Testing for inequality constraints in singular models by trimming or winsorizing the variance matrix

There are many applications in which a statistic follows, at least asymptotically, a normal distribution with a singular or nearly singular variance matrix. A classic example occurs in linear regression models under multicollinearity but there are many more such examples. There is well-developed theory for testing linear equality constraints when the alternative is two-sided and the variance matrix is either singular or non-singular. In recent years there is considerable, and growing, interest in developing methods for situations in which the estimated variance matrix is nearly singular. However, there is no corresponding methodology for addressing one-sided, i.e., constrained or ordered alternatives. In this paper we develop a unified framework for analyzing such problems. Our approach may be viewed as the trimming or winsorizing of the eigenvalues of the corresponding variance matrix. The proposed methodology is applicable to a wide range of scientific problems and to a variety of statistical models in which inequality constraints arise. We illustrate the methodology using data from a gene expression microarray experiment obtained from the NIEHS' Fibroid Growth Study.