中文摘要：

随机系数回归模型是一类广泛应用于社会学、经济学、生物学、心理学、药代动力学等科学研究领域的线性模型．本文讨论了闭区间[a，b]上随机系数回归模型基于D-，G-，A-，Dβ-和I-最优准则下的几类最优设计问题．证明了在设计准则满足实值单调的条件下最优设计可在设计域端点处获得的结论，并给出了这几类设计准则下最优设计的解析表达式．

英文摘要：

The random coefficient models have been widely used in the researching in the area of sociology, economics, biosciences, psychology and population pharmacokinetics. The purpose of this paper is to present the optimal design based on D-, A-, G-, Dβ and I- optimality criteria for the random coefficient regression models on the interval [a, b]. A lemma is proved that the search of optimal designs can be confined at the extreme settings of the design region when the design criteria satisfy the assumption of the real valued monotone design criteria. Analytical solutions of D-, A-, G-, Dβ- and I- optimal design for the random coefficient regression models on the interval [a, b] are constructed in the paper.