Scope of the sampling, propagation and inference interfaces

For a repeated-observation sampling model with the IID and second-moment hypotheses of Repeated-observation sampling models with IID and finite-variance hypotheses, Finite IID sample-mean variance and Chebyshev bound states that $E\mid \bar{Y}-\mu {\mid }^2=\frac{{\sigma }^2}{n},\mathbb{P}(\mid \bar{Y}-\mu \mid \ge a)\le \frac{{\sigma }^2}{n{a}^2}(a>0),$ If the variables are additionally specified as an infinite IID sequence, IID finite-variance weak law also gives convergence of its sample means to $\mu$ in $L^2$ and in probability. The factorization of mixed moments used in its proof is Expectations factor over finite products of independent random variables. The bound is distribution-free in the law of the observations, but it is conditional on the two model hypotheses and on the finite number ${\sigma }^2$; when the hypotheses fail, the identity and the bound are not available from these items, and ${\sigma }^2$ is model data rather than a measured quantity of an apparatus.

The covariance of the linearization is $A\mathrm{\Sigma }{A}^{\mathsf{T}}$ computes the mean vector and covariance matrix of the affine map $T(x)=h(\mu )+Dh(\mu )(x-\mu )$ applied to the random vector $X$: they are $h(\mu )$ and $Dh(\mu )\mathrm{Cov}(X)Dh(\mu {)}^{\mathsf{T}}$ (Mean vector and covariance matrix of a square-integrable random vector). That covariance is the covariance of the linearization, not of $h\circ X$; they agree if $h$ is equal to its linearization on $U$, that is, if $h(x)=h(\mu )+Dh(\mu )(x-\mu )$ for every $x\in U$. Under the convexity, $C^2$ and Hessian-bound hypotheses of Mean, mean-square and covariance bounds for second-order linearization error the scalar remainder mean is bounded using the second moment, and the covariance difference is bounded using the fourth moment. For a vector map $f=(f_i)$ with component Hessian bounds $B_i$, put $S=Df(\mu )\mathrm{Cov}(X)Df(\mu {)}^{\mathsf{T}}$ and $q_i=B_i^{2}E\mathrm{\mid }X-\mu {\mathrm{\mid }}^4\mathrm{/}4$. The explicit bound is $\mathrm{\mid }\mathrm{Cov}(f(X){)}_{ij}-S_{ij}\mathrm{\mid }\le \sqrt{S_{ii}{q}_j}+\sqrt{S_{jj}{q}_i}+\sqrt{q_i{q}_j}\mathrm{.}$ Without those hypotheses no such domination is asserted, and any use of the propagated covariance as an uncertainty of a nonlinear observation carries the hypotheses and moment conditions of that corollary.

The size of a test is ${\sup }_{\theta \in {\mathrm{\Theta }}_0}{\mathbb{P}}_{\theta }(R)$ (Statistical tests, rejection regions, power functions and size); it is a supremum over a null set of a specified model family, and no distribution on the parameter set enters its definition. The level of a confidence set is ${\inf }_{\theta }{\mathbb{P}}_{\theta }(\theta \in C)$ (Confidence sets and their coverage level); it is a pre-data infimum of coverage probabilities, not the probability, given the realized datum, that the realized set contains the parameter. A likelihood $L_s(\theta )$ is the value at the datum $s$ of one density $p_{\theta }$ of the family (Likelihood functions for a dominated parametric family); it is not a probability of the parameter value. The specification of the family, the rejection region, the confidence map and the dominating measure are data of each construction. The cited secondary sources record the corresponding interpretive warnings, which this remark states as properties of the definitions above.

The measurement model of Measurement models and additive-error observation models and the sampling model of Repeated-observation sampling models with IID and finite-variance hypotheses are mathematical structures; neither contains an apparatus, a preparation, a sampling protocol, a measured value or a measured uncertainty. The statements proved here are conditional on the stated hypotheses and are silent about whether any physical sequence of measurements satisfies them. A physical use must state preparation, apparatus, calibration, sampling and uncertainty separately, and must distinguish its model-conditional prediction from the reported observations it compares against them.

No Gaussian error model, normal density, convolution identity or quantile table appears in these items; the quantitative bound recorded above is the distribution-free inequality of IID finite-variance weak law. Any exact interval claim from a normal error model would require its normalized density and a separate convolution, quantile or tail supplier, and is not made here.