Empirical likelihood inference for a common mean in the presence of heteroscedasticity

The authors develop empirical likelihood (EL) based methods of inference for a common mean using data from several independent but nonhomogeneous populations. For point estimation, they propose a maximum empirical likelihood (MEL) estimator and show that it is n‐consistent and asymptotically optimal. For confidence intervals, they consider two EL based methods and show that both intervals have approximately correct coverage probabilities under large samples. Finite‐sample performances of the MEL estimator and the EL based confidence intervals are evaluated through a simulation study. The results indicate that overall the MEL estimator and the weighted EL confidence interval are superior alternatives to the existing methods.     

Pseudo‐empirical likelihood ratio confidence intervals for complex surveys

The authors show how an adjusted pseudo‐empirical likelihood ratio statistic that is asymptotically distributed as a chi‐square random variable can be used to construct confidence intervals for a finite population mean or a finite population distribution function from complex survey samples. They consider both non‐stratified and stratified sampling designs, with or without auxiliary information. They examine the behaviour of estimates of the mean and the distribution function at specific points using simulations calling on the Rao‐Sampford method of unequal probability sampling without replacement. They conclude that the pseudo‐empirical likelihood ratio confidence intervals are superior to those based on the normal approximation, whether in terms of coverage probability, tail error rates or average length of the intervals.     

Weighted empirical likelihood inference for multiple samples

We propose a weighted empirical likelihood approach to inference with multiple samples, including stratified sampling, the estimation of a common mean using several independent and non-homogeneous samples and inference on a particular population using other related samples. The weighting scheme and the basic result are motivated and established under stratified sampling. We show that the proposed method can ideally be applied to the common mean problem and problems with related samples. The proposed weighted approach not only provides a unified framework for inference with multiple samples, including two-sample problems, but also facilitates asymptotic derivations and computational methods. A bootstrap procedure is also proposed in conjunction with the weighted approach to provide better coverage probabilities for the weighted empirical likelihood ratio confidence intervals. Simulation studies show that the weighted empirical likelihood confidence intervals perform better than existing ones.     

Symmetric maximum kernel likelihood estimation

We introduce an estimator for the population mean based on maximizing likelihoods formed from a symmetric kernel density estimate. Due to these origins, we have dubbed the estimator the symmetric maximum kernel likelihood estimate (smkle). A speedy computational method to compute the smkle based on binning is implemented in a simulation study which shows that the smkle at an optimal bandwidth is decidedly superior in terms of efficiency to the sample mean and other measures of location for heavy-tailed symmetric distributions. An empirical rule and a computational method to estimate this optimal bandwidth are developed and used to construct bootstrap confidence intervals for the population mean. We show that the intervals have approximately nominal coverage and have significantly smaller average width than the corresponding intervals for other measures of location.     

Empirical Likelihood Intervals for Conditional Value‐at‐Risk in Heteroscedastic Regression Models

Abstract. Non‐parametric regression models have been studied well including estimating the conditional mean function, the conditional variance function and the distribution function of errors. In addition, empirical likelihood methods have been proposed to construct confidence intervals for the conditional mean and variance. Motivated by applications in risk management, we propose an empirical likelihood method for constructing a confidence interval for the pth conditional value‐at‐risk based on the non‐parametric regression model. A simulation study shows the advantages of the proposed method.     

Some Further Issues Concerning Likelihood Inference for Left Truncated and Right Censored Lognormal Data

The maximum likelihood estimates (MLEs) of the parameters of a two-parameter lognormal distribution with left truncation and right censoring are developed through the Expectation Maximization (EM) algorithm. For comparative purpose, the MLEs are also obtained by the Newton–Raphson method. The asymptotic variance-covariance matrix of the MLEs is obtained by using the missing information principle, under the EM framework. Then, using asymptotic normality of the MLEs, asymptotic confidence intervals for the parameters are constructed. Asymptotic confidence intervals are also obtained using the estimated variance of the MLEs by the observed information matrix, and by using parametric bootstrap technique. Different confidence intervals are then compared in terms of coverage probabilities, through a Monte Carlo simulation study. A prediction problem concerning the future lifetime of a right censored unit is also considered. A numerical example is given to illustrate all the inferential methods developed here.     

Semi-empirical likelihood inference for the ROC curve with missing data

The receiver operating characteristic (ROC) curve is one of the most commonly used methods to compare the diagnostic performance of two or more laboratory or diagnostic tests. In this paper, we propose semi-empirical likelihood based confidence intervals for ROC curves of two populations, where one population is parametric and the other one is non-parametric and both have missing data. After imputing missing values, we derive the semi-empirical likelihood ratio statistic and the corresponding likelihood equations. It is shown that the log-semi-empirical likelihood ratio statistic is asymptotically scaled chi-squared. The estimating equations are solved simultaneously to obtain the estimated lower and upper bounds of semi-empirical likelihood confidence intervals. We conduct extensive simulation studies to evaluate the finite sample performance of the proposed empirical likelihood confidence intervals with various sample sizes and different missing probabilities.     

Confidence intervals for the closed population size under inverse sampling without replacement

Inverse sampling is an appropriate design for the second phase of capture-recapture experiments which provides an exactly unbiased estimator of the population size. However, the sampling distribution of the resulting estimator tends to be highly right skewed for small recapture samples, so, the traditional Wald-type confidence intervals appear to be inappropriate. The objective of this paper is to study the performance of interval estimators for the population size under inverse recapture sampling without replacement. To this aim, we consider the Wald-type, the logarithmic transformation-based, the Wilson score, the likelihood ratio and the exact methods. Also, we propose some bootstrap confidence intervals for the population size, including the with-replacement bootstrap (BWR), the without replacement bootstrap (BWO), and the Rao–Wu’s rescaling method. A Monte Carlo simulation is employed to evaluate the performance of suggested methods in terms of the coverage probability, error rates and standardized average length. Our results show that the likelihood ratio and exact confidence intervals are preferred to other competitors, having the coverage probabilities close to the desired nominal level for any sample size, with more balanced error rate for exact method and shorter length for likelihood ratio method. It is notable that the BWO and Rao–Wu’s rescaling methods also may provide good intervals for some situations, however, those coverage probabilities are not invariant with respect to the population arguments, so one must be careful to use them.     

Confidence intervals for the mean of a population containing many zero values under unequal‐probability sampling

In many applications, a finite population contains a large proportion of zero values that make the population distribution severely skewed. An unequal‐probability sampling plan compounds the problem, and as a result the normal approximation to the distribution of various estimators has poor precision. The central‐limit‐theorem‐based confidence intervals for the population mean are hence unsatisfactory. Complex designs also make it hard to pin down useful likelihood functions, hence a direct likelihood approach is not an option. In this paper, we propose a pseudo‐likelihood approach. The proposed pseudo‐log‐likelihood function is an unbiased estimator of the log‐likelihood function when the entire population is sampled. Simulations have been carried out. When the inclusion probabilities are related to the unit values, the pseudo‐likelihood intervals are superior to existing methods in terms of the coverage probability, the balance of non‐coverage rates on the lower and upper sides, and the interval length. An application with a data set from the Canadian Labour Force Survey‐2000 also shows that the pseudo‐likelihood method performs more appropriately than other methods. The Canadian Journal of Statistics 38: 582–597; 2010 © 2010 Statistical Society of Canada     

ON THE ANALYSIS AND APPLICATION OF MEASURES OF LINKAGE DISEQUILIBRIUM

The maximum likelihood, jackknife and bootstrap estimators of linkage disequilibrium, a measure of association in population genetics, are derived and compared. It is found that for point estimation, the resampling methods generate almost identical mean square errors. The maximum likelihood estimator could have bigger or smaller mean square errors depending on the parameters of the underlying population. However the bootstrap confidence interval is superior to the other two as the length of the intervals is shorter or the probability that the 95% confidence intervals include the true parameter is closer to 0.95. Although the standardised measure of linkage disequilibrium has a range from -1 to 1 regardless of marginal frequencies, it is shown that the distribution of this standardised measure is still not allele frequency independent under the multinomial sampling scheme.     

Confidence intervals for estimating the population signal-to-noise ratio: a simulation study

This paper considered several confidence intervals for estimating the population signal-to-noise ratio based on parametric, non-parametric and modified methods. A simulation study has been conducted to compare the performance of the interval estimators under both symmetric and skewed distributions. We reported coverage probability and average width of the interval estimators. Based on the simulation study, we observed that some of our proposed interval estimators are performing better in the sense of smaller width and coverage probability and have been recommended for the researchers.     

New bootstrap confidence intervals for means of positively skewed distributions

ABSTRACT

In this paper, we consider the problem of constructing non parametric confidence intervals for the mean of a positively skewed distribution. We suggest calibrated, smoothed bootstrap upper and lower percentile confidence intervals. For the theoretical properties, we show that the proposed one-sided confidence intervals have coverage probability α + O(n^{? 3/2}). This is an improvement upon the traditional bootstrap confidence intervals in terms of coverage probability. A version smoothed approach is also considered for constructing a two-sided confidence interval and its theoretical properties are also studied. A simulation study is performed to illustrate the performance of our confidence interval methods. We then apply the methods to a real data set.     

Bootstrap confidence intervals for smoothing splines and their comparison to bayesian confidence intervals

We construct bootstrap confidence intervals for smoothing spline estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from .exponential families. Several vari- ations of bootstrap confidence intervals are considered and compared. We find that the commonly used ootstrap percentile intervals are inferior to the T intervals and to intervals based on bootstrap estimation of mean squared errors. The best variations of the bootstrap confidence intervals behave similar to the well known Bayesian confidence intervals. These bootstrap confidence intervals have an average coverage probability across the function being estimated, as opposed to a pointwise property.     

Empirical likelihood for generalized partially linear varying-coefficient models

Generalized partially linear varying-coefficient models (GPLVCM) are frequently used in statistical modeling. However, the statistical inference of the GPLVCM, such as confidence region/interval construction, has not been very well developed. In this article, empirical likelihood-based inference for the parametric components in the GPLVCM is investigated. Based on the local linear estimators of the GPLVCM, an estimated empirical likelihood-based statistic is proposed. We show that the resulting statistic is asymptotically non-standard chi-squared. By the proposed empirical likelihood method, the confidence regions for the parametric components are constructed. In addition, when some components of the parameter are of particular interest, the construction of their confidence intervals is also considered. A simulation study is undertaken to compare the empirical likelihood and the other existing methods in terms of coverage accuracies and average lengths. The proposed method is applied to a real example.     

Empirical Likelihood Local Polynomial Regression Analysis of Clustered Data

Abstract. In this article, a naive empirical likelihood ratio is constructed for a non‐parametric regression model with clustered data, by combining the empirical likelihood method and local polynomial fitting. The maximum empirical likelihood estimates for the regression functions and their derivatives are obtained. The asymptotic distributions for the proposed ratio and estimators are established. A bias‐corrected empirical likelihood approach to inference for the parameters of interest is developed, and the residual‐adjusted empirical log‐likelihood ratio is shown to be asymptotically chi‐squared. These results can be used to construct a class of approximate pointwise confidence intervals and simultaneous bands for the regression functions and their derivatives. Owing to our bias correction for the empirical likelihood ratio, the accuracy of the obtained confidence region is not only improved, but also a data‐driven algorithm can be used for selecting an optimal bandwidth to estimate the regression functions and their derivatives. A simulation study is conducted to compare the empirical likelihood method with the normal approximation‐based method in terms of coverage accuracies and average widths of the confidence intervals/bands. An application of this method is illustrated using a real data set.     

Statistical Inference in Partially Linear Varying-Coefficient Models with Missing Responses at Random

This article considers statistical inference for partially linear varying-coefficient models when the responses are missing at random. We propose a profile least-squares estimator for the parametric component with complete-case data and show that the resulting estimator is asymptotically normal. To avoid to estimate the asymptotic covariance in establishing confidence region of the parametric component with the normal-approximation method, we define an empirical likelihood based statistic and show that its limiting distribution is chi-squared distribution. Then, the confidence regions of the parametric component with asymptotically correct coverage probabilities can be constructed by the result. To check the validity of the linear constraints on the parametric component, we construct a modified generalized likelihood ratio test statistic and demonstrate that it follows asymptotically chi-squared distribution under the null hypothesis. Then, we extend the generalized likelihood ratio technique to the context of missing data. Finally, some simulations are conducted to illustrate the proposed methods.     

Constrained estimation and likelihood intervals for censored data

The authors propose a reduction technique and versions of the EM algorithm and the vertex exchange method to perform constrained nonparametric maximum likelihood estimation of the cumulative distribution function given interval censored data. The constrained vertex exchange method can be used in practice to produce likelihood intervals for the cumulative distribution function. In particular, the authors show how to produce a confidence interval with known asymptotic coverage for the survival function given current status data.     

On the simple step-stress model for two-parameter exponential distribution

In this paper, we consider the simple step-stress model for a two-parameter exponential distribution, when both the parameters are unknown and the data are Type-II censored. It is assumed that under two different stress levels, the scale parameter only changes but the location parameter remains unchanged. It is observed that the maximum likelihood estimators do not always exist. We obtain the maximum likelihood estimates of the unknown parameters whenever they exist. We provide the exact conditional distributions of the maximum likelihood estimators of the scale parameters. Since the construction of the exact confidence intervals is very difficult from the conditional distributions, we propose to use the observed Fisher Information matrix for this purpose. We have suggested to use the bootstrap method for constructing confidence intervals. Bayes estimates and associated credible intervals are obtained using the importance sampling technique. Extensive simulations are performed to compare the performances of the different confidence and credible intervals in terms of their coverage percentages and average lengths. The performances of the bootstrap confidence intervals are quite satisfactory even for small sample sizes.     

Efficient mean estimation in log-normal linear models

Log-normal linear models are widely used in applications, and many times it is of interest to predict the response variable or to estimate the mean of the response variable at the original scale for a new set of covariate values. In this paper we consider the problem of efficient estimation of the conditional mean of the response variable at the original scale for log-normal linear models. Several existing estimators are reviewed first, including the maximum likelihood (ML) estimator, the restricted ML (REML) estimator, the uniformly minimum variance unbiased (UMVU) estimator, and a bias-corrected REML estimator. We then propose two estimators that minimize the asymptotic mean squared error and the asymptotic bias, respectively. A parametric bootstrap procedure is also described to obtain confidence intervals for the proposed estimators. Both the new estimators and the bootstrap procedure are very easy to implement. Comparisons of the estimators using simulation studies suggest that our estimators perform better than the existing ones, and the bootstrap procedure yields confidence intervals with good coverage properties. A real application of estimating the mean sediment discharge is used to illustrate the methodology.     

Interval estimation via tail functions

The authors describe a new method for constructing confidence intervals. Their idea consists in specifying the cutoff points in terms of a function of the target parameter rather than as constants. When it is suitably chosen, this so‐called tail function yields shorter confidence intervals in the presence of prior information. It can also be used to improve the coverage properties of approximate confidence intervals. The authors illustrate their technique by application to interval estimation of the mean of Bernoulli and normal populations. They further suggest guidelines for choosing the optimal tail function and discuss the relationship with Bayesian inference.