ELSEVIER Statistics & Probability Letters 34 (1997) 67-73 STATIITICS t PROBAIIILITY LETll[R$ Consistent estimation for the non-normal ultrastructural model Anil K. Srivastava a'*, Shalabh b aDepartment of Statistics, University of Lucknow, 15 Shastri Nagar, Lucknow 226004, India b Department of Statistics, University of Jammu, Jammu, India Abstract The present article considers the linear ultrastructural model which encompasses the two popular forms of measure- ment error models, viz., the functional and structural models. The measurement error variance associated with the explanatory variable is assumed to be known which leads to consistent estimators of parameters. The efficiency properties of the thus obtained estimators of slope parameter, intercept term and the measurement error variance of study variable are derived under non-normal error distributions and the effects of departures from symmetry and peakedness of the error distributions are studied. Keywords: Immaculate estimator; Measurement errors; Ultrastructural model 1. Introduction Dolby (1976) introduced the ultrastructural model which includes as special cases the functional and structural measurement error models. Assuming normal measurement errors, he derived the maximum likelihood estimates of the parameters. These parameters were found to be unidentifiable due to the lack of sufficient information. However, he obtained the maximum likelihood estimates of the parameters assuming the two variance ratios to be known - one is the variance of measurement errors associated with observed study variable divided by the variance of observed explanatory variable while the other is variance of true explanatory variable divided by measurement error variance of observed explanatory variable. Cheng and Van Ness (1991) incorporated additional information in the form of known measurement error variance associated with explanatory variable and derived asymptotic properties of the maximum likelihood estimators under the assumption of normal errors. These estimators were found to be consistent. However, the assumption of normal errors restricts the utility of the asymptotic results in many applications. This article provides asymptotic results for such estimators when the measurement errors are not necessarily normal. Such an investigation can shed light on the robustness of the results under non-normal measurement errors. * Corresponding author. 0167-7152/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S0 1 67-7 1 52(96)001 67-8