TABLE 6:

Summary of 3 approaches when Y and B are Gaussian

Approach	Method for including external information	Available form of the external information	$\mathrm{ARE}{(\widehat{\gamma })}^{*}$
			${\widehat{\gamma }}_{\text{X}}$	${\widehat{\gamma }}_{\text{B}}$
Standard MLE (ref)	None	NA	1	1
1: Synthetic data method	m additional synthetic data observations	Ability to draw Y values from Y|X distribution, regardless of the form	$1-{\text{A}}^{†}-\frac{{\sigma }_{\text{X}}^2{\theta }^{*2}}{{\sigma }_{\text{X}}^2{\theta }^{*2}+{\sigma }_{\theta }^{*2}}{\text{D}}^{‡}$	1 − D
2: Constrained MLE (Cheng et al., 2018)	Constraint	2.1: The estimated coefficient β is known.	1 − A	1
		2.2: Both of the estimated coefficient β and the standard deviation σβ are known.	$1-\text{A}-\frac{{\sigma }_{\text{X}}^2{\theta }^{*2}}{{\sigma }_{\text{X}}^2{\theta }^{*2}+{\sigma }_{\theta }^{*2}}\text{D}$	1 − D
3: CSPML (Chatterjee et al., 2016)	Constraint	Known expectation of Y| X	1 − A	1

^*$\mathrm{ARE}(\widehat{\gamma })={\mathrm{Var}}_{\text{M}}(\widehat{\gamma })/{\mathrm{Var}}_{\text{MLE}}(\widehat{\gamma }),\text{M}\in$ {Synthetic Data, CML, CSPML}

^†$\text{A}=\frac{{\sigma }_{\theta }^2}{{\sigma }_{\theta }^2+{\sigma }_{\text{X}}^2{\theta }^{*2}}\frac{{\sigma }_{\gamma }^2}{{\sigma }_{\gamma }^2+{\gamma }_{\text{B}}^2{\sigma }_{\theta }^2},\text{where}{\theta }^{*}=\frac{{\beta }^{*}-{\gamma }_{\text{X}}}{{\gamma }_{\text{B}}}$

^‡$\text{D}=\frac{2{\sigma }_{\gamma }^2{\gamma }_{\text{B}}^2{\sigma }_{\theta }^{*2}}{{\sigma }_{\beta }^{*4}},\text{where}{\sigma }_{\theta }^{*2}=\frac{{\sigma }_{\beta }^{*2}-{\sigma }_{\gamma }^2}{{\gamma }_{\text{B}}^2}$