Krylov Sequences as a Tool for Analysing Iterated Regression Algorithms

Abstract.  We use Krylov sequences to analyse a class of regression methods based on successive identification of latent factors. Some results already proved for partial least squares regression (PLSR) are shown to hold for other methods also. We prove that the well-known peculiar pattern of alternating shrinkage and inflation of the principal components is not unique for PLSR. We also show that for any method in the class under study, the coefficient of determination is always at least as high as for principal components regression with the same number of factors.     

Lagged Regression Residuals and Serial-Correlation Tests

A new family of statistics is proposed to test for the presence of serial correlation in linear regression models. The tests are based on partial sums of lagged cross-products of regression residuals that define a class of interesting Gaussian processes. These processes are characterized in terms of regressor functions, the serial-correlation structure, the distribution of the noise process, and the order of the lag of the cross-products of residuals. It is shown that these four factors affect the lagged residual processes independently. Large-sample distributional results are presented for test statistics under the null hypothesis of no serial correlation or for alternatives from a range of interesting hypotheses. Some indication of the circumstances to which the asymptotic results apply in finite-sample situations and of those to which they should be applied with some caution are obtained through a simulation study. Tables of selected quantiles of the proposed tests are also given. The tests are illustrated with two examples taken from the empirical literature. It is also proposed that plots of lagged residual processes be used as diagnostic tools to gain insight into the correlation structure of residuals derived from regression fits.     

Change-Point Detection in Two-Phase Regression with Inequality Constraints on the Regression Parameters

Two-phase regression models with inequality constraints on the regression coefficients and with a small number of measurements is considered. A new test based on the likelihood ratio in linear model with inequality constraints for the presence of a change-point is proposed. Numerical approximations to the powers against various alternatives are given and compared with the powers of the likelihood ratio test in the two-phase regression models without inequality constraints, the backwards CUSUM test, and the k-linear-r-ahead recursive residuals tests. Performance of related likelihood based estimators of the change-point is briefly studied in a Monte Carlo experiment.     

Tests for Symmetric Error Distribution in Linear and Nonparametric Regression Models

In linear and nonparametric regression models, the problem of testing for symmetry of the distribution of errors is considered. We propose a test statistic which utilizes the empirical characteristic function of the corresponding residuals. The asymptotic null distribution of the test statistic as well as its behavior under alternatives is investigated. A simulation study compares bootstrap versions of the proposed test to other more standard procedures.     

Preliminary Test Estimators Induced by Three Large Sample Tests for Stochastic Constraints in a Regression Model with Multivariate Student-t Error

In this article, we consider the preliminary test approach to the estimation of the regression parameter in a multiple regression model with multivariate Student-t distribution. The preliminary test estimators (PTE) based on the Wald (W), Likelihood Ratio (LR), and Lagrangian Multiplier (LM) tests are given under the suspicion of stochastic constraints occurring. The bias, mean square error matr ix (MSEM), and weighted mean square error (WMSE) of the proposed estimators are derived and compared. The conditions of superiority of the proposed estimators are obtained. Finally, we conclude that the optimum choice of the level of significance becomes the traditional choice by using the W test.     

On Permutation Tests in Multiple Regression and Analysis of Covariance Problems

We introduce distribution-free permutation tests and corresponding estimates for studying the effect of a treatment variable x on a response y. The methods apply in the presence of a multivariate covariate z. They are based on the assumption that the treatment values are assigned randomly to the subjects.     

Statistical Inference in Orthogonal Regression for Three-Part Compositional Data Using a Linear Model with Type-II Constraints

Orthogonal regression is a proper tool to analyze relations between two variables when three-part compositional data, i.e., three-part observations carrying relative information (like proportions or percentages), are under examination. When linear statistical models with type-II constraints (constraints involving other parameters besides the ones of the unknown model) are employed for estimating the parameters of the regression line, approximate variances and covariances of the estimated line coefficients can be determined. Moreover, the additional assumption of normality enables to construct confidence domains and perform hypotheses testing. The theoretical results are applied to a real-world example.     

Law of iterated logarithm and almost sure limit points for random sums

Let {X_n,n≥1} be a sequence of independent identically distributed (i.i.d) random variables with a common distribution function F. When F belongs to the domain of partial attraction of a Semi-Stable law with index ,0<<2, we give complete solution to the results of R. Vasudeva and G. Divanji [Law of iterated logarithm for random subsequences, Statist. Probab. Lett. 12 (1991) 189–194], where they obtained Chover’s form of the law of iterated logarithm for random subsequences. Further, we extended the situation in obtaining almost sure limit points for random subsequences.     

城乡居民消费结构变动影响因素的比较分析——基于PLSR的阶段性证据

居民消费结构变动的影响因素可以根据新兴古典经济学的分析框架进行分析。虽然没有与专业化水平、交易效率和经济绩效直接对应的数据，但是可以选择一组涵盖三个方面的指标，运用偏最小二乘回归方法进行实证研究。基于1978～2005的数据，对中国城乡居民消费结构变动的影响因素进行比较分析，其结果显示了影响因素的阶段性差异。     

Laws of the Iterated Logarithm and a Moderate Deviation of MLE for the Proportional Hazards Model with Incomplete Information

In this paper, we obtain a law of iterated logarithm, a Chung-type law of iterated logarithm, and a moderate deviation result of the maximum likelihood estimator (MLE) for the unknown regression parameter vector in a proportional hazards model with incomplete information.     

Tests of heteroscedasticity and correlation in multivariate t regression models with AR and ARMA errors

Heteroscedasticity checking in regression analysis plays an important role in modelling. It is of great interest when random errors are correlated, including autocorrelated and partial autocorrelated errors. In this paper, we consider multivariate t linear regression models, and construct the score test for the case of AR(1) errors, and ARMA(s,d) errors. The asymptotic properties, including asymptotic chi-square and approximate powers under local alternatives of the score tests, are studied. Based on modified profile likelihood, the adjusted score test is also developed. The finite sample performance of the tests is investigated through Monte Carlo simulations, and also the tests are illustrated with two real data sets.     

Complete convergence for weighted sums of i.i.d. random variables with applications in regression estimation and EV model

In this paper, we obtain complete convergence results for Stout type weighted sums of i.i.d. random variables. A strong law for weighted sums of i.i.d. random variables is also obtained. As the applications of the strong law, the strong consistency and rate of the nonparametric regression estimations and the rates of the strong consistency of LS estimators for the unknown parameters of the simple linear errors in variables (EV) model are given.     

Estimation of Slope for Linear Regression Model with Uncertain Prior Information and Student-t Error

This article considers estimation of the slope parameter of the linear regression model with Student-t errors in the presence of uncertain prior information on the value of the unknown slope. Incorporating uncertain non sample prior information with the sample data the unrestricted, restricted, preliminary test, and shrinkage estimators are defined. The performances of the estimators are compared based on the criteria of unbiasedness and mean squared errors. Both analytical and graphical methods are explored. Although none of the estimators is uniformly superior to the others, if the non sample information is close to its true value, the shrinkage estimator over performs the rest of the estimators.     

A comparison of equally spaced designs with different correlation structures in one and more dimensions

Equally spaced designs are compared using the generalized variance as a measure of efficiency. Results for polynomial models are derived on the increased efficiency arising from increasing the number of design points when the regions are fixed and when the regions are expanded. The effects of dependence among the observations on these results are studied by considering a particular family of stationary correlated error structures.     

Tests and confidence intervals for the shape parameter of a gamma distribution

Inference based on the Central Limit Theorem has only first order accuracy. We give tests and confidence intervals (CIs) of second orderaccuracy for the shape parameter ρ of a gamma distribution for both the unscaled and scaled cases.

Tests and CIs based on moment and cumulant estimates are considered as well as those based on the maximum likelihood estimate (MLE).

For the unscaled case the MLE is the moment estimate of order zero; the most efficient moment estimate of integral order is the sample mean, having asymptotic relative efficiency (ARE) .61 when ρ= 1.

For the scaled case the most efficient moment estimate is a functionof the mean and variance. Its ARE is .39 when ρ = 1.

Our motivation for constructing these tests of ρ = 1 and CIs forρ is to provide a simple and convenient method for testing whether a distribution is exponential in situations such as rainfall models where such an assumption is commonly made.     

Exact and approximate distributions of the chi-square statistic for equiprobability

The distribution of the chi-square goodness-of-fit statistic is studied in the equiprobable case. Tables of exact critical values are given for a = .1, .05, .01, .005; k = 2(1)4, N = 26(1)50; k = 5, N = 26(1)40; k = 6(1)10, N = 26(1)30, where a is the desired significance level, k is the number of cells and N is the sample size. Methods of fitting the true distribution are compared. If k> 3, it is found that a simple additive adjustment to the asymptotic chi-square fit leads to high accuracy even for N between 10 and 20. For k = 2, the Yates corrected chi-square statistic is very accurately fitted by the usual chi-square distribution.     

A comparison of two approaches for power and sample size calculations in logistic regression models

Whittemore (1981) proposed an approach for calculating the sample size needed to test hypotheses with specified significance and power against a given alternative for logistic regression with small response probability. Based on the distribution of covariate, which could be either discrete or continuous, this approach first provides a simple closed-form approximation to the asymptotic covariance matrix of the maximum likelihood estimates, and then uses it to calculate the sample size needed to test a hypothesis about the parameter. Self et al. (1992) described a general approach for power and sample size calculations within the framework of generalized linear models, which include logistic regression as a special case. Their approach is based on an approximation to the distribution of the likelihood ratio statistic. Unlike the Whittemore approach, their approach is not limited to situations of small response probability. However, it is restricted to models with a finite number of covariate configurations. This study compares these two approaches to see how accurate they would be for the calculations of power and sample size in logistic regression models with various response probabilities and covariate distributions. The results indicate that the Whittemore approach has a slight advantage in achieving the nominal power only for one case with small response probability. It is outperformed for all other cases with larger response probabilities. In general, the approach proposed in Self et al. (1992) is recommended for all values of the response probability. However, its extension for logistic regression models with an infinite number of covariate configurations involves an arbitrary decision for categorization and leads to a discrete approximation. As shown in this paper, the examined discrete approximations appear to be sufficiently accurate for practical purpose.