Difficulties Involved in the Estimation of a Population Mean Using Transformed Sample Data

A problem of frequent occurrence in the statistical analysis of experimental data is the estimation of the first two moments of a population with continuous but unknown c.d.f. F(x), based on a random sample (x , * xn) - (x,). Standard estimates for E(x) and Var (x) are x or x ?- t,_as/ n and s , respectively. If the population {x} is non-normal x and S2 are not minimum variance. For the sake of obtaining efficient estimates of E(x) or var (x) the sample (xi) is often transformed by a function F: xi --, y, = F(xi)*, where F is chosen so that (y,) satisfies tests of normality. Estimates for E(y) and var (y) are then computed as y or y ?t tl-,s,/s/ and s2 respectively. (It is assumed that such estimates are efficient; they are, in fact sufficient provided we know that the population {y} is normal). Since these estimates are sufficient, a function of them alone is an efficient estimate of that function's expected value. A rule therefore commonly given for estimating E(x) is E(x) = F-(y) (1) or