Peter M. Aronow
= Peter M. Aronow
{c}
{tag=Mathematics author}
\b[Peter M. Aronow] develops statistical methods for <causal inference>, particularly <design-based inference>. Yale's institutional biography identifies research on complex experimental designs, network settings and incomplete survey data alongside appointments spanning <political science> and <statistics>.<Peter M. Aronow profile by Yale School of Public Health>[Titles and Biography, design-based methodological research]{ref}
His joint work with <Cyrus Samii> studies how to estimate <sampling variance> when a <sampling design> never observes some pairs together. A missing product cannot simply be deleted from the population <variance> formula: doing so can produce either sign of <bias>. Their construction uses <Young inequality> to bound unobserved products by individually estimable powers and obtain a <conservative variance estimator>.<Conservative variance estimation for sampling designs with zero pairwise inclusion probabilities by Peter M. Aronow and Cyrus Samii>[Section2.2, Proposition1; Section3.1, Proposition2; Section3.2, quadratic special case]{ref}
For an original exact illustration, choose one of two fixed values $y_1,y_2$ with equal <probabilities>. As in the <single-draw variance identifiability obstruction>, the <Horvitz-Thompson estimator> $2y_J$ has true <variance> $V_p=(y_1-y_2)^2$. Retaining only diagonal contributions gives $D=2y_J^2$, with
$$
\mathbb E_pD=y_1^2+y_2^2,
\qquad
\mathbb E_pD-V_p=2y_1y_2.
$$
Opposite signs therefore produce downward <bias>. The elementary <inequality> $-2ab\le a^2+b^2$, obtained from $(a+b)^2\ge0$, instead gives
$$
V_p\le2(y_1^2+y_2^2)=\mathbb E_p[4y_J^2].
$$
Consequently $C=4y_J^2$ is a <conservative variance estimator> for this stipulated <sampling design>; its excess in <expected value> is exactly $(y_1+y_2)^2$.
With $(y_1,y_2)=(3,-1)$, the true <variance> is16. The two possible values of $D$ are18 and2, averaging10; those of $C$ are36 and4, averaging20. Although $\mathbb E_pC\ge16$, a realized value of $C$ can be below16. \b[Conservativeness here is an expectation inequality], rather than a bound for every sample. By itself it also does not prove a specified <confidence interval> coverage probability.
\Image[/single-draw-conservative-variance.png]
{title=A conservative variance estimator can still fall below the true variance in an individual sample}
{description=Original exact two-unit example with fixed values 3 and -1, each selected with probability one half. Diagonal-only estimates are 18 and 2; corrected estimates are 36 and 4. Their means are 10 and 20, while the true variance of the Horvitz-Thompson total is16.}