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The bootstrap works without needing assumptions like normality, but it can be highly variable when the sample size is small and the population is not normal. So it can be better in the sense of the assumptions holding, but it is not better in all ways. The bootstrap samples with replacement, permutation tests sample without replacement. The Mann-Whitney ...

5

You could try to directly estimate the CDF via a binomial rate smoother ? Here is an idealized example for x stemming from a normal distribution: ci = seq(from=-3,to=3,length=500) X = rnorm(500) Y = rep(NA, 500) for (i in 1:500) Y[i] = as.numeric(X[i] < ci[i] ) plot(ci,Y, type="s") library(mgcv) library(boot) fit=gam(Y~s(ci), ...

4

I'm not an expert on this, but most text books on non-parametric methods have at least a short discussion on the 'root-n consistency' problem, see: http://www.ucl.ac.uk/~uctppca/aarhus/robinson.pdf The basic result is that non-parametric estimators can't achieve a $O(n^{-1/2})$ order of convergence. Kernel methods, for example, give $O(n^{-2/5})$ ...

2

I like to use heterogeneous Mixture Models to describe combined effects from fundamentally different sources. You might look at something like a "Zero Inflated Poisson" model in the style of Diane Lambert. "Zero-Inflated Poisson Regression, With an Application to Defects in Manufacturing", Diane Lambert, Technometrics, Vol. 34, Iss. 1, 1992 I find this ...

2

You are correct that a K-S test deals with the shape of the populations and not the location. Before I suggest a strategy to try, consider this: if there is a substantial change in the shape of the distribution then is the location really what you want to know about? Is a test of a hypothesis relating to the location of any inferential utility? If you ...

1

With so little information on what the numbers represent, how you obtained or generated them and what kind of differences you are interested in, it's difficult to give meaningful advice. But even before thinking about tests, distributions, and assumptions, it seems to me that the two examples in the question are visibly different: For now, the most likely ...

1

What about the Kolmogorov-Smirnov test? It's a non-parametric test that tests the differences in distributions. Here's some sample R code that'll run the test. set.seed(56945) n <- 1000 x1 <- rnorm(n,mean=5,sd=2) +rnorm(n,mean=0.0,sd=0.4) x2 <- rnorm(n,mean=5,sd=2) plot(ecdf(x1),col="blue") ;grid(4,5,"black") lines(ecdf(x2),col="red") ...

1

My first advice would be to create some plots to visualize your data. Exactly what plot will depend on what software is available to you, what you are comfortable with, and also on the question you want to address. To know if stores tend to carry products with different price points, I would just plot the price of the various products side-by-side, a bit ...

1

If you want to compare the distribution shapes, you might use Kolmogorov-Smirnov. I didn't quite understand: that if one sample size is 20 and the other one is thousands? Please be clear in your question. Explain your study more clearly please. What is timestamping? And what are these duplicates? What are your study goals? What are your variables? How many ...

1

Regarding your comment to @jerad's solution, I believe that you don't have to get disappointed because you cannot prove formula 12. It needs some theory of Stochastic Processes. If you want to know how formula 12 is derived check at Ferguson's paper, A bayesian analysis of some nonparametric problems (The Annals of Statistics 1973, 1(2):209), who first ...

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