We consider two variations on a Lehmann alternatives to symmetry-at-zero semiparametric model, with a real parameter $\theta$ quantifying skewness and a symmetric-at-0 distribution as a nuisance function. We show that a test of symmetry based on the signed log-rank statistic [A signed log-rank test of symmetry at zero (2011) University of Rochester] is asymptotically efficient in these models, derive its properties under local alternatives and present efficiency results relative to other signed-rank tests. We develop efficient estimation of the primary parameter in each model, using model-specific estimates of the nuisance function, and provide a method for choosing between the two models. All inference methods proposed are based solely on the signed ranks of the absolute values of the observations, the invariantly sufficient statistic. A simulation study is summarized and an example presented. Extensions to regression modeling are envisaged.
Digital Object Identifier: 10.1214/12-IMSCOLL914