How precise is the finite sample approximation of the asymptotic distribution of realised variation measures in the presence of jumps?

This paper studies the impact of jumps on volatility estimation and inference based on various realised variation measures such as realised variance, realised multipower variation and truncated realised multipower variation. We review the asymptotic theory of those realised variation measures and present a new estimator for the asymptotic ‘variance’ of the centered realised variance in the presence of jumps. Next, we compare the finite sample performance of the various estimators by means of detailed Monte Carlo studies. Here we study the impact of the jump activity, of the jump size of the jumps in the price and of the presence of additional independent or dependent jumps in the volatility. We find that the finite sample performance of realised variance and, in particular, of log-transformed realised variance is generally good, whereas the jump-robust statistics tend to struggle in the presence of a highly active jump process.     

Penalized Empirical Likelihood via Bridge Estimator in Cox's Proportional Hazard Model

We propose the penalized empirical likelihood method via bridge estimator in Cox's proportional hazard model for parameter estimation and variable selection. Under reasonable conditions, we show that penalized empirical likelihood in Cox's proportional hazard model has oracle property. A penalized empirical likelihood ratio for the vector of regression coefficients is defined and its limiting distribution is a chi-square distributions. The advantage of penalized empirical likelihood as a nonparametric likelihood approach is illustrated in testing hypothesis and constructing confidence sets. The method is illustrated by extensive simulation studies and a real example.     

连续贝塔、非连续贝塔与股票风险溢酬

本文利用高频数据将传统的CAPM贝塔分解为连续贝塔和非连续贝塔（跳跃贝塔和隔夜贝塔），并在此基础上进一步考虑正向市场和负向市场的非对称性，将跳跃贝塔又细分为正向跳跃贝塔和负向跳跃贝塔，以探讨不同类型系统性风险的特征差异及其所对应的风险溢酬。实证结果表明，个股对市场发生的非连续变动比连续变动更加敏感，投资者对市场发生的负向跳跃比正向跳跃反应更加强烈；中国股票市场上的系统性非连续风险溢酬（跳跃风险溢酬和隔夜风险溢酬）显著为正，但系统性连续风险并没有得到定价；其中，跳跃风险溢酬则主要来源于对系统性负向跳跃风险的补偿，而正向跳跃风险对股票横截面收益率没有显著的影响。     

Empirical likelihood for single index model with missing covariates at random

In this paper, we investigate the empirical-likelihood-based inference for the construction of confidence intervals and regions of the parameters of interest in single index models with missing covariates at random. An augmented inverse probability weighted-type empirical likelihood ratio for the parameters of interest is defined such that this ratio is asymptotically standard chi-squared. Our approach is to directly calibrate the empirical log-likelihood ratio, and does not need multiplication by an adjustment factor for the original ratio. Our bias-corrected empirical likelihood is self-scale invariant and no plug-in estimator for the limiting variance is needed. Some simulation studies are carried out to assess the performance of our proposed method.     

Penalized empirical likelihood inference for sparse additive hazards regression with a diverging number of covariates

High-dimensional sparse modeling with censored survival data is of great practical importance, as exemplified by applications in high-throughput genomic data analysis. In this paper, we propose a class of regularization methods, integrating both the penalized empirical likelihood and pseudoscore approaches, for variable selection and estimation in sparse and high-dimensional additive hazards regression models. When the number of covariates grows with the sample size, we establish asymptotic properties of the resulting estimator and the oracle property of the proposed method. It is shown that the proposed estimator is more efficient than that obtained from the non-concave penalized likelihood approach in the literature. Based on a penalized empirical likelihood ratio statistic, we further develop a nonparametric likelihood approach for testing the linear hypothesis of regression coefficients and constructing confidence regions consequently. Simulation studies are carried out to evaluate the performance of the proposed methodology and also two real data sets are analyzed.     

Non‐parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps

Abstract. We consider a stochastic process driven by diffusions and jumps. Given a discrete record of observations, we devise a technique for identifying the times when jumps larger than a suitably defined threshold occurred. This allows us to determine a consistent non‐parametric estimator of the integrated volatility when the infinite activity jump component is Lévy. Jump size estimation and central limit results are proved in the case of finite activity jumps. Some simulations illustrate the applicability of the methodology in finite samples and its superiority on the multipower variations especially when it is not possible to use high frequency data.     

Contrast function estimation for the drift parameter of ergodic jump diffusion process

In this paper, we consider an ergodic diffusion process with jumps whose drift coefficient depends on an unknown parameter. We suppose that the process is discretely observed. We introduce an estimator based on a contrast function, which is efficient without requiring any conditions on the rate at which the step discretization goes to zero, and where we allow the observed process to have nonsummable jumps. This extends earlier results where the condition on the step discretization was needed and where the process was supposed to have summable jumps. In general situations, our contrast function is not explicit and one has to resort to some approximation. In the case of a finite jump activity, we propose explicit approximations of the contrast function such that the efficient estimation of the drift parameter is feasible. This extends the results obtained by Kessler in the case of continuous processes.     

Correlation analysis with additive distortion measurement errors

This paper studies the estimation of correlation coefficient between unobserved variables of interest. These unobservable variables are distorted in a additive fashion by an observed confounding variable. Two estimators, a direct-plug-in estimator and a residual-based estimator, are proposed. Their asymptotical results are obtained, and the residual-based estimator is shown asymptotically efficient. Moreover, we suggest an asymptotic normal approximation and an empirical likelihood-based statistic to construct the confidence interval. The empirical likelihood statistic is shown to be asymptotically chi-squared. Simulation studies are conducted to examine the performance of the proposed estimators. These methods are applied to analyse the Boston housing price data for an illustration.     

Full likelihood inference in the Cox model with LTRC data when covariates are discrete

For the regression parameter β in the Cox model, there have been several estimates based on different types of approximated likelihood. For right-censored data, Ren and Zhou [Full likelihood inferences in the Cox model: an empirical approach. Ann Inst Statist Math. 2011;63:1005–1018] derive the full likelihood function for (β, F₀), where F₀ is the baseline distribution function in the Cox model. In this article, we extend their results to left-truncated and right-censored data with discrete covariates. Using the empirical likelihood parameterization, we obtain the full-profile likelihood function for β when covariates are discrete. Simulation results indicate that the maximum likelihood estimator outperforms Cox's partial likelihood estimator in finite samples.     

Nonparametric estimation of jump characteristics under market microstructure noise

This article proposes a simple nonparametric method to estimate the jump characteristics in asset price with noisy high-frequency data. We combine the pre-averaging approach and the threshold technique to identify the jumps, and then propose the pre-averaging threshold estimators for the number and sizes of jumps occurred. We further present the asymptotic properties of the proposed estimators. The Monte Carlo simulation shows that the estimators are robust to microstructure noise and work very well especially when the data frequency is ultra-high. Finally, an empirical example further demonstrates the power of the proposed method.     

Pricing power options with a generalized jump diffusion

In this article, the valuation of power option is investigated when the dynamic of the stock price is governed by a generalized jump-diffusion Markov-modulated model. The systematic risk is characterized by the diffusion part, and the non systematic risk is characterized by the pure jump process. The jumps are described by a generalized renewal process with generalized jump amplitude. By introducing NASDAQ Index Model, their risk premium is identified respectively. A risk-neutral measure is identified by employing Esscher transform with two families of parameters, which represent the two parts risk premium. In this article, the non systematic risk premium is considered, based on which the price of power option is studied under the generalized jump-diffusion Markov-modulated model. In the case of a special renewal process with log double exponential jump amplitude, the accurate expressions for the Esscher parameters and the pricing formula are provided. By numerical simulation, the influence of the non systematic risk’s price and the index of the power options on the price of the option is depicted.     

The Empirical Likelihood for First-Order Random Coefficient Integer-Valued Autoregressive Processes

This article studies the empirical likelihood method for the first-order random coefficient integer-valued autoregressive process. The limiting distribution of the log empirical likelihood ratio statistic is established. Confidence region for the parameter of interest and its coverage probabilities are given, and hypothesis testing is considered. The maximum empirical likelihood estimator for the parameter is derived and its asymptotic properties are established. The performances of the estimator are compared with the conditional least squares estimator via simulation.     

Estimating the Parameters of Stochastic Volatility Models Using Option Price Data

This article describes a maximum likelihood method for estimating the parameters of the standard square-root stochastic volatility model and a variant of the model that includes jumps in equity prices. The model is fitted to data on the S&P 500 Index and the prices of vanilla options written on the index, for the period 1990 to 2011. The method is able to estimate both the parameters of the physical measure (associated with the index) and the parameters of the risk-neutral measure (associated with the options), including the volatility and jump risk premia. The estimation is implemented using a particle filter whose efficacy is demonstrated under simulation. The computational load of this estimation method, which previously has been prohibitive, is managed by the effective use of parallel computing using graphics processing units (GPUs). The empirical results indicate that the parameters of the models are reliably estimated and consistent with values reported in previous work. In particular, both the volatility risk premium and the jump risk premium are found to be significant.     

Empirical likelihood methods for discretely observed Gaussian moving averages

This paper is concerned with parametric estimation, model specification and autocorrelation diagnosis for stationary moving averages driven by a Wiener process. By incorporating the analysis of the spectral densities of the discretely observed trajectory, empirical likelihood methods based on moment conditions are developed to the dependent sequences in this paper for estimation and test. Theoretical properties of the empirical likelihood estimator for parameters are provided. Empirical likelihood ratio tests for model specification of the moving averages are proposed by means of the bootstrap strategy. Simulation and empirical case studies are carried out to confirm the effectiveness of the proposed estimation and test.     

Asymptotic normality of locally modelled regression estimator for functional data

We focus on the nonparametric regression of a scalar response on a functional explanatory variable. As an alternative to the well-known Nadaraya-Watson estimator for regression function in this framework, the locally modelled regression estimator performs very well [cf. [Barrientos-Marin, J., Ferraty, F., and Vieu, P. (2010), ‘Locally Modelled Regression and Functional Data’, Journal of Nonparametric Statistics, 22, 617–632]. In this paper, the asymptotic properties of locally modelled regression estimator for functional data are considered. The mean-squared convergence as well as asymptotic normality for the estimator are established. We also adapt the empirical likelihood method to construct the point-wise confidence intervals for the regression function and derive the Wilk's phenomenon for the empirical likelihood inference. Furthermore, a simulation study is presented to illustrate our theoretical results.     

A new characterization of the jump rate for piecewise-deterministic Markov processes with discrete transitions

Piecewise-deterministic Markov processes form a general class of non diffusion stochastic models that involve both deterministic trajectories and random jumps at random times. In this paper, we state a new characterization of the jump rate of such a process with discrete transitions. We deduce from this result a non parametric technique for estimating this feature of interest. We state the uniform convergence in probability of the estimator. The methodology is illustrated on a numerical example.     

Two-sample high-dimensional empirical likelihood

In this paper, we apply empirical likelihood for two-sample problems with growing high dimensionality. Our results are demonstrated for constructing confidence regions for the difference of the means of two p-dimensional samples and the difference in value between coefficients of two p-dimensional sample linear model. We show that empirical likelihood based estimator has the efficient property. That is, as p → ∞ for high-dimensional data, the limit distribution of the EL ratio statistic for the difference of the means of two samples and the difference in value between coefficients of two-sample linear model is asymptotic normal distribution. Furthermore, empirical likelihood (EL) gives efficient estimator for regression coefficients in linear models, and can be as efficient as a parametric approach. The performance of the proposed method is illustrated via numerical simulations.     

A stochastic process approach to multilayer neutron detectors

The sparsity of the isotope Helium‐3, ongoing since 2009, has initiated a new generation of neutron detectors. One particularly promising development line for detectors is the multilayer gaseous detector. In this paper, a stochastic process approach is used to determine the neutron energy from the additional data afforded by the multilayer nature of these novel detectors. The data from a multilayer detector consist of counts of the number of absorbed neutrons along the sequence of the detector's layers, in which the neutron absorption probability is unknown. We study the maximum likelihood estimator for the intensity and absorption probability and show its consistency and asymptotic normality, as the number of incoming neutrons goes to infinity. We combine these results with known results on the relation between the absorption probability and the wavelength to derive an estimator of the wavelength and to show its consistency and asymptotic normality.     

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     

Hill's estimator for the tail index of an ARMA model

This paper investigates Hill's estimator for the tail index of an ARMA model with i.i.d. residuals. Based on the estimated residuals, it is shown that Hill's estimator is asymptotically normal. This method can achieve a smaller asymptotic variance than applying Hill's estimator to the original data. These results are the same as those in Resnick and Starica (Commun. Statist.—Stochastic Models 13 (4) (1997) 703) for an AR model. However, Resnick and Starica (Commun. Statist.—Stochastic Models 13 (4) (1997) 703) imposed one more condition on the choice of sample fraction than the i.i.d. case. This condition is removed in this paper so that data-driven methods for choosing optimal sample fraction based on i.i.d. data can be applied to our case. As an auxiliary theorem, we establish the weak convergence of the tail empirical process of the estimated residuals, which may be of independent interest.