Efficient quantile regression for heteroscedastic models

Quantile regression (QR) provides estimates of a range of conditional quantiles. This stands in contrast to traditional regression techniques, which focus on a single conditional mean function. Lee et al. [Regularization of case-specific parameters for robustness and efficiency. Statist Sci. 2012;27(3):350–372] proposed efficient QR by rounding the sharp corner of the loss. The main modification generally involves an asymmetric ℓ₂ adjustment of the loss function around zero. We extend the idea of ℓ₂ adjusted QR to linear heterogeneous models. The ℓ₂ adjustment is constructed to diminish as sample size grows. Conditions to retain consistency properties are also provided

Asymptotics of Lower Dimensional Zero-Density Regions

Topological data analysis (TDA) allows us to explore the topological features of a dataset. Among topological features, lower dimensional ones have recently drawn the attention of practitioners in mathematics and statistics due to their potential to aid the discovery of low dimensional structure in a data set. However, lower dimensional features are usually challenging to detect from a probabilistic perspective. In this paper, lower dimensional topological features occurring as zero-density regions of density functions are introduced and thoroughly investigated. Specifically, we consider sequences of coverings for the support of a density function in which the coverings are comprised of balls with shrinking radii. We show that, when these coverings satisfy certain sufficient conditions as the sample size goes to infinity, we can detect lower dimensional, zero-density regions with increasingly higher probability while guarding against false detection. We supplement the theoretical developments with the discussion of simulated experiments that elucidate the behavior of the methodology for different choices of the tuning parameters that govern the construction of the covering sequences and characterize the asymptotic results.Comment: 28 page

Application of site and haplotype-frequency based approaches for detecting selection signatures in cattle

<p>Abstract</p> <p>Background</p> <p>'Selection signatures' delimit regions of the genome that are, or have been, functionally important and have therefore been under either natural or artificial selection. In this study, two different and complementary methods--integrated Haplotype Homozygosity Score (|iHS|) and population differentiation index (F_ST)--were applied to identify traces of decades of intensive artificial selection for traits of economic importance in modern cattle.</p> <p>Results</p> <p>We scanned the genome of a diverse set of dairy and beef breeds from Germany, Canada and Australia genotyped with a 50 K SNP panel. Across breeds, a total of 109 extreme |iHS| values exceeded the empirical threshold level of 5% with 19, 27, 9, 10 and 17 outliers in Holstein, Brown Swiss, Australian Angus, Hereford and Simmental, respectively. Annotating the regions harboring clustered |iHS| signals revealed a panel of interesting candidate genes like SPATA17, MGAT1, PGRMC2 and ACTC1, COL23A1, MATN2, respectively, in the context of reproduction and muscle formation. In a further step, a new Bayesian F_ST-based approach was applied with a set of geographically separated populations including Holstein, Brown Swiss, Simmental, North American Angus and Piedmontese for detecting differentiated loci. In total, 127 regions exceeding the 2.5 per cent threshold of the empirical posterior distribution were identified as extremely differentiated. In a substantial number (56 out of 127 cases) the extreme F_STvalues were found to be positioned in poor gene content regions which deviated significantly (p < 0.05) from the expectation assuming a random distribution. However, significant F_STvalues were found in regions of some relevant genes such as SMCP and FGF1.</p> <p>Conclusions</p> <p>Overall, 236 regions putatively subject to recent positive selection in the cattle genome were detected. Both |iHS| and F_STsuggested selection in the vicinity of the Sialic acid binding Ig-like lectin 5 gene on BTA18. This region was recently reported to be a major QTL with strong effects on productive life and fertility traits in Holstein cattle. We conclude that high-resolution genome scans of selection signatures can be used to identify genomic regions contributing to within- and inter-breed phenotypic variation.</p