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 Statistics , 2014, Abstract: We propose a two-sample extended empirical likelihood for inference on the difference between two p-dimensional parameters defined by estimating equations. The standard two-sample empirical likelihood for the difference is Bartlett correctable but its domain is a bounded subset of the parameter space. We expand its domain through a composite similarity transformation to derive the two-sample extended empirical likelihood which is defined on the full parameter space. The extended empirical likelihood has the same asymptotic distribution as the standard one and can also achieve the second order accuracy of the Bartlett correction. We include two applications to illustrate the use of two-sample empirical likelihood methods and to demonstrate the superior coverage accuracy of the extended empirical likelihood confidence regions.
 Statistics , 2013, Abstract: We derive an extended empirical likelihood for parameters defined by estimating equations which generalizes the original empirical likelihood for such parameters to the full parameter space. Under mild conditions, the extended empirical likelihood has all asymptotic properties of the original empirical likelihood. Its contours retain the data-driven shape of the latter. It can also attain the second order accuracy. The first order extended empirical likelihood is easy-to-use yet it is substantially more accurate than other empirical likelihoods, including second order ones. We recommend it for practical applications of the empirical likelihood method.
 Statistics , 2013, DOI: 10.1214/13-AOS1143 Abstract: We extend the empirical likelihood of Owen [Ann. Statist. 18 (1990) 90-120] by partitioning its domain into the collection of its contours and mapping the contours through a continuous sequence of similarity transformations onto the full parameter space. The resulting extended empirical likelihood is a natural generalization of the original empirical likelihood to the full parameter space; it has the same asymptotic properties and identically shaped contours as the original empirical likelihood. It can also attain the second order accuracy of the Bartlett corrected empirical likelihood of DiCiccio, Hall and Romano [Ann. Statist. 19 (1991) 1053-1061]. A simple first order extended empirical likelihood is found to be substantially more accurate than the original empirical likelihood. It is also more accurate than available second order empirical likelihood methods in most small sample situations and competitive in accuracy in large sample situations. Importantly, in many one-dimensional applications this first order extended empirical likelihood is accurate for sample sizes as small as ten, making it a practical and reliable choice for small sample empirical likelihood inference.
 Statistics , 2015, Abstract: A non parametric method based on the empirical likelihood is proposed for detecting the change in the coefficients of high-dimensional linear model where the number of model variables may increase as the sample size increases. This amounts to testing the null hypothesis of no change against the alternative of one change in the regression coefficients. Based on the theoretical asymptotic behaviour of the empirical likelihood ratio statistic, we propose, for a fixed design, a simpler test statistic, easier to use in practice. The asymptotic normality of the proposed test statistic under the null hypothesis is proved, a result which is different from the $\chi^2$ law for a model with a fixed variable number. Under alternative hypothesis, the test statistic diverges. We can then find the asymptotic confidence region for the difference of parameters of the two phases. Some Monte-Carlo simulations study the behaviour of the proposed test statistic.
 Statistics , 2014, DOI: 10.3150/14-BEJ636 Abstract: Empirical likelihood approach is one of non-parametric statistical methods, which is applied to the hypothesis testing or construction of confidence regions for pivotal unknown quantities. This method has been applied to the case of independent identically distributed random variables and second order stationary processes. In recent years, we observe heavy-tailed data in many fields. To model such data suitably, we consider symmetric scalar and multivariate $\alpha$-stable linear processes generated by infinite variance innovation sequence. We use a Whittle likelihood type estimating function in the empirical likelihood ratio function and derive the asymptotic distribution of the empirical likelihood ratio statistic for $\alpha$-stable linear processes. With the empirical likelihood statistic approach, the theory of estimation and testing for second order stationary processes is nicely extended to heavy-tailed data analyses, not straightforward, and applicable to a lot of financial statistical analyses.
 Jian-Jian Ren Mathematics , 2008, DOI: 10.1214/009053607000000695 Abstract: In this article, the weighted empirical likelihood is applied to a general setting of two-sample semiparametric models, which includes biased sampling models and case-control logistic regression models as special cases. For various types of censored data, such as right censored data, doubly censored data, interval censored data and partly interval-censored data, the weighted empirical likelihood-based semiparametric maximum likelihood estimator $(\tilde{\theta}_n,\tilde{F}_n)$ for the underlying parameter $\theta_0$ and distribution $F_0$ is derived, and the strong consistency of $(\tilde{\theta}_n,\tilde{F}_n)$ and the asymptotic normality of $\tilde{\theta}_n$ are established. Under biased sampling models, the weighted empirical log-likelihood ratio is shown to have an asymptotic scaled chi-squared distribution for censored data aforementioned. For right censored data, doubly censored data and partly interval-censored data, it is shown that $\sqrt{n}(\tilde{F}_n-F_0)$ weakly converges to a centered Gaussian process, which leads to a consistent goodness-of-fit test for the case-control logistic regression models.
 Entropy , 2013, DOI: 10.3390/e15093458 Abstract: In monitoring the depth of anesthesia (DOA), the electroencephalography (EEG) signals of patients have been utilized during surgeries to diagnose their level of consciousness. Different entropy methods were applied to analyze the EEG signal and measure its complexity, such as spectral entropy, approximate entropy (ApEn) and sample entropy (SampEn). However, as a weak physiological signal, EEG is easily subject to interference from external sources such as the electric power, electric knives and other electrophysiological signal sources, which lead to a reduction in the accuracy of DOA determination. In this study, we adopt the multivariate empirical mode decomposition (MEMD) to decompose and reconstruct the EEG recorded from clinical surgeries according to its best performance among the empirical mode decomposition (EMD), the ensemble EMD (EEMD), and the complementary EEMD (CEEMD) and the MEMD. Moreover, according to the comparison between SampEn and ApEn in measuring DOA, the SampEn is a practical and efficient method to monitor the DOA during surgeries at real time.
 Hiroaki Ogata Advances in Decision Sciences , 2012, DOI: 10.1155/2012/704693 Abstract: An application of the empirical likelihood method to non-Gaussian locally stationary processes is presented. Based on the central limit theorem for locally stationary processes, we give the asymptotic distributions of the maximum empirical likelihood estimator and the empirical likelihood ratio statistics, respectively. It is shown that the empirical likelihood method enables us to make inferences on various important indices in a time series analysis. Furthermore, we give a numerical study and investigate a finite sample property. 1. Introduction The empirical likelihood is one of the nonparametric methods for a statistical inference proposed by Owen [1, 2]. It is used for constructing confidence regions for a mean, for a class of M-estimates that includes quantile, and for differentiable statistical functionals. The empirical likelihood method has been applied to various problems because of its good properties: generality of the nonparametric method and effectiveness of the likelihood method. For example, we can name applications to the general estimating equations, [3] the regression models [4–6], the biased sample models [7], and so forth. Applications are also extended to dependent observations. Kitamura [8] developed the blockwise empirical likelihood for estimating equations and for smooth functions of means. Monti [9] applied the empirical likelihood method to linear processes, essentially under the circular Gaussian assumption, using a spectral method. For short- and long-range dependence, Nordman and Lahiri [10] gave the asymptotic properties of the frequency domain empirical likelihood. As we named above, some applications to time series analysis can be found but it seems that they were mainly for stationary processes. Although stationarity is the most fundamental assumption when we are engaged in a time series analysis, it is also known that real time series data are generally nonstationary (e.g., economics analysis). Therefore we need to use nonstationary models in order to describe the real world. Recently Dahlhaus [11–13] proposed an important class of nonstationary processes, called locally stationary processes. They have so-called time-varying spectral densities whose spectral structures smoothly change in time. In this paper we extend the empirical likelihood method to non-Gaussian locally stationary processes with time-varying spectra. First, We derive the asymptotic normality of the maximum empirical likelihood estimator based on the central limit theorem for locally stationary processes, which is stated in Dahlhaus [13, Theorem
 Entropy , 2013, DOI: 10.3390/e15093325 Abstract: EEG (Electroencephalography) signals can express the human awareness activities and consequently it can indicate the depth of anesthesia. On the other hand, Bispectral-index (BIS) is often used as an indicator to assess the depth of anesthesia. This study is aimed at using an advanced signal processing method to analyze EEG signals and compare them with existing BIS indexes from a commercial product ( i.e., IntelliVue MP60 BIS module). Multivariate empirical mode decomposition (MEMD) algorithm is utilized to filter the EEG signals. A combination of two MEMD components (IMF2 + IMF3) is used to express the raw EEG. Then, sample entropy algorithm is used to calculate the complexity of the patients’ EEG signal. Furthermore, linear regression and artificial neural network (ANN) methods were used to model the sample entropy using BIS index as the gold standard. ANN can produce better target value than linear regression. The correlation coefficient is 0.790 ± 0.069 and MAE is 8.448 ± 1.887. In conclusion, the area under the receiver operating characteristic (ROC) curve (AUC) of sample entropy value using ANN and MEMD is 0.969 ± 0.028 while the AUC of sample entropy value without filter is 0.733 ± 0.123. It means the MEMD method can filter out noise of the brain waves, so that the sample entropy of EEG can be closely related to the depth of anesthesia. Therefore, the resulting index can be adopted as the reference for the physician, in order to reduce the risk of surgery.
 Statistics , 2015, Abstract: The multivariate extended skew-normal distribution allows for accommodating raw data which are skewed and heavy tailed, and has at least three appealing statistical properties, namely closure under conditioning, affine transformations, and marginalization. In this paper we propose a Bayesian computational approach based on a sequential Monte Carlo (SMC) sampler to estimate such distributions. The practical implementation of each step of the algorithm is discussed and the elicitation of prior distributions takes into consideration some unusual behaviour of the likelihood function and the corresponding Fisher information matrix. Using Monte Carlo simulations, we provide strong evidence regarding the performances of the SMC sampler as well as some new insights regarding the parametrizations of the extended skew-normal distribution. A generalization to the extended skew-normal sample selection model is also presented. Finally we proceed with the analysis of two real datasets.
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