Abstract
In this paper, we investigate empirical likelihood (EL) inference for density-weighted average derivatives in nonparametric multiple regression models. A simply adjusted empirical log-likelihood ratio for the vector of density-weighted average derivatives is defined and its limiting distribution is shown to be a standard Chi-square distribution. To increase the accuracy and coverage probability of confidence regions, an EL inference procedure for the rescaled parameter vector is proposed by using a linear instrumental variables regression. The new method shares the same properties of the regular EL method with i.i.d. samples. For example, estimation of limiting variances and covariances is not needed. A Monte Carlo simulation study is presented to compare the new method with the normal approximation method and an existing EL method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen SX, Qin YS (2000) Empirical likelihood confidence intervals for local linear smoothers. Biometrika 87: 946–953
Chen SX, Hall P (1993) Smoothed empirical likelihood confidence intervals for quantiles. Ann Stat 21: 1166–1181
Geenens G, Delecroix M (2006) A survey about single-index models theory. Int J Stat Syst 2: 203–230
Hall P, La Scala B (1990) Methodology and algorithms of empirical likelihood. Int Stat Rev 58: 109–127
Härdle W, Stoker TM (1989) Investigating smooth multiple regression by the method of average derivatives. J Am Stat Assoc 84: 986–995
Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 19: 293–325
Horowitz JL, Härdle W (1996) Direct semiparametric estimation of single-index models with discrete covariates. J Am Stat Assoc 91: 1632–1640
Hristache M, Juditsky A, Spokoiny V (2001) Direct estimation of the index coeffients in a single-index model. Ann Stat 29: 595–623
Ichimura H (1993) Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J Econ 58: 71–120
Kitamura Y (1997) Empirical likelihood methods with weakly dependent processes. Ann Stat 25: 2084–2102
Kolaczyk ED (1994) Empirical likelihood for generalized linear models. Stat Sinica 4: 199–218
Li Q, Lu X, Ullah A (2003) Multivariate local polynomial regression for estimating average derivatives. J Nonparametr Stat 4(5): 607–624
Liang H, Wang S, Carroll RJ (2007) Partially linear models with missing response variables and error-prone covariates. Biometrika 94: 185–198
Lu X, Qi Y (2004) Empirical likelihood for the additive risk model. Probab Math Stat 24: 419–431
Lu X, Qi Y (2006) Empirical likelihood for average derivatives. J Nonparametr Stat 18: 315–331
Newey WK, Stoker TM (1993) Efficiency of weighted average derivative estimators and index models. Econometrica 61: 1199–1223
Owen A (1988) Empirical likelihood ratio confidence intervals for single functional. Biometrika 75: 237–249
Owen A (1990) Empirical likelihood ratio confidence regions. Ann Stat 18: 90–120
Owen A (1991) Empirical likelihood for linear models. Ann Stat 19: 1725–1747
Owen A (2001) Empirical likelihood. Chapman & Hall/CRC, Boca Raton
Powell JL, Stock JH, Stoker TM (1989) Semiparametric estimation of index coefficients. Econometrica 57: 1403–1430
Powell JL, Stoker TM (1996) Optimal bandwidth choice for density-weighted average. J Econ 75: 291–316
Qin G, Jing BY (2001) Empirical likelihood for Cox regression model under random censorship. Commun Stat Simul Comput 30: 79–90
Qin J, Lawless JF (1994) Empirical likelihood and general estimating equations. Ann Stat 22: 300–325
Serfling RJ (1980) Approximation theorems of mathematical statistics. Wiley, New York
Stoker TM (1986) Consistent estimation of scaled coefficients. Econometrica 54: 1461–1481
Wang QH, Jing BY (2003) Empirical likelihood for partial linear models. Ann Inst Stat Math 55: 585–595
Whang Y-J (2006) Smoothed empirical likelihood methods for quantile regression models. Econ Theory 22: 173–205
Xia Y (2006) Asymptotic distributions for two estimators of the single-index models. Econ Theory 22: 1112–1137
Zhu LX, Xue LG (2006) Empirical likelihood confidence regions in a partially linear single-index model. J R Stat Soc B 68: 549–570
Zhou M (2005) Empirical likelihood analysis of the rank estimator for the censored accelerated failure time model. Biometrika 92: 492–498
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, W., Lu, X. Empirical likelihood for density-weighted average derivatives. Stat Papers 52, 391–412 (2011). https://doi.org/10.1007/s00362-009-0237-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-009-0237-5