Behrens–Fisher problem

Builds on Statistical hypothesis test Normal distribution Confidence interval

Problem

Let $X_1, \dots, X_m$ be i.i.d. normal $\mathcal{N}(\mu_1, \sigma_1^2)$ and let $Y_1, \dots, Y_n$ be i.i.d. $\mathcal{N}(\mu_2, \sigma_2^2)$ , independent of the $X_i$ . All four parameters are unknown, and $\sigma_1^2 = \sigma_2^2$ is not assumed. The problem asks for a test of $H_0 \colon \mu_1 = \mu_2$ whose type I error probability is exactly $\alpha$ for all values of $\sigma_1^2$ and $\sigma_2^2$ . Equivalently, it asks for a confidence interval for $\mu_1 - \mu_2$ with exact coverage $1 - \alpha$ for all values of the variances.

Remarks

The natural statistic is $(\bar X - \bar Y)/\sqrt{S_1^2/m + S_2^2/n}$ , where $S_1^2$ and $S_2^2$ are the sample variances. Its law under $H_0$ depends on the unknown ratio $\sigma_1^2/\sigma_2^2$ , which is the source of the difficulty. If the ratio $\sigma_1^2/\sigma_2^2$ is known, a two-sample Student's t-test solves the problem exactly. Welch's test compares the statistic with a t-distribution whose degrees of freedom are estimated from the data; its level is close to $\alpha$ but not exactly $\alpha$ in general. Fisher proposed a fiducial solution, and its level as a frequentist test is not exactly $\alpha$ either.

Reading path

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The rationals form an ordered field statement
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Statistic definition
Variance of a sum of independent random variables statement
Statistical hypothesis test concept
Sample mean definition
Confidence interval definition
Behrens–Fisher problem problem

Tag	`EJ3W3836VZ`
URI	`m2://EJ3W3836VZ`
Kind	problem
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MSC	62F03, 62F25
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About	Statistical hypothesis test
Uses	Statistical hypothesis test, Normal distribution, Confidence interval
Same as	Wikidata Q814556 Wikipedia Behrens–Fisher problem

Behrens–Fisher problem

Problem

Remarks

Reading path

Further reading

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