Bootstrap inference in local polynomial regression of time series

In this paper we consider the inferential aspect of the nonparametric estimation of a conditional function , where X _t,m represents the vector containing the m conditioning lagged values of the series. Here is an arbitrary measurable function. The local polynomial estimator of order p is used for the estimation of the function g, and of its partial derivatives up to a total order p. We consider α-mixing processes, and we propose the use of a particular resampling method, the local polynomial bootstrap, for the approximation of the sampling distribution of the estimator. After analyzing the consistency of the proposed method, we present a simulation study which gives evidence of its finite sample behaviour.     

Variable selection for partially linear proportional hazards model with covariate measurement error

In survival analysis, we may encounter the following three problems: nonlinear covariate effect, variable selection and measurement error. Existing studies only address one or two of these problems. The goal of this study is to fill the knowledge gap and develop a novel approach to simultaneously address all three problems. Specifically, a partially time-varying coefficient proportional hazards model is proposed to more flexibly describe covariate effects. Corrected score and conditional score approaches are employed to accommodate potential measurement error. For the selection of relevant variables and regularised estimation, a penalisation approach is adopted. It is shown that the proposed approach has satisfactory asymptotic properties. It can be effectively realised using an iterative algorithm. The performance of the proposed approach is assessed via simulation studies and further illustrated by application to data from an AIDS clinical trial.     

Generalized growth curve models for longitudinal data in application to a randomized controlled trial

Growth curve analysis is beneficial in longitudinal studies, where the pattern of response variables measured repeatedly over time is of interest, yet unknown. In this article, we propose generalized growth curve models under a polynomial regression framework and offer a complete process that identifies the parsimonious growth curves for different groups of interest, as well as compares the curves. A higher order of a polynomial degree generally provides more flexible regression, yet it may suffer from the complicated and overfitted model in practice. Therefore, we employ the model selection procedure that chooses the optimal degree of a polynomial consistently. Consideration of a quadratic inference function (Qu et al., 2000) for estimation on regression parameters is addressed and estimation efficiency is improved by incorporating the within-subject correlation commonly existing in longitudinal data. In biomedical studies, it is of particular interest to compare multiple treatments and provide an effective one. We further conduct the hypothesis test that assesses the equality of the growth curves through an asymptotic chi-square test statistic. The proposed methodology is employed on a randomized controlled longitudinal dataset on depression. The effectiveness of our procedure is also confirmed with simulation studies.     

On approximating distribution of the quadratic discriminant function

The quadratic discriminant function (QDF) with known parameters has been represented in terms of a weighted sum of independent noncentral chi-square variables. To approximate the density function of the QDF as m-dimensional exponential family, its moments in each order have been calculated. This is done using the recursive formula for the moments via the Stein's identity in the exponential family. We validate the performance of our method using simulation study and compare with other methods in the literature based on the real data. The finding results reveal better estimation of misclassification probabilities, and less computation time with our method.     

Bimatrix variate gamma-beta distributions

Two bimatrix distributions with beta and gamma marginals are introduced. Various properties (including product moments of determinants and traces, entropies, marginal distributions) are derived. Parameter estimation by the method of maximum likelihood is discussed. The performance and efficiencies of the maximum likelihood estimators and the associated confidence intervals are assessed by simulation. The efficiencies are compared versus those for the maximum likelihood estimators and the associated confidence intervals based on matrix variate gamma distributions. A discussion of possible applications of the bimatrix distributions is given.     

Orthogonal polynomials generated by random vectors

Every random q-vector with finite moments generates a set of orthonormal polynomials. These are generated from the basis functions xn = xn11…xnqq using Gram–Schmidt orthogonalization. One can cycle through these basis functions using any number of ways. Here, we give results using minimum cycling. The polynomials look simpler when centered about the mean of X, and still simpler form when X is symmetric about zero. This leads to an extension of the multivariate Hermite polynomial for a general random vector symmetric about zero. As an example, the results are applied to the multivariate normal distribution.     

Fitting polynomial trend to time series by the method of Buys-Ballot estimators

The statistical properties of two closed-form estimators of the parameters of the quadratic time trend model are derived. The estimators are based on the derived variables from Buys-Ballot table. The estimators are derived by assuming that error term is identically and independently distributed. However, the validity of this assumption is sometimes difficult to verify. We also study, through simulations, the impact of misspecifying the error distribution on the estimation and prediction accuracy in the quadratic time trend model. It is shown that the estimators are inconsistent in the presence of misspecification. T methods are illustrated with real-life examples.     

用级数法导出各向同性谐振子径向矩阵元的通项

利用广义拉盖尔多项式的级数表达式和分步积分法,给出了各向同性谐振子径向矩阵元的另一种表达式.     

On boosting kernel regression

In this paper we propose a simple multistep regression smoother which is constructed in an iterative manner, by learning the Nadaraya–Watson estimator with _L2$L_2$ boosting. We find, in both theoretical analysis and simulation experiments, that the bias converges exponentially fast, and the variance diverges exponentially slow. The first boosting step is analysed in more detail, giving asymptotic expressions as functions of the smoothing parameter, and relationships with previous work are explored. Practical performance is illustrated by both simulated and real data.     

Practical Automated Reasoning on Inequalities: Generic Programs for Inequality Proving and Discovering

１Ａｌｇｅｂｒａｉｃａｐｐｒｏａｃｈｅｓ：ａｃｈｉｅｖｅｍｅｎｔｓａｎｄｄｉｆｉｃｕｌｔｉｅｓＷｅｈａｖｅｒｅａｓｏｎｔｏｂｅｌｉｅｖｅｔｈａｔｃｏｍｐｕｔｅｒｗｉｌｐｌａｙａｍｕｃｈｍｏｒｅｉｍｐｏｒｔａｎｔｒｏｌｅｉｎｒｅａｓｏｎｉｎｇｓｃｉｅ...