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Independent Rademacher errors need not give the Gaussian regression t pivot

Statement refuted

Full column rank, $n > p$ , independent errors with mean zero and common positive variance suffice for the exact Gaussian-regression Student $t_{n - p}$ contrast pivot.

Facts & Assumptions

Given: The Rademacher-error model described below, and the Student $t_{1}$ law.

[A1]

The Axiom of Choice is assumed for the normal and Student $t$ law definitions; the finite enumeration below uses no choice. The Axiom of Choice

[F1]

The $t_{1}$ CDF is continuous and strictly increasing on $R$ , so $t_{1}$ is not a point mass at zero. Student t and Fisher F CDFs are continuous and strictly increasing on their supports

Counterexample

Take $n = 2$ , $p = 1$ , $X = (1, 1)^{T}$ , $β = 0$ , and independent errors $ϵ_{1}, ϵ_{2}$ with $P (ϵ_{i} = 1) = P (ϵ_{i} = - 1) = 1 / 2$ . They have mean zero and variance $σ^{2} = 1$ , while $X$ has rank one and $n > p$ . Let $\hat{β} = (X^{T} X)^{- 1} X^{T} Y$ , $R S S = ∥ (I - P_{X}) Y ∥^{2}$ , $s^{2} = R S S / (n - p)$ , and use the nonzero contrast $c = 1$ with pivot set to zero when $s = 0$ .

Given: All four equally likely pairs $(ϵ_{1}, ϵ_{2}) \in {(- 1, - 1), (- 1, 1), (1, - 1), (1, 1)}$ .

1.1givenconstructalgebra

Since $X^{T} X = 2$ , $\hat{β} = (ϵ_{1} + ϵ_{2}) / 2$ and $R S S = (ϵ_{1} - ϵ_{2})^{2} / 2$ . If the errors agree, then $R S S = 0$ and the pivot is zero by its definition. If they disagree, then $\hat{β} = 0$ , $R S S = 2$ , and the pivot denominator is $s \sqrt{(X^{T} X)^{- 1}} = \sqrt{2} \sqrt{1 / 2} = 1$ , so the pivot is again zero. These cases exhaust the four outcomes.

2.1A1F1step 1.1algebra∎

Thus the pivot equals zero almost surely. By [F1], a $t_{1}$ random variable has a continuous law and hence assigns probability zero to the singleton ${0}$ , whereas this pivot assigns it probability one. Its law is not $t_{1}$ , refuting the statement.

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Sources

Dmitry Panchenko, MIT 18.443 Statistics for Applications, Fall 2006, Section 15: Multiple Linear Regression, PDF pp. 5–7 (standard reference, not scraped)
Charlotte Wickham, Statistics: Lecture Three — Linear Models, PDF pp. 6–7 (standard reference, not scraped)