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Some Extensions of the Idea of Bias
H. R. van der Vaart
The Annals of Mathematical Statistics
Vol. 32, No. 2 (Jun., 1961), pp. 436447
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2237754
Page Count: 12
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Topics: Estimators, Estimation bias, Estimators for the mean, Point estimators, Statistical estimation, Unbiased estimators, Statistics, Correlation coefficients, Random variables
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Abstract
Laplace ([13], p. 44, lines 5 and 6), in his statement concerning the "milieu de probabilité", seems to have referred to a probability distribution of the true value of a certain quantity ("le véritable instant du phénomène"), or, as we would say at present, to a probability distribution of a certain parameter. Thereby he differs from the attitude adopted in most of the work discussed in the present paper. Yet, one might hold that he possessed the idea of medianunbiased estimators. At any rate, when applying his notions to what Todhunter ([26] p. 469, art. 875) calls a case of no practical value, Laplace ([13], p. 48, lines 11 and 12 from the bottom) virtually rejected the use of arithmetic means of observations. Judging from innumerable texts, one finds that after him emphasis has long been mainly on meanunbiasedness (see, however, Pitman (]20], bottom of p. 215), who mentions the existence of bias in the sense that the probability that a certain meanunbiased estimator is less than the parameter in question is $>\frac{1}{2}$). Yet it is hard to find the requirement of meanunbiasedness justified in print (cf., Brown ([3], lines 68 of Section 3): the average of independent meanunbiased estimates is consistent; Lehmann ([14], lines 410 from bottom of p. 588): meanunbiasedness flows from his general concept in the case of a quadratic loss function; Birnbaum ([2], p. 32): meanunbiasedness is merely a technically useful property of the classical estimators in the linear estimation problem, which, at least in the case of normal errors, could equally well or preferably be justified on the basis of medianunbiasedness), much harder, in fact, than to find warnings against the hope that much is gained if an estimator be meanunbiased (cf., Kendall ([12], Vol. 2, Section 17.9); the examples provided by Girshick, Mosteller and Savage ([9], middle of p. 20), Halmos ([10], the end of p. 43), Savage ([23], bottom of p. 244); lack of invariance under certain transformations being stressed by Halmos ([10], bottom of p. 42), Brown ([3], lines 1316 of Section 3), Fisher ([7], p. 143, line 13 from bottom)). All the same, much interesting work has been devoted to meanunbiased estimators, some of it investigating the conversion of biased estimators into unbiased ones (e.g., Quenouille [21], Olkin and Pratt [17]), or deriving unbiased estimators ab initio (e.g. Tate [25]). It is not the purpose of this paper to provide a bibliography that is at all near completeness, but it is interesting that the last two references illustrate a statement, made by Schmetterer ([24], middle of p. 215), to the effect that a close connection exists between integral equations and linear operators on the one hand, and the theory of meanunbiased estimators on the other. This suggests that part of the motivation for the research in this field is of a mathematical, rather than a statistical nature. This view seems to be corroborated by Fraser's statement ([8], lines 1214 from bottom of p. 49) to the effect that medianunbiasedness does not seem to lend itself to the mathematical analysis needed to find minimum risk estimates, and hence has found little application. The present paper seeks to extend the notion of unbiasedness (and the notion of bias) in a direction different from Lehmann [14] (who gave a definition within the framework of general decision theory), and from Brown [3] (who was primarily concerned with types of unbiasedness, among them medianunbiasedness, that are invariant "under simultaneous onetoone transformations of the parameter and (its) estimate", or rather under simultaneous strictly monotone transformations of the parameter and its estimate), and from Peterson's [19] densityunbiasedness. It originated in work by the author [29], [30], on the estimation of the latent roots of certain matrices occurring in response surface theory. It had become clear that in this case it was of primary interest whether or not the frequency of obtaining too small (or too large, respectively) estimates would be unduly large. The present paper will make this notion more precise. Several types of bias (or of unbiasedness, respectively) will emerge, all of them clearly invariant in the sense of Brown. Medianunbiasedness will turn out to be a special case of this larger concept. Finally, certain seemingly unfamiliar properties of the sample median, of the productmoment correlation coefficient, and of Olkin and Pratt's function of the latter [17] will be proved and used to illustrate some of the concepts discussed.
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The Annals of Mathematical Statistics © 1961 Institute of Mathematical Statistics