Calcutta University Readership Lectures. Statistical Theory of Estimation

The variance of the median is equal to 1 which is approximately equal to 1 4nys (2s -i- 1) !16 STATISTICAL THEoRY oF ESTIMATION In the case of the Cauchy distribution under consideration Therefore, the variance of the median V (median) = 7:2 4n Thus the median has increasing precision as the size of the sample is increased. [...] In a large and important class of can-istent statistics the random sampling distri- bution tends to the normal (Gails:ian) form as the size of the sample is increased, and in such a way that the variance (the square of the standard deviation) falls off inversely to the size of the sample. [...] The variance of the mean of a sample of n is n Hence /a times the variance is equal to v. If on the other hand we use the median then the variance is where now Yu = V 2rry so that the variance of the median is iry 2vs 3 —miss)18 STATISTICA L OF ESTIMATION Therefore n times the variance is :Tv Hence the median is only — times as efficient as the mean. [...] Of the methods of estimation based on linear functions of the frequencies, that with smallest limiting variance is the method of maximal likelihood, and for this the limit in large samples of —1 is equal to i. nV Let x stand for the frequency observed of observatidns baying probability of occurrence f and let in 222 nf, the expected frequency in a sample of n. Consider any linear function of the [...] In general, the problem of recognising the character of configuration to be used in this way, is the Problem of the Nile, which T stated in the Mathematical Conference of the Tricentenary Celebrations at Harvard in 1936 :— " The agricultural land of a pre-dynastic Egyptian village is of unequal fertility.