Covariance estimation

Bayesian

Inverse Wishart priors. 🏗 Other?

Precision estimation

The workhorse of learning graphical models under linearity and Gaussianity. See precision estimation for a more complete treatment.

Continuous

See kernel learning.

Parametric

Unordered

Thanks to Rothman (2010) I now think about covariance estimates as being different in ordered versus exchangeable data.

Sandwich estimators

For robust covariances of vector data. AKA Heteroskedasticity-consistent covariance estimators. Incorporating Eicker-Huber-White sandwich estimator, Andrews kernel HAC estimator, Newey-West and others. For an intro see Achim Zeileis, Open-Source Econometric Computing in R.

Incoming

• Basic inference using Inverse Wishart by having a basic “process model” that increases uncertainty of the covariance estimate.
• general moment combination tricks
• John Cook’s comparison of standard deviation estimation tricks

Bounding by harmonic and arithmetic means

There are some known bounds for the univariate case. Wikipedia says, in Relations with the harmonic and arithmetic means that it has been shown (Mercer 2000) that for a sample \(\left\{y_i\right\}\) of positive real numbers, \[ \sigma_y^2 \leq 2 y_{\max }(A-H) \] where \(y_{\max }\) is the maximum of the sample, \(A\) is the arithmetic mean, \(H\) is the harmonic mean of the sample and \(\sigma_y^2\) is the (biased) variance of the sample. This bound has been improved, and it is known that variance is bounded by \[ \begin{gathered} \sigma_y^2 \leq \frac{y_{\max }(A-H)\left(y_{\max }-A\right)}{y_{\max }-H}, \\ \sigma_y^2 \geq \frac{y_{\min }(A-H)\left(A-y_{\min }\right)}{H-y_{\min }}, \end{gathered} \] where \(y_{\min }\) is the minimum of the sample (Sharma 2008).

Mond and Pec̆arić (1996) says

Let us define the arithmetic mean of \(A\) with weight \(w\) as \[ A_n(A ; w)=\sum_{i=1}^n w_i A_i \] and the harmonic mean of \(A\) with weight \(w\) as \[ H_n(A ; w)=\left(\sum_{i=1}^n w_i A_i^{-1}\right)^{-1} \] It is well known \([2,5]\) that \[ H_n(A ; w) \leqslant A_n(A ; w) \] Moreover, if \(A_{i j}(i, j=1, \ldots, n)\) are positive definite matrices from \(H_m\), then the following inequality is also valid [1]: \[ \frac{1}{n} \sum_{j=1}^n\left(\frac{1}{n} \sum_{i=1}^n A_{i j}^{-1}\right)^{-1} \leqslant\left[\frac{1}{n} \sum_{i=1}^n\left(\frac{1}{n} \sum_{j=1}^n A_{i j}\right)^{-1}\right]^{-1} \]

For multivariate covariance we are interested in the PSD matrix version of this.

References