Is there any statistical test that is parametric and non-parametric? This question was asked by an interview panel. Is it valid question?
It is fundamentally difficult to tell exactly what is meant by a "parametric test" and a "non-parametric test", though there are many concrete examples where most will agree on whether a test is parametric or non-parametric (but never both). A quick search gave this table, which I imagine represents a common practical distinction in some areas between parametric and non-parametric tests.
Just above the table referred to there is a remark:
"... parametric data has an underlying normal distribution .... Anything else is non-parametric."
It may be a accepted criterion in some areas that either we assume normality and use ANOVA, and this is parametric, or we don't assume normality and use non-parametric alternatives.
It's perhaps not a very good definition, and it's not really correct in my opinion, but it may be a practical rule of thumb. Mostly because the end goal in the social sciences, say, is to analyze data, and what good is it to be able to formulate a parametric model based on a non-normal distribution and then not be able to analyze the data?
An alternative definition, is to define "non-parametric tests" as tests that do not rely on distributional assumptions and parametric tests as anything else.
The former as well as the latter definition presented defines one class of tests and then defines the other class as the complement (anything else). By definition, this rules out that a test can be parametric as well as non-parametric.
The truth is that also the latter definition is problematic. What if there are certain natural "non-parametric" assumptions, such as symmetry, that can be imposed? Will that turn a test statistic that does otherwise not rely on any distributional assumptions into a parametric test? Most would say no!
Hence there are tests in the class of non-parametric tests that are allowed to make some distributional assumptions $-$ as long as they are not "too parametric". The borderline between the "parametric" and the "non-parametric" tests has become blurred, but I believe that most will uphold that either a test is parametric or it is non-parametric, perhaps it can be neither but saying that it is both makes little sense.
Taking a different point of view, many parametric tests are (equivalent to) likelihood ratio tests. This makes a general theory possible, and we have a unified understanding of the distributional properties of likelihood ratio tests under suitable regularity conditions. Non-parametric tests are, on the contrary, not equivalent to likelihood ratio tests per se $-$ there is no likelihood $-$ and without the unifying methodology based on the likelihood we have to derive distributional results on a case-by-case basis. The theory of empirical likelihood developed mainly by Art Owen at Stanford is, however, a very interesting compromise. It offers a likelihood based approach to statistics (an important point to me, as I regard the likelihood as a more important object than a $p$-value, say) without the need of typical parametric distributional assumptions. The fundamental idea is a clever use of the multinomial distribution on the empirical data, the methods are very "parametric" yet valid without restricting parametric assumptions.
Tests based on empirical likelihood have, IMHO, the virtues of parametric tests and the generality of non-parametric tests, hence among the tests I can think of, they come closest to qualify for being parametric as well as non-parametric, though I would not use this terminology.
Parametric is used in (at least) two meanings: A- To declare you are assuming the family of the noise distribution up to it's parameters. B- To declare you are assuming the specific functional relationship between the explanatory variables and the outcome.
- A quantile regression with a linear link would qualify as B-parametric and A-non-parametric.
- Spline smoothing of a time series with Gaussian noise can quality as A-non-parametric and B-parametric.
The term "semi-parametric" usually refers to case B and means you are not assuming the whole functional relation, but rather you have milder assumptions such as "additive in some smooth transformation of the predictors".
You could also have milder assumptions on the distribution of the noise- such as "all moments are finite", without specifically specifying the shape of the distribution. To the best of my knowledge, there is no term for this type of assumption.
Note that the answer relates to the underlying assumptions behind the data generating process. When saying "a-parametric test", one usually refers to non-parametric in sense A. In this is what you meant, then I would answer "no". It would be impossible to be parametric and non-parametric in the same sense at the same time.
I suppose that depends on what they mean by "parametric and non-parametric"? At the same time exactly both, or a blend of the two?
Many consider the Cox proportional hazards model to be semi-parametric, as it doesn't parametrically estimate the baseline hazard.
Or you might choose to view many non-parametric statistics as actually massively parametric.
Bradley, in his classic Distribution-Free Statistical Tests (1968, p. 15–16 - see this question for a quote) clarifies the difference between distribution-free and nonparametric tests, which he says are often conflated with each other, and gives an example of a parametric distribution-free test as the Sign test for the median. This test makes no assumption about the underlying distribution of the sampled population of variate values, so it is distribution-free. However, if the selected median is correct, values above and below it should be selected at equal probability, testing random samples from the sampled variates as to whether they are above or below the median estimate should be binomial with $p=0.5$ so it is simultaneously parametric.
Based on the discussion in the comments (thank you, whuber), it seems as if Bradley is in the minority, and what Bradley calls distribution-free, most others call parametric. And while nothing can really be $(A \cap \neg A)$ simultaneously, the answer to the question may just well depend on how you define the term, whether you make Bradley's distinction or call both elements of Bradley "parametric".