Alphabeta MathHow statement and proof provenance work
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Sample covariance estimators and squared operator loss
Definition
Let be integers and let be independent identically distributed random vectors in with finite second moments, mean , and covariance matrix The oracle-centered sample covariance and the sample-centered covariance are respectively For a matrix estimate , squared operator loss is , with the operator norm of The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||. This definition distinguishes the known-mean raw second moment from the estimator that removes an unknown mean.
Depends on
Used by
- Entrywise-thresholded sample covariance Algorithm
- The stated sparse-covariance minimax rate is unbounded for super-exponential dimension Counterexample
- Covariance and correlation principal component analysis Definition
- Sample covariance is singular when dimension exceeds sample size Example
- Expected squared operator error of thresholded sparse covariance Lemma
- Sample-centered covariance is a measurable statistic and estimator Lemma
- Capped thresholded minimax rate for weak-lq sparse covariance Theorem
- Dense sample covariance concentration for sub-Gaussian vectors Theorem
- Entrywise deviation for sample-centered covariance under directional Gaussian tails Theorem
- Finite-fourth-moment root-n consistency of sample-centered covariance Theorem
Dependency tree · two levels
12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Philippe Rigollet, 18.S997 High-Dimensional Statistics, Chapter 4, Section 4.3 (standard reference, not scraped)
- Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, Chapter 4 (standard reference, not scraped)