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Wikpedia defines "parametric statistics" as:

...a branch of statistics which assumes that sample data comes from a population that follows a probability distribution based on a fixed set of parameters.

Many hypothesis tests usually characterized as parametric, such as the classical one-sample $t$-test for the sample mean, are based on knowledge of a parametric distribution for the statistic. In terms of assumptions on the data, the $t$-test requires that the data be normally distributed only if the sample is small, but once the CLT kicks in, no distributional assumptions on the data are required.

Would it therefore be fair to describe the $t$-test and similar as "parametric in finite samples but asymptotically nonparametric"? This would seem to be a very unusual characterization.

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    $\begingroup$ no. A nonparametric test is nonparametric for finite sample size (such as the permutation test, the log-rank test, the Wilcoxon rank sum test), etc. $\endgroup$ – DeltaIV Mar 11 '18 at 21:10
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    $\begingroup$ Once CLT kicks in you still have that distributional assumptions are required. But the difference is that this requirement has reduced to only the mean and variance of the distribution, which is now being approximated by a normal distribution. $\endgroup$ – Sextus Empiricus Jan 16 at 15:55
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NO. One could take the stand as in the comment by @DeltaIV, or one could accept the possibility of an asymptotically nonparametric test, but then point out that even asymptotically, the power of the t-test depends on parametric assumptions.

This is discussed in Lehmann & Romano, Testing Statistical Hypothesis, around page 462. Specifically, the asymptotic properties of power depends on the skewness of the underlying sampling distribution.

For asymptotic power, see Asymptotic power

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