Exact Fiducial Limits in Non-Linear Estimation

SUMMARY In the problem of estimating an unknown parameter of a distribution (in normal regression or in more general situations), the determination of a point estimate and its distribution is contrasted with the setting of fiducial limits for the parameter value. The use of a point estimate is effective for setting fiducial limits only when the estimate is a sufficient statistic for the parameter. Exact fiducial limits can be determined for the non-linear parameters in a normal regression function. Let the regression of y on x be linear in f(x, γ), which is a non-linear function of γ. If f′(x, γ) is the derivative of f(x, γ) with respect to γ, the partial regression of y on f vanishes when γ takes the least-squares value. Exact fiducial limits are defined as those values making this partial regression coefficient significant at the required level of probability. A pathological example of fiducial limits is discussed. The extension of the principle to general estimation problems is indicated.