Smooth expectiles for panel data using penalized splines

Expectile regression is a topic which became popular in the last years. It includes ordinary mean regression as special case but is more general as it offers the possibility to also model non-central parts of a distribution. Semi-parametric expectile models have recently been developed and it is easy to perform flexible expectile estimation with modern software like R. We extend the model class by allowing for panel observations, i.e. clustered data with repeated measurements taken at the same individual. A random (individual) effect is incorporated in the model which accounts for the dependence structure in the data. We fit expectile sheets, meaning that not a single expectile is estimated but a whole range of expectiles is estimated simultaneously. The presented model allows for multiple covariates, where a semi-parametric approach with penalized splines is pursued to fit smooth expectile curves. We apply our methods to panel data from the German Socio-Economic Panel.     

Bayesian expectile regression with asymmetric normal distribution

In this paper, we adopt the Bayesian approach to expectile regression employing a likelihood function that is based on an asymmetric normal distribution. We demonstrate that improper uniform priors for the unknown model parameters yield a proper joint posterior. Three simulated data sets were generated to evaluate the proposed method which show that Bayesian expectile regression performs well and has different characteristics comparing with Bayesian quantile regression. We also apply this approach into two real data analysis.     

An elastic-net penalized expectile regression with applications

To perform variable selection in expectile regression, we introduce the elastic-net penalty into expectile regression and propose an elastic-net penalized expectile regression (ER-EN) model. We then adopt the semismooth Newton coordinate descent (SNCD) algorithm to solve the proposed ER-EN model in high-dimensional settings. The advantages of ER-EN model are illustrated via extensive Monte Carlo simulations. The numerical results show that the ER-EN model outperforms the elastic-net penalized least squares regression (LSR-EN), the elastic-net penalized Huber regression (HR-EN), the elastic-net penalized quantile regression (QR-EN) and conventional expectile regression (ER) in terms of variable selection and predictive ability, especially for asymmetric distributions. We also apply the ER-EN model to two real-world applications: relative location of CT slices on the axial axis and metabolism of tacrolimus (Tac) drug. Empirical results also demonstrate the superiority of the ER-EN model.     

Nonparametric multiple expectile regression via ER-Boost

Expectile regression [Newey W, Powell J. Asymmetric least squares estimation and testing, Econometrica. 1987;55:819–847] is a nice tool for estimating the conditional expectiles of a response variable given a set of covariates. Expectile regression at 50% level is the classical conditional mean regression. In many real applications having multiple expectiles at different levels provides a more complete picture of the conditional distribution of the response variable. Multiple linear expectile regression model has been well studied [Newey W, Powell J. Asymmetric least squares estimation and testing, Econometrica. 1987;55:819–847; Efron B. Regression percentiles using asymmetric squared error loss, Stat Sin. 1991;1(93):125.], but it can be too restrictive for many real applications. In this paper, we derive a regression tree-based gradient boosting estimator for nonparametric multiple expectile regression. The new estimator, referred to as ER-Boost, is implemented in an R package erboost publicly available at http://cran.r-project.org/web/packages/erboost/index.html. We use two homoscedastic/heteroscedastic random-function-generator models in simulation to show the high predictive accuracy of ER-Boost. As an application, we apply ER-Boost to analyse North Carolina County crime data. From the nonparametric expectile regression analysis of this dataset, we draw several interesting conclusions that are consistent with the previous study using the economic model of crime. This real data example also provides a good demonstration of some nice features of ER-Boost, such as its ability to handle different types of covariates and its model interpretation tools.     

Penalized image averaging and discrimination with facial and fishery applications

In this paper we use a penalized likelihood approach to image warping in the context of discrimination and averaging. The choice of average image is formulated statistically by minimizing a penalized likelihood, where the likelihood measures the similarity between images after warping and the penalty is a measure of distortion of a warping. The notions of measures of similarity are given in terms of normalized image information. The measures of distortion are landmark based. Thus we use a combination of landmark and normalized image information. The average defined in the paper is also extended by allowing random perturbation of the landmarks. This strategy improves averages for discrimination purposes. We give here real applications from medical and biological areas.     

The effect of dependence on distribution of the functions of random variables

In this article, we study the effect of dependence on the distributional properties of functions of two random variables. Expressions for the cumulative distribution functions of the linear combinations, products, and ratios of two dependent random variables in terms of their associated copula are derived. We discuss the effect of dependence on quantities such as the variances of linear combinations of functions, the value-at-risk measure, and the stress–strength parameter. Several examples, a simulation study, and a real data analysis are provided to illustrate the result.     

On some sampling distributions for skew-normal population

The distribution of weighted function of independent skew-normal random variables, which includes the sample mean, is useful in many applications. In this paper, we derive this distribution and study the null distribution of a linear form and a quadratic form. Finally, we discuss some of its applications in control charts, in which the skew-normal model plays a key role.     

On confidence intervals for semiparametric expectile regression

In regression scenarios there is a growing demand for information on the conditional distribution of the response beyond the mean. In this scenario quantile regression is an established method of tail analysis. It is well understood in terms of asymptotic properties and estimation quality. Another way to look at the tail of a distribution is via expectiles. They provide a valuable alternative since they come with a combination of preferable attributes. The easy weighted least squares estimation of expectiles and the quadratic penalties often used in flexible regression models are natural partners. Also, in a similar way as quantiles can be seen as a generalisation of median regression, expectiles offer a generalisation of mean regression. In addition to regression estimates, confidence intervals are essential for interpretational purposes and to assess the variability of the estimate, but there is a lack of knowledge regarding the asymptotic properties of a semiparametric expectile regression estimate. Therefore confidence intervals for expectiles based on an asymptotic normal distribution are introduced. Their properties are investigated by a simulation study and compared to a boostrap-based gold standard method. Finally the introduced confidence intervals help to evaluate a geoadditive expectile regression model on childhood malnutrition data from India.     

Nonparametric estimation of multivariate extreme-value copulas

Extreme-value copulas arise in the asymptotic theory for componentwise maxima of independent random samples. An extreme-value copula is determined by its Pickands dependence function, which is a function on the unit simplex subject to certain shape constraints that arise from an integral transform of an underlying measure called spectral measure. Multivariate extensions are provided of certain rank-based nonparametric estimators of the Pickands dependence function. The shape constraint that the estimator should itself be a Pickands dependence function is enforced by replacing an initial estimator by its best least-squares approximation in the set of Pickands dependence functions having a discrete spectral measure supported on a sufficiently fine grid. Weak convergence of the standardized estimators is demonstrated and the finite-sample performance of the estimators is investigated by means of a simulation experiment.     

Quantile-based cumulative Kullback–Leibler divergence

The paper introduces a quantile-based cumulative Kullback–Leibler divergence and study its various properties. Unlike the distribution function approach, the quantile-based measure possesses some unique properties. The quantile functions used in many applied works do not have any tractable distribution functions where the proposed measure is a useful tool to compute the distance between two random variables. Some useful bounds are obtained for quantile-based residual cumulative Kullback–Leibler divergence and quantile-based reliability measures. Characterization results based on the functional forms of quantile-based residual Kullback–Leibler divergence are obtained for some well-known life distributions, namely exponential, Pareto II and beta.     

Statistical inference for a relaxation index of stochastic dominance under density ratio model

Stochastic dominance is usually used to rank random variables by comparing their distributions, so it is widely applied in economics and finance. In actual applications, complete stochastic dominance is too demanding to meet, so relaxation indexes of stochastic dominance have attracted more attention. The π index, the biggest gap between two distributions, can be a measure of the degree of deviation from complete dominance. The traditional estimation method is to use the empirical distribution functions to estimate it. Considering the populations under comparison are generally of the same nature, we can link the populations through density ratio model under certain condition. Based on this model, we propose a new estimator and establish its statistical inference theory. Simulation results show that the proposed estimator substantially improves estimation efficiency and power of the tests and coverage probabilities satisfactorily match the confidence levels of the tests, which show the superiority of the proposed estimator. Finally we apply our method to a real example of the Chinese household incomes.     

A Note on the Innovation Distribution of a Gamma Distributed Autoregressive Process

A representation of the innovation random variable for a gamma distributed first-order autoregressive process was found by Lawrance (1982) in the form of a compound Poisson distribution, connected with a shot-noise process. In this note we simplify the representation of Lawrance by providing a direct representation in terms of density functions.     

Measuring association between nominal categorical variables: an alternative to the Goodman–Kruskal lambda

As a measure of association between two nominal categorical variables, the lambda coefficient or Goodman–Kruskal's lambda has become a most popular measure. Its popularity is primarily due to its simple and meaningful definition and interpretation in terms of the proportional reduction in error when predicting a random observation's category for one variable given (versus not knowing) its category for the other variable. It is an asymmetric measure, although a symmetric version is available. The lambda coefficient does, however, have a widely recognized limitation: it can equal zero even when there is no independence between the variables and when all other measures take on positive values. In order to mitigate this problem, an alternative lambda coefficient is introduced in this paper as a slight modification of the Goodman–Kruskal lambda. The properties of the new measure are discussed and a symmetric form is introduced. A statistical inference procedure is developed and a numerical example is provided.     

A generalized linear mixed model for longitudinal binary data with a marginal logit link function

Longitudinal studies of a binary outcome are common in the health, social, and behavioral sciences. In general, a feature of random effects logistic regression models for longitudinal binary data is that the marginal functional form, when integrated over the distribution of the random effects, is no longer of logistic form. Recently, Wang and Louis (2003) proposed a random intercept model in the clustered binary data setting where the marginal model has a logistic form. An acknowledged limitation of their model is that it allows only a single random effect that varies from cluster to cluster. In this paper, we propose a modification of their model to handle longitudinal data, allowing separate, but correlated, random intercepts at each measurement occasion. The proposed model allows for a flexible correlation structure among the random intercepts, where the correlations can be interpreted in terms of Kendall's τ. For example, the marginal correlations among the repeated binary outcomes can decline with increasing time separation, while the model retains the property of having matching conditional and marginal logit link functions. Finally, the proposed method is used to analyze data from a longitudinal study designed to monitor cardiac abnormalities in children born to HIV-infected women.     

On the use of quasi-likelihood for estimation in hidden-Markov random fields

This paper illustrates the use of quasi-likelihood methods of inference for hidden Markov random fields. These are simple to use and can be employed under circumstances where only the model form and its covariance structure are specified. In particular they can be used to derive the same estimating equations as the E-M algorithm or change of measure methods, which make full distributional assumptions.     

A Dynamic Discrimination Information Based On Cumulative Residual Entropy And Its Properties

In this article, we introduce a measure of discrepancy between two life-time distributions based on cumulative residual entropy. The dynamic form of this measure is considered and some of its properties are obtained. The relations between dynamic form and some well-known concepts in reliability such as mean residual life-time, hazard rate order, and new better (worse) than used are studied.     

Uniform Representation of Product-Limit Integrals with Applications

Abstract.  Let X be a d -variate random vector that is completely observed, and let Y be a random variable that is subject to right censoring and left truncation. For arbitrary functions φ we consider expectations of the form E [ φ ( X ,  Y )], which appear in many statistical problems, and we estimate these expectations by using a product-limit estimator for censored and truncated data, extended to the context where covariates are present. An almost sure representation for these estimators is obtained, with a remainder term that is of a certain negligible order, uniformly over a class of φ -functions. This uniformity is important for the application to goodness-of-fit testing in regression and to inference for the regression depth, which we consider in more detail.     

Targeted estimation of state occupation probabilities for the non-Markov illness-death model

We use semi-parametric efficiency theory to derive a class of estimators for the state occupation probabilities of the continuous-time irreversible illness-death model. We consider both the setting with and without additional baseline information available, where we impose no specific functional form on the intensity functions of the model. We show that any estimator in the class is asymptotically linear under suitable assumptions about the estimators of the intensity functions. In particular, the assumptions are weak enough to allow the use of data-adaptive methods, which is important for making the identifying assumption of coarsening at random plausible in realistic settings. We suggest a flexible method for estimating the transition intensity functions of the illness-death model based on penalized Poisson regression. We apply this method to estimate the nuisance parameters of an illness-death model in a simulation study and a real-world application.     

Joint Gait-Cadence Analysis for Human Identification Using an Elastic Shape Framework

We perform human identification by gait recognition where subjects' gait is represented by silhouettes which are elements of a manifold of Square-Root Velocity functions. Gait cycles become stochastic processes on this manifold; cadence its rate of execution. Using geometry of this manifold, we compute mean gait cycle templates for subjects. An observation model, where test sequences are random perturbations of templates, produces likelihood functions for classification. We perform temporal registration—linear and nonlinear—of cycles with templates, removing cadence effects. In an experiment on 26 individuals, linear registration, preserving cadence, performs better than nonlinear registration, which removes cadence.