Random Sieve Likelihood and General Regression Models

Abstract Consider a semiparametric regression model Y = f(θ, X, ϵ), where f is a known function, θ is an unknown vector, ϵ consists of a random error and possibly of some unobserved variables, and the distribution F(·) of (ϵ, X) is unspecified. This article introduces, in a general setting, new methodology for estimating θ and F(·). The proposed method constructs a profile likelihood defined on random-level sets (a random sieve). The proposed method is related to empirical likelihood but is more generally applicable. Four examples are discussed, including a quadratic model, high-dimensional semiparametric regression, a nonparametric random-effects model, and linear regression with right-censored data. Simulation results and asymptotic analysis support the utility and effectiveness of the proposed method.

[1]  Art B. Owen,et al.  Empirical Likelihood for Linear Models , 1991 .

[2]  D. Harrington,et al.  Counting Processes and Survival Analysis , 1991 .

[3]  Joseph P. Romano,et al.  Nonparametric confidence limits by resampling methods and least favorable families , 1990 .

[4]  I. James,et al.  Linear regression with censored data , 1979 .

[5]  J. Lawless,et al.  Empirical Likelihood and General Estimating Equations , 1994 .

[6]  W. Wong,et al.  Probability inequalities for likelihood ratios and convergence rates of sieve MLEs , 1995 .

[7]  W. Wong,et al.  Convergence Rate of Sieve Estimates , 1994 .

[8]  G. Robinson.  That BLUP is a Good Thing: The Estimation of Random Effects , 1991 .

[9]  W. Wong,et al.  Profile Likelihood and Conditionally Parametric Models , 1992 .

[10]  Zhiliang Ying,et al.  Linear rank statistics in regression analysis with censored or truncated data , 1992 .

[11]  Xiaotong Shen,et al.  On methods of sieves and penalization , 1997 .

[12]  John A. Nelder,et al.  Generalized linear models. 2nd ed. , 1993 .

[13]  P. Massart,et al.  Minimum contrast estimators on sieves: exponential bounds and rates of convergence , 1998 .

[14]  P. Massart,et al.  Rates of convergence for minimum contrast estimators , 1993 .

[15]  A. Owen.  Empirical Likelihood Ratio Confidence Regions , 1990 .

[16]  Jing Qin,et al.  Empirical Likelihood in Biased Sample Problems , 1993 .

[17]  E. Kaplan,et al.  Nonparametric Estimation from Incomplete Observations , 1958 .

[18]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[19]  Jiahua Chen,et al.  Empirical likelihood estimation for ?nite populations and the e?ective usage of auxiliary informatio , 1993 .

[20]  Art B. Owen,et al.  Empirical Likelihood Confidence Bands in Density Estimation , 1993 .

[21]  Rupert G. Miller.  Least squares regression with censored data , 1976 .

[22]  A. Owen.  Empirical Likelihood and Small Samples , 1992 .

[23]  A. Owen.  Empirical likelihood ratio confidence intervals for a single functional , 1988 .