View source for Asymptotically-unbiased test

A concept indicating that the test is unbiased in the limit. For example, in the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013830/a0138301.png" /> independent samples from a one-dimensional distribution depending on a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013830/a0138302.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013830/a0138303.png" /> be the null hypothesis: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013830/a0138304.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013830/a0138305.png" /> be the alternative:

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The critical set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013830/a0138307.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013830/a0138308.png" />-dimensional Euclidean space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013830/a0138309.png" /> is an asymptotically-unbiased test of the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013830/a01383010.png" /> with level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013830/a01383011.png" /> if

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The function

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is called the asymptotic power function of the test <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013830/a01383015.png" />.