## Institute of Mathematical Statistics Collections

### Efficient testing and estimation in two Lehmann alternatives to symmetry-at-zero models

#### Abstract

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.

#### Chapter information

Source
Banerjee, M., Bunea, F., Huang, J., Koltchinskii, V., and Maathuis, M. H., eds., From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2013) , 197-212

Dates
First available in Project Euclid: 8 March 2013

https://projecteuclid.org/euclid.imsc/1362751188

Digital Object Identifier
doi:10.1214/12-IMSCOLL914

Mathematical Reviews number (MathSciNet)
MR3202634

Zentralblatt MATH identifier
1325.62083

Subjects
Primary: 62G07: Density estimation 62H12: Estimation
Secondary: 62G05: Estimation 62G20: Asymptotic properties

Rights

#### Citation

Hall, W. J.; Wellner, Jon A. Efficient testing and estimation in two Lehmann alternatives to symmetry-at-zero models. From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, 197--212, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2013. doi:10.1214/12-IMSCOLL914. https://projecteuclid.org/euclid.imsc/1362751188

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