In this paper, we develop statistical inference techniques for the unknown coefficient functions and single-index parameters in single-index varying-coefficient models. We first estimate the nonparametric component via the local linear fitting, then construct an estimated empirical likelihood ratio function and hence obtain a maximum empirical likelihood estimator for the parametric component. Our estimator for parametric component is asymptotically efficient, and the estimator of nonparametric component has an optimal convergence rate. Our results provide ways to construct the confidence region for the involved unknown parameter. We also develop an adjusted empirical likelihood ratio for constructing the confidence regions of parameters of interest. A simulation study is conducted to evaluate the finite sample behaviors of the proposed methods.
Statistical learning evolves quickly with more and more sophisticated models proposed to incorporate the complicated data structure from modern scientific and business problems. Varying index coefficient models extend varying coefficient models and single index models, becoming the latest state-of-the-art for semiparametric regression. This new class of models offers greater flexibility to characterize complicated nonlinear interaction effects in regression analysis. To safeguard against outliers and extreme observations, we consider a robust quantile regression approach to estimate the model …
Empirical likelihood is a well-known nonparametric method in statistics and has been widely applied in statistical inference. The method has been employed by Lu and Peng (2002) to constructing confidence intervals for the tail index of a heavy-tailed distribution. It is demonstrated in Lu and Peng (2002) that the empirical likelihood-based confidence intervals performs better than confidence intervals based on normal approximation in terms of the coverage probability. In general, the empirical likelihood method can be hindered by its imprecision in the coverage probability when the sample size…
We re-consider Leadbetter's extremal index for stationary sequences. It has interpretation as reciprocal of the expected size of an extremal cluster above high thresholds. We focus on heavy-tailed time series, in particular on regularly varying stationary sequences, and discuss recent research in extreme value theory for these models. A regularly varying time series has multivariate regularly varying finite-dimensional distributions. Thanks to results by Basrak and Segers we have explicit representations of the limiting cluster structure of extremes, leading to explicit expressions of the limi…
Motivated by better modeling of intra-individual variability in longitudinal data, we propose a class of location-scale mixed effects models, in which the data of each individual is modeled by a parameter-varying generalized hyperbolic distribution. We first study the local maximum-likelihood asymptotics and reveal the instability in the numerical optimization of the log-likelihood. Then, we construct an asymptotically efficient estimator based on the Newton-Raphson method based on the original log-likelihood function with the initial estimator being naive least-squares-type. Numerical experim…
For the conditional mean function of panel count model with time-varying coefficients, we propose to use local kernel regression method for estimation. Partial log-likelihood with local polynomial is formed for estimation. Under some regularity conditions, strong uniform consistency rates are obtained for the local estimator. At target time point, we show that the local estimator converges in distribution to normal distribution. The baseline mean function estimator is also shown to be consistent. Simulation studies show that the time-varying coefficient estimator is close to the true value, th…
The MDL two-part coding $ \textit{index of resolvability} $ provides a finite-sample upper bound on the statistical risk of penalized likelihood estimators over countable models. However, the bound does not apply to unpenalized maximum likelihood estimation or procedures with exceedingly small penalties. In this paper, we point out a more general inequality that holds for arbitrary penalties. In addition, this approach makes it possible to derive exact risk bounds of order $1/n$ for iid parametric models, which improves on the order $(\log n)/n$ resolvability bounds. We conclude by discussing …
In regression models for spatial data, it is often assumed that the marginal effects of covariates on the response are constant over space. In practice, this assumption might often be questionable. In this article, we show how a Gaussian process-based spatially varying coefficient (SVC) model can be estimated using maximum likelihood estimation (MLE). In addition, we present an approach that scales to large data by applying covariance tapering. We compare our methodology to existing methods such as a Bayesian approach using the stochastic partial differential equation (SPDE) link, geographical…
We propose a class of weighted least squares estimators for the tail index of a distribution function with a regularly varying upper tail. Our approach is based on the method developed by \cite{Holan2010} for the Parzen tail index. Asymptotic normality of the estimators is proved. Through a simulation study, these and earlier estimators are compared in the Pareto and Hall models using the mean squared error as criterion. The results show that the weighted least squares estimator is better than the other estimators investigated.
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
This paper proposes a formal approach to online learning and planning for agents operating in a priori unknown, time-varying environments. The proposed method computes the maximally likely model of the environment, given the observations about the environment made by an agent earlier in the system run and assuming knowledge of a bound on the maximal rate of change of system dynamics. Such an approach generalizes the estimation method commonly used in learning algorithms for unknown Markov decision processes with time-invariant transition probabilities, but is also able to quickly and correctly…
Rooted in genetics, human complex diseases are largely influenced by environmental factors. Existing literature has shown the power of integrative gene-environment interaction analysis by considering the joint effect of environmental mixtures on a disease risk. In this work, we propose a functional varying index coefficient model for longitudinal measurements of a phenotypic trait together with multiple environmental variables, and assess how the genetic effects on a longitudinal disease trait are nonlinearly modified by a mixture of environmental influences. We derive an estimation procedure …
We develop a new statistical model to analyse time-varying ranking data. The model can be used with a large number of ranked items, accommodates exogenous time-varying covariates and partial rankings, and is estimated via the maximum likelihood in a straightforward manner. Rankings are modelled using the Plackett-Luce distribution with time-varying worth parameters that follow a mean-reverting time series process. To capture the dependence of the worth parameters on past rankings, we utilise the conditional score in the fashion of the generalised autoregressive score (GAS) models. Simulation e…
The additive hazards model specifies the effect of covariates on the hazard in an additive way, in contrast to the popular Cox model, in which it is multiplicative. As non-parametric model, it offers a very flexible way of modeling time-varying covariate effects. It is most commonly estimated by ordinary least squares. In this paper we consider the case where covariates are bounded, and derive the maximum likelihood estimator under the constraint that the hazard is non-negative for all covariate values in their domain. We describe an efficient algorithm to find the maximum likelihood estimator…
National Chiao Tung University（国立交通大学）；Texas A&M University（德克萨斯农工大学）
Modifying the reward-biased maximum likelihood method originally proposed in the adaptive control literature, we propose novel learning algorithms to handle the explore-exploit trade-off in linear bandits problems as well as generalized linear bandits problems. We develop novel index policies that we prove achieve order-optimality, and show that they achieve empirical performance competitive with the state-of-the-art benchmark methods in extensive experiments. The new policies achieve this with low computation time per pull for linear bandits, and thereby resulting in both favorable regret as …
Competing risk data appear widely in modern biomedical research. Cause-specific hazard models are often used to deal with competing risk data in the past two decades. There is no current study on the kernel likelihood method for the cause-specific hazard model with time-varying coefficients. We propose to use the local partial log-likelihood approach for nonparametric time-varying coefficient estimation. Simulation studies demonstrate that our proposed nonparametric kernel estimator has a good performance under assumed finite sample settings. Finally, we apply the proposed method to analyze a …
We address the issue of performing inference on the parameters that index a bimodal extension of the Birnbaum-Saunders distribution (BS). We show that maximum likelihood point estimation can be problematic since the standard nonlinear optimization algorithms may fail to converge. To deal with this problem, we penalize the log-likelihood function. The numerical evidence we present show that maximum likelihood estimation based on such penalized function is made considerably more reliable. We also consider hypothesis testing inference based on the penalized log-likelihood function. In particular,…
Tissues are often heterogeneous in their single-cell molecular expression, and this can govern the regulation of cell fate. For the understanding of development and disease, it is important to quantify heterogeneity in a given tissue. We introduce the \proglang{R} package \pkg{stochprofML} which is designed to parameterize heterogeneity from the cumulative expression of small random pools of cells. This method outweighs the demixing of mixed samples with a saving in cost and effort and less measurement error. The approach uses the maximum likelihood principle and was originally presented in Ba…
We consider the problem of tracking an unknown time varying parameter that characterizes the probabilistic evolution of a sequence of independent observations. To this aim, we propose a stochastic gradient descent-based recursive scheme in which the log-likelihood of the observations acts as time varying gain function. We prove convergence in mean-square error in a suitable neighbourhood of the unknown time varying parameter and illustrate the details of our findings in the case where data are generated from distributions belonging to the exponential family.
We propose an empirical likelihood ratio test for nonparametric model selection, where the competing models may be nested, nonnested, overlapping, misspecified, or correctly specified. It compares the squared prediction errors of models based on the cross-validation and allows for heteroscedasticity of the errors of models. We develop its asymptotic distributions for comparing additive models and varying-coefficient models and extend it to test significance of variables in additive models with massive data. The method is applicable to model selection among supervised learning models. To facili…
We propose a novel class of time-varying nonparanormal graphical models, which allows us to model high dimensional heavy-tailed systems and the evolution of their latent network structures. Under this model, we develop statistical tests for presence of edges both locally at a fixed index value and globally over a range of values. The tests are developed for a high-dimensional regime, are robust to model selection mistakes and do not require commonly assumed minimum signal strength. The testing procedures are based on a high dimensional, debiasing-free moment estimator, which uses a novel kerne…
National Yang Ming Chiao Tung University（国立阳明交通大学）
Reward-biased maximum likelihood estimation (RBMLE) is a classic principle in the adaptive control literature for tackling explore-exploit trade-offs. This paper studies the stochastic contextual bandit problem with general bounded reward functions and proposes NeuralRBMLE, which adapts the RBMLE principle by adding a bias term to the log-likelihood to enforce exploration. NeuralRBMLE leverages the representation power of neural networks and directly encodes exploratory behavior in the parameter space, without constructing confidence intervals of the estimated rewards. We propose two variants …
In this paper we propose a new time-varying econometric model, called Time-Varying Poisson AutoRegressive with eXogenous covariates (TV-PARX), suited to model and forecast time series of counts. {We show that the score-driven framework is particularly suitable to recover the evolution of time-varying parameters and provides the required flexibility to model and forecast time series of counts characterized by convoluted nonlinear dynamics and structural breaks.} We study the asymptotic properties of the TV-PARX model and prove that, under mild conditions, maximum likelihood estimation (MLE) yie…
We consider the clustering of extremes for stationary regularly varying random fields over arbitrary growing index sets. We study sufficient assumptions on the index set such that the limit of the point random fields of the exceedances above a high threshold exists. Under the so-called anti-clustering condition, the extremal dependence is only local. Thus the index set can have a general form compared to previous literature [3, 21]. However, we cannot describe the clustering of extreme values in terms of the usual spectral tail measure [23] except for hyperrectangles or index sets in the latti…
This paper is an attempt to set a justification for making use of some dicrepancy indexes, starting from the classical Maximum Likelihood definition, and adapting the corresponding basic principle of inference to situations where minimization of those indexes between a model and some extension of the empirical measure of the data appears as its natural extension. This leads to the so called generalized bootstrap setting for which minimum divergence inference seems to replace Maximum Likelihood one. 1 Motivation and context Divergences between probability measures are widely used in Statistics …
