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Re: twosample nonparametric test on quantiles
Posted:
Jan 27, 2013 8:08 PM


On Jan 27, 2:19 am, Anonymous wrote: > Hello, > I have two random samples (each of them i.i.d. with continuous > distribution) and I need to test, whether they come from > distributions which have the same 100p% quantile (for p=5%). > What I need is some generalisation of twosample MannWhitney > test on equality of medians. > > I would also need to have nonparametric confidence intervals > for empirical quantiles of some sort. > > I intuitively understand, that I would need to have quite large > samples for p close to zero to reject the null (q1=q2) hypothesis. > > Any reference to literature and/or software implementation that > would solve these problems would be appreciated.
Let x_1,...,x_m and y_1,...,y_n be the two sets of observations. Unless you make some assumptions about the forms of their true distributions, you have no basis for distinguishing among values that lie between successive order statistics of the pooled data. Let Z refer to the set of midpoints of the intervals between successive order statistics of the pooled data.
Now suppose you want test the hypothesis that some particular value z is the q'th quantile of both the X and Y parent distributions. (Note that you must specify both z and q.) Compute
t[z,q] = (n*(#{x < z}  m*q)^2 + m*(#{y < z}  n*q)^2)/(m*n*q*(1q)).
If the hypothesis is true and if min{m,n}*min{q,1q} is "sufficiently large"  say >= 5, certainly >= 1  then t should be distributed approximately as chisquare with 2 df.
To get a 100p% CI for the q'th quantile, find the subset of Z for which t[z,q] < the p'th quantile of the chisquare(2) distribution. (The subset may be empty if the sample x and y distributions are very different from one another.)



