol. 85, no 13, p. 2619-2631Article in journal (Refereed)
    Abstract [en]

    Test statistics for sphericity and identity of the covariance matrix are presented, when the data are multivariate normal and the dimension, p, can exceed the sample size, n. Under certain mild conditions mainly on the traces of the unknown covariance matrix, and using the asymptotic theory of U-statistics, the test statistics are shown to follow an approximate normal distribution for large p, also when p and#8811;n. The accuracy of the statistics is shown through simulation results, particularly emphasizing the case when p can be much larger than n. A real data set is used to illustrate the application of the proposed test statistics.

  • 2.
    Sysoev, Oleg
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Computer and Information Science, Statistics.
    Grimvall, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Computer and Information Science, Statistics.
    Burdakov, Oleg
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .
    Bootstrap estimation of the variance of the error term in monotonic regression models2013In: Journal of Statistical Computation and Simulation, ISSN 0094-9655, E-ISSN 1563-5163, Vol. 83, no 4, p. 625-638Article in journal (Refereed)
    Abstract [en]

    The variance of the error term in ordinary regression models and linear smoothers is usually estimated by adjusting the average squared residual for the trace of the smoothing matrix (the degrees of freedom of the predicted response). However, other types of variance estimators are needed when using monotonic regression (MR) models, which are particularly suitable for estimating response functions with pronounced thresholds. Here, we propose a simple bootstrap estimator to compensate for the over-fitting that occurs when MR models are estimated from empirical data. Furthermore, we show that, in the case of one or two predictors, the performance of this estimator can be enhanced by introducing adjustment factors that take into account the slope of the response function and characteristics of the distribution of the explanatory variables. Extensive simulations show that our estimators perform satisfactorily for a great variety of monotonic functions and error distributions.

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