# Computes a shrunken version of the maximum likelihood estimator for the sample covariance matrix under the assumption of multivariate normality.

### Description

For a sample matrix, x, we compute the sample covariance matrix as the maximum likelihood estimator (MLE) of the population covariance matrix and shrink it towards its diagonal.

### Usage

 1 cov_shrink_diag(x, gamma = 1) 

### Arguments

 x data matrix with n observations and p feature vectors gamma the shrinkage parameter. Must be between 0 and 1, inclusively. By default, the shrinkage parameter is 1, which simply yields the MLE.

### Details

Let \widehat{Σ} be the MLE of the covariance matrix Σ. Then, we shrink the MLE towards its diagonal by computing

\widehat{Σ}(γ) = γ \widehat{Σ} + (1 - γ) \widehat{Σ} \circ I_p,

where \circ denotes the Hadamard product and γ \in [0,1].

For γ < 1, the resulting shrunken covariance matrix estimator is positive definite, and for γ = 1, we simply have the MLE, which can potentially be positive semidefinite (singular).

The estimator given here is based on Section 18.3.1 of the Hastie et al. (2008) text.

### Value

shrunken sample covariance matrix of size p \times p

### References

Hastie, T., Tibshirani, R., and Friedman, J. (2008), "The Elements of Statistical Learning: Data Mining, Inference, and Prediction," 2nd edition. http://statweb.stanford.edu/~tibs/ElemStatLearn/

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