An investigation of some alternative estimation procedures

Estimators are investigated which provide alternatives to the usual unbiased estimators for the mean and variance. It is assumed that the experimental situation is one in which the statistician has at his disposal some information concerning the parameter space. In particular, the experimenter may have reason to believe that the parameter is in a certain region of the parameter space.

The first estimator examined is reasonably close to the natural origin. The admissibility of θ̂ with respect to x̅ is also exhibited.

The final estimator examined is a preliminary test estimator for the variance when the mean is also unknown. The estimator chooses between functions of the form cs², where s² is the usual unbiased estimator, and functions of the form kw², where w² is a crude sum of squares, as the final estimator of the variance on the basis of a preliminary t-test on the mean. S² is apparently made inadmissible by several of these intermediate values of θ, while the MSE(θ̂p|θ) and MSE(x̅|θ) are shown to be approximately the same for relatively large values of θ. θ̂p is shown to be inadmissible universally but admissible with respect to x̅.

In order to improve upon θ̂p in the areas in which it performs poorly a new estimator, θ̂, is proposed. θ̂ is a weighted estimator between x̅ and kx̅ and its weighting functions are fiducial probabilities. The MSE(θ̂|θ) function is shown to be smaller than the MSE(θ̂p|θ) for values of θ so-called preliminary test estimator for an unknown mean θ with the variance known and is defined by

θ̂p = x̅ , if x̅ ε R

 = kx̅,  if  x̅ ε R̅, 

where 0 < k < 1, R is the region of θ̂ in which MSE(x̅|θ) < MSE(kx̅|θ̂) and R̅ is the complement of R. Procedures are given for determining optimal values of k. Bias and MSE for θ̂p are found. The MSE(θ̂p|θ) function is found to be smaller than that for x̅ around the "natural origin", larger than MSE(x̅|θ) for estimators as is

(n-1/n+1) s²

which has uniformly minimum MSE among estimators of the form c². Formulas are given for calculation of the MSE in closed form when the mean is actually the same as the hypothesized value.