The multicategory support vector machine (MSVM) has been widely used for multicategory classification. Despite its widespread popularity, regular MSVM cannot provide direct probabilistic results and suffers from excessive computational cost, as it is formulated on the hinge loss function and it solves a sum-to-zero constrained quadratic programming problem. In this study, we propose a general refinement of regular MSVM, termed as the simplex-based proximal MSVM (SPMSVM). Our SPMSVM uses a novel family of squared error loss functions in place of the hinge loss and it removes the explicit sum-to-zero constraint by the simplex structure. Consequently, the SPMSVM only requires solving an unconstrained linear system, leading to closed-form solutions. In addition, the SPMSVM can be cast into a weighted regression problem so that it is scalable for large-scale applications. Moreover, the SPMSVM naturally yields an estimate of the conditional category probability, which is more informative than regular MSVM. Theoretically, the SPMSVM is shown to include many existing MSVMs as its special cases, and its asymptotic and finite-sample statistical properties are well established. Simulations and real examples show that the proposed SPMSVM is a stable, scalable and competitive classifier.

The rate reduction models have been widely used to model the recurrent failure data for their capabilities in quantifying the repair effects. Despite the widespread popularity, there have been limited studies on statistical inference of most failure rate reduction models. In view of this fact, this study proposes a semiparametric estimation framework for a general class of such models, called extended geometric failure rate reduction (EGFRR) models. Covariates are considered in our analysis and their effects are modeled as a log-linear factor on the baseline failure rate. Unlike the existing inference methods for the EGFRR models that assume the failure data are censored at a fixed number of failures, our study considers covariates and time-censoring, which are more common in practice. The semiparametric maximum likelihood (ML) estimators are obtained by carefully constructing the likelihood function. Asymptotic properties including consistency and weak convergence of the ML estimators are established by using the properties of the martingale process. In addition, we show that the semiparametric estimators are asymptotically efficient. A real example from the automobile industry illustrates the usefulness of the proposed framework and extensive simulations show its outstanding performance when comparing with the existing methods.