The generalized sigmoidal quantile function

References

• Abdous, B., and B. Remillard. 1995. Relating quantiles and expectiles under weighted-wymmetry. Annals of the Institute of Statistical Mathematics 47 (2):371–84. doi:10.1007/BF00773468.
   Web of Science ®Google Scholar
• Abdous, B., and R. Theodorescu. 1992. Note on the spatial quantile of a random vector. Statistics & Probability Letters 13 (4):333–6. doi:10.1016/0167-7152(92)90043-5.
   Web of Science ®Google Scholar
• Bassett, G., Jr., and R. Koenker. 1982. An empirical quantile function for linear models with i.i.d. errors. Journal of the American Statistical Association 77:407–15.
   Web of Science ®Google Scholar
• Chen, C., and O. L. Mangasarian. 1995. Smoothing methods for convex inequalities and linear complementarity problems. Mathematical Programming 71 (1):51–69. doi:10.1007/BF01592244.
   Web of Science ®Google Scholar
• Cheng, C. 1995. The Bernstein polynomial estimator of a smooth quantile function. Statistics & Probability Letters 24 (4):321–30. doi:10.1016/0167-7152(94)00190-J.
   Web of Science ®Google Scholar
• Davison, A. C., and D. V. Hinkley. 1997. Bootstrap methods and their applications. New York: Cambridge University Press.
   Google Scholar
• Efron, B., and R. Tibshirani. 1993. An introduction to the bootstrap. New York: Chapman & Hall.
   Google Scholar
• Hampel, F., C. Hennig, and E. Ronchetti. 2011. A smoothing principle for the Huber and other location M-estimators. Computational Statistics & Data Analysis 55 (1):324–37. doi:10.1016/j.csda.2010.05.001.
   Web of Science ®Google Scholar
• Harrell, F., and C. E. Davis. 1982. A new distribution-free quantile estimator. Biometrika 69 (3):635–40. doi:10.1093/biomet/69.3.635.
   Web of Science ®Google Scholar
• Hutson, A. D. 2000. A composite quantile function estimator with applications in bootstrapping. Journal of Applied Statistics 27 (5):567–77. doi:10.1080/02664760050076407.
   Web of Science ®Google Scholar
• Hutson, A. D. 2002. A semiparametric quantile function estimator for use in bootstrap estimation procedures. Statistics and Computing 12 (4):331–8. doi:10.1023/A:1020783911574.
   Web of Science ®Google Scholar
• Hutson, A. D. 2018. Quantile planes without crossing via nonlinear programming. Computer Methods and Programs in Biomedicine 153:185–90. doi:10.1016/j.cmpb.2017.10.019.
   PubMed Web of Science ®Google Scholar
• Hutson, A. D., and M. D. Ernst. 2000. The exact bootstrap mean and variance of an L-estimator. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 62 (1):89–94. doi:10.1111/1467-9868.00221.
   Web of Science ®Google Scholar
• Jones, M. C. 1994. Expectiles and M-quantiles are quantiles. Statistics & Probability Letters 20 (2):149–53. doi:10.1016/0167-7152(94)90031-0.
   Web of Science ®Google Scholar
• Koenecker, R. 1993. When are expectiles percentiles? Econometric Theory 9:526–7.
   Web of Science ®Google Scholar
• Mao, T., K. W. Ng, and T. Hu. 2015. Asymptotic expansions of generalized quantiles and expectiles for extreme risks. Probability in the Engineering and Informational Sciences 29 (3):309–27. doi:10.1017/S0269964815000017.
   Web of Science ®Google Scholar
• Mudholkar, G. S., A. D. Hutson, and C. E. Marchetti. 2001. A simple Hermitian estimator of the quantile function. Journal of Nonparametric Statistics 13 (2):229–43. doi:10.1080/10485250108832851.
   Web of Science ®Google Scholar
• Newey, W. K., and J. L. Powell. 1987. Asymmetric least squares estimation and testing. Econometrica 55 (4):819–47. doi:10.2307/1911031.
   Web of Science ®Google Scholar
• Parzen, E. 1993. Change PP plot and continuous sample quantile function. Communications in Statistics - Theory and Methods 22 (12):3287–304. doi:10.1080/03610929308831216.
   Web of Science ®Google Scholar
• Polansky, A. M. 1999. Upper Bounds on the True Coverage of Bootstrap Percentile Type Confidence Interval. The American Statistician 53:362–9.
   Google Scholar
• Polansky, A. M., and W. R. Schucany. 1997. Kernel smoothing to improve bootstrap confidence intervals. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 59 (4):821–38. doi:10.1111/1467-9868.00099.
   Google Scholar
• Reiss, R.-D. 1980. Estimation of quantiles in certain nonparametric models. Annals of Mathematical Statists 8:87–105.
   Google Scholar
• Reiss, R.-D. 1989. Approximate distributions of order statistics: With applications to nonparametric statistics. New York: Springer-Verlag.
   Google Scholar
• Schmidt, M., G. Fung, and R. Rosales. 2007. Fast optimization methods for L1 regularization: A comparative study and two new approaches. In Machine learning: ECML 2007. Lecture notes in computer science, Vol. 4701, ed. J. N. Kok, J. Koronacki, R. L. Mantaras, S. Matwin, D. Mladenič, and A. Skowron. Berlin, Heidelberg: Springer.
   Google Scholar
• Serfling, R. J. 1980. Approximation theorems of mathematical statistics. New York: John Wiley & Sons.
   Google Scholar
• Shao, J., and D. Tu. 1995. The Jackknife and bootstrap. New York: Springer-Verlag.
   Google Scholar
• Sheather, S. J., and J. S. Marron. 1990. Kernel quantile estimators. Journal of the American Statistical Association 85 (410):410–6. doi:10.1080/01621459.1990.10476214.
   Web of Science ®Google Scholar
• Stigler, S. M. 1977. Fractional order statistics, with applications. Journal of the American Statistical Association 72 (359):544–50. doi:10.1080/01621459.1977.10480611.
   Web of Science ®Google Scholar
• Young, G. A. 1994. Bootstrap: More than a stab in the dark. Statistical Science 9 (3):382–415. doi:10.1214/ss/1177010383.
   Web of Science ®Google Scholar