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As estimating equations, Brass (1958) employed 1-L = x, !J-2 = s2 , and fr(1) = n/n. Estimators of the parameters p and k follow from these equations. From the third equation of (13.3.4), we write

and (13.3.10)

p = ~(l-fT(l)] and IJ-2

(13.3.11)

and k* = p*x-(n 11n). l - p* (13.3.12)

Brass demonstrated that these estimators are consistent, although they are not unbiased. However, when n is large, the effect of bias is slight. The efficiency for most combinations of parameter values is above 90%. Accordingly, these easily calculated estimators are satisfactory as final estimates in many applications, and when the situation calls for maximum likelihood estimates, they provide excellent first approximations from which to begin iterations to the MLE. Variances and covariances of estimates were given by Brass as

v( :~) = Jr(1)[1; Jr(l)J, Cov(!J-*,s2) = ~3 , Cov (IJ-*, ~) = fr( 1 )(~- !J-)'

( 2 ~) _ fT(l)(1 - IJ-2 + IJ-2 - 2 j.L] Cov s, - . n n

(13.3.13)