Define KolmogorovSmirnov Two Sample test and PermutationSolu

Define Kolmogorov-Smirnov Two- Sample test and Permutation.

Solution

The Kolmogorov-Smirnov test is such a test.

It tests

H0:The distributions are the same

H1:The distributions are not the same

Suppose we have two samples, X1,...,Xm and Y1,...,Yn.

Denote F₁(x) the cumulative distribution function (CDF) for the distribution of the first population

and F₂(y) the CDF for the second.

Next, denote F₁^ the sample CDF for the first treatment

and F₂^ be the sample CDF for the second treatment.

The Kolmogorov-Smirnov statistic, denotes KS

    KS=max_w|F1^(w)F2^(w)|.

                   OR

D_n=supsup_1jk|_i=1j(f_exp,(i)f_obs,(i))|

where is a permutation of the order of the categories, f_.,i are the observed and expected frequencies in category i. This can be written equivalently as:

D_n=12_i=1|f_exp,if_obs,i|

I would like to extend this to a two-sample case using a randomising/permutation procedure, such:

D^(r)_n=12_i=1|f^(r)_group1,if^(r)_group2,i|,r=1,…,R

where .^(r) denotes a statistic calculated based on the rth permutation of the categorical variable.

Reject if the value of the original statistic is larger than the value of 95% of the permuted statistics.