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For simplification and motivated by the data set from the National Longitudinal Survey of Youth (NLSY79), we only consider a bivariate linear regression model with β = (β 1 , β 2 ), where β 1 and β 2 are both p × 1 vectors. We show that if β 1 ≠ β 2 , then it degenerates to a univariate linear regression model which is classified as case 1 in here. On the other hand, we show that if there is a linear restriction on β, for instance, the two column vectors in β are the same, that is β 1 = β 2 , it is a bivariate linear regression model which does not degenerate to a univariate linear regression model and is classified as case 2.

In this thesis, we propose to estimate the regression coefficients by a modified Buckley-James estimator and show that the modified BJE is consistent and has asymptotic normality under certain discontinuous regularity conditions for both case 1 and case 2. Moreover, in case 1, various non-normal asymptotic distributions of the BJE are presented when the regularity conditions are violated. In case 2, we further carry out simulation studies to compare the asymptotic properties of the modified BJE under different sample sizes and various continuous underlying distributions. We also perform data analysis to the NLSY79 data.

We further consider the estimation problem in case 1 with missing covariates involved, which is denoted as case 3. The extension of the BJE based on the generalized maximum likelihood estimator (GMLE) of the underlying distribution is discussed. The Newton-Raphson (NR) method is not feasible in computing the GMLE due to the large sample size of the real data example. We propose a self-consistent algorithm to bypass this difficulty.