Partial sum process to check regression models with multiple correlated response: With an application for testing a change-point in profile data

We consider regression models with multiple correlated responses for each design point. Under the null hypothesis, a linear regression is assumed. For the least-squares residuals of this linear regression, we establish the limit of the partial sums. This limit is a projection on a certain subspace of the reproducing Kernel Hilbert space of a multivariate Brownian motion. Based on this limit, we propose a significance test of Kolmogorov-Smirnov type to test the null hypothesis and show that this result can be used to study a change-point problem in the case of linear profile data (panel data). We compare our proposed method, which does not rely on any distributional assumptions, with the likelihood ratio test in a simulation study.     

异方差回归中的广义方差比检验

在同方差假设之下,线性模型在回归分析的理论与应用方面起着突出的作用,很受许多研究工作者的青睐.然而,回归模型中同方差性这一标准假设不一定总是成立的.因此我们考虑了用一类基于似残差的方法来检验异方差情形下线性模型拟合观测数据的情况.本文既给出了大量的模拟,又给出了实际数据作为应用的例子.效果都很好.     

线性回归模型多个离群点的向前逐步诊断方法

当线性回归模型中存在多个离群点时,经典的诊断方法常常因掩盖和淹没现象而失效,导致模型误用。针对此问题,本文在回顾有关文献的基础上,将稳健回归技术与经典诊断量相结合,提出一种向前逐步诊断方法。通过对模拟数据的分析,说明该法可有效地识别回归数据中潜在的离群点,并作正式的统计检验。     

Asymptotics of the goodness-of-fit test for a partial linear model with randomly censored data

In this paper, we discuss the problem of testing the hypothesis that the underlying regression is a partial linear model. A test statistic, which is based on the quadratic form of a cusum process of residuals, is proposed. The asymptotic distributions of the test statistic under null hypothesis and the local alternative hypothesis are given. The number simulation shows that the test is available.     

Testing for a constant coefficient of variation in nonparametric regression by empirical processes

In the common nonparametric regression model, we consider the problem of testing the hypothesis that the coefficient of the scale and location function is constant. The test is based on a comparison of the standardized (by a local linear estimate of the scale function) observations with their mean. We show weak convergence of a centered version of this process to a Gaussian process under the null hypothesis and the alternative and use this result to construct a test for the hypothesis of a constant coefficient of variation in the nonparametric regression model. A small simulation study is also presented to investigate the finite sample properties of the new test.     

A Monte Carlo evaluation of the performance of two new tests for symmetry

We propose two new tests for symmetry based on well-known characterisations of symmetric distributions. The performance of the new tests is evaluated and compared to that of other existing tests by means of a Monte Carlo study. All tests are carried out in a regression setup where we test whether the error distribution in a linear regression model is symmetric. It is found that the newly proposed tests perform favourably compared to the other tests.     

医学研究中标准曲线修正的初探

目的:在医学研究和临床检验中经常使用定量检测法,在该方法中标准曲线被广泛应用,且其准确性直接影响检测结果.从统计学角度对其进行修正,使结果更为可靠.方法:应用直线回归法对直线型标准曲线进行处理,求出直线回归方程;对曲线型标准曲线,先将指数方程取对数后变成直线方程,然后求直线回归方程后再回代,求出曲线回归方程.结果:以蛋白量与其对应的OD值间关系求得直线回归方程y=0 .0 0 6x;以TNFα浓度与对应的OD值间关系求得指数回归方程y=0 .0 1 5 x0 .732 ,对两者进行F检验,回归效果均极显著.结论:应用回归法可获得定量检测法中常用的直线型及曲线型标准曲线的方程,据此可直接计算出待测样本所需数据,使其更为简便、准确、可靠.     

基于灰色理论的线性回归模型的参数理论及应用

线性回归分析是数理统计学的基本内容之一,但传统数理统计学中的线性回归分析,是建立在非模糊的随机数据上的线性回归估计和回归系数检验。而现实经济社会中大量存在含有模糊或灰色等不分明性的数据,面对这类不分明性数据,简单地使用传统的统计分析方法显然是不足取的。要想较为科学合理地分析与决策,需要利用灰色系统的相关理论,应用于随机系统信息,从而建立灰色线性回归估计、预测和灰色回归系数检验的基本理论方法,并把该方法应用于金融分析实例中,与经典线性回归分析方法进行对比,足见灰色线性回归方法能够提供比经典线性回归较多的有效信息,从而提出处理不分明数据新的方法。     

A Simple Goodness-of-fit Test for Linear Models Under a Random Design Assumption

Let (X,Y) denote a random vector with decomposition Y = f(X) + where f(x) = E[Y ¦ X = x] is the regression of Y on X. In this paper we propose a test for the hypothesis that f is a linear combination of given linearly independent regression functions g1,..,gd. The test is based on an estimator of the minimal L2-distance between f and the subspace spanned by the regression functions. More precisely, the method is based on the estimation of certain integrals of the regression function and therefore does not require an explicit estimation of the regression. For this reason the test proposed in this paper does not depend on the subjective choice of a smoothing parameter. Differences between the problem of regression diagnostics in the nonrandom and random design case are also discussed.     

Normal Linear Regression Models With Recursive Graphical Markov Structure

A multivariate normal statistical model defined by the Markov properties determined by an acyclic digraph admits a recursive factorization of its likelihood function (LF) into the product of conditional LFs, each factor having the form of a classical multivariate linear regression model (≡WMANOVA model). Here these models are extended in a natural way to normal linear regression models whose LFs continue to admit such recursive factorizations, from which maximum likelihood estimators and likelihood ratio (LR) test statistics can be derived by classical linear methods. The central distribution of the LR test statistic for testing one such multivariate normal linear regression model against another is derived, and the relation of these regression models to block-recursive normal linear systems is established. It is shown how a collection of nonnested dependent normal linear regression models (≡Wseemingly unrelated regressions) can be combined into a single multivariate normal linear regression model by imposing a parsimonious set of graphical Markov (≡Wconditional independence) restrictions.     

基于多元线性回归模型的东北地区需水量分析

多元线性回归模型在社会、经济、技术以及众多自然科学研究领域中已被广泛使用,某个地区需水量应与该地区多种因素有关,故选取东北地区的GDP、水库蓄水总量、人均可支配收入、城市绿地面积和工业用水量等5个因素,借助MATLAB软件阐明了多元线性回归模型在东北地区需水量分析中的应用.并通过皮尔森相关性检验、拟合优度检验、F检验、t检验和残差分析的方法对模型进行优化,得到了准确可靠的多元线性回归模型,此模型具有拟合程度高、简易、直观等优势,为多元线性回归模型在需水量分析中的应用提供了有力参考.     

Heteroscedasticity checks for regression models

For checking on heteroscedasticity in regression models, a unified approach is proposed to constructing test statistics in parametric and nonparametric regression models. For nonparametric regression, the test is not affected sensitively by the choice of smoothing parameters which are involved in estimation of the nonparametric regression function. The limiting null distribution of the test statistic remains the same in a wide range of the smoothing parameters. When the covariate is one-dimensional, the tests are, under some conditions, asymptotically distribution-free. In the high-dimensional cases, the validity of bootstrap approximations is investigated. It is shown that a variant of the wild bootstrap is consistent while the classical bootstrap is not in the general case, but is applicable if some extra assumption on conditional variance of the squared error is imposed. A simulation study is performed to provide evidence of how the tests work and compare with tests that have appeared in the literature. The approach may readily be extended to handle partial linear, and linear autoregressive models.     

Partially linear single-index beta regression model and score test

An important model in handling the multivariate data is the partially linear single-index regression model with a very flexible distribution—beta distribution, which is commonly used to model data restricted to some open intervals on the line. In this paper, the score test is extended to the partially linear single-index beta regression model. The penalized likelihood estimation based on P-spline is proposed. Based on the estimation, the score test statistics about varying dispersion parameter is given. Its asymptotical property is investigated. Both simulated examples are used to illustrate our proposed methods.     

Conditionally specified models and dimension reduction in the exponential families

We consider informative dimension reduction for regression problems with random predictors. Based on the conditional specification of the model, we develop a methodology for replacing the predictors with a smaller number of functions of the predictors. We apply the method to the case where the inverse conditional model is in the linear exponential family. For such an inverse model and the usual Normal forward regression model it is shown that, for any number of predictors, the sufficient summary has dimension two or less. In addition, we develop a test of dimensionality. The relationship of our method with the existing dimension reduction theory based on the marginal distribution of the predictors is discussed.     

Quantile regression for index tracking and enhanced indexation

Quantile regression differs from traditional least-squares regression in that one constructs regression lines for the quantiles of the dependent variable in terms of the independent variable. In this paper we apply quantile regression to two problems in financial portfolio construction, index tracking and enhanced indexation. Index tracking is the problem of reproducing the performance of a stock market index, but without purchasing all of the stocks that make up the index. Enhanced indexation deals with the problem of out-performing the index. We present a mixed-integer linear programming formulation of these problems based on quantile regression. Our formulation includes transaction costs, a constraint limiting the number of stocks that can be in the portfolio and a limit on the total transaction cost that can be incurred. Numeric results are presented for eight test problems drawn from major world markets, where the largest of these test problems involves over 2000 stocks.     

带有约束条件的方差扩大模型的Score检验

本对带有的约束条件的约性模型方差扩大模型的假检测问题给出了Score检验统计量,指出两个主要的结果。     

Testing Regression Coefficients in High-Dimensional and Sparse Settings

In the high-dimensional setting, this article considers a canonical testing problem in multivariate analysis, namely testing coefficients in linear regression models. Several tests for highdimensional regression coefficients have been proposed in the recent literature. However, these tests are based on the sum of squares type statistics, that perform well under the dense alternatives and suffer from low power under the sparse alternatives. In order to attack this issue, we introduce a new test statistic which is based on the maximum type statistic and magnifies the sparse signals. The limiting null distribution of the test statistic is shown to be the extreme value distribution of type I and the power of the test is analysed. In particular, it is shown theoretically and numerically that the test is powerful against sparse alternatives. Numerical studies are carried out to examine the numerical performance of the test and to compare it with other tests available in the literature.     

Post-J test inference in non-nested linear regression models

This paper considers the post-J test inference in non-nested linear regression models. Post-J test inference means that the inference problem is considered by taking the first stage J test into account. We first propose a post-J test estimator and derive its asymptotic distribution. We then consider the test problem of the unknown parameters, and a Wald statistic based on the post-J test estimator is proposed. A simulation study shows that the proposed Wald statistic works perfectly as well as the two-stage test from the view of the empirical size and power in large-sample cases, and when the sample size is small, it is even better. As a result,the new Wald statistic can be used directly to test the hypotheses on the unknown parameters in non-nested linear regression models.     

Testing the equality of several linear regression models

The linear regression models are widely used in different research fields, and often there is the need to analyze if there are similarities between two or more different linear models or to verify if a given relation between two variables remains the same in different intervals of time, in particular in cases where small differences might make a big difference. Motivated by these problems the authors consider a test of equality of k linear regression models which is a simultaneous test of equality of slopes, intercepts and variances. In order to overcome the extreme difficulties that exist in the use of the exact distribution of the likelihood ratio test (LRT) statistic and to make this test reliable and easy to use, we propose the use of near-exact distributions to approximate the distribution of the LRT statistic, under \(H_0\), in the balanced case, and of new asymptotic approximations for the unbalanced case. The near-exact approximations are built by approximating one factor of an adequate factorization of the characteristic function of the logarithm of the LRT statistic and may be easily implemented. The asymptotic approximations are developed using an expansion for the ratio of gamma functions. The quality of these approximations is analyzed and confirmed. Power studies are conducted in order to better assess the performance of the test. Finally to illustrate the applicability of the test we consider a real data set of gross domestic product at market prices and final consumption expenditure in European countries and one tests the existence of similarities between countries.     

Forecasting Loss Given Default models: impact of account characteristics and the macroeconomic state

On the basis of two data sets containing Loss Given Default (LGD) observations of home equity and corporate loans, we consider non-linear and non-parametric techniques to model and forecast LGD. These techniques include non-linear Support Vector Regression (SVR), a regression tree, a transformed linear model and a two-stage model combining a linear regression with SVR. We compare these models with an ordinary least squares linear regression. In addition, we incorporate several variants of 11 macroeconomic indicators to estimate the influence of the economic state on loan losses. The out-of-time set-up is complemented with an out-of-sample set-up to mitigate the limited number of credit crisis observations available in credit risk data sets. The two-stage/transformed model outperforms the other techniques when forecasting out-of-time for the home equity/corporate data set, while the non-parametric regression tree is the best performer when forecasting out-of-sample. The incorporation of macroeconomic variables significantly improves the prediction performance. The downturn impact ranges up to 5% depending on the data set and the macroeconomic conditions defining the downturn. These conclusions can help financial institutions when estimating LGD under the internal ratings-based approach of the Basel Accords in order to estimate the downturn LGD needed to calculate the capital requirements. Banks are also required as part of stress test exercises to assess the impact of stressed macroeconomic scenarios on their Profit and Loss (P&L) and banking book, which favours the accurate identification of relevant macroeconomic variables driving LGD evolutions.