Bernoulli

  • Bernoulli
  • Volume 15, Number 4 (2009), 1259-1286.

Evaluation for moments of a ratio with application to regression estimation

Paul Doukhan and Gabriel Lang

Full-text: Open access

Abstract

Ratios of random variables often appear in probability and statistical applications. We aim to approximate the moments of such ratios under several dependence assumptions. Extending the ideas in Collomb [C. R. Acad. Sci. Paris 285 (1977) 289–292], we propose sharper bounds for the moments of randomly weighted sums and for the $L^p$-deviations from the asymptotic normal law when the central limit theorem holds. We indicate suitable applications in finance and censored data analysis and focus on the applications in the field of functional estimation.

Article information

Source
Bernoulli, Volume 15, Number 4 (2009), 1259-1286.

Dates
First available in Project Euclid: 8 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.bj/1262962235

Digital Object Identifier
doi:10.3150/09-BEJ190

Mathematical Reviews number (MathSciNet)
MR2597592

Zentralblatt MATH identifier
1200.62035

Keywords
division ratio regression estimation weighted sums

Citation

Doukhan, Paul; Lang, Gabriel. Evaluation for moments of a ratio with application to regression estimation. Bernoulli 15 (2009), no. 4, 1259--1286. doi:10.3150/09-BEJ190. https://projecteuclid.org/euclid.bj/1262962235


Export citation

References

  • [1] Ango Nze, P. and Doukhan, P. (1996). Nonparametric minimax estimation in a weakly dependent framework I: Quadratic properties. Math. Methods Statist. 5 404–423.
  • [2] Ango Nze, P. and Doukhan, P. (1998). Nonparametric minimax estimation in a weakly dependent framework II: Uniform properties. J. Statist. Plann. Inference 68 5–29.
  • [3] Ango Nze, P. and Doukhan, P. (2004). Weak dependence: Models and applications to econometrics. Econom. Theory 20 995–1045.
  • [4] Ango Nze, P., Bühlman, P. and Doukhan, P. (2002). Weak dependence beyond mixing and asymptotics for nonparametric regression. Ann. Statist. 30 397–430.
  • [5] Bensaïd, N. and Fabre, J.-P. (2007). Optimal asymptotic quadratic errors of kernel estimators of Radon–Nikodym derivative for strong mixing data. J. Nonparametr. Stat. 19 77–88.
  • [6] Collomb, G. (1977). Quelques propriétés de la méthode du noyau pour l’estimation non paramétrique de la régression en un point fixé. C. R. Acad. Sci. Paris 285 289–292.
  • [7] Collomb, G. and Doukhan, P. (1983). Estimation nonparamétrique de la fonction d’autorégression d’un processus stationnaire et ϕ-mélangeant. C. R. Acad. Sci. Paris 296 859–863.
  • [8] Dedecker, J. (2001). Exponential inequalities and functional central limit theorems for random fields. ESAIM Probab. Stat. 5 77–104. Available at http://www.esaim-ps.org.
  • [9] Dedecker, J., Doukhan, P., Lang, G., León, J.R., Louhichi, S. and Prieur, C. (2007). Weak Dependence: Models, Theory and Applications. Lecture Notes in Statist. 190. New York: Springer.
  • [10] Dedecker, J. and Prieur, C. (2005). New dependence coefficients. Examples and applications to statistics. Probab. Theory Related Fields 132 203–236.
  • [11] de la Peña, V.H., Klass, M.J. and Lai, T.L. (2007). Pseudo-maximization and self-normalized processes. Probab. Surv. 4 172–192.
  • [12] Del Moral, P. and Miclo, L. (2000). Branching and interacting particle systems approximations of Feynman–Kac formulae with applications to non-linear filtering. In Séminaire de Probabilités XXXIV (J. Azéma, M. Emery, M. Ledoux and M. Yor, eds.). Lecture Notes in Math. 1729 1–145. New York: Springer.
  • [13] Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statist. 85. New York: Springer.
  • [14] Doukhan, P. and Louhichi, S. (2001). Functional density estimation under a new weak dependence condition. Scand. J. Statist. 28 325–342.
  • [15] Doukhan, P. and Neumann, M. (2007). A Bernstein type inequality for times series. Stochastic Process. Appl. 117 878–903.
  • [16] Doukhan, P. and Portal, F. (1983). Moments de variables aléatoires mélangeantes. C. R. Acad. Sci. Paris 297 129–132.
  • [17] Doukhan, P. and Wintenberger, O. (2007). Invariance principle for new weakly dependent stationary models under sharp moment assumptions. Probab. Math. Statist. 27 45–73.
  • [18] Fearnhead, P. and Liu, Z. (2007). On-line inference for multiple changepoint problems. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 589–605.
  • [19] Figiel, T., Hitczenko, P., Johnson, W.B., Schechtmann, G. and Zinn, J. (1997). Extremal properties of Rademacher functions with applications to the Khinchine and Rosenthal inequalities. Trans. Amer. Math. Soc. 349 997–1027.
  • [20] Jaber, M.Y. and Salameh, M.K. (1995). Optimal lot sizing under learning considerations: Shortages allowed and backordered. Appl. Math. Model. 19 307–310.
  • [21] Li, G. and Rabitz, H. (2006). Ratio control variate method for efficiently determining high-dimensional model representations. J. Comput. Chem. 27 1112–1118.
  • [22] Pisier, G. (1978). Some results on Banach spaces without local unconditional structure. Compos. Math. 37 3–19.
  • [23] Robert, C.P. and Casella, G. (1999). Monte Carlo Statistical Methods. New York: Springer.
  • [24] Rio, E. (2000). Théorie asymptotique pour des processus aléatoires faiblement dépendants. In SMAI, Mathématiques et Applications 31. Berlin: Springer.
  • [25] Spiegelmann, C. and Sachs, J. (1980). Consistent window estimation in nonparametric regression. Ann. Statist. 8 240–246.
  • [26] Stone, C.J. (1980). Optimal rates of convergence for nonparametric estimators. Ann. Statist. 8 1348–1360.
  • [27] Stone, C.J. (1982). Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040–1053.
  • [28] Tsybakov, A.B. (2004). Introduction à l’estimation non-paramétrique. In SMAI, Mathématiques et Applications 41. Berlin: Springer.
  • [29] Viennet, G. (1997). Inequalities for absolutely regular sequences: Application to density estimation. Probab. Theory Related Fields 107 467–492.