Integral transform methods in goodness-of-fit testing, II: the Wishart distributions
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Integral transform methods in goodness-of-fit testing, II: the Wishart distributions Elena Hadjicosta1 · Donald Richards1 Received: 1 May 2019 / Revised: 12 September 2019 © The Institute of Statistical Mathematics, Tokyo 2019
Abstract We initiate the study of goodness-of-fit testing for data consisting of positive definite matrices. Motivated by the appearance of positive definite matrices in numerous applications, including factor analysis, diffusion tensor imaging, volatility models for financial time series, wireless communication systems, and polarimetric radar imaging, we apply the method of Hankel transforms of matrix argument to develop goodnessof-fit tests for Wishart distributions with given shape parameter and unknown scale matrix. We obtain the limiting null distribution of the test statistic and a corresponding covariance operator, show that the eigenvalues of the operator satisfy an interlacing property, and apply our test to some financial data. We establish the consistency of the test against a large class of alternative distributions and derive the asymptotic distribution of the test statistic under a sequence of contiguous alternatives. We obtain the Bahadur and Pitman efficiency properties of the test statistic and establish a modified version of Wieand’s condition. Keywords Bahadur slope · Bessel function of matrix argument · Contiguous alternative · Diffusion tensor imaging · Factor analysis · Gaussian random field · Pitman efficiency · Zonal polynomial
1 Introduction The problem of testing that a random sample of positive definite matrices follows a Wishart distribution arose in factor analysis over fifty years ago; Browne (1968, p. 278)
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10463019-00737-z) contains supplementary material, which is available to authorized users.
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Donald Richards [email protected] Elena Hadjicosta [email protected]
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Department of Statistics, Pennsylvania State University, University Park, PA 16801, USA
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E. Hadjicosta, D. Richards
noted the difficulty of performing such a test, but no such results have appeared since then. More recently, random positive definite matrix data have appeared in numerous applications, e.g., diffusion tensor imaging, financial time series, wireless communication, and polarimetric radar images. Positive definite random matrix data are especially important in medical research, specifically in diffusion tensor imaging (DTI) (Dryden et al. 2009; Jian et al. 2007; Jian and Vemuri 2007; Kim et al. 2011; Lee and Schwartzman 2017; Schwartzman 2006; Schwartzman et al. 2005, 2008). DTI is a magnetic resonance imaging method that has attracted much interest in the study of brain diseases. DTI is based on the observation that water molecules in vivo are always in motion; by modeling the diffusion of the water molecules at any location by a three-dimensional Brownian motion, the resulting diffusion tensor image is represented by the 3 × 3 positive definite matrix of the local diffusion pro
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