document: An approximation to the F distribution using the chi-square distribution For the cumulative distribution function -LRB- c.d.f. -RRB- of the F distribution , F -LRB- x ; k , n -RRB- , with associated degrees of freedom , k and n , a shrinking factor approximation -LRB- SFA -RRB- , G -LRB- lambda kx ; k -RRB- , is proposed for large n and any fixed k , where G -LRB- x ; k -RRB- is the chi-square c.d.f. with degrees of freedom , k , and lambda = lambda -LRB- kx ; n -RRB- is the shrinking factor . Numerical analysis indicates that for n/k > or = 3 , approximation accuracy of the SFA is to the fourth decimal place for most small values of k . This is a substantial improvement on the accuracy that is achievable using the normal , ordinary chi-square , and Scheffe-Tukey approximations . In addition , it is shown that the theoretical approximation error of the SFA , | F -LRB- x ; k , n -RRB- - G -LRB- lambda kx ; k -RRB- | , is O -LRB- 1/n/sup 2 / -RRB- uniformly over x
doc_bio_tags: O O O O B I O O O O O O B I I O O O O O B I O O O O O O O O O O O O B I I O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
extractive_keyphrases: f distribution cumulative distribution function degrees of freedom shrinking factor approximation numerical analysis
abstractive_keyphrases: chi-square distribution