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Sample-centered covariance is a measurable statistic and estimator

Statement

Let $(P_{θ})_{θ \in Θ}$ be a family of Borel probability laws on $R^{p}$ with $E_{θ} ∥ X ∥_{2}^{2} < \infty$ . For each $n \geq 1$ , observe an iid sample $X_{1}, \dots, X_{n}$ with common law $P_{θ}$ and sample law $P_{θ}^{\otimes n}$ on the Borel data space $(R^{p})^{n}$ . Put $Σ_{θ} = {Cov}_{θ} (X)$ and let $ψ (θ) = Σ_{θ}$ be the covariance-matrix estimand with target the Borel space $S^{p}$ of real symmetric $p \times p$ matrices. The sample-centered covariance map $T_{n} (x_{1}, \dots, x_{n}) : = \frac{1}{n} \sum_{i = 1}^{n} (x_{i} - \overset{ˉ}{x}) (x_{i} - \overset{ˉ}{x})^{T}, \overset{ˉ}{x} = \frac{1}{n} \sum_{i = 1}^{n} x_{i},$ is a statistic with target the Borel space $S^{p}$ , and hence is an estimator of $ψ$ in the sense of Statistics and estimators. This statement asserts measurability and target type only; it asserts no unbiasedness or consistency.

Facts & Assumptions

Given: A finite $p \geq 1$ , $n \geq 1$ , a family of Borel laws $(P_{θ})_{θ \in Θ}$ on $R^{p}$ with finite second moments, and an iid sample experiment on Borel sample space $(R^{p})^{n}$ .

[F1]

Statistical experiments and Parametric statistical models: the sample experiment is a parameter-indexed family of probability laws on its measurable data space.

[F2]

Identical distribution and IID families: under each $P_{θ}^{\otimes n}$ , the sample coordinates are iid with common law $P_{θ}$ .

[F3]

Covariance matrix of a finite-dimensional random vector and A finite second moment defines a positive semidefinite covariance matrix: the covariance $Σ_{θ}$ is finite and symmetric, so it belongs to $S^{p}$ .

[F4]

Estimands on a statistical model: $ψ (θ) = Σ_{θ}$ is the stated target function on the parameter set.

[F5]

Sample covariance estimators and squared operator loss: the sample-centered covariance is $n^{- 1} \sum_{i} (X_{i} - \overset{ˉ}{X}) (X_{i} - \overset{ˉ}{X})^{T}$ .

[F6]

Statistics and estimators: a statistic is a measurable map from the data space to a measurable target, and it is an estimator when its target space is that of the stated estimand.

[F7]

A continuous map has Borel preimages of Borel sets: a continuous map between topological spaces has Borel preimages of Borel sets.

Proof

technique · direct

1.1F5F7givenalgebra

Regard $S^{p}$ as the subspace of $R^{p^{2}}$ with its Borel sigma-algebra. Each matrix coordinate of $T_{n}$ is a polynomial in the $n p$ real sample coordinates, so $T_{n} : (R^{p})^{n} \to R^{p^{2}}$ is continuous. Its values are symmetric because every outer product is symmetric, hence it is also continuous as a map into the subspace $S^{p}$ . By [F7] it is Borel measurable.

2.1F1F2F3F4F5F6step 1.1∎

By [F1] and [F2], the sample experiment is the stated parameter-indexed iid experiment; [F3] places $Σ_{θ}$ in $S^{p}$ , and [F4] identifies it as the estimand target. The formula in [F5] is exactly $T_{n}$ . Thus [F6] and step 1.1 make $T_{n}$ a statistic and, since its target space is the same $S^{p}$ , an estimator of $ψ$ .

Depends on

Used by

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Sources

Online Bayesian Data Analysis, Lecture Notes 8: Learning Signal Representations (11 April 2016) (standard reference, not scraped)
Soumendu Sundar Mukherjee, Multivariate Analysis, Lecture 8: Principal Component Analysis (standard reference, not scraped)