  Linear structural error-in-variables models with univariate observations are
revisited for studying modified least squares estimators of the slope and
intercept. New marginal central limit theorems (CLT's) are established for
these estimators, assuming the existence of four moments for the measurement
errors and that the explanatory variables are in the domain of attraction of
the normal law. The latter condition for the explanatory variables is used the
first time, and is so far the most general in this context. It is also optimal,
or nearly optimal, for our CLT's. Moreover, due to the obtained CLT's being in
Studentized and self-normalized forms to begin with, they are a priori nearly,
or completely, data-based, and free of unknown parameters of the joint
distribution of the error and explanatory variables. Consequently, they lead to
a variety of readily available, or easily derivable, large-sample approximate
confidence intervals (CI's) for the slope and intercept. In contrast, in
related CLT's in the literature so far, the variances of the limiting normal
distributions, in general, are complicated and depend on various, typically
unknown, moments of the error and explanatory variables. Thus, the
corresponding CI's for the slope and intercept in the literature, unlike those
of the present paper, are available only under some additional model
assumptions.
