Descriptive and Inferential vs Parametric and Non-Parametric Statistics Are the concepts of descriptive vs inferential statistics and parametric vs non-parametric statistics orthogonal? As in, can we have a descriptive parametric statistic or a descriptive non-parametric statistic? if so what are some examples of? I know you can have an inferential parametric or non-parametric statistic. 
My original thought was that parametric vs non-parametric statistics falls in a subcategory of inferential statistics, and that inferential statistics makes use of descriptive statistics. but some sources I read seem to say that you can have descriptive non-parametric stats.
Anyone can shed some light on this?
 A: The "parametric" part of "parametric statistics" refers to the fact that in parametric statistics you are assuming/inferring a (descriptive) statistic has a known distribution (e.g Normal distribution). For example, you may infer that the mean is normally distributed as the sample size increases. 
In non-parametric statistics, you are not making any assumptions about the distribution of the mean (or any other descriptive statistic). As an example, Chebyshev's inequality guarantees that no matter what the distribution, no more than 1/k^2 of the values can be greater than k standard deviations away from the mean.
The sticking point between parametric and non-parametric approaches is that you can do hypothesis testing quite easily if you go the parametric approach. The road is not as smooth if you go the non-parametric route. Which road you choose depends, in part, on how you view the validity of the Central Limit Theorem and convergence in general.
A: Here are some examples:
Descriptive Parametric: (I'm having trouble coming up with an example here. I guess the idea is that "descriptive" statistics, by nature, are non-parametric, hence they let us see the shape of the data prior to our making assumptions about it.)
Descriptive Non-Parametric: A histogram of the data.
Inferential Parametric: A first order ordinary least squares linear regression, which assumes a particular shape in the data (i.e. a linear fit) is an appropriate model.
Inferential Non-Parametric: Fitting the data using an ensemble of regression trees to develop a predictive model. The model does not make any assumptions about the shape of the data.
A: You can have any combination of nonparametric/parametric and descriptive/inferential.
In plain language:
Descriptive statistics describe a sample. 
Inferential statistics infer from a sample to a population.
Nonparametric vs. parametric is trickier. See http://en.wikipedia.org/wiki/Non-parametric_statistics
B.S. Everitt, in Dictionary of Statistics, defines "parametric methods" as "procedures for testing hypotheses about parameters in a population described by a specified distributional form...." and contrast them with "distribution free methods". However, I think that if you estimated an OLS regression on a sample without making any inferences to a population, that would be a parametric descriptive statistic. 
