equally spaced intervals and consider the following estimators for the unknown parameters u and az (Xr -XR-)=6(X,n-Xo 6=1∑(x.-x-1-p (4c) The estimators A and aZ correspond to the maximum-likelihood estimators of the u and o? parameters; ab is also an estimator of of but uses subset of n 1 observations Xo, X2, X4 X and corre sponds formally to i times the variance estimator for increments of even-numbered observations Under standard asymptotic theory, all three estimators are strongly consistent; that is, holding all other parameters constant as the total number of observations 2n increases without bound the estimators converge almost surely to their population values In addi tion, it is well known that both a2 and ab possess the following gaussian limiting distributions 2n(G2-a2)gN(0,2oa) (5a) /2n(G3-a2)gN(0,4oa where a indicates that the distributional equivalence is asymptotic Of course, it is the limiting distribution of the difference of the variances that interests us. Although it may readily be shown that such a difference is so asymptotically gaussian with zero mean, the variance of the limiting distribution is not apparent since the two variance estimators are clearly not asymptotically uncorrelated. However, since the estimator a2 is asymp totically efficient under the null hypothesis H, we may apply Hausman's ( 1978)result, which shows that the asymptotic variance of the difference is simply the difference of the asymptotic variances. If we define Ja =a- 02. then we have the result 2n/a a N(o, 2o) Using any consistent estimator of the asymptotic variance of Ja, a standard any other estimator of 0. If not, then there exists a linear combination of A, and 4.-0, that is more efficient than i. contradicting the assumed efficiency of 0. The result follows directly, then, since where avar( ) denotes the asymptotic variance operator