Fixed-design linear regression model

Let $n,p\ge 1$ be integers and let $X\in {\mathbb{R}}^{n\times p}$ be a fixed matrix. A fixed-design first-moment linear regression model is a parametric statistical experiment on $({\mathbb{R}}^n,\mathcal{B}({\mathbb{R}}^n))$ with parameter set ${\mathbb{R}}^p$, laws $(P_{\beta }{)}_{\beta \in {\mathbb{R}}^p}$, and observed vector $Y(y)=y$, such that every coordinate of $Y$ is integrable under $P_{\beta }$ and ${\mathbb{E}}_{\beta }Y=X\beta (\beta \in {\mathbb{R}}^p)\mathrm{.}$ Here $\mathcal{B}({\mathbb{R}}^n)$ is the Borel sigma-algebra, and expectation is taken under the particular law $P_{\beta }$. The residual vector ${ϵ}_{\beta }:=Y-X\beta$ therefore has coordinatewise mean zero. The phrase fixed design means that $X$ is held constant in this experiment; it carries no probability law or assignment mechanism for $X$.

This definition states only the first-moment model. It imposes no covariance, independence, or Gaussian assumption. Such assumptions must be stated where used.

Residual mean. For each fixed $\beta$ and coordinate $j$, put $c_j=(X\beta {)}_j$. Both $Y_j$ and the constant $c_j$ are integrable under the probability law $P_{\beta }$, and the constant has expectation $c_j$. By Linearity, monotonicity, and the modulus bound for expectation, ${\mathbb{E}}_{\beta }[{ϵ}_{\beta ,j}]={\mathbb{E}}_{\beta }[Y_j-c_j]={\mathbb{E}}_{\beta }[Y_j]-c_j=(X\beta {)}_j-c_j=0.$ Thus the residual-mean conclusion in the Definition uses expectation linearity coordinatewise under each law; it adds no covariance or distributional assumption.

This is an abstract mathematical regression model, not a complete scientific model contract for an application. The Definition of Scientific statistical model contracts requires a field record for a specified analysis, including its target population; this pair has no specified application or target population. The following notes record the model-side fields that can be stated at this abstract level.

• Borel convention. $\mathcal{B}({\mathbb{R}}^n)$ is generated by the metric topology of the standard Euclidean metric $d_2$: $d_2$ is a metric by ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it, and its metric-open sets form a topology by The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed, and Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison. Thus this is the Borel sigma-algebra specified by The Borel sigma-algebra of a topological space.

• Record, units, and indices. One observed data object is the vector $Y\in {\mathbb{R}}^n$ on the Borel space ${\mathbb{R}}^n$. The analysis unit is this vector; coordinate $i$ is an abstract response slot, not a specified biological, physical, sampling, or assignment unit. No enrollment process is modeled, so enrollment status is not applicable; this says nothing about enrollment in a later application. There is no time index, and distinct coordinates need not be independent or distinct scientific units.

• Fixed, random, and observed quantities. $X$ is fixed; $\beta$ is a fixed parameter indexing $P_{\beta }$; $Y$ is random under that law and observed as $y$; ${ϵ}_{\beta }$ is its random residual. No latent quantity or randomized assignment is specified.

• Target, mechanism, law, and conditioning. $\beta$ indexes the formal parameter. For a contrast $c$, ${\psi }_c(\beta )=c^T\beta$ is a formal model estimand; no target population is specified. The law is directly $P_{\beta }$ on ${\mathbb{R}}^n$, with no separate mechanism from a fuller scientific state and no additional support, density, or common-support assumption. All moments are under $P_{\beta }$ with $X$ fixed; there is no averaging over a random design.

• Adequacy. In a specified application, the mean relation ${\mathbb{E}}_{\beta }Y=X\beta$ is empirically assessable: data or validation can challenge it but cannot prove it. This pair supplies no application data and does not assert adequacy. Additional moment, covariance, independence, or Gaussian conditions are theorem-specific mathematical hypotheses, not part of this first-moment model.

The scientific application, including its target population and substantive adequacy evidence, remains unspecified.