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#1 07/03/18 03:59:46

Registered: 19/07/17
Posts: 599


Suppose we want to test the null hypothesis H0 that the data X1 … co-Jersey/ , . . . , XN is a sample from some unspecified distribution. Permutation tests are tests of this hypothesis in which the p-value is computed conditional on knowing the set S of data values observed but without knowing which data value corresponds to X1, which corresponds to X2 and so on. For instance, if N = 3 and X1 = 5, X2 = 7, X3 = 2, then the p-value is computed conditional on the information that the set of data values is S = {2 … ds-Jersey/ , 5, 7}.

P-value makes use of the fact that, conditional on the set of data values S, each of the N! Possible ways of assigning these N values to the original data is equally likely when the null hypothesis is true. That is, suppose that N = 3 and the set of data values is, as in the preceding, S = {2 … ey-Jersey/ , 5, 7}. Now the null hypothesis H0 states that X1, X2, X3 are independent and identically distributed. Consequently, if H0 is true then, given the data set S, it follows that the vector (X1 , X2, and X3) is equally likely to equal any of the 3! permutations of the values 2, 5, 7.

Permutation tests are also useful in the two-sample problems where we test whether samples from two populations have the same underlying distribution. Specifically, let X1, . . . , Xn be a sample from an unknown population distribution F , and let Xn+1, . . . , Xn+m be an independent sample from an unknown population distribution G, and suppose we want to use these data to test the hypothesis that the two population distributions are identical against the alternative hypothesis that data from the second distribution tend to be larger than those from the first. That is, we want to use these data to test the null hypothesis

H0: F = G
, against the alternative
H1: data from G tend to be larger than data from F

If the data values are Xi = xi, i = 1 … do-Jersey/ , . . . , n+m, then a permutation test of the preceding null hypothesis is done conditional on knowing S = {x1, . . . , xn+m}, the set of these n + m numbers in no particular order. Then if H0 is true, and so all n + m random variables X1 … et-Jersey/ , . . . , Xn+m are independent and identically distributed, then given the set of values S, each subset of size n of this set is equally likely to be the set of the data values of X1, . . . , Xn. Because the alternative hypothesis is that data from the population distribution F tend to be smaller than data from the population distribution G, a reasonable test would be to reject the null hypothesis if the sum of the data values from the population distribution F is smaller than might be expected by chance when n values are randomly chosen from the data set S.
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