Estimation Of The Constrained Parameters Via The EM-type Algorithm

Because accounting for the constraints on model parameters lead to more efficient and interpretable estimators. Therefore, constrained parameter problems arise in a variety of statistical applications. For example, if θ_i is the toxic of a certain medicine at dose i(i = 1,2,… ,k) , then it is well known that the toxic is increasing as the dose increasing, it is reasonable to assume the simple ordering, that is θ₁ ≤ θ₂ ≤ … ≤ θ_k. Another example, if μ_i denote the efficiency of some medicine at dose i(i = 1,2, … , k), we know that the efficiency is usually increasing up to some dose m, and then is decreasing. Thus an umbrella ordering with change-point m seems more reasonable. Let μ₁ ≤ … ≤ μm ≥ … ≥ μk . Furthermore, in life insurance , let p_i denote the true mortality rate at age t_i, it is reasonable toassume that {p_i} are increasing convex, that is reasonable to assume {p_i} to beThis paper concerns the method for estimating the restricted parameters in multivariate normal via the EM-type algorithms. The model is y_l = Xθ + e_l, e_l N_m(0, ∑), l = 1,2, … ,p. where y_l is an m-vector of responses for individual l, X_m*q is a known matrix, regression coefficients θ = (θ₁, … , θ_q) are restricted to the simple ordering θ₁ ≤ … ≤ θ_q, the umbrella ordering θ₁ ≤ … ≤ θ_h ≥ θ_h+1 ≥ … ≥θ_q, and the increasing convex ordering 0where {d_k} are known and d₁ < … < d_q .