SEMIPARAMETRIC LIKELIHOOD RATIO INFERENCE

Likelihood ratio tests and related con dence intervals for a real parameter in the presence of an in nite dimensional nuisance parameter are considered. In all cases, the estimator of the real parameter has an asymptotic normal distribution. However, the estimator of the nuisance parameter may not be asymptotically Gaussian or may converge to the true parameter value at a slower rate than the square root of the sample size. Nevertheless the likelihood ratio statistic is shown to possess an asymptotic chisquared distribution. The examples considered are tests concerning survival probabilities based on doubly censored data, a test for presence of heterogeneity in the gamma frailty model, a test for signi cance of the regression coe cient in Cox's regression model for current status data and a test for a ratio of hazards rates in an exponential mixture model. In both of the last examples the rate of convergence of the estimator of the nuisance parameter is less than the square root of the sample size.

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

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

[3]  R. Gill.  Non- and semi-parametric maximum likelihood estimators and the Von Mises method , 1986 .

[4]  E. Giné,et al.  Lectures on the central limit theorem for empirical processes , 1986 .

[5]  Susan A. Murphy,et al.  Asymptotic Theory for the Frailty Model , 1995 .

[6]  M. Kendall.  Theoretical Statistics , 1956, Nature.

[7]  DavidR . Thomas,et al.  Confidence Interval Estimation of Survival Probabilities for Censored Data , 1975 .

[8]  J. Qin,et al.  Empirical likelihood in a semi-parametric model , 1996 .

[9]  Susan A. Murphy,et al.  Consistency in a Proportional Hazards Model Incorporating a Random Effect , 1994 .

[10]  The Law of Large Numbers for $D[0,1]$-Valued Random Variables , 1963 .

[11]  K. Roeder,et al.  A Semiparametric Mixture Approach to Case-Control Studies with Errors in Covariables , 1996 .

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

[13]  Gang Li,et al.  On nonparametric likelihood ratio estimation of survival probabilities for censored data , 1995 .

[14]  Peter Hall,et al.  Methodology and algorithms of empirical likelihood , 1990 .

[15]  Myron N. Chang.  Weak Convergence of a Self-Consistent Estimator of the Survival Function with Doubly Censored Data , 1990 .

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

[17]  Grace L. Yang,et al.  Strong Consistency of a Nonparametric Estimator of the Survival Function with Doubly Censored Data , 1987 .

[18]  Aad van der Vaart,et al.  Bracketing smooth functions , 1994 .

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

[20]  M. Ossiander,et al.  A Central Limit Theorem Under Metric Entropy with $L_2$ Bracketing , 1987 .

[21]  S. Murphy.  Likelihood Ratio-Based Confidence Intervals in Survival Analysis , 1995 .

[22]  Niels Keiding,et al.  Statistical Models Based on Counting Processes , 1993 .

[23]  James L. Powell,et al.  Estimation of semiparametric models , 1994 .

[24]  Richard D. Gill,et al.  A counting process approach to maximum likelihood estimation in frailty models , 1992 .

[25]  P. Bickel.  Efficient and Adaptive Estimation for Semiparametric Models , 1993 .

[26]  J. Wellner,et al.  Information Bounds and Nonparametric Maximum Likelihood Estimation , 1992 .

[27]  B. Wijers.  Nonparametric estimation for a windowed line-segment process , 1997 .