Empirical likelihood for mixed-effects error-in-variables model

This paper mainly introduces the method of empirical likelihood and its applications on two different models. We discuss the empirical likelihood inference on fixed-effect parameter in mixed-effects model with error-in-variables. We first consider a linear mixed-effects model with measurement errors in both fixed and random effects. We construct the empirical likelihood confidence regions for the fixed-effects parameters and the mean parameters of random-effects. The limiting distribution of the empirical log likelihood ratio at the true parameter is X2p＋q, where p, q are dimension of fixed and random effects respectively. Then we discuss empirical likelihood inference in a semi-linear error-in-variable mixed-effects model. Under certain conditions, it is shown that the empirical log likelihood ratio at the true parameter also converges to X2p＋q. Simulations illustrate that the proposed confidence region has a coverage probability more closer to the nominal level than normal approximation based confidence region.     

Semiparametric inference for transformation models via empirical likelihood

Recent advances in the transformation model have made it possible to use this model for analyzing a variety of censored survival data. For inference on the regression parameters, there are semiparametric procedures based on the normal approximation. However, the accuracy of such procedures can be quite low when the censoring rate is heavy. In this paper, we apply an empirical likelihood ratio method and derive its limiting distribution via U-statistics. We obtain confidence regions for the regression parameters and compare the proposed method with the normal approximation based method in terms of coverage probability. The simulation results demonstrate that the proposed empirical likelihood method overcomes the under-coverage problem substantially and outperforms the normal approximation based method. The proposed method is illustrated with a real data example. Finally, our method can be applied to general U-statistic type estimating equations.     

Empirical likelihood analysis of longitudinal data involving within-subject correlation

In this paper we use profile empirical likelihood to construct confidence regions for regression coefficients in partially linear model with longitudinal data. The main contribution is that the within-subject correlation is considered to improve estimation efficiency. We suppose a semi-parametric structure for the covariances of observation errors in each subject and employ both the first order and the second order moment conditions of the observation errors to construct the estimating equations. Although there are nonparametric estimators, the empirical log-likelihood ratio statistic still tends to a standard ?? _p ² variable in distribution after the nuisance parameters are profiled away. A data simulation is also conducted.     

Empirical Likelihood for a Class of Functionals of Survival Distribution with Censored Data

The empirical likelihood was introduced by Owen, although its idea originated from survival analysis in the context of estimating the survival probabilities given by Thomas and Grunkemeier. In this paper, we investigate how to apply the empirical likelihood method to a class of functionals of survival function in the presence of censoring. We define an adjusted empirical likelihood and show that it follows a chi-square distribution. Some simulation studies are presented to compare the empirical likelihood method with the Studentized-t method. These results indicate that the empirical likelihood method works better than or equally to the Studentized-t method, depending on the situations.     

An Inversion-Free Estimating Equations Approach for Gaussian Process Models

One of the scalability bottlenecks for the large-scale usage of Gaussian processes is the computation of the maximum likelihood estimates of the parameters of the covariance matrix. The classical approach requires a Cholesky factorization of the dense covariance matrix for each optimization iteration. In this work, we present an estimating equations approach for the parameters of zero-mean Gaussian processes. The distinguishing feature of this approach is that no linear system needs to be solved with the covariance matrix. Our approach requires solving an optimization problem for which the main computational expense for the calculation of its objective and gradient is the evaluation of traces of products of the covariance matrix with itself and with its derivatives. For many problems, this is an O(nlog?n) effort, and it is always no larger than O(n²). We prove that when the covariance matrix has a bounded condition number, our approach has the same convergence rate as does maximum likelihood in that the Godambe information matrix of the resulting estimator is at least as large as a fixed fraction of the Fisher information matrix. We demonstrate the effectiveness of the proposed approach on two synthetic examples, one of which involves more than 1 million data points.     

Empirical likelihood inference for estimating equation with missing data

In this article, empirical likelihood inference for estimating equation with missing data is considered. Based on the weighted-corrected estimating function, an empirical log-likelihood ratio is proved to be a standard chi-square distribution asymptotically under some suitable conditions. This result is different from those derived before. So it is convenient to construct confidence regions for the parameters of interest. We also prove that our proposed maximum empirical likelihood estimator θ is asymptotically normal and attains the semiparametric efficiency bound of missing data. Some simulations indicate that the proposed method performs the best.     

Balanced augmented jackknife empirical likelihood for two sample <Emphasis Type="Italic">U</Emphasis>-statistics

In this paper, we investigate the two sample U-statistics by jackknife empirical likelihood (JEL), a versatile nonparametric approach. More precisely, we propose the method of balanced augmented jackknife empirical likelihood (BAJEL) by adding two articial points to the original pseudo-value dataset, and we prove that the log likelihood ratio based on the expanded dataset tends to the χ² distribution.     

Empirical characteristic function approach to goodness-of-fit tests for the Cauchy distribution with parameters estimated by MLE or EISE

We consider goodness-of-fit tests of the Cauchy distribution based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard Cauchy distribution. For standardization of data Gürtler and Henze (2000,Annals of the Institute of Statistical Mathematics,52, 267–286) used the median and the interquartile range. In this paper we use the maximum likelihood estimator (MLE) and an equivariant integrated squared error estimator (EISE), which minimizes the weighted integral. We derive an explicit form of the asymptotic covariance function of the characteristic function process with parameters estimated by the MLE or the EISE. The eigenvalues of the covariance function are numerically evaluated and the asymptotic distributions of the test statistics are obtained by the residue theorem. A simulation study shows that the proposed tests compare well to tests proposed by Gürtler and Henze and more traditional tests based on the empirical distribution function.     

Partial penalized empirical likelihood ratio test under sparse case

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.     

A test for the presence of stochastic ordering under censoring: the k-sample case

In this paper, we develop an empirical likelihood-based test for the presence of stochastic ordering under censoring in the k-sample case. The proposed test statistic is formed by taking the supremum of localized empirical likelihood ratio test statistics. Its asymptotic null distribution has a simple representation in terms of a standard Brownian motion process. Through simulations, we show that it outperforms in terms of power existing methods for the same problem at all the distributions that we consider. A real-life example is used to illustrate the applicability of this new test.

     

Bayesian estimation ofg-and-k distributions using MCMC

Summary  In this paper we investigate a Bayesian procedure for the estimation of a flexible generalised distribution, notably the MacGillivray adaptation of theg-and-k distribution. This distribution, described through its inverse cdf or quantile function, generalises the standard normal through extra parameters which together describe skewness and kurtosis. The standard quantile-based methods for estimating the parameters of generalised distributions are often arbitrary and do not rely on computation of the likelihood. MCMC, however, provides a simulation-based alternative for obtaining the maximum likelihood estimates of parameters of these distributions or for deriving posterior estimates of the parameters through a Bayesian framework. In this paper we adopt the latter approach. The proposed methodology is illustrated through an application in which the parameter of interest is slightly skewed.     

ML Estimation of the MultivariatetDistribution and the EM Algorithm

Maximum likelihood estimation of the multivariatetdistribution, especially with unknown degrees of freedom, has been an interesting topic in the development of the EM algorithm. After a brief review of the EM algorithm and its application to finding the maximum likelihood estimates of the parameters of thetdistribution, this paper provides new versions of the ECME algorithm for maximum likelihood estimation of the multivariatetdistribution from data with possibly missing values. The results show that the new versions of the ECME algorithm converge faster than the previous procedures. Most important, the idea of this new implementation is quite general and useful for the development of the EM algorithm. Comparisons of different methods based on two datasets are presented.     

Finite-sample inference with monotone incomplete multivariate normal data, I

We consider problems in finite-sample inference with two-step, monotone incomplete data drawn from , a multivariate normal population with mean and covariance matrix . We derive a stochastic representation for the exact distribution of , the maximum likelihood estimator of . We obtain ellipsoidal confidence regions for through T², a generalization of Hotelling’s statistic. We derive the asymptotic distribution of, and probability inequalities for, T² under various assumptions on the sizes of the complete and incomplete samples. Further, we establish an upper bound for the supremum distance between the probability density functions of and , a normal approximation to .     

Goodness-of-fit for a concentrated von Mises-Fisher distribution

The von Mises-Fisher distribution is widely used for modelling directional data. In this paper we propose goodness-of-fit methods for a concentrated von Mises-Fisher distribution and we analyse by simulation some questions concerning the application of these tests. We analyse the empirical power of the Kolmogorov-Smirnov test for several dimensions of the sphere, supposing as alternative hypothesis a mixture of two von Mises-Fisher distributions with known parameters. We also compare the empirical power of the Kolmogorov-Smirnov test with the Rao’s score test for data on the sphere, supposing as alternative hypothesis, a mixture of two Fisher distributions with unknown parameters replaced by their maximum likelihood estimates or a 5-parameter Fisher-Bingham distribution. Finally, we give an example with real spherical data.     

Empirical likelihood for AR-ARCH models based on LAD estimation

By employing the empirical likelihood method,confidence regions for the stationary AR(p)-ARCH(q) models are constructed.A self-weighted LAD estimator is proposed under weak moment conditions.An empirical log-likelihood ratio statistic is derived and its asymptotic distribution is obtained.Simulation studies show that the performance of empirical likelihood method is better than that of normal approximation of the LAD estimator in terms of the coverage accuracy,especially for relative small size of observation.     

A Partial Empirical Likelihood Based Score Test Under a Semiparametric Finite Mixture Model

We propose a score statistic to test the null hypothesis that the two-component density functions are equal under a semiparametric finite mixture model. The proposed score test is based on a partial empirical likelihood function under an I-sample semiparametric model. The proposed score statistic has an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to a partial empirical likelihood ratio test and a Wald test. We present some results on a simulation study.     

弱相依数据中的分组经验Cressie-Read似然方法

本文考虑一般的弱相依数据, 提出了分组经验Cressie-Read似然方法. 得到了分组经验Cressie-Read似然参数估计的强收敛性、渐近正态性和其分组经验Cressie-Read统计量的渐近$\chi^{2}$性.     

Distribution asymptotique de l'affinité L2 de densités gaussiennes

After recalling the L² affinity measure between two multidimensionnal Gaussian density functions, we prove its asymptotic normality when parameters of one density are replaced by their maximum likelihood estimators. We extend this result to the vectorial case and we use it to allocate an estimated density to a class of densities as does the discriminant analysis method. As an application, a dating method for archeological data is proposed.     

The excess-mass ellipsoid

The excess-mass ellipsoid is the ellipsoid that maximizes the difference between its probability content and a constant multiple of its volume, over all ellipsoids. When an empirical distribution determines the probability content, the sample excess-mass ellipsoid is a random set that can be used in contour estimation and tests for multimodality. Algorithms for computing the ellipsoid are provided, as well as comparative simulations. The asymptotic distribution of the parameters for the sample excess-mass ellipsoid are derived. It is found that a n^1/3 normalization of the center of the ellipsoid and lengths of its axes converge in distribution to the maximizer of a Gaussian process with quadratic drift. The generalization of ellipsoids to convex sets is discussed.     

Parameter estimation for von Mises–Fisher distributions

When analyzing high-dimensional data, it is often appropriate to pay attention only to the direction of each datum, disregarding its norm. The von Mises–Fisher (vMF) distribution is a natural probability distribution for such data. When we estimate the parameters of vMF distributions, parameter κ which corresponds to the degree of concentration is difficult to obtain, and some approximations are necessary. In this article, we propose an iterative algorithm using fixed points to obtain the maximum likelihood estimate (m.l.e.) for κ. We prove that there is a unique local maximum for κ. Besides, using a specific function to calculate the m.l.e., we obtain the upper and lower bounds of the interval in which the exact m.l.e. exists. In addition, based on these bounds, a new and good approximation is derived. The results of numerical experiments demonstrate the new approximation exhibits higher precision than traditional ones.