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This paper provides a framework for estimating the mean and variance of ahigh-dimensional normal density. The main setting considered is a fixed numberof vector following a high-dimensional normal distribution with unknown meanand diagonal covariance matrix. The diagonal covariance matrix can be known orunknown. If the covariance matrix is unknown, the sample size can be as smallas $2$. The proposed estimator is based on the idea that the unobserved pairsof mean and variance for each dimension are drawn from an unknown bivariatedistribution, which we model as a mixture of normal-inverse gammas. The mixtureof normal-inverse gamma distributions provides advantages over more traditionalempirical Bayes methods, which are based on a normal-normal model. When fittinga mixture model, we are essentially clustering the unobserved mean and variancepairs for each dimension into different groups, with each group having adifferent normal-inverse gamma distribution. The proposed estimator of eachmean is the posterior mean of shrinkage estimates, each of which shrinks asample mean towards a different component of the mixture distribution.Similarly, the proposed estimator of variance has an analogous interpretationin terms of sample variances and components of the mixture distribution. Ifdiagonal covariance matrix is known, then the sample size can be as small as$1$, and we treat the pairs of known variance and unknown mean for eachdimension as random observations coming from a flexible mixture ofnormal-inverse gamma distributions.

Estimating the Mean and Variance of a High-dimensional Normal Distribution Using a Mixture Prior