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.

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.

Abstract:
Jing (1995) and Liu et al. (2008) studied the two-sample empirical likelihood and showed it is Bartlett correctable for the univariate and multivariate cases, respectively. We expand its domain to the full parameter space and obtain a two-sample extended empirical likelihood which is more accurate and can also achieve the second-order accuracy of the Bartlett correction.

Abstract:
During the last decade Levy processes with jumps have received increasing popularity for modelling market behaviour for both derviative pricing and risk management purposes. Chan et al. (2009) introduced the use of empirical likelihood methods to estimate the parameters of various diffusion processes via their characteristic functions which are readily avaiable in most cases. Return series from the market are used for estimation. In addition to the return series, there are many derivatives actively traded in the market whose prices also contain information about parameters of the underlying process. This observation motivates us, in this paper, to combine the return series and the associated derivative prices observed at the market so as to provide a more reflective estimation with respect to the market movement and achieve a gain of effciency. The usual asymptotic properties, including consistency and asymptotic normality, are established under suitable regularity conditions. Simulation and case studies are performed to demonstrate the feasibility and effectiveness of the proposed method.

Abstract:
Dynamical properties of the elliptical stadium billiard, which is a generalization of the stadium billiard and a special case of the recently introduced mushroom billiards, are investigated analytically and numerically. In dependence on two shape parameters delta and gamma, this system reveals a rich interplay of integrable, mixed and fully chaotic behavior. Poincare sections, the box counting method and the stability analysis determine the structure of the parameter space and the borders between regions with different behavior. Results confirm the existence of a large fully chaotic region surrounding the straight line delta=1-gamma corresponding to the Bunimovich circular stadium billiard. Bifurcations due to the hour-glass and multidiamond orbits are described. For the quantal elliptical stadium billiard, statistical properties of the level spacing fluctuations are examined and compared with classical results.

Abstract:
This paper addresses the problems of parameter estimation of multivariable stationary stochastic systems on the basis of observed output data. The main contribution is to employ the expectation-maximisation (EM) method as a means for computation of the maximum-likelihood (ML) parameter estimation of the system. Closed form of the expectation of the studied system subjected to Gaussian distribution noise is derived and parameter choice that maximizes the expectation is also proposed. This results in an iterative algorithm for parameter estimation and the robust algorithm implementation based on technique of QR-factorization and Cholesky factorization is also discussed. Moreover, algorithmic properties such as non-decreasing likelihood value, necessary and sufficient conditions for the algorithm to arrive at a local stationary parameter, the convergence rate and the factors affecting the convergence rate are analyzed. Simulation study shows that the proposed algorithm has attractive properties such as numerical stability, and avoidance of difficult initial conditions.

Abstract:
Bayesian inference provides a flexible way of combining data with prior information. However, quantile regression is not equipped with a parametric likelihood, and therefore, Bayesian inference for quantile regression demands careful investigation. This paper considers the Bayesian empirical likelihood approach to quantile regression. Taking the empirical likelihood into a Bayesian framework, we show that the resultant posterior from any fixed prior is asymptotically normal; its mean shrinks toward the true parameter values, and its variance approaches that of the maximum empirical likelihood estimator. A more interesting case can be made for the Bayesian empirical likelihood when informative priors are used to explore commonality across quantiles. Regression quantiles that are computed separately at each percentile level tend to be highly variable in the data sparse areas (e.g., high or low percentile levels). Through empirical likelihood, the proposed method enables us to explore various forms of commonality across quantiles for efficiency gains. By using an MCMC algorithm in the computation, we avoid the daunting task of directly maximizing empirical likelihood. The finite sample performance of the proposed method is investigated empirically, where substantial efficiency gains are demonstrated with informative priors on common features across several percentile levels. A theoretical framework of shrinking priors is used in the paper to better understand the power of the proposed method.

Abstract:
We consider an empirical likelihood inference for parameters defined by general estimating equations when some components of the random observations are subject to missingness. As the nature of the estimating equations is wide-ranging, we propose a nonparametric imputation of the missing values from a kernel estimator of the conditional distribution of the missing variable given the always observable variable. The empirical likelihood is used to construct a profile likelihood for the parameter of interest. We demonstrate that the proposed nonparametric imputation can remove the selection bias in the missingness and the empirical likelihood leads to more efficient parameter estimation. The proposed method is further evaluated by simulation and an empirical study on a genetic dataset on recombinant inbred mice.

Abstract:
We use the effective field theory of dark energy to explore the space of modified gravity models which are capable of driving the present cosmic acceleration. We identify five universal functions of cosmic time that are enough to describe a wide range of theories containing a single scalar degree of freedom in addition to the metric. The first function (the effective equation of state) uniquely controls the expansion history of the universe. The remaining four functions appear in the linear cosmological perturbation equations, but only three of them regulate the growth history of large scale structures. We propose a specific parameterization of such functions in terms of characteristic coefficients that serve as coordinates in the space of modified gravity theories and can be effectively constrained by the next generation of cosmological experiments. We address in full generality the problem of the soundness of the theory against ghost-like and gradient instabilities and show how the space of non-pathological models shrinks when a more negative equation of state parameter is considered. This analysis allows us to locate a large class of stable theories that violate the null energy condition (i.e. super-acceleration models) and to recover, as particular subsets, various models considered so far. Finally, under the assumption that the true underlying cosmological model is the $\Lambda$ Cold Dark Matter ($\Lambda$CDM) scenario, and relying on the figure of merit of EUCLID-like observations, we demonstrate that the theoretical requirement of stability significantly narrows the empirical likelihood, increasing the discriminatory power of data. We also find that the vast majority of these non-pathological theories generating the same expansion history as the $\Lambda$CDM model predict a different, lower, growth rate of cosmic structures.

Abstract:
The likelihood function plays a pivotal role in statistical inference; it is adaptable to a wide range of models and the resultant estimators are known to have good properties. However, these results hinge on correct specification of the data generating mechanism. Many modern problems involve extremely complicated distribution functions, which may be difficult -- if not impossible -- to express explicitly. This is a serious barrier to the likelihood approach, which requires not only the specification of a distribution, but the correct distribution. Non-parametric methods are one way to avoid the problem of having to specify a particular data generating mechanism, but can be computationally intensive, reducing their accessibility for large data problems. We propose a new approach that combines multiple non-parametric likelihood-type components to build a data-driven approximation of the true function. The new construct builds on empirical and composite likelihood, taking advantage of the strengths of each. Specifically, from empirical likelihood we borrow the ability to avoid a parametric specification, and from composite likelihood we utilize multiple likelihood components. We will examine the theoretical properties of this composite empirical likelihood, both for purposes of application and to compare properties to other established likelihood methods.