### 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 goodness-of-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.

Original language | English (US) |
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Journal | Annals of the Institute of Statistical Mathematics |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability

### Cite this

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**Integral transform methods in goodness-of-fit testing, II : the Wishart distributions.** / Hadjicosta, Elena; Richards, Donald.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Integral transform methods in goodness-of-fit testing, II

T2 - the Wishart distributions

AU - Hadjicosta, Elena

AU - Richards, Donald

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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 goodness-of-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.

AB - 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 goodness-of-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.

UR - http://www.scopus.com/inward/record.url?scp=85075380838&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075380838&partnerID=8YFLogxK

U2 - 10.1007/s10463-019-00737-z

DO - 10.1007/s10463-019-00737-z

M3 - Article

AN - SCOPUS:85075380838

JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

SN - 0020-3157

ER -