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Asymptotics for Lasso-type estimators. (English) Zbl 1105.62357
Summary: We consider the asymptotic behavior ofregression estimators that minimize the residual sum of squares plus a penalty proportional to \(\sum| \beta_j|^{\gamma}\), for some \(\gamma>0\). These estimators include the Lasso as a special case when \(\gamma=1\). Under appropriate conditions, we show that the limiting distributions can have positive probability mass at 0 when the true value of the parameter is 0. We also consider asymptotics for “nearly singular” designs.

MSC:
62J05 Linear regression; mixed models
62E20 Asymptotic distribution theory in statistics
62J07 Ridge regression; shrinkage estimators (Lasso)
Software:
PDCO
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[1] Anderson, P. K. and Gill, R. D. (1982). Cox’s regression model for counting processes: a large sample study. Ann. Statist. 10 1100-1120. · Zbl 0526.62026 · doi:10.1214/aos/1176345976
[2] Besag, J. (1986). On the statistical analysis ofdirty pictures (with discussion). J. Roy. Statist. Soc. Ser. B 48 259-302. JSTOR: · Zbl 0609.62150 · links.jstor.org
[3] Chen, S. S., Donoho, D. L. and Saunders, M. A. (1999). Atomic decomposition by basis pursuit. SIAM J. Scientific Computing 20 33-61. · Zbl 0919.94002 · doi:10.1137/S1064827596304010
[4] Fan, J. and Li, R. (1999). Variable selection via penalized likelihood. Unpublished manuscript.
[5] Fiacco, A. V. and McCormick, G. P. (1990). Nonlinear Programming: Sequential Unconstrained Minimization Techniques. SIAM, Philadelphia. · Zbl 0713.90043
[6] Frank, I. E. and Friedman, J. H. (1993). A statistical view ofsome chemometrics regression tools (with discussion). Technometrics 35 109-148. · Zbl 0775.62288 · doi:10.2307/1269656
[7] Fu, W. J. (1998). Penalized regressions: the Bridge versus the Lasso. J. Comput. Graph. Statist. 7 397-416. JSTOR: · doi:10.2307/1390712 · links.jstor.org
[8] Fu, W. J. (1999). Estimating the effective trends in age-period-cohort studies. Unpublished manuscript.
[9] Geyer, C. J. (1994). On the asymptotics ofconstrained M-estimation. Ann. Statist. 22 1993-2010. · Zbl 0829.62029 · doi:10.1214/aos/1176325768
[10] Geyer, C. J. (1996). On the asymptotics ofconvex stochastic optimization. Unpublished manuscript.
[11] Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statistics 18 191-219. · Zbl 0703.62063 · doi:10.1214/aos/1176347498
[12] Kupper, L. L., Janis, J. M., Karmous, A. and Greenberg, B. G. (1985). Statistical age-periodcohort analysis: a review and critique. J. Chronic Disease 38 811-830.
[13] Linhart, H. and Zucchini, W. (1986). Model Selection. Wiley, New York. · Zbl 0665.62003
[14] Osborne, M. R., Presnell, B. and Turlach, B. A. (1998). On the Lasso and its dual. Research Report 98/1, Dept. Statistics, Univ. Adelaide.
[15] Pflug, G. Ch. (1995). Asymptotic stochastic programs. Math. Oper. Res. 20 769-789. JSTOR: · Zbl 0847.90107 · doi:10.1287/moor.20.4.769 · links.jstor.org
[16] Pollard, D. (1991). Asymptotics for least absolute deviation regression estimators. Econometric Theory 7 186-199. JSTOR: · Zbl 04504753 · doi:10.1017/S0266466600004394 · links.jstor.org
[17] Sardy, S., Bruce, A. G. and Tseng, P. (1998). Block coordinate relaxation methods for nonparametric signal denoising with wavelet dictionaries. Technical Report, Dept. Mathematics, EPFL, Lausanne.
[18] Srivastava, M. S. (1971). On fixed width confidence bounds for regression parameters. Ann. Math. Statist. 42 1403-1411. · Zbl 0224.62034 · doi:10.1214/aoms/1177693251
[19] Tibshirani, R. (1996). Regression shrinkage and selection via the Lasso. J. Roy. Statist. Soc. Ser. B 58 267-288. JSTOR: · Zbl 0850.62538 · links.jstor.org
[20] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes with Applications to Statistics. Springer, New York. · Zbl 0862.60002
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