Empirical likelihood-based inference for parameter and nonparametric function in partially nonlinear models

This paper is concerned with statistical inference for partially nonlinear models. Empirical likelihood method for parameter in nonlinear function and nonparametric function is investigated. The empirical log-likelihood ratios are shown to be asymptotically chi-square and then the corresponding confidence intervals are constructed. By the empirical likelihood ratio functions, we also obtain the maximum empirical likelihood estimators of the parameter in nonlinear function and nonparametric function, and prove the asymptotic normality. A simulation study indicates that, compared with normal approximation-based method and the bootstrap method, the empirical likelihood method performs better in terms of coverage probabilities and average length/widths of confidence intervals/bands. An application to a real dataset is illustrated.     

Short notes on maximum likelihood inference for control‐based pattern‐mixture models

A likelihood‐based analytical approach has been proposed for the control‐based pattern‐mixture model and its extension. In this note, we derive equivalent but simpler analytical expressions for the treatment effect and its variance for these control‐based pattern mixture models. Our formulae are easier to use and interpret. An application of our formulae to an antidepressant trial is provided, in which the likelihood‐based analysis is compared with the multiple imputation approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Empirical likelihood‐based inferences for a low income proportion

Low income proportion is an important index in comparisons of poverty in countries around the world. The stability of a society depends heavily on this index. An accurate and reliable estimation of this index plays an important role for government's economic policies. In this paper, the authors study empirical likelihood‐based inferences for a low income proportion under the simple random sampling and stratified random sampling designs. It is shown that the limiting distributions of the empirical likelihood ratios for the low income proportion are the scaled chi‐square distributions. The authors propose various empirical likelihood‐based confidence intervals for the low income proportion. Extensive simulation studies are conducted to evaluate the relative performance of the normal approximation‐based interval, bootstrap‐based intervals, and the empirical likelihood‐based intervals. The proposed methods are also applied to analyzing a real economic survey income dataset. The Canadian Journal of Statistics 39: 1–16; 2011 ©2011 Statistical Society of Canada     

Discretisation for inference on normal mixture models

The problem of inference in Bayesian Normal mixture models is known to be difficult. In particular, direct Bayesian inference (via quadrature) suffers from a combinatorial explosion in having to consider every possible partition of n observations into k mixture components, resulting in a computation time which is O(k ⁿ). This paper explores the use of discretised parameters and shows that for equal-variance mixture models, direct computation time can be reduced to O(D ^k n ^k), where relevant continuous parameters are each divided into D regions. As a consequence, direct inference is now possible on genuine data sets for small k, where the quality of approximation is determined by the level of discretisation. For large problems, where the computational complexity is still too great in O(D ^k n ^k) time, discretisation can provide a convergence diagnostic for a Markov chain Monte Carlo analysis.     

Empirical likelihood inference for error density estimators in first-order autoregression models

In this paper we apply empirical likelihood method to the error density estimators in first-order autoregressive models under some mild conditions. The log-likelihood ratio statistic is shown to be asymptotically chi-squared distributed at a fixed point. In simulation, we show that the empirical likelihood produces confidence intervals having theoretical coverage accuracy which is better than normal approximation.     

Empirical likelihood inference for probability density functions under association

In this paper, we consider to apply the empirical likelihood method to a probability density function under an associated sample. It is shown that the empirical likelihood ratio statistic is asymptotically χ²-type distributed under some mild conditions. The result is used to construct empirical likelihood-based confidence intervals on the probability density function.     

Wavelet‐based estimators for mixture regression

We consider a process that is observed as a mixture of two random distributions, where the mixing probability is an unknown function of time. The setup is built upon a wavelet‐based mixture regression. Two linear wavelet estimators are proposed. Furthermore, we consider three regularizing procedures for each of the two wavelet methods. We also discuss regularity conditions under which the consistency of the wavelet methods is attained and derive rates of convergence for the proposed estimators. A Monte Carlo simulation study is conducted to illustrate the performance of the estimators. Various scenarios for the mixing probability function are used in the simulations, in addition to a range of sample sizes and resolution levels. We apply the proposed methods to a data set consisting of array Comparative Genomic Hybridization from glioblastoma cancer studies.     

Order-restricted hypothesis testing in a variation of the normal mixture model

This paper examines likelihood-ratio tests concerning the relationships among a fixed number of univariate normal means given a sample of normal observations whose population membership is uncertain. The asymptotic null distributions of likelihood-ratio test statistics are derived for a class of tests including hypotheses which place linear inequality constraints on the normal means. The use of such tests in the interval mapping of quantitative trait loci is addressed.     

Empirical likelihood inference for partial functional linear model with missing responses

In this paper, we consider the empirical likelihood inferences of the partial functional linear model with missing responses. Two empirical log-likelihood ratios of the parameters of interest are constructed, and the corresponding maximum empirical likelihood estimators of parameters are derived. Under some regularity conditions, we show that the proposed two empirical log-likelihood ratios are asymptotic standard Chi-squared. Thus, the asymptotic results can be used to construct the confidence intervals/regions for the parameters of interest. We also establish the asymptotic distribution theory of corresponding maximum empirical likelihood estimators. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals. An example of real data is also used to illustrate our proposed methods.     

Empirical likelihood dimension reduction inference for partially non-linear error-in-responses models with validation data

ABSTRACT

In this article, partially non linear models when the response variable is measured with error and explanatory variables are measured exactly are considered. Without specifying any error structure equation, a semiparametric dimension reduction technique is employed. Two estimators of unknown parameter in non linear function are obtained and asymptotic normality is proved. In addition, empirical likelihood method for parameter vector is provided. It is shown that the estimated empirical log-likelihood ratio has asymptotic Chi-square distribution. A simulation study indicates that, compared with normal approximation method, empirical likelihood method performs better in terms of coverage probabilities and average length of the confidence intervals.     

Empirical likelihood inference in partially linear single-index models with endogenous covariates

In this article, we consider how to construct the confidence regions of the unknown parameters for partially linear single-index models with endogenous covariates. To eliminate the influence of the endogenous covariates, an empirical likelihood method is proposed based on instrumental variables. Under some regularly conditions, the asymptotic distribution of the proposed empirical log-likelihood ratio is proved to be a Chi-squared distribution. We investigate the finite-sample performance of the proposed method via simulation studies.     

Empirical likelihood inference for the bivariate survival function under univariate censoring

In this article, we apply the empirical likelihood method to make inference on the bivariate survival function of paired failure times by estimating the survival function of censored time with the Kaplan–Meier estimator. Adjusted empirical likelihood (AEL) confidence intervals for the bivariate survival function are developed. We conduct a simulation study to compare the proposed AEL method with other methods. The simulation study shows the proposed AEL method has better performance than other existing methods. We illustrate the proposed method by analyzing the skin graft data.     

Uniqueness of estimation and identifiability in mixture models

Further properties of the nonparametric maximum-likelihood estimator of a mixing distribution are obtained by exploiting the properties of totally positive kernels. Sufficient conditions for uniqueness of the estimator are given. This result is more general, and the proof is substantially simpler, than given previously. When the component density has support on N points, it is shown that all identifiable mixing distributions have support on no more than N/2 points. Identifiable mixtures are shown to lie on the boundary of the mixture model space. The maximum-likelihood estimate is shown to be unique if the vector of observations lies outside this space.     

A mixture model using likelihood‐based and Bayesian approaches for identifying responders and non‐responders in longitudinal clinical trials

A longitudinal mixture model for classifying patients into responders and non‐responders is established using both likelihood‐based and Bayesian approaches. The model takes into consideration responders in the control group. Therefore, it is especially useful in situations where the placebo response is strong, or in equivalence trials where the drug in development is compared with a standard treatment. Under our model, a treatment shows evidence of being effective if it increases the proportion of responders or increases the response rate among responders in the treated group compared with the control group. Therefore, the model has flexibility to accommodate different situations. The proposed method is illustrated using simulation and a depression clinical trial dataset for the likelihood‐based approach, and the same depression clinical trial dataset for the Bayesian approach. The likelihood‐based and Bayesian approaches generated consistent results for the depression trial data. In both the placebo group and the treated group, patients are classified into two components with distinct response rate. The proportion of responders is shown to be significantly higher in the treated group compared with the control group, suggesting the treatment paroxetine is effective. Copyright © 2014 John Wiley & Sons, Ltd.     

Constrained nonparametric maximum-likelihood estimation for mixture models

A nonparametric mixture model specifies that observations arise from a mixture distribution, ∫ f(x, θ) dG(θ), where the mixing distribution G is completely unspecified. A number of algorithms have been developed to obtain unconstrained maximum-likelihood estimates of G, but none of these algorithms lead to estimates when functional constraints are present. In many cases, there is a natural interest in functional ?(G), such as the mean and variance, of the mixing distribution, and profile likelihoods and confidence intervals for ?(G) are desired. In this paper we develop a penalized generalization of the ISDM algorithm of Kalbfleisch and Lesperance (1992) that can be used to solve the problem of constrained estimation. We also discuss its use in various different applications. Convergence results and numerical examples are given for the generalized ISDM algorithm, and asymptotic results are developed for the likelihood-ratio test statistics in the multinomial case.     

Empirical likelihood inferences for varying coefficient partially nonlinear models

In this article, empirical likelihood inferences for the varying coefficient partially nonlinear models are investigated. An empirical log-likelihood ratio function for the unknown parameter vector in the nonlinear function part and a residual-adjusted empirical log-likelihood ratio function for the nonparametric component are proposed. The corresponding Wilks phenomena are proved and the confidence regions for parametric component and nonparametric component are constructed. Simulation studies indicate that, in terms of coverage probabilities and average areas of the confidence regions, the empirical likelihood method performs better than the normal approximation-based method. Furthermore, a real data set application is also provided to illustrate the proposed empirical likelihood estimation technique.     

The likelihood ratio test for homogeneity in finite mixture models

The authors study the asymptotic behaviour of the likelihood ratio statistic for testing homogeneity in the finite mixture models of a general parametric distribution family. They prove that the limiting distribution of this statistic is the squared supremum of a truncated standard Gaussian process. The autocorrelation function of the Gaussian process is explicitly presented. A re‐sampling procedure is recommended to obtain the asymptotic p‐value. Three kernel functions, normal, binomial and Poisson, are used in a simulation study which illustrates the procedure.     

Likelihood‐based Inference with Missing Data Under Missing‐at‐Random

Likelihood‐based inference with missing data is challenging because the observed log likelihood is often an (intractable) integration over the missing data distribution, which also depends on the unknown parameter. Approximating the integral by Monte Carlo sampling does not necessarily lead to a valid likelihood over the entire parameter space because the Monte Carlo samples are generated from a distribution with a fixed parameter value. We consider approximating the observed log likelihood based on importance sampling. In the proposed method, the dependency of the integral on the parameter is properly reflected through fractional weights. We discuss constructing a confidence interval using the profile likelihood ratio test. A Newton–Raphson algorithm is employed to find the interval end points. Two limited simulation studies show the advantage of the Wilks inference over the Wald inference in terms of power, parameter space conformity and computational efficiency. A real data example on salamander mating shows that our method also works well with high‐dimensional missing data.     

Empirical Likelihood for Linear Models Under Linear Process Errors

In this article, we study the construction of confidence intervals for regression parameters in a linear model under linear process errors by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically χ²-type distributed. The result is used to obtain EL based confidence regions for regression parameters. The finite-sample performance of the method is evaluated through a simulation study.