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AMDA-2008-Session 9

# AMDA-2008-Session 9 - samples these among T time of...

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Permutation Tests

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Permutation Tests It is a non-parametric method for testing whether two distributions are the same. Suppose we have random sample of size m from a population with cdf F X and another independent random sample of size n from a population with cdf F Y i.e. X 1 ,…,X m are iid F X and Y 1 ,…,Y n are iid F Y and the two samples are independent. We are interested to test H 0 : F X =F Y versus H 1 :F X ≠ F Y
Permutation Tests T. of on distributi n permutatio the called is T each on N! 1 mass puts that P on distributi The data the of ns permutatio N! over the d distribute uniformly is ,..., , ,..., X values, data ordered given the , hypothesis null Under the ,..., T by values these Denote T. statistic test the compute, n permutatio each For . ,..., , ,..., X data the of ns permutatio N! all forming consider and n m N Let . | X | ) ,..., , ,..., T(X e.g. For statistic. test some be ) ,..., , ,..., T(X Let j 0 1 1 ! 1 1 1 m 1 1 1 1 + = - = n m N n m n n m n m Y Y X T Y Y X Y Y Y X Y Y X

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Permutation Tests value. - p the es approximat
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Unformatted text preview: samples these among T time of fraction The ns. permutatio of set the from randomly sampling by value-p the e approximat can We ns permutatio N! all evaluate to practical not is it Usually ) I(T N! 1 ) t (T P value-p is value-p the large, is T n reject whe we Assuming statistic. test the of value observed the be Let t j N! 1 j j obs obs obs obs t t • • = = • • ∑ = Algorithm for Permutation Test ∑ = = B j obs j B n m t T I T Y Y X X T 1 1 1 1 obs ) ( B 1 is value-p e approximat The 4. values. resulting the be ,..., T let and times B step previous Repeat the 3. data. permuted the using again statistic test the Compute data. the permute Randomly . 2 ) ,..., , ,..., ( t statistic test the of value observed the Compute . 1...
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AMDA-2008-Session 9 - samples these among T time of...

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