Theorem 1. Assume that β ∈ R and that X is a n × k matrix of rank k < n. Let Y1, . . . , Yn are independent normally distributed random variables with mean vector μ = Xβ. Then, the likelihood ratio test of the hypothesis H0 : Aβ = 0 versus H1 : Aβ 6= 0. where A is a q × k matrix has critical region C = {y;F (y) ≥ F0}. F is given by F (y) = ∑n i=1(yi − ̂̂ μi) −∑ni=1(yi − μ̂i) ∑n i=1(yi − μ̂i) n− k q...

Since there are more equations than unknowns (m > n), this is an overdetermined set of equations. If the measurements of the independent variables xi are known precisely, then the only difference between the model, ŷ(xi; a), and the measured data yi, must be due to measurement error in yi and natural variability in the data that the model is unable to reflect. In such cases, the ‘best’ approxim...

We discuss semiparametric regression when only the ranks of responses are observed. The model is Yi = F (x ′ iβ0 + εi), where Yi is the unobserved response, F is a monotone increasing function, xi is a known p−vector of covariates, β0 is an unknown p-vector of interest, and εi is an error term independent of xi. We observe {(xi, Rn(Yi)) : i = 1, . . . , n}, where Rn is the ordinal rank function...

Since there are more equations than unknowns (m > n), this is an overdetermined set of equations. If the measurements of the independent variables xi are known precisely, then the only difference between the model, ŷ(xi; a), and the measured data yi, must be due to measurement error in yi and natural variability in the data that the model is unable to reflect. In such cases, the ‘best’ approxim...