Variance matrix estimation in multivariate classical measurement error models

Measurement errors are often encountered in several continuous variables in a data set, and various methods have been proposed to handle these measurement errors when they are supposed to be independent. Recently, Kekeç and Van Keilegom (Electron J Stat 16(1):1831–1854, 2022) considered bivariate classical measurement error models with correlated measurement errors, and estimated their variance matrix when no validation data nor auxiliary variables are available. However, in practice, one often observes more than two variables with correlated measurement errors. In this paper, we introduce a flexible and practical method to estimate the variance matrix of multivariate classical additive Gaussian measurement errors, without additional information. We show that the error variance matrix is identifiable under certain circumstances and introduce an estimation method of the variance matrix and of the multivariate density of the variables of interest by means of Bernstein polynomials. Asymptotic properties and finite sample characteristics of the proposed methodology are also examined. Furthermore, we consider in a simulation study a multiple linear regression model with measurement errors in multiple covariates, and use the proposed estimator of the variance matrix to estimate the model parameters by means of the simulation extrapolation (SIMEX) algorithm. Finally, the method is illustrated on a dataset on fetal biometry measurements.