when the U values for the two independently obtained estimates of the spec- 

 trum are added together to get the best final estimate, the results will be 

 in terms of E values. 



Degrees of Freedom 

 In the single variable case, each of the final spectral estimates has a 

 Chi Square distribution with f degrees of freedom where f as given by Tukey 

 is determined by equation (8,23). 



(8.23) f=2(^4) 



This result is obtained from rather complex considerations of all of 

 the operations on the original time series which have led to the final values 

 of the U's. In the case of an electronic analogue analyzer, the procedure is 

 described by Pierson [1954], and the results depend on the shape of the 

 smoothing filter. 



In the two variable case under consideration here, the smoothing filter 

 is known only at 9 points as given in Table 8.1. 



The values of U(r, s) for a particular r and s is a random variable 

 with a Chi Square distribution with an as yet to be determined number of de- 

 grees of freedom. When U is considered as a random variable the degrees 

 of freedom can be found from the following equation, 



(8.24) {- 2(E(U))^ _ (SWk)^ M^M^ 



E(U2) 4(SWj^)2 m^m^^ 



