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Abstract:
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Bayesian estimation of mean and covariance matrix under the joint constraint ?µ = µ of p-dimensional Gaussian distribution is not well explored. The entanglement of the mean-covariance, viewed as a multivariate analog of the estimation of ? in N(?,?^2 ), is a source of many inferential challenges. We propose a novel structured covariance through reparameterization of ? involving µ, which addresses the issue of positive definiteness, accommodates the constraint and reduces the number of parameters from quadratic to linear function of dimension. Inspired by the shrinkage inverse Wishart (Berger et al., 2020), we use normal and inverse gamma prior on the mean and eigenvalues of this model respectively. We have proposed a fast method for approximation of the MLE and MAP estimators that bypasses the time-consuming Gibbs sampling. We propose a lower bound for the likelihood, which is concave under certain mild conditions and maximize it to approximate MLE and MAP both. A simulation study shows good performance of our estimates.
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