D. E. Lea and C. A. Coulson 281 



where d = (l-b + \og m)ja. (48) 



Thus \M-=-{- x (x + c) 2 d-x (x + c) + 2 (x + c) d + 1}. (49) 



fdm m K 



Now L, the log likelihood, is (apart from irrelevant terms) S [log/], the summation being 

 for the N observations of r, and the maximal likelihood condition is 



oAs 



"If 



fdm 

 i.e. S[x(x + c) 2 d + x(x + c)-2(x + c)d-l] = 0. (50) 



The routine for applying this method is as follows. Employing the preliminary estimate 

 of m given by the median method, (48) is used to calculate d, and then (46) is used to 

 calculate a value of x from each of the N experimental observations of r. For each of these 

 N values of x the expression 



x (x + c) 2 d + x (x + c) -2 (x + c) d-l 



is evaluated and the N quantities added. The sum is similarly evaluated for several 

 adjacent values of m, and by plotting against m (or otherwise) the value of m which satisfies 

 (50) is deduced. 



The variance to be attached to the maximal likelihood estimate of a parameter m is 

 given by Fisher's formula (cp. e.g. Fisher, 1938) 



Ni' 



(51) 



where i = E 



(1 df \ 2 ] /l df\ 2 



-rj—\ is the expectation of (->-/-) • Hence, using (49), 



im 2 = E[{xH + x 2 (2cd+\) + x(c 2 d + c-2d)-(2cd + \)} 2 } 

 = d 2 E [x 6 ] + (&c 2 d 2 + Gcd-4:d 2 + l)E [z 4 ] 

 + (c*d 2 + 2cH - 12c 2 d 2 -\2cd + U 2 + c 2 -2)E [x 2 ] 

 + (ic 2 d 2 + ±cd + l) 

 + terms involving odd powers of a;. 



Now it is readily shown that x being distributed normally about zero with unit variance, 

 E [x n ] vanishes for odd n, and 



E [x°] = E [x 2 ] =1, E O 4 ] = 3, E [x*] = 15. 



Inserting these values in (52) we obtain 



im 2 = d 2 (c* + 10c 2 + 7)+d(2c 3 + 10c) + (c 2 + 2), 



1 



J 2 (c 4 + 10c 2 + 7)+d(2c 3 + 10c) + (c 2 + 2) ' 



with a = 11-6, 6 = 4-5, c = 2-02, d = (l-b + \ogm)/a. 



The part of Fig. 3D to the right of ra = 10 is a plot of I — I JN against m. Having 



determined the maximal likelihood estimate of m, as described in this section, the standard 

 deviation to attach to it is read off from Fig. 3 D. 



sothat ft) a 4,- 



4i 



