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Exact confidence and prediction intervals in Gaussian regression

Statement

Use the fixed full-column-rank Gaussian regression model and notation of Exact t and partial F inference in full-rank Gaussian regression, with $ν = n - p$ . Let $0 < α < 1$ , let $t^{*} = t_{ν, 1 - α / 2}$ be the unique $(1 - α / 2)$ quantile of $t_{ν}$ , and put $v_{c} = c^{T} (X^{T} X)^{- 1} c$ for $c \neq 0$ . Then $C_{c} (Y) = [c^{T} \hat{β} - t^{*} s \sqrt{v_{c}}, c^{T} \hat{β} + t^{*} s \sqrt{v_{c}}]$ has exact coverage $1 - α$ for $c^{T} β$ .

For any $x_{0} \in R^{p}$ , put $h_{0} = x_{0}^{T} (X^{T} X)^{- 1} x_{0}$ . The mean-response interval $C_{m e a n} (Y; x_{0}) = [x_{0}^{T} \hat{β} - t^{*} s \sqrt{h_{0}}, x_{0}^{T} \hat{β} + t^{*} s \sqrt{h_{0}}]$ has exact coverage $1 - α$ for $x_{0}^{T} β$ when $x_{0} \neq 0$ . If $x_{0} = 0$ , it is the singleton ${0}$ and has coverage $1$ .

If $Y_{0}^{*} = x_{0}^{T} β + ϵ_{0}^{*}$ is one future response with $ϵ_{0}^{*} \sim N (0, σ^{2})$ independent of the training data, then $C_{p r e d} (Y; x_{0}) = [x_{0}^{T} \hat{β} - t^{*} s \sqrt{1 + h_{0}}, x_{0}^{T} \hat{β} + t^{*} s \sqrt{1 + h_{0}}]$ satisfies $P_{β, σ^{2}} {Y_{0}^{*} \in C_{p r e d} (Y; x_{0})} = 1 - α$ . The added $1$ is the variance contribution of the future response's own error.

Facts & Assumptions

Given: The Axiom of Choice, the Gaussian regression experiment in Exact t and partial F inference in full-rank Gaussian regression, $0 < α < 1$ , a contrast $c \neq 0$ , a predictor vector $x_{0} \in R^{p}$ , and, for prediction, an independent future Gaussian error $ϵ_{0}^{*} \sim N (0, σ^{2})$ .

[A1]

The Axiom of Choice is inherited from the Gaussian model and its normal-law suppliers. The Axiom of Choice

[F1]

In this model, $\hat{β} \sim N_{p} (β, σ^{2} (X^{T} X)^{- 1})$ , $\hat{β}$ is independent of $R S S_{F}$ , and the stated nonzero-contrast pivot has law $t_{ν}$ . Exact t and partial F inference in full-rank Gaussian regression

[F2]

A pivot is a parameter-indexed family of measurable statistics with one common sampling law. Pivotal quantities

[F3]

A confidence-set procedure is measurable when each parameter-membership section is measurable, and coverage is the probability that the true parameter belongs to the set. Confidence sets and coverage

[F4]

Inverting a pivot acceptance set of probability $1 - α$ gives a confidence set with exact coverage $1 - α$ . Coverage from pivot inversion and exact calibration

[F5]

The $t_{ν}$ CDF is symmetric, continuous, and strictly increasing, so $[- t_{ν, 1 - α / 2}, t_{ν, 1 - α / 2}]$ has probability $1 - α$ . Student t and Fisher F CDFs are continuous and strictly increasing on their supports

[F6]

For a multivariate normal vector, every linear projection is normal with variance given by its covariance quadratic form; independent normal coordinates give the corresponding product normal law. Multivariate normal law, including singular covariance

[F7]

Independent random elements have product joint law, and measurable functions of independent random elements remain independent. Independent random elements have product joint law Measurable coordinatewise functions preserve independence

[F8]

The future error has the centered normal law with variance $σ^{2}$ , independently of the training data. Standard normal and normal laws

[F9]

For $ν \geq 1$ , $t_{ν}$ is the law of an independent standard normal divided by $\sqrt{χ_{ν}^{2} / ν}$ . Student t and Fisher F laws Chi-square laws, including zero degrees

[F10]

A positive-degree chi-square denominator is positive almost surely. Positive-degree chi-square laws have a positive density and regular CDF

Proof

Given: The hypotheses in the Statement.

1.1A1F1F2F3F4F5algebra

Fix $c \neq 0$ . Then $v_{c} > 0$ because $X^{T} X$ is positive definite. For each full parameter $θ = (β, σ^{2})$ , the statistic $q_{θ} (Y) = (c^{T} \hat{β} - c^{T} β) / (s \sqrt{v_{c}})$ , assigned value $0$ when $s = 0$ , is a pivot by [F1] and [F2], with common law $t_{ν}$ . By [F5], $B = [- t^{*}, t^{*}]$ has probability $1 - α$ . Define $D_{c} (Y) = {(β^{'}, σ^{' 2}) \in Θ : c^{T} β^{'} \in C_{c} (Y)}$ . For each fixed $(β^{'}, σ^{' 2}) \in Θ$ , its membership event is $∣ c^{T} \hat{β} - c^{T} β^{'} ∣ \leq t^{*} s \sqrt{v_{c}}$ , a closed, hence Borel, set because $\hat{β}$ and $s$ are continuous functions of $Y$ . Thus [F3] applies to $D_{c}$ . Pivot inversion [F4] gives exact coverage for $D_{c}^{'} (Y) = {θ^{'} \in Θ : ∣ q_{θ^{'}} (Y) ∣ \leq t^{*}}$ . On $s > 0$ , $D_{c} (Y) = D_{c}^{'} (Y)$ ; the exceptional event $s = 0$ has probability zero for every parameter by [F1]. Therefore $D_{c}$ has exact coverage $1 - α$ , which is precisely $P_{β, σ^{2}} {c^{T} β \in C_{c} (Y)} = 1 - α$ .

2.1F1step 1.1algebra

If $x_{0} \neq 0$ , the same argument with $c = x_{0}$ gives the displayed mean-response interval because $h_{0} = v_{x_{0}}$ . If $x_{0} = 0$ , then $x_{0}^{T} β = x_{0}^{T} \hat{β} = h_{0} = 0$ for every parameter and data vector, so the interval is ${0}$ and contains the true mean with probability $1$ .

3.1A1F1F5F6F7F8F9F10step 2.1algebra∎

Write $Z = (\hat{β} - β) / σ$ and $G_{*} = ϵ_{0}^{*} / σ$ . By [F1] and [F8], $Z$ and $G_{*}$ are independent centered normal vectors/scalars with covariance $(X^{T} X)^{- 1}$ and variance $1$ . Their joint law is the product law by [F7], so [F6] gives $\frac{Y_{0}^{*} - x_{0}^{T} \hat{β}}{σ \sqrt{1 + h_{0}}} = \frac{G_{*} - x_{0}^{T} Z}{\sqrt{1 + h_{0}}} \sim N (0, 1)$ , since the numerator has variance $1 + x_{0}^{T} (X^{T} X)^{- 1} x_{0} = 1 + h_{0}$ . Because $G_{*}$ is independent of the training data, [F7] gives $L (Z, R S S_{F}, G_{*}) = L (Z, R S S_{F}) \otimes L (G_{*})$ . Since $Z$ is a measurable affine function of $\hat{β}$ , [F1] and [F7] imply that $Z$ is independent of $R S S_{F}$ ; the joint law therefore factors as $L (Z) \otimes L (R S S_{F}) \otimes L (G_{*})$ , so $(Z, G_{*})$ is independent of $R S S_{F}$ . The displayed standard normal is a measurable function of $(Z, G_{*})$ , so it is independent of $R S S_{F} / σ^{2} \sim χ_{ν}^{2}$ by [F7]. By [F9] and [F10], the ratio $(Y_{0}^{*} - x_{0}^{T} \hat{β}) / (s \sqrt{1 + h_{0}})$ has law $t_{ν}$ with positive denominator almost surely. Its central acceptance interval has probability $1 - α$ by [F5], which is exactly the event $Y_{0}^{*} \in C_{p r e d} (Y; x_{0})$ .

Depends on

Used by

Mean-response and prediction intervals from one Gaussian regression fit Example

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Sources

Dmitry Panchenko, MIT 18.443 Statistics for Applications, Fall 2006, Section 15: Multiple Linear Regression, PDF pp. 4–7 (standard reference, not scraped)
Dmitry Panchenko, MIT 18.443 Statistics for Applications, Fall 2006, Lecture 4: Multivariate Normal Distribution, PDF pp. 1–4 (standard reference, not scraped)
Charlotte Wickham, Statistics: Lecture Three — Linear Models, PDF pp. 6–7 (standard reference, not scraped)
Stanford University, Stats 203, Inference for Linear Regression, Lecture 4, sections on mean-response and prediction intervals (standard reference, not scraped)