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A Gaussian error contrast has the induced multivariate-normal law

Statement

Assume AC and the Gaussian branch of the finite-dimensional mixed model. Let $X \in R^{n \times p}$ have full column rank with $p < n$ , let $K \in R^{n \times (n - p)}$ be an error-contrast matrix, and let $V = Z G Z^{T} + R = Cov (Y)$ . Then $K^{T} Y \sim N_{n - p} (0, K^{T} V K) .$ The contrast law does not depend on $β$ . The matrix $K^{T} V K$ may be singular; positive definiteness is a separate hypothesis when a density is needed.

Facts & Assumptions

Given: The centered Gaussian mixed model, AC, a full-rank fixed design, and an error-contrast matrix $K$ .

[F1]

$Y = X β + Z b + ε$ with centered $b, ε$ , covariance matrices $G, R$ , and $b$ independent of $ε$ . Finite-dimensional linear mixed model with explicit longitudinal roles.

[F2]

Under AC, the concatenated vector $(b, ε)$ has a multivariate-normal law. Gaussian branch of the linear mixed model.

[F3]

The response covariance is $V = Z G Z^{T} + R$ . Marginal covariance of the linear mixed model.

[F4]

An error-contrast matrix has shape $n \times (n - p)$ and satisfies $X^{T} K = 0$ . Error-contrast matrix for a full-rank fixed design.

[F5]

Every scalar linear projection of a multivariate-normal vector has the specified univariate normal law, including variance zero. Multivariate normal law, including singular covariance.

[F6]

Expectation is linear on finite sums of integrable random variables. The Lebesgue integral is linear on $L^{1} (μ)$ .

[F7]

Covariance is a centered second moment and is bilinear in finite linear combinations. Moments, variance, and covariance on a probability space, Covariance is symmetric and bilinear in finite linear combinations.

[F8]

A fixed finite-dimensional linear map is continuous and Borel measurable, so its composition with a Borel random vector is a Borel random vector. Every Euclidean linear map has a unique matrix and satisfies $∥ L h ∥_{2} \leq K ∥ h ∥_{2}$ for some $K \geq 0$ , A continuous map has Borel preimages of Borel sets.

[F9]

A univariate normal law has characteristic function $\exp (i t m - t^{2} σ^{2} / 2)$ . Characteristic function of a normal law.

[F10]

The characteristic function of $N_{d} (m, Σ)$ is $\exp (i t^{T} m - t^{T} Σ t / 2)$ . Characteristic function of a multivariate normal law.

[F11]

The characteristic function of a multivariate-normal law uniquely determines its Borel probability law, including singular covariance. Characteristic function of a multivariate normal law.

[A1]

AC holds. The Axiom of Choice.

Proof

Given: The hypotheses in the Statement.

1.1F1F3F6

By [F1], $Y - X β = Z b + ε$ is centered. Linearity [F6] gives $E Y = X β$ , and [F3] gives $Cov (Y) = V$ .

2.1A1F1F2F4F5F6F7step 1.1

Fix $u \in R^{n - p}$ . Since $X^{T} K = 0$ by [F4], $u^{T} K^{T} Y = u^{T} K^{T} (Z b + ε)$ , a scalar linear projection of the jointly Gaussian vector in [F2]; it is normal by [F5]. Its mean is zero by [F1] and [F6], and its variance is $u^{T} K^{T} V K u$ by step 1.1 and covariance bilinearity [F7].

3.1F8F9F10F11step 2.1∎

By [F8], $Q = K^{T} Y$ is a Borel random vector. For each $u$ , [F9] and step 2.1 give $E e^{i u^{T} Q} = \exp (- u^{T} K^{T} V K u / 2)$ . This is the characteristic function of $N_{n - p} (0, K^{T} V K)$ by [F10]; uniqueness [F11] gives the stated law. Its mean and covariance do not depend on $β$ , so neither does its law.

Depends on

Used by

Restricted maximum likelihood from error contrasts Definition

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Sources

Fan Li, STA610 Lecture 4: Maximum Likelihood Estimation and Inference for the Hierarchical Normal Model, PDF pp. 4–5 (standard reference, not scraped)
Duchateau, Janssen, and Rowlands, Linear Mixed Models, Chapter 2 §2.4.2 (standard reference, not scraped)