Improved estimation of the mean in one-parameter exponential families with known coefficient of variation

The value for which the mean square error of a biased estimator 14 aT for the mean mu is less than the variance of an unbiased estimator T is derived by minimizing MSE(aT). The resulting optimal value is 1/[1+c(n)upsilon(2)], where upsilon = sigma/mu, is the coefficient of variation. When T is the UMVUE (X) over bar, then c(n) = 1/n, and the optimal value becomes 1/(n + upsilon(2)) (Searls, 1964). Whenever prior information about the size of v is available the shrinkage procedure is useful. In fact for some members of the one-parameter exponential families it is known that the variance is at most a quadratic function of the mean. If we identify the pertinent coefficients in the quadratic function, it becomes easy to determine upsilon.