Robert J. MacG. Dawson wrote: > > One might also ask, what exactly is a parameter? It's usual to > suppose that (for instance) the mean is always a parameter; but is > it, if one doesn't have the rest of a parametric model? Conversely, > if one has a parametric model, the median is usually a valid > parameter. (If the models in the family are symmetric, it is the same > as the mean, and the sample mean and sample median are both valid > estimators.)
These are good questions. There has been a move in some applications to use the terms "distibution-free" and "non-parametric" to refer to two different classes of tests. Here the first obviously relates to assumptions about distributions, while the second relates to the model-structure (of location) implied by the null and alternative hypotheses. For example suppozse the question were detecting trend or change in a time series. Then one possibility is to explicitly parameterise the altenative hypothesis to have a linear-in-time trend ... but, since it is then possible to construct a "distribution-free" test based on resampling, this give a distribution-free test of a parametric hypothesis. Of course there some standard tests in the "non-parametric" literature for "change" that don't involve either a parametric hypothesis about location or assumption about distributional form. And it is possible to think of tests of change that are non-parametric in the location hypothesis by distribution dependent ... for example, if a variance were assumed known, a test might be constructed based on the range, with the distribution used for the test derived from the distribution of the range in Normal samples. > > One might, I think, argue that (for instance) the t test, done on a > sample large enough that one is not concerned about close > approximation to normality, is in fact nonparametric, in that no > specific parametric model is ever assumed. > There seems to be a strong parametric assumption about about location in this case which leads to the particular test-statistic being used, but the distribution of the test-statistic may be effictively close to being distribution free. David Jones . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
