Empirical likelihood intervals for the population mean and quantiles based on balanced ranked set samples

We consider nonparametric interval estimation for the population mean and quantiles based on a ranked set sample. The asymptotic distributions of the empirical log likelihood ratio statistic for the mean and quantiles are derived. Interval estimates of the population mean and quantiles are obtained by inverting the likelihood ratio statistic. Simulations are carried out to investigate and compare the performance of the empirical likelihood intervals with other known intervals.     

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 Current Status Data

Abstract.  The likelihood ratio statistic for testing pointwise hypotheses about the survival time distribution in the current status model can be inverted to yield confidence intervals (CIs). One advantage of this procedure is that CIs can be formed without estimating the unknown parameters that figure in the asymptotic distribution of the maximum likelihood estimator (MLE) of the distribution function. We discuss the likelihood ratio-based CIs for the distribution function and the quantile function and compare these intervals to several different intervals based on the MLE. The quantiles of the limiting distribution of the MLE are estimated using various methods including parametric fitting, kernel smoothing and subsampling techniques. Comparisons are carried out both for simulated data and on a data set involving time to immunization against rubella. The comparisons indicate that the likelihood ratio-based intervals are preferable from several perspectives.     

Empirical phi-divergence test statistics for testing simple and composite null hypotheses

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest is defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood (EL) on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the EL ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the ‘empirical modified likelihood ratio test statistic’ provides a competitive alternative to the EL ratio test statistic, and is also more robust than the EL ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.     

Empirical likelihood for probability density functions under negatively associated samples

In this paper, we study the construction of confidence intervals for a probability density function under a negatively associated sample 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 interval on the probability density function.     

Optimal detection of a jump in the intensity of a Poisson process or in a density with likelihood ratio statistics

We consider the problem of detecting a ‘bump’ in the intensity of a Poisson process or in a density. We analyze two types of likelihood ratio‐based statistics, which allow for exact finite sample inference and asymptotically optimal detection: The maximum of the penalized square root of log likelihood ratios (‘penalized scan’) evaluated over a certain sparse set of intervals and a certain average of log likelihood ratios (‘condensed average likelihood ratio’). We show that penalizing the square root of the log likelihood ratio — rather than the log likelihood ratio itself — leads to a simple penalty term that yields optimal power. The thus derived penalty may prove useful for other problems that involve a Brownian bridge in the limit. The second key tool is an approximating set of intervals that is rich enough to allow for optimal detection, but which is also sparse enough to allow justifying the validity of the penalization scheme simply via the union bound. This results in a considerable simplification in the theoretical treatment compared with the usual approach for this type of penalization technique, which requires establishing an exponential inequality for the variation of the test statistic. Another advantage of using the sparse approximating set is that it allows fast computation in nearly linear time. We present a simulation study that illustrates the superior performance of the penalized scan and of the condensed average likelihood ratio compared with the standard scan statistic.     

Quantile inference based on partially rank-ordered set samples

This paper develops statistical inference for population quantiles based on a partially rank-ordered set (PROS) sample design. A PROS sample design is similar to a ranked set sample with some clear differences. This design first creates partially rank-ordered subsets by allowing ties whenever the units in a set cannot be ranked with high confidence. It then selects a unit for full measurement at random from one of these partially rank-ordered subsets. The paper develops a point estimator, confidence interval and hypothesis testing procedure for the population quantile of order p. Exact, as well as asymptotic, distribution of the test statistic is derived. It is shown that the null distribution of the test statistic is distribution-free, and statistical inference is reasonably robust against possible ranking errors in ranking process.     

Censored median regression and profile empirical likelihood

We implement profile empirical likelihood-based inference for censored median regression models. Inference for any specified subvector is carried out by profiling out the nuisance parameters from the “plug-in” empirical likelihood ratio function proposed by Qin and Tsao. To obtain the critical value of the profile empirical likelihood ratio statistic, we first investigate its asymptotic distribution. The limiting distribution is a sum of weighted chi square distributions. Unlike for the full empirical likelihood, however, the derived asymptotic distribution has intractable covariance structure. Therefore, we employ the bootstrap to obtain the critical value, and compare the resulting confidence intervals with the ones obtained through Basawa and Koul’s minimum dispersion statistic. Furthermore, we obtain confidence intervals for the age and treatment effects in a lung cancer data set.     

Comparisons of approximate confidence intervals for the smallest extreme value distribution simple linear regression model under time censoring

Exact confidence interval estimation for accelerated life regression models with censored smallest extreme value (or Weibull) data is often impractical. This paper evaluates the accuracy of approximate confidence intervals based on the asymptotic normality of the maximum likelihood estimator, the asymptotic X²distribution of the likelihood ratio statistic, mean and variance correction to the likelihood ratio statistic, and the so-called Bartlett correction to the likelihood ratio statistic. The Monte Carlo evaluations under various degrees of time censoring show that uncorrected likelihood ratio intervals are very accurate in situations with heavy censoring. The benefits of mean and variance correction to the likelihood ratio statistic are only realized with light or no censoring. Bartlett correction tends to result in conservative intervals. Intervals based on the asymptotic normality of maximum likelihood estimators are anticonservative and should be used with much caution.     

Adjusted empirical likelihood for right censored lifetime data

Adjusted empirical likelihood (AEL) is a method to improve the performance of the empirical likelihood (EL) particularly in the construction of the confidence interval based on completely observed data. In this paper, we extend AEL approach to the analysis of right censored data by adopting an influence function method. The main results include that the adjusted log-likelihood ratio is asymptotically Chi-squared distributed. Simulation results indicate that the proposed AEL-based confidence intervals perform better compared with normality-based or EL-based confidence intervals specifically for small sample size within the right-censoring setting. The proposed method is illustrated by analysis of survival time of patients after operation for spinal tumors.     

A robust estimate of the correlation coefficient for bivariate normal distribution using ranked set sampling

A robust estimate of the correlation coefficient for a bivariate normal distribution using balanced ranked set sampling is studied. We show that this estimate is at least as efficient as the corresponding estimate based on simple random sampling and highly efficient compared to the maximum likelihood estimate using balanced ranked set sampling. The estimate is robust to common ranking errors. Small sample performance of the estimate is studied by simulation under imperfect and perfect ranking. A variance stabilizing transformation for the confidence interval of the correlation coefficient is obtained.     

Testing a block exchangeable covariance matrix

Testing hypotheses about the structure of a covariance matrix for doubly multivariate data is often considered in the literature. In this paper the Rao's score test (RST) is derived to test the block exchangeable covariance matrix or block compound symmetry (BCS) covariance structure under the assumption of multivariate normality. It is shown that the empirical distribution of the RST statistic under the null hypothesis is independent of the true values of the mean and the matrix components of a BCS structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Simulation studies are performed for the sample size consideration, and for the estimation of the empirical quantiles of the null distribution of the test statistic. The RST procedure is illustrated on a real data set from the medical studies.     

EMPIRICAL LIKELIHOOD METHODS FOR THE GINI INDEX

The Gini index and its generalizations have been used extensively for measuring inequality and poverty in the social sciences. Recently, interval estimation based on nonparametric statistics has been proposed in the literature, for example the naive bootstrap method, the iterated bootstrap method and the bootstrap method via a pivotal statistic. In this paper, we propose empirical likelihood methods to construct confidence intervals for the Gini index or the difference of two Gini indices. Simulation studies show that the proposed empirical likelihood method performs slightly worse than the bootstrap method based on a pivotal statistic in terms of coverage accuracy, but it requires less computation. However, the bootstrap calibration of the empirical likelihood method performs better than the bootstrap method based on a pivotal statistic.     

Sign test based on k-tuple general ranked set samples

ABSTRACT

The sign test based on the k-tuple ranked set samples is discussed here. We first derive the distribution of the k-tuple ranked set sample sign test statistic, and then the asymptotic distribution is also obtained. We then compare its performance with its counterparts based on simple random sample and classical ranked set sample. The asymptotic relative efficiency and the power are then derived. Finally, the effect of imperfect ranking on the procedure is assessed.     

Empirical likelihood for truncation parameter in random truncation model

In this paper we use the empirical likelihood method to construct confidence interval for truncation parameter in random truncation model. The empirical log-likelihood ratio is derived and its asymptotic distribution is shown to be a weighted chi-square. Simulation studies are used to compare the confidence intervals based on empirical likelihood and those based on normal approximation. It is found that the empirical likelihood method provides improved confidence interval.     

Statistical analysis of competing risks model from Marshall–Olkin extended Chen distribution under adaptive progressively interval censoring with random removals

In this paper, a new censoring scheme named by adaptive progressively interval censoring scheme is introduced. The competing risks data come from Marshall–Olkin extended Chen distribution under the new censoring scheme with random removals. We obtain the maximum likelihood estimators of the unknown parameters and the reliability function by using the EM algorithm based on the failure data. In addition, the bootstrap percentile confidence intervals and bootstrap-t confidence intervals of the unknown parameters are obtained. To test the equality of the competing risks model, the likelihood ratio tests are performed. Then, Monte Carlo simulation is conducted to evaluate the performance of the estimators under different sample sizes and removal schemes. Finally, a real data set is analyzed for illustration purpose.     

Statistical inference of risk difference in K correlated 2×2 tables with structural zero

K correlated 2×2 tables with structural zero are commonly encountered in infectious disease studies. A hypothesis test for risk difference is considered in K independent 2×2 tables with structural zero in this paper. Score statistic, likelihood ratio statistic and Wald‐type statistic are proposed to test the hypothesis on the basis of stratified data and pooled data. Sample size formulae are derived for controlling a pre‐specified power or a pre‐determined confidence interval width. Our empirical results show that score statistic and likelihood ratio statistic behave better than Wald‐type statistic in terms of type I error rate and coverage probability, sample sizes based on stratified test are smaller than those based on the pooled test in the same design. A real example is used to illustrate the proposed methodologies. Copyright © 2009 John Wiley & Sons, Ltd.     

A note on empirical likelihoods derived from pairwise score functions

Pairwise likelihood functions are convenient surrogates for the ordinary likelihood, useful when the latter is too difficult or even impractical to compute. One drawback of pairwise likelihood inference is that, for a multidimensional parameter of interest, the pairwise likelihood analogue of the likelihood ratio statistic does not have the standard chi-square asymptotic distribution. Invoking the theory of unbiased estimating functions, this paper proposes and discusses a computationally and theoretically attractive approach based on the derivation of empirical likelihood functions from the pairwise scores. This approach produces alternatives to the pairwise likelihood ratio statistic, which allow reference to the usual asymptotic chi-square distribution and which are useful when the elements of the Godambe information are troublesome to evaluate or in the presence of large data sets with relative small sample sizes. Two Monte Carlo studies are performed in order to assess the finite-sample performance of the proposed empirical pairwise likelihoods.     

Empirical likelihood inference in mixture of semiparametric varying-coefficient models for longitudinal data with non-ignorable dropout

In this paper, empirical likelihood inference in mixture of semiparametric varying-coefficient models for longitudinal data with non-ignorable dropout is investigated. We estimate the non-parametric function based on the estimating equations and the local linear profile-kernel method. An empirical log-likelihood ratio statistic for parametric components is proposed to construct confidence regions and is shown to be an asymptotically chi-squared distribution. The non-parametric version of Wilk's theorem is also derived. A simulation study is undertaken to illustrate the finite sample performance of the proposed method.     

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.