A consistent test via the partial penalized empirical likelihood approach for the parametric hypothesis testing under the sparse case, called the partial penalized empirical likelihood ratio(PPELR) test, is proposed in this paper. Our results are demonstrated for the mean vector in multivariate analysis and regression coefficients in linear models, respectively. And we establish its asymptotic distributions under the null hypothesis and the local alternatives of order n^(-1/2) under regularity conditions. Meanwhile, the oracle property of the partial penalized empirical likelihood estimator also holds. The proposed PPELR test statistic performs as well as the ordinary empirical likelihood ratio test statistic and outperforms the full penalized empirical likelihood ratio test statistic in term of size and power when the null parameter is zero. Moreover, the proposed method obtains the variable selection as well as the p-values of testing. Numerical simulations and an analysis of Prostate Cancer data confirm our theoretical findings and demonstrate the promising performance of the proposed method in hypothesis testing and variable selection.
The loglinear model under product-multinomial sampling with constraints is considered. The asymptotic expansion and normality of the restricted minimum φ-divergence estimator (RMφDE) which is a generalization of the maximum likelihood estimator is presented. Then various statistics based on φ-divergence and RMφDE are used to test various hypothesis test problems under the model considered. These statistics contain the classical loglikelihood ratio test statistics and Pearson chi-squared test statistics. In the last section, a simulation study is implemented.
For complete observation and p-dimensional parameterθdefined by an estimation equation,empirical likelihood method of construction of confidence region is based on the asymptoticχ2pdistribution of-2 log(EL ratio).For right censored lifetime data with covariables,however,it is shown in literature that-2 log(EL ratio)converges weakly to a scaledχ2pdistribution,where the scale parameter is a function of unknown asymptotic covariance matrix.The construction of confidence region requires estimation of this scale parameter.In this paper,by using influence functions in the estimating equation,we show that-2 log(EL ratio)converges weakly to a standardχ2pdistribution and hence eliminates the procedure of estimating the scale parameter.
We consider the problem of parameter estimation in both linear and nonlinear ordinary differential equation(ODE) models. Nonlinear ODE models are widely used in applications. But their analytic solutions are usually not available. Thus regular methods usually depend on repetitive use of numerical solutions which bring huge computational cost. We proposed a new two-stage approach which includes a smoothing method(kernel smoothing or local polynomial fitting) in the first stage, and a numerical discretization method(Eulers discretization method, the trapezoidal discretization method,or the Runge–Kutta discretization method) in the second stage. Through numerical simulations, we find the proposed method gains a proper balance between estimation accuracy and computational cost.Asymptotic properties are also presented, which show the consistency and asymptotic normality of estimators under some mild conditions. The proposed method is compared to existing methods in term of accuracy and computational cost. The simulation results show that the estimators with local linear smoothing in the first stage and trapezoidal discretization in the second stage have the lowest average relative errors. We apply the proposed method to HIV dynamics data to illustrate the practicability of the estimator.
Case-cohort sampling is a commonly used and efficient method for studying large cohorts.In many situations,some covariates are easily measured on all cohort subjects,and surrogate measurements of the expensive covariates also may be observed.In this paper,to make full use of the covariate data collected outside the case-cohort sample,we propose a class of weighted estimators with general time-varying weights for the additive hazards model,and the estimators are shown to be consistent and asymptotically normal.We also identify the estimator within this class that maximizes efficiency,and simulation studies show that the efficiency gains of the proposed estimator over the existing ones can be substantial in practical situations.A real example is provided.
In this paper, the parameters of a p-dimensional linear structural EV(error-in-variable)model are estimated when the coefficients vary with a real variable and the model error is time series.The adjust weighted least squares(AWLS) method is used to estimate the parameters. It is shown that the estimators are weakly consistent and asymptotically normal, and the optimal convergence rate is also obtained. Simulations study are undertaken to illustrate our AWLSEs have good performance.
In lifetime data analysis,naturally recorded observations are length-biased data if the probability to select an item is proportional to its length.Based on i.i.d.observations of the true distribution,empirical likelihood(EL) procedure is proposed for the inference on mean residual life(MRL) of naturally recorded item.The limit distribution of the EL based log-likelihood ratio is proved to be the chi-square distribution.Under right censorship,since the EL based log-likelihood ratio leads to a scaled chi-square distribution and estimating the scale parameter leads to lower coverage of confidence interval,we propose an algorithm to calculate the likelihood ratio(LR) directly.The corresponding log-likelihood ratio converges to the standard chi-square distribution and the corresponding confidence interval has a better coverage.Simulation studies are used to support the theoretical results.
The maximum entropy method has been widely used in many fields, such as statistical mechanics,economics, etc. Its crucial idea is that when we make inference based on partial information, we must use the distribution with maximum entropy subject to whatever is known. In this paper, we investigate the empirical entropy method for right censored data and use simulation to compare the empirical entropy method with the empirical likelihood method. Simulations indicate that the empirical entropy method gives better coverage probability than that of the empirical likelihood method for contaminated and censored lifetime data.
