On Inverse Estimation in Linear Regression

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Abstract
A procedure suggested by Krutchkoff (1967) f or inverse estimation in linear regression is compared with the classical procedure from other points of view than that taken by Krutchkoff, i.e. comparative mean square error. In particular, comparisons are made on the basis of “closeness” in estimation (Pitman, 1937), consistency (in a setting where this concept is relevant), and mean square error of the relevant asymptotic distributions. It is found that, for large samples, Krutchkoff's estimate is superior in the sense of “closeness” if values of the independent variable are restricted to a certain closed interval around the mean of the independent variates in the experiment and inferior elsewhere. However, the width of this interval varies inversely as the product of the absolute value of the standardized slope (i.e. scaled by the error standard deviation) and the standard deviation of the independent variables used in the experiment. As a practical matter the parameter tends to be large so that the interval where the Krutchkoff estimate is superior will be trivially small. In addition large values of this product parameter imply that the two estimates being compared are virtually indistinguishable. Coupling these latter remarks with the fact that the classical procedure allows an exact confidence interval for the parameter under estimate while the Krutchkoff procedure does not, suggests the classical estimate is to be preferred using the “closeness” criterion. If one uses the criterion of mean square error applied to the relevant asymptotic distribution, one reaches conclusions similar to the above, except that the interval of superiority of the Krutchkoff estimate is no longer trivially small even at best. However, the mean square error criterion fails to take into account the fact that the estimates are correlated and so should be considered an intrinsically less appropriate criterion than closeness. We also find that, in circumstances where the concept is applicable, Krutchkoff's estimate is not consistent whereas the classical estimate is; Krutchkoff's estimate can be trivially modified so that it is consistent but will then tend to never be better in the sense of closeness than the classical estimate.

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