Parametric estimation of discretely sampled Gamma-OU processes

The stationary Gamma-OU processes are recommended to be the volatility of the financial assets. A parametric estimation for the Gamma-OU processes based on the discrete observations is considered in this paper. The estimator of an intensity parameter A and its convergence result are given, and the simulations show that the estimation is quite accurate. Assuming that the parameter A is estimated, the maximum likelihood estimation of shape parameter c and scale parameter a, whose likelihood function is not explicitly computable, is considered. By means of the Gaver-Stehfest algorithm, we construct an explicit sequence of approximations to the likelihood function and show that it converges the true (but unkown) one. Maximizing the sequence results in an estimator that converges to the true maximum likelihood estimator and the approximation shares the asymptotic properties of the true maximum likelihood estimator. Some simulation experiments reveal that this method is still quite accurate in most of rational situations for the background of volatility.     

Empirical likelihood for balanced ranked-set sampled data

Ranked-set sampling (RSS) often provides more efficient inference than simple random sampling (SRS). In this article, we propose a systematic nonparametric technique, RSS-EL, for hypothesis testing and interval estimation with balanced RSS data using empirical likelihood (EL). We detail the approach for interval estimation and hypothesis testing in one-sample and two-sample problems and general estimating equations. In all three cases, RSS is shown to provide more efficient inference than SRS of the same size. Moreover, the RSS-EL method does not require any easily violated assumptions needed by existing rank-based nonparametric methods for RSS data, such as perfect ranking, identical ranking scheme in two groups, and location shift between two population distributions. The merit of the RSS-EL method is also demonstrated through simulation studies. This work was supported by National Natural Science Foundation of China (Grant No. 10871037)     

Empirical likelihood for AR-ARCH models based on LAD estimation

By employing the empirical likelihood method,confidence regions for the stationary AR(p)-ARCH(q) models are constructed.A self-weighted LAD estimator is proposed under weak moment conditions.An empirical log-likelihood ratio statistic is derived and its asymptotic distribution is obtained.Simulation studies show that the performance of empirical likelihood method is better than that of normal approximation of the LAD estimator in terms of the coverage accuracy,especially for relative small size of observation.     

二阶扩散模型的经验似然推断

本文利用经验似然方法得到了二阶扩散模型的漂移系数和扩散系数的经验似然估计量, 并研究这些估计量的相合性和渐近正态性. 进一步在经验似然方法的基础上给出了漂移系数和扩散系数的非对称的置信区间, 并且在一定的条件下证明了调整的对数似然比是渐近卡方分布的.     

Maximum likelihood estimation for noncausal autoregressive processes

We discuss a maximum likelihood procedure for estimating parameters in possibly noncausal autoregressive processes driven by i.i.d. non-Gaussian noise. Under appropriate conditions, estimates of the parameters that are solutions to the likelihood equations exist and are asymptotically normal. The estimation procedure is illustrated with a simulation study for AR(2) processes.     

Empirical likelihood estimators for the error distribution in nonparametric regression models

The aim of this paper is to show that existing estimators for the error distribution in non-parametric regression models can be improved when additional information about the distribution is included by the empirical likelihood method. The weak convergence of the resulting new estimator to a Gaussian process is shown and the performance is investigated by comparison of asymptotic mean squared errors and by means of a simulation study.      

BOOTSTRAP MAXIMUM LIKELIHOOD ESTIMATION OF THE PARAMETER IN SPECTRAL DENSITY OF STATIONARY PROCESSES

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Conditional Empirical Processes Defined by Nonstationary Absolutely Regular Sequences

K. I. Yoshihara (1990,Comput. Math. Appl.19, No. 1, 149–158) proved the weak invariance of the conditional nearest neighbor regression function estimator called the conditional empirical process based on-mixing observations. In this paper, we extend the result for nonstationary and absolutely regular random variables which have applications for Markov processes, for which the initial measure is not necessary, the invariant measure.     

截断情况下生存函数的函数的经验似然

Owen首次在完全样本下提出了经验似然的方法,WangQihua将该方法应用到带有截断情况的生存函数的函数估计问题．本文给出了更为一般的调整似然比统计量,证明了在适当的情况下该统计量仍渐近服从χ^2分布,同时模拟的结果也表明该统计量具有良好的性质。     

Maximum likelihood estimation in processes of Ornstein-Uhlenbeck type

In this article we propose a maximum likelihood methodology to estimate the parameters of a one-dimensional stationary process of Ornstein-Uhlenbeck type that is constructed via a self-decomposable distribution D. Our approach is based on the inversion of the characteristic function and the use of the classical or fractional discrete fast Fourier transform. The results are illustrated throughout an extensive simulation study. This includes the cases where D belongs to the gamma, tempered stable and normal inverse Gaussian family of distributions.      

条件经验随机过程的渐近性质

设(Xi,Yi)(i=1,2,…,n)是来自总体(X,Y)的样本(独立同分布),其中X∈R1,Y∈Rq.M(x y)是Y=y时X的条件分布,Mnkn(x y)为M(x y)的第kn个最近邻域的经验分布估计量,讨论条件经验过程Sn(t,x,y)=kn12(Mnkn(x y)-M(x y))的渐近性质,得出在适当条件下,对固定的y,Sn(t,x,y)(x,t为参数)弱收敛于某一G aussian过程S(.).     

随机受控分支过程中繁衍变量均值的经验似然估计

文章研究受控分支过程在随机环境下的繁衍变量均值的估计问题.我们基于加权条件最小二乘法构造估计方程,发展了一个经验似然比检验,并证明了这个检验统计量的极限分布是χ^2分布.最后通过随机模拟验证了经验似然方法有较高的覆盖概率.     

Empirical processes indexed by smooth functions

In this paper we derive a general invariance principle for empirical processes indexed by smooth functions. The method is applied to prove bounds for the convergence of the empirical distributions which might be useful to verify asymptotic normality of smooth statistical functionals. As one further application we get the convergence of the so-called empirical characteristic function process.     

Empirical Bayes estimation for queueing systems and networks

Empirical Bayes estimators are derived for standardM/M/1 queues,M/M/1 queues with state-dependent arrival and service rates, finite capacityM/M/1 queues with state-dependent rates and for open Jackson networks. The asymptotic properties of the empirical Bayes estimators are derived both with respect to the conditional distribution of the observations given the parameters, and with respect to the joint distribution of the observations and the parameters.     

分数填补下两非参数总体一般差异的经验似然置信区间

在一定的条件下证明了缺失数据情形基于分数填补方法得到的两非参数总体一般差异指标的经验似然比统计量的渐近分布为加权χ21,由此可构造差异指标的经验似然置信区间.     

不等式约束条件下经验似然法在两总体中位数检验中的应用

本文中, Owen 引入的经验似然方法被用于参数空间带不等式约束的两总体中位数的比较. 迄今为止, 还没有人研究过该问题. 这是因为, 在构造经验似然函数过程中所使用的辅助函数不是光滑函数, 因而不是凸函数, 从而使研究难度大大增加. 然而, 通过引入经验过程的办法, 本文很巧妙地解决了此问题. 根据经验过程, 本文证明了两中位数比较的经验似然比检验统计量的极限分布要么是单一的卡方分布, 要么是两个卡方分布的等权混合分布. 这一理论结果得到了模拟运算结果的有力支持.     

相依序列密度函数的经验似然推断

本文利用了强平稳$m-$相依序列的特殊性质,讨论了$m-$相依序列密度函数的经验似然推断, 给出了似然比统计量的极限分布,可构造参数的经验似然置信区间. 并且通过模拟计算来说明有限样本下应用经验似然方法的合理性.     

Convergence results for maximum likelihood type estimators in multivariable ARMA models II

The consistency proof for the (Gaussian quasi) maximum likelihood estimator in multivariable ARMA models as given in Dunsmuir and Hannan (1976, Adv, in Appl. Probab. 8, 339–364) rests on a certain property of the underlying parameter space, called B6 in their paper. It is not known whether the usual parameter spaces like the manifold M(n) or the parameter spaces corresponding to echelon forms satisfy condition B6, since the argument given by Dunsmuir and Hannan to establish this fact is inconclusive. In Pötscher (1987, J. Multivariate Anal. 21 29–52) it was shown how consistency can be proved without relying on B6 if the data generating process is Gaussian. In this note we show that the Gaussianity assumption can be replaced by ergodicity thus restoring Dunsmuir and Hannan's consistency proof to its full generality and extending it to parameter spaces which do not satisfy condition B6.     

Empirical likelihood confidence regions of the parameters in a partially linear single-index model

In this paper,a partially linear single-index model is investigated,and three empirical log-likelihood ratio statistics for the unknown parameters in the model are sug- gested.It is proved that the proposed statistics are asymptotically standard chi-square un- der some suitable conditions,and hence can be used to construct the confidence regions of the parameters.Our methods can also deal with the confidence region construction for the index in the pure single-index model.A simulation study indicates that,in terms of cov- erage probabilities and average areas of the confidence regions,the proposed methods perform better than the least-squares method.