  In this paper, we study the classification problem by estimating the
conditional probability function of the given data. Different from the
traditional expected risk estimation theory on empirical data, we calculate the
probability via Fredholm equation, this leads to estimate the distribution of
the data. Based on the Fredholm equation, a new expected risk estimation theory
by estimating the cumulative distribution function is presented. The main
characteristics of the new expected risk estimation is to measure the risk on
the distribution of the input space. The corresponding empirical risk
estimation is also presented, and an $\varepsilon$-insensitive $L_{1}$
cumulative support vector machines ($\varepsilon$-$L_{1}VSVM$) is proposed by
introducing an insensitive loss. It is worth mentioning that the classification
models and the classification evaluation indicators based on the new mechanism
are different from the traditional one. Experimental results show the
effectiveness of the proposed $\varepsilon$-$L_{1}VSVM$ and the corresponding
cumulative distribution function indicator on validity and interpretability of
small data classification.
