The likelihood ratio criterion and the asymptotic expansion of its distribution

Summary Asymptotic expansion of the distribution of the likelihood ratio criterion (LRC) for testing a composite hypothesis is derived under null hypothesis and a correction factor ρ which makes the term of order 1/n in the asymptotic expansion of the distribution of it vanish is obtained. The problem is extended to the case of a general composite hypothesis and of Pitman's local alternatives. The asymptotic distribution of LRC for a simple hypothesis is studied under a fixed alternative. The Institute of Statistical Mathematics     

On tests for the mean direction of the Langevin distribution

The asymptotic expansions of the distribution of a sum of independent random vectors with Langevin distribution are given. The power functions of the likelihood ratio criterion, Watson statistic, Rao statistic and the modified Wald statistic for testing the hypothesis of the mean direction are obtained asymptotically and a numerical comparison is made.     

生长曲线模型中的球性检验问题

本文给出了生长曲线模型中球性检验似然比准则在原假设相接近的两类备择假设下的非零渐近分布。     

Likelihood ratio tests for goodness-of-fit of a nonlinear regression model

We propose likelihood and restricted likelihood ratio tests for goodness-of-fit of nonlinear regression. The first-order Taylor approximation around the MLE of the regression parameters is used to approximate the null hypothesis and the alternative is modeled nonparametrically using penalized splines. The exact finite sample distribution of the test statistics is obtained for the linear model approximation and can be easily simulated. We recommend using the restricted likelihood instead of the likelihood ratio test because restricted maximum-likelihood estimates are not as severely biased as the maximum-likelihood estimates in the penalized splines framework.     

Large deviation theorems in the problem of testing two simple hypotheses

Limit theorems on large deviations of the logarithm of the likelihood ratio are proved for the problem of distinguishing two simple hypotheses in the general scheme of statistical experiments under the null hypothesis and under an alternative hypothesis. The theorems obtained are applied to the investigation of a decrease in the probability of errors of the first and second kind for the Neumann-Pearson criterion.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 227–235, February, 1995.     

Propriétés asymptotiques d'un estimateur dans un modèle non linéaire à effets mixtes

We propose an estimator of nonlinear mixed effects model's parameters, obtained by maximization of simulated pseudo likelihood. This simulated criterion is constructed from the likelihood of a Gaussian model whose means and variances are given by Monte Carlo approximations of means and variances of the true model. If the number of experimental units and the sample size of Monte Carlo simulations are respectively-denoted by N and K, we obtain the strong consistency and asymptotic normality of the estimator when N tends to infinity and the ratio √N/K tends to zero.     

Empirical likelihood inference for linear transformation models

Empirical likelihood inference is developed for censored survival data under the linear transformation models, which generalize Cox's [Regression models and life tables (with Discussion), J. Roy. Statist. Soc. Ser. B 34 (1972) 187-220] proportional hazards model. We show that the limiting distribution of the empirical likelihood ratio is a weighted sum of standard chi-squared distribution. Empirical likelihood ratio tests for the regression parameters with and without covariate adjustments are also derived. Simulation studies suggest that the empirical likelihood ratio tests are more accurate (under the null hypothesis) and powerful (under the alternative hypothesis) than the normal approximation based tests of Chen et al. [Semiparametric of transformation models with censored data, Biometrika 89 (2002) 659-668] when the model is different from the proportional hazards model and the proportion of censoring is high.     

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.     

A note on the simple structural regression model

In this paper we investigate some aspects like estimation and hypothesis testing in the simple structural regression model with measurement errors. Use is made of orthogonal parametrizations obtained in the literature. Emphasis is placed on some properties of the maximum likelihood estimators and also on the distribution of the likelihood ratio statistics.     

Asymptotic tests of composite hypotheses for non-ergodic type stochastic processes

Limiting distributions of a score statistic and the likelihood ratio statistic for testing a composite hypothesis involving several parameters in non-ergodic type stochastic processes are obtained. It is shown that, unlike in the usual theory (ergodic type processes), the limiting distributions of these statistics are different both under the null and a contiguous sequence of alternative hypotheses. The results are applied to a regression model with explosive autoregressive Gaussian errors. In the discussion of this example a modified score statistic is suggested where the limiting null and non-null distributions are the same as those of the likelihood ratio statistic.     

Likelihood-based Inference for the Ratios of Regression Coefficients in Linear Models

We consider the standard linear multiple regression model in which the parameter of interest is the ratio of two regression coefficients. Our setup includes a broad range of applications. We show that the 1− α confidence interval for the interest parameter based on the profile, conditional profile, modified profile or adjusted profile likelihood can potentially become the entire real line, while appropriately chosen integrated likelihoods do not suffer from this drawback. We further explore the asymptotic length of confidence intervals in order to compare integrated likelihood-based proposals. The analysis is facilitated by an orthogonal parameterization.     

Partial sum process to check regression models with multiple correlated response: With an application for testing a change-point in profile data

We consider regression models with multiple correlated responses for each design point. Under the null hypothesis, a linear regression is assumed. For the least-squares residuals of this linear regression, we establish the limit of the partial sums. This limit is a projection on a certain subspace of the reproducing Kernel Hilbert space of a multivariate Brownian motion. Based on this limit, we propose a significance test of Kolmogorov-Smirnov type to test the null hypothesis and show that this result can be used to study a change-point problem in the case of linear profile data (panel data). We compare our proposed method, which does not rely on any distributional assumptions, with the likelihood ratio test in a simulation study.     

正态条件下带AR(1)-型方差结构GMANOVA-MANOVA模型极大似然估计的小样本特征

研究了一类带一阶自回归(AR(1))-型方差结构的广义多元方差分析-多元方差分析(GMANO VA-MANOVA)模型参数极大似然估计的小样本特征.对带AR(1)-型方差结构GMANOVA-MANOVA模型,文章在正态条件下给出了参数极大似然估计存在的一个充分必要条件,讨论了极大似然估计唯一的充分条件.在该充分条件下,文章证明了相关系数极大似然估计的精确分布只与相关系数有关,并依此给出了自相关系数简单假设H0:ρ=0v.s.H1:ρ≠0的一个不需要叠代计算估计的检验,同时模拟表明该检验为无偏检验且势函数与似然比检验势函数无太大差异.     

Empirical likelihood-based subset selection for partially linear autoregressive models

Based on the empirical likelihood method, the subset selection and hypothesis test for parameters in a partially linear autoregressive model are investigated. We show that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. We then present the definitions of the empirical likelihood-based Bayes information criteria （EBIC） and Akaike information criteria （EAIC）. The results show that EBIC is consistent at selecting subset variables while EAIC is not. Simulation studies demonstrate that the proposed empirical likelihood confidence regions have better coverage probabilities than the least square method, while EBIC has a higher chance to select the true model than EAIC.     

A Wald-type variance estimation for the nonparametric distribution estimators for doubly censored data

We discuss the variance estimation for the nonparametric distribution estimator for doubly censored data. We first provide another view of Kuhn–Tucker’s conditions to construct the profile likelihood, and lead a Newton–Raphson algorithm as an optimization technique unlike the EM algorithm. The main proposal is an iteration-free Wald-type variance estimate based on the chain rule of differentiating conditions to construct the profile likelihood, which generalizes the variance formula in only right- or left-censored data. In this estimation procedure, we overcome some difficulties caused in directly applying Turnbull’s formula to large samples and avoid a load with computationally heavy iterations, such as solving the Fredholm equations, computing the profile likelihood ratio or using the bootstrap. Also, we establish the consistency of the formulated Wald-type variance estimator. In addition, simulation studies are performed to investigate the properties of the Wald-type variance estimates in finite samples in comparison with those from the profile likelihood ratio.     

Jackknife-blockwise empirical likelihood methods under dependence

Empirical likelihood for general estimating equations is a method for testing hypothesis or constructing confidence regions on parameters of interest. If the number of parameters of interest is smaller than that of estimating equations, a profile empirical likelihood has to be employed. In case of dependent data, a profile blockwise empirical likelihood method can be used. However, if too many nuisance parameters are involved, a computational difficulty in optimizing the profile empirical likelihood arises. Recently, Li et al. (2011) [9] proposed a jackknife empirical likelihood method to reduce the computation in the profile empirical likelihood methods for independent data. In this paper, we propose a jackknife-blockwise empirical likelihood method to overcome the computational burden in the profile blockwise empirical likelihood method for weakly dependent data.     

一类相合的分位点型检验

广义非参数似然比检验统计量是一类很广的统计量,包含了众多重要的检验统计量,如Anderson-Darling(AD)等.利用Rubin的随机经验分布函数替代经验分布函数的方法,得到了广义非参数似然比检验统计量的新版本,构造了新的检验统计量.由于新的检验统计量在给定样本下仍然是随机变量,选择了它的分位点和期望作为检验统计量,分别称之为分位点型检验统计量和期望型检验统计量.在简单假设情况下,证明了分位点型检验统计量和期望型检验统计量在固定备择下的相合性.模拟结果显示,在某些备择下,新的检验的功效明显高于原有的基于经验分布函数的检验的功效.     

Large deviations for M-estimators

We study the large deviation principle for M-estimators (and maximum likelihood estimators in particular). We obtain the rate function of the large deviation principle for M-estimators. For exponential families, this rate function agrees with the Kullback–Leibler information number. However, for location or scale families this rate function is smaller than the Kullback–Leibler information number. We apply our results to obtain confidence regions of minimum size whose coverage probability converges to one exponentially. In the case of full exponential families, the constructed confidence regions agree with the ones obtained by inverting the likelihood ratio test with a simple null hypothesis.     

A test for additional information in canonical correlation analysis

Summary In canonical correlation analysis a hypothesis concerning the relevance of a subset of variables from each of the two given variable sets is formulated. The likelihood ratio statistic for the hypothesis and an asymptotic expansion for its null distribution are obtained. In discriminant analysis various alternative forms of a hypothesis concerning the relevance of a specified variable subset are also discussed.     

A modification of profile empirical likelihood for the exponential-tilt model

We consider semiparametric models whose infinite-dimensional parameter corresponds to a probability distribution. The NPMLE based on the profile empirical likelihood for this kind of semiparametric model has attracted considerable interest. We propose the use of a modified profile empirical likelihood to improve the accuracy of this estimation. We consider applications to the exponential-tilt model and show that the accuracy of the proposed estimator is better than that of the conventional NPMLE by numerical study.