Abstract: Test difficulty coefficient plays important roles in building examination paper and in the measurement of the quality of exam questions.But how to accurately obtain coefficient is difficult during the development of exam questions.In this paper，the test difficulty coefficient is partitioned into the academic difficulty coefficient and the sampling difficulty coefficient.And the series of the sampling difficulty coefficient is made in the Hilbert space.Using the self-contained of the space，it is obtained that the series is convergent to the academic difficulty coefficient.Finally，using the empirical risk minimization in statistical learning theory，a learning model of the sampling difficulty coefficient is made by rigorously mathematic theory.The model can self-learn from the exam sampling data and converge rapidly.

Key words: difficulty coefficient, statistical learning, empirical risk minimization, algorithm

摘要： 试题的难度系数是自动生成考试试卷、影响考试质量的一个重要因素。然而，如何准确确定试题的难度系数是考试系统的一个难点。提出将试题的难度系数分为理论难度系数G和样本难度系数G_k，并将序列{G_k}构造到Hilbert空间上，利用Hilbert空间的完备性，得出了序列{G_k}收敛于G。最后利用统计学习理论的经验风险最小准则，构造了G_k的一个学习器模型，该模型建立在严密的数学理论基础之上，具有收敛速度快和根据考试样本数据进行学习修正的特点。

关键词: 难度系数, 统计学习, 经验风险最小准则, 算法