One elicitation diagnostic identifies a family of prior distributions that are so diffuse that they are practically equivalent to the completely diffuse prior. Another elicitation diagnostic identifies a family of prior distributions that concentrate enough mass in the neighborhood of zero that they are practically equivalent to the dogmatic prior that sets a parameter exactly equal to zero. If either question thus posed can be answered in the affirmative then there is no need to go to the expense of a more accurate elicitation of the prior distribution.

This paper extends the simple errors-in-variable bound to the setting of systems of equations. Both diagonal and nondiagonal measurement error covariance matrices are considered. In the nondiagonal case, the analogue of the simple errors-in-variable interval of estimates is an ellipsoid with diagonal equal to the line segment connecting the direct least squares with a two-stage least squares estimate. For the diagonal case, the set of estimates under some conditions must lie within the convex hull of 2k points.

[The matrix weighted average (H = V extasciicircum-1) extasciicircum-1Hb, where H and V are symmetric positive definite matrices and b is a vector, is shown to lie in one ellipsoid if V is bounded from below, V "* @ extless V, another ellipsoid if V is bounded from above, V @ extless V*, and another ellipsoid if V is bounded from above and below, V "*@ extless V @ extless V*. These results are applied to bound the posterior mean vector of the normal linear regression model.]

[If independent observations x are drawn from the distribution located at @m, f (x; @m)=c"3 exp[-g(x -@m)], and if g is symmetric and strictly convex, then the maximum likelihood estimate of μ lies between the smallest and largest folded sample observations. If the distribution has fatter tails than a normal distribution, then the maximum likelihood estimate lies between the smallest and largest means of trimmed subsamples. If the distribution is assumed to be symmetric and unimodal, the centers of tight clusters of observations can be maximum likelihood estimates. If observations are not independent, then there is no bound: given any example any number is a maximum likelihood estimate for some sampling distribution. Stationary is not sufficient to bound the estimate between the minimum and maximum observations.]

IF ONE OF THE EXPLANATORY VARIABLES in a linear regression model is measured with error, the ordinary least squares estimator is known to be biased and inconsistent. Given suitable assumptions, an instrumental variables estimator is known to be consistent. In a large sample, the instrumental variables estimator is thus unambiguously preferred, but the choice of an estimator in a small sample remains a puzzle. The method of maximum likelihood sheds light on this puzzle. It will be shown below that the instrumental variables estimate is the maximum likelihood estimate if, and only if, it lies between the ordinary least squares estimate and the reverse least squares estimate, that is, if and only if it satisfies the bounds implied by the simple errors in variables model. The letters Y, x, and z will indicate, respectively, the vector of observations of the dependent variable, the vector of error-ridden measurements of the explanatory variable and the vector of observations of an instrumental variable, each measured around its mean. The ordinary least squares estimate is then

[This paper discusses in general the use of a priori information in regression analysis and explores in detail the use of mixtures of normal prior distributions. It includes an application to distributed lag analysis.]

[We consider the nature of the inferences that can be made when all variables in a linear regression are measured with error. Assuming that the measurement errors are orthogonal to each other and the unobserved correctly measured regressors, we demonstrate that the true regression coefficient vector can be restricted to the convex hull of all possible regressions iff all these regressions yield coefficient vectors lying in the same orthant. Otherwise, the set of feasible coefficient vectors is unbounded. For the unbounded case, we demonstrate that prior information concerning the "seriousness" of the measurement errors in the variables can bound the feasible region. Two diagnostics are proposed to indicate the sensitivity of conventional inferences to measurement error in the regressors, and an illustrative example is presented.]

[In most statistical estimation problems, the distribution of errors is unknown, and the traditional assumption of normality is used for convenience. We investigate here the fragility of the inferences based on normality by hypothesizing a neighborhood of distributions around the normal distribution, and by identifying the set of alternative maximum likelihood estimates corresponding to the set of error distributions.]