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Department of Statistical and Actuarial Sciences
Center of Actuarial Excellence (CAE)
Convocation June 2014
Convocation June 2014

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  • Colloquium (Thursday, October 23, 2014)
    Time: 12:00pm-1:00pm and Room: WSC 248
    Speaker: Dr. Jean-Marie Dufour - McGill University
    Title: Wald-type tests when rank conditions fail: a smooth regularization approach

    This paper examines Wald-type tests in the presence of a possibly singular asymptotic covariance matrix. Two different types of singularity are considered. First, the covariance matrix estimator has full rank but converges to a singular covariance matrix, so the Wald statistic can be computed as usual, but regularity conditions for the standard asymptotic chi-square distribution do not hold. This type of singularity is not easily corrected by using a generalized inverse. Second, the sample matrix does not have full rank, but converges to a possibly nonsingular matrix whose rank may differ from the finite-sample rank of the covariance matrix estimate. Locally redundant restrictions can lead to this type of situation. To address such difficulties, we introduce a novel mathematical object: the regularized inverse which is related but different of usual generalized inverses. Generalized inverses are motivated by exploiting results on eigenspaces along with a variance regularizing function (VRF) which modifies small eigenvalues (using a certain threshold c) so a unique inverse is always available. The proposed class of regularized inverse matrices include as special cases several regularization methods such as spectral cut-off approaches and Tikhonov-type inverses, mainly for estimation purposes. Under general regularity conditions, we show that sample regularized inverse matrices converge to their regularized population counterpart. In the context of testing based on non-regular Wald statistics, we propose regularized Wald statistics obtained by replacing the usual inverse of the estimated covariance matrix (or the generalized inverse) by a regularized inverse, allowing for both Gaussian and non-Gaussian parameter estimates. Three general classes ...

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