Empirical likelihood based confidence intervals for copulas

Copula as an effective way of modeling dependence has become more or less a standard tool in risk management, and a wide range of applications of copula models appear in the literature of economics, econometrics, insurance, finance, etc. How to estimate and test a copula plays an important role in practice, and both parametric and nonparametric methods have been studied in the literature. In this paper, we focus on interval estimation and propose an empirical likelihood based confidence interval for a copula. A simulation study and a real data analysis are conducted to compare the finite sample behavior of the proposed empirical likelihood method with the bootstrap method based on either the empirical copula estimator or the kernel smoothing copula estimator.     

Sieve maximum likelihood estimation for doubly semiparametric zero-inflated Poisson models

For nonnegative measurements such as income or sick days, zero counts often have special status. Furthermore, the incidence of zero counts is often greater than expected for the Poisson model. This article considers a doubly semiparametric zero-inflated Poisson model to fit data of this type, which assumes two partially linear link functions in both the mean of the Poisson component and the probability of zero. We study a sieve maximum likelihood estimator for both the regression parameters and the nonparametric functions. We show, under routine conditions, that the estimators are strongly consistent. Moreover, the parameter estimators are asymptotically normal and first order efficient, while the nonparametric components achieve the optimal convergence rates. Simulation studies suggest that the extra flexibility inherent from the doubly semiparametric model is gained with little loss in statistical efficiency. We also illustrate our approach with a dataset from a public health study.     

Variable selection for semiparametric varying coefficient partially linear errors-in-variables models

This paper focuses on the variable selections for semiparametric varying coefficient partially linear models when the covariates in the parametric and nonparametric components are all measured with errors. A bias-corrected variable selection procedure is proposed by combining basis function approximations with shrinkage estimations. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and the oracle property of the regularized estimators are established. A simulation study and a real data application are undertaken to evaluate the finite sample performance of the proposed method.     

Empirical likelihood inference in partially linear single-index models for longitudinal data

The empirical likelihood method is especially useful for constructing confidence intervals or regions of parameters of interest. Yet, the technique cannot be directly applied to partially linear single-index models for longitudinal data due to the within-subject correlation. In this paper, a bias-corrected block empirical likelihood (BCBEL) method is suggested to study the models by accounting for the within-subject correlation. BCBEL shares some desired features: unlike any normal approximation based method for confidence region, the estimation of parameters with the iterative algorithm is avoided and a consistent estimator of the asymptotic covariance matrix is not needed. Because of bias correction, the BCBEL ratio is asymptotically chi-squared, and hence it can be directly used to construct confidence regions of the parameters without any extra Monte Carlo approximation that is needed when bias correction is not applied. The proposed method can naturally be applied to deal with pure single-index models and partially linear models for longitudinal data. Some simulation studies are carried out and an example in epidemiology is given for illustration.     

Empirical likelihood for linear models under negatively associated errors

In this paper, we discuss the construction of the confidence intervals for the regression vector β in a linear model under negatively associated errors. It is shown that the blockwise empirical likelihood (EL) ratio statistic for β is asymptotically χ²-type distributed. The result is used to obtain an EL based confidence region for β.     

Improving on the maximum likelihood estimators of the means in Poisson decomposable graphical models

In this article we study the simultaneous estimation of the means in Poisson decomposable graphical models. We derive some classes of estimators which improve on the maximum likelihood estimator under the normalized squared losses. Our estimators are based on the argument in Chou [Simultaneous estimation in discrete multivariate exponential families, Ann. Statist. 19 (1991) 314-328.] and shrink the maximum likelihood estimator depending on the marginal frequencies of variables forming a complete subgraph of the conditional independence graph.     

Empirical likelihood for single-index models

The empirical likelihood method is especially useful for constructing confidence intervals or regions of the parameter of interest. This method has been extensively applied to linear regression and generalized linear regression models. In this paper, the empirical likelihood method for single-index regression models is studied. An estimated empirical log-likelihood approach to construct the confidence region of the regression parameter is developed. An adjusted empirical log-likelihood ratio is proved to be asymptotically standard chi-square. A simulation study indicates that compared with a normal approximation-based approach, the proposed method described herein works better in terms of coverage probabilities and areas (lengths) of confidence regions (intervals).     

Empirical likelihood analysis of the Buckley-James estimator

The censored linear regression model, also referred to as the accelerated failure time (AFT) model when the logarithm of the survival time is used as the response variable, is widely seen as an alternative to the popular Cox model when the assumption of proportional hazards is questionable. Buckley and James [Linear regression with censored data, Biometrika 66 (1979) 429-436] extended the least squares estimator to the semiparametric censored linear regression model in which the error distribution is completely unspecified. The Buckley-James estimator performs well in many simulation studies and examples. The direct interpretation of the AFT model is also more attractive than the Cox model, as Cox has pointed out, in practical situations. However, the application of the Buckley-James estimation was limited in practice mainly due to its illusive variance. In this paper, we use the empirical likelihood method to derive a new test and confidence interval based on the Buckley-James estimator of the regression coefficient. A standard chi-square distribution is used to calculate the P-value and the confidence interval. The proposed empirical likelihood method does not involve variance estimation. It also shows much better small sample performance than some existing methods in our simulation studies.     

Empirical likelihood for the parametric part in partially linear errors-in-function models

Partially linear errors-in-function models were proposed by Liang (2000), but their inferences have not been systematically studied. This article proposes an empirical likelihood method to construct confidence regions of the parametric components. Under mild regularity conditions, the nonparametric version of the Wilk’s theorem is derived. Simulation studies show that the proposed empirical likelihood method provides narrower confidence regions, as well as higher coverage probabilities than those based on the traditional normal approximation method.     

Empirical likelihood for semiparametric regression model with missing response data

A bias-corrected technique for constructing the empirical likelihood ratio is used to study a semiparametric regression model with missing response data. We are interested in inference for the regression coefficients, the baseline function and the response mean. A class of empirical likelihood ratio functions for the parameters of interest is defined so that undersmoothing for estimating the baseline function is avoided. The existing data-driven algorithm is also valid for selecting an optimal bandwidth. Our approach is to directly calibrate the empirical log-likelihood ratio so that the resulting ratio is asymptotically chi-squared. Also, a class of estimators for the parameters of interest is constructed, their asymptotic distributions are obtained, and consistent estimators of asymptotic bias and variance are provided. Our results can be used to construct confidence intervals and bands for the parameters of interest. A simulation study is undertaken to compare the empirical likelihood with the normal approximation-based method in terms of coverage accuracies and average lengths of confidence intervals. An example for an AIDS clinical trial data set is used for illustrating our methods.     

Estimation in generalized linear models for functional data via penalized likelihood

We analyze in a regression setting the link between a scalar response and a functional predictor by means of a Functional Generalized Linear Model. We first give a theoretical framework and then discuss identifiability of the model. The functional coefficient of the model is estimated via penalized likelihood with spline approximation. The L² rate of convergence of this estimator is given under smoothness assumption on the functional coefficient. Heuristic arguments show how these rates may be improved for some particular frameworks.     

Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin-Watanabe

Summary The asymptotic expansions of the probability distributions of statistics for the small diffusion are derived by means of the Malliavin calculus. From this the second order efficiency of the maximum likelihood estimator is proved.The research was supported in part by Grant-in-Aid for Encouragement of Young Scientists from the Ministry of Education, Science and Culture     

Depth estimators and tests based on the likelihood principle with application to regression

We investigate depth notions for general models which are derived via the likelihood principle. We show that the so-called likelihood depth for regression in generalized linear models coincides with the regression depth of Rousseeuw and Hubert (J. Amer. Statist. Assoc. 94 (1999) 388) if the dependent observations are appropriately transformed. For deriving tests, the likelihood depth is extended to simplicial likelihood depth. The simplicial likelihood depth is always a U-statistic which is in some cases not degenerated. Since the U-statistic is degenerated in the most cases, we demonstrate that nevertheless the asymptotic distribution of the simplicial likelihood depth and thus asymptotic α-level tests for general types of hypotheses can be derived. The tests are distribution-free. We work out the method for linear and quadratic regression.     

Estimators of scale parameters in linear regression

This note discusses the asymptotic distribution of two scale and location invariant estimators of two scale parameters in the multiple linear regression model. Both of these estimators need an initial estimator of the regression parameter vector. The asymptotic distribution of one of these estimators does not depend on this initial estimator. Both of these estimators are useful in the computation of scale and translation invariant adaptive estimators and M-estimators of the regression parameter vector.     

Empirical likelihood for linear models under m -dependent errors

In this paper, the empirical likelihood confidence regions for the regression coefficient in a linear model are constructed under m-dependent errors. It is shown that the blockwise empirical likelihood is a good way to deal with dependent samples. Partly supported by the National Natural Science Foundation of China and the SF of Guangxi Normal University.     

Empirical likelihood for LAD estimators in infinite variance ARMA models

In this paper, we use an empirical likelihood method to construct confidence regions for the stationary ARMA(p,q) models with infinite variance. An empirical log-likelihood ratio is derived by the estimating equation of the self-weighted LAD estimator. It is proved that the proposed statistic has an asymptotic standard chi-squared distribution. Simulation studies show that in a small sample case, the performance of empirical likelihood method is better than that of normal approximation of the LAD estimator in terms of the coverage accuracy.     

Maximum likelihood estimator consistency for a ballistic random walk in a parametric random environment

We consider a one dimensional ballistic random walk evolving in an i.i.d. parametric random environment. We provide a maximum likelihood estimation procedure of the parameters based on a single observation of the path till the time it reaches a distant site, and prove that the estimator is consistent as the distant site tends to infinity. Our main tool consists in using the link between random walks and branching processes in random environments and explicitly characterising the limiting distribution of the process that arises. We also explore the numerical performance of our estimation procedure.     

Selecting mixed-effects models based on a generalized information criterion

The generalized information criterion (GIC) proposed by Rao and Wu [A strongly consistent procedure for model selection in a regression problem, Biometrika 76 (1989) 369-374] is a generalization of Akaike's information criterion (AIC) and the Bayesian information criterion (BIC). In this paper, we extend the GIC to select linear mixed-effects models that are widely applied in analyzing longitudinal data. The procedure for selecting fixed effects and random effects based on the extended GIC is provided. The asymptotic behavior of the extended GIC method for selecting fixed effects is studied. We prove that, under mild conditions, the selection procedure is asymptotically loss efficient regardless of the existence of a true model and consistent if a true model exists. A simulation study is carried out to empirically evaluate the performance of the extended GIC procedure. The results from the simulation show that if the signal-to-noise ratio is moderate or high, the percentages of choosing the correct fixed effects by the GIC procedure are close to one for finite samples, while the procedure performs relatively poorly when it is used to select random effects.