Nonparametric regression for dependent data in the errors-in-variables problem

We consider the nonparametric estimation of the regression functions for dependent data. Suppose that the covariates are observed with additive errors in the data and we employ nonparametric deconvolution kernel techniques to estimate the regression functions in this paper. We investigate how the strength of time dependence affects the asymptotic properties of the local constant and linear estimators. We treat both short-range dependent and long-range dependent linear processes in a unified way and demonstrate that the long-range dependence (LRD) of the covariates affects the asymptotic properties of the nonparametric estimators as well as the LRD of regression errors does.     

Nonparametric estimation of extreme conditional quantiles

The estimation of extreme conditional quantiles is an important issue in different scientific disciplines. Up to now, the extreme value literature focused mainly on estimation procedures based on independent and identically distributed samples. Our contribution is a two-step procedure for estimating extreme conditional quantiles. In a first step nonextreme conditional quantiles are estimated nonparametrically using a local version of [Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33–50.] regression quantile methodology. Next, these nonparametric quantile estimates are used as analogues of univariate order statistics in procedures for extreme quantile estimation. The performance of the method is evaluated for both heavy tailed distributions and distributions with a finite right endpoint using a small sample simulation study. A bootstrap procedure is developed to guide in the selection of an optimal local bandwidth. Finally the procedure is illustrated in two case studies.     

A Laguerre polynomial approximation for a goodness-of-fit test for exponential distribution based on progressively censored data

This paper proposes an approximation to the distribution of a goodness-of-fit statistic proposed recently by Balakrishnan et al. [Balakrishnan, N., Ng, H.K.T. and Kannan, N., 2002, A test of exponentiality based on spacings for progressively Type-II censored data. In: C. Huber-Carol et al. (Eds.), Goodness-of-Fit Tests and Model Validity (Boston: Birkhäuser), pp. 89–111.] for testing exponentiality based on progressively Type-II right censored data. The moments of this statistic can be easily calculated, but its distribution is not known in an explicit form. We first obtain the exact moments of the statistic using Basu's theorem and then the density approximants based on these exact moments of the statistic, expressed in terms of Laguerre polynomials, are proposed. A comparative study of the proposed approximation to the exact critical values, computed by Balakrishnan and Lin [Balakrishnan, N. and Lin, C.T., 2003, On the distribution of a test for exponentiality based on progressively Type-II right censored spacings. Journal of Statistical Computation and Simulation, 73 (4), 277–283.], is carried out. This reveals that the proposed approximation is very accurate.     

Investigating the sensitivity of Gaussian processes to the choice of their correlation function and prior specifications

A Gaussian process (GP) can be thought of as an infinite collection of random variables with the property that any subset, say of dimension n, of these variables have a multivariate normal distribution of dimension n, mean vector β and covariance matrix Σ [O'Hagan, A., 1994, Kendall's Advanced Theory of Statistics, Vol. 2B, Bayesian Inference (John Wiley & Sons, Inc.)]. The elements of the covariance matrix are routinely specified through the multiplication of a common variance by a correlation function. It is important to use a correlation function that provides a valid covariance matrix (positive definite). Further, it is well known that the smoothness of a GP is directly related to the specification of its correlation function. Also, from a Bayesian point of view, a prior distribution must be assigned to the unknowns of the model. Therefore, when using a GP to model a phenomenon, the researcher faces two challenges: the need of specifying a correlation function and a prior distribution for its parameters. In the literature there are many classes of correlation functions which provide a valid covariance structure. Also, there are many suggestions of prior distributions to be used for the parameters involved in these functions. We aim to investigate how sensitive the GPs are to the (sometimes arbitrary) choices of their correlation functions. For this, we have simulated 25 sets of data each of size 64 over the square [0, 5]×[0, 5] with a specific correlation function and fixed values of the GP's parameters. We then fit different correlation structures to these data, with different prior specifications and check the performance of the adjusted models using different model comparison criteria.     

Robust non-parametric smoothing of non-stationary time series

Motivated by the need of extracting local trends and low frequency components in non-stationary time series, this paper discusses methods of robust non-parametric smoothing. Basic approach is the combination of the parametric M-estimation with kernel and local polynomial regression methods. The result is an iterative estimator that retains a linear structure, but has kernel weights also in the direction of the prediction errors. The design of smoothing coefficients is carried out with robust cross-validation criteria and rules of thumb. The method works well both to remove the influence of patches of outliers and to detect the local breaks and persistent structural change in time series.     

矩阵多项式的逆

本文证明了数域Ｋ上矩阵Ａ的多项式的逆仍是Ｋ上矩阵Ａ的多项式,并给出了求矩阵多项式的逆的方法．     

关于完全三部图的色等价性

设Ｇ为简单图,Ｐ（Ｇ,λ）为Ｇ的色多项式。若简单图Ｈ满足Ｐ（Ｈ,λ）＝Ｐ（Ｇ,λ）,则称Ｈ与Ｇ色等价。     

Cramer法则与Vandermonde行列式的应用

本文利用Cramer法则与Vandermonde行列式的结果,推导出了Lagrange插值公式以及一些缺项的多项式的表达式。这是对Cramer法则与Vander－monde行列式行列工的一个很好的综合运用。     

Joint detection for functional polynomial regression with autoregressive errors

In this article, we are concerned with detecting the true structure of a functional polynomial regression with autoregressive (AR) errors. The first issue is to detect which orders of the polynomial are significant in functional polynomial regression. The second issue is to detect which orders of the AR process in the AR errors are significant. We propose a shrinkage method to deal with the two problems: polynomial order selection and autoregressive order selection. Simulation studies demonstrate that the new method can identify the true structure. One empirical example is also presented to illustrate the usefulness of our method.     

Fractional Moments and Maximum Entropy: Geometric Meaning

We consider the problem of recovering a probability density on a bounded or unbounded subset D of [0, ∞), from the knowledge of its sequence of fractional moments within a maximum entropy (MaxEnt) setup. Based upon entropy convergence results previously formulated, the fractional moments are selected so that the entropy of the MaxEnt approximation be minimum. A geometric interpretation of the reconstruction procedure is formulated as follows: the two moment curves generated by the unknown density and its MaxEnt approximation are interpolating in Hermite-Birkoff sense; that is, they are both interpolating and tangent at the selected nodes.