The Efficiencies of Alternative Estimators for an Asymptotic Regression Equation

Methods for the estimation of the parameter p in equation (1) have been discussed, this equation representing the expectation of observations on a quantity y for a specified value of x. The methods all involve taking as the estimator the ratio of either two linear functions or two quadratic functions of the observed y. Their relative efficiencies and biases are considered under two models for the generation of observations, the y either being independent and normally distributed about their expectations with constant variance or arising from a continuous autoregressive process in which the variance increases and successive values of y for an individual are correlated. Detailed investigation has been possible only for the very simple situation of four observations equally spaced in x. As is well known, the Patterson estimator, a ratio of two linear functions, is of high asymptotic efficiency for the first model, and it proves to be also highly efficient for the second. An interesting alternative is the calculation of a linear regression of yi+1on yi. This is also highly efficient, and has the advantage of simultaneously estimating the parameter a; moreover, under the second model, the estimators of p and α maximize the likelihood for any number of equally spaced observations. However, the estimator of p appears likely to have a considerable negative bias if the variance of observations about their expectation is at all large. Other quadratic estimators have been examined, but showed no special merits. The indications are that the Patterson estimator is always a fairly safe one to use, for any number of equally spaced observations; its efficiency is never low, and it is unlikely to be seriously biased. The estimator calculated autoregressively is likely to be more efficient (in the narrow sense of having a smaller asymptotic variance), especially for the second model and for a larger number of observations, and if the variance per observation is low this advantage will not be offset by its greater expectation of bias.