where C2V7+2 (^) i^ ^^ cumulative distribution function for the continuous 

 X^-distribution with 2w+2 degrees of freedom. The H^ , which are non- 

 negative numbers and satisfy j; H = 1 , are determined by the algebraic 

 identity (in z) ^'^ 



V-V2t, _ (1 _ i/v).]-V2 . J H z- , 



w=0 ^ 

 for the expression that occurs for |z| sufficiently sr.all. Here, 



[1 - (1 - 1/V)z]~-^^^ = 1 + i(l - 1/V)z + (3/8) (1 - lA)^z^ 



+ (5/16) (1 - 1/V)"^z"^ + (35/128) (1 - l/\7)^z'^ + 

 Thus, 



Hq = V"-^/^ , H^ = i(l - 1A)V"-^/^ , H^ = (3/8) (1 - 1/V)V^/^ , 



H^ = (5/16) (1 - 1/V)\~-^^^ , H^ = (35/128) (1 - 1/V)\~-^^'^ , etc. 



For practical applications, the inequalities (W >_ 0) 



W 00 W W 



y H c ^(x) < y H c ^(x) < y h c ^(x) + [i - y h ]c^„ ^(x) 



^„ w w+2 — ^^ w w+2 — ^„ w w+2 ^„ w 2W+4 



w=0 w=0 w=0 w=0 



are helpful. Using these results, 



w 22" 



I «wC„+9(R^/^) lP(ti +Vt2 < r2/v) < I HC (r2/v) 



„ w w+2 ''— ^J- ^— ' ' — L .^ 2w+2 

 w=0 w=0 



W 



1 

 w=0 



^ ^^- I «w]Sw+4(^'/^> 



2 2 2 

 Ordinarily, the value of P(ti + Vt2 ^ R /v) can be approximated to reason- 

 able accuracy without using a veiry large value for W . 



REFERENCES 



1. S. N. Roy, Some Aspects of Multivariate Analysis , John Wiley and Sons, 

 1957, Appendix. 



2. Herbert Robbins and E. J. G. Pitman, "Application of the method of 

 mixtures to quadratic forms in normal variates , " Annals of Mathematical 

 Statistics, Vol. 20 (1949) , pp. 552-560. 



455 



