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Standard error of a ratio of two means

Asked at G-Research

You estimate a ratio R=μX/μYR = \mu_X / \mu_Y from two independent samples: Xˉ\bar X (size nn, variance σX2/n\sigma_X^2/n) estimates the numerator mean μX\mu_X, and Yˉ\bar Y (size mm, variance σY2/m\sigma_Y^2/m) estimates the denominator mean μY\mu_Y.

Use the delta method to find the approximate variance of R^=Xˉ/Yˉ\hat R = \bar X / \bar Y.

Show a hint

For a function of two estimators, add the squared-partial-derivative contributions. Independence means no cross term.

Your answer

This one is open-ended. Work it through, then check your reasoning against the full solution.

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