One Likelihood Adjustment May Be Inadequate
H. W. Norton · Biometrics · 1956
Fisher (1925) says . . since the equations of maximum likelihood do not always lend themselves to direct solution, it is of importance that, starting with an inefficient estimate, we can, by a single process of approximation, obtain an efficient estimate .... It is sufficient for our purpose that the error of estimation is of the order n-2 and starting with an inefficient statistic, a single process of approximation will ordinary cases give an efficient statistic differing from the maximum likelihood solution, by a quantity which with increasing samples decreases as n1. The problem is more complicated than Fisher made it appear. This paper uses a simple example to show that, even in ordinary cases, one likelihood adjustment is sometimes quite inadequate. Fisher (1950, chapter 9) takes the genetical problem of the frequency of crossing over to illustrate the problem of statistical estimation. He indicates five different statistics which might be adopted as solutions of the problem, each being consistent and each having sampling variance inversely proportional to the size of the sample. The first and second of these five estimates are inefficient, and might be used as starting points for the adjustment Fisher proposed. The first of these estimates and its successive improved values appear Table 1. The first entry the second column is the value