Duality and local sensitivity analysis in least squares,minimax, and least absolute values regressions

Abstract

This paper deals with the problem of local sensitivity analysis in regression, i.e., how sensitive the results of a regression model (objective function, parameters, and dual variables) are to changes in the data. We use a general formula for local sensitivities in optimization problems to calculate the sensitivities in three standard regression problems (least squares, minimax, and least absolute values). Closed formulas for all sensitivities are derived. Sensitivity contours are presented to help in assessing the sensitivity of each observation in the sample. The dual problems of the minimax and least absolute values are obtained and interpreted. The proposed sensitivity measures are shown to deal more effectively with the masking problem than the existing methods. The methods are illustrated by their application to some examples and graphical illustrations are given.     

Fitting linear regression models to censored data by least squares and maximum likelihood methods

Several approaches have been suggested for fitting linear regression models to censored data. These include Cox's propor­tional hazard models based on quasi-likelihoods. Methods of fitting based on least squares and maximum likelihoods have also been proposed. The methods proposed so far all require special purpose optimization routines. We describe an approach here which requires only a modified standard least squares routine.

We present methods for fitting a linear regression model to censored data by least squares and method of maximum likelihood. In the least squares method, the censored values are replaced by their expectations, and the residual sum of squares is minimized. Several variants are suggested in the ways in which the expect­ation is calculated. A parametric (assuming a normal error model) and two non-parametric approaches are described. We also present a method for solving the maximum likelihood equations in the estimation of the regression parameters in the censored regression situation. It is shown that the solutions can be obtained by a recursive algorithm which needs only a least squares routine for optimization. The suggested procesures gain considerably in computational officiency. The Stanford Heart Transplant data is used to illustrate the various methods.     

空间回归模型估计中的最小二乘法

空间回归模型由于引入了空间地理信息而使得其参数估计变得复杂,因为主要采用最大似然法,致使一般人认为在空间回归模型参数估计中不存在最小二乘法。通过分析空间回归模型的参数估计技术,研究发现,最小二乘法和最大似然法分别用于估计空间回归模型的不同的参数,只有将两者结合起来才能快速有效地完成全部的参数估计。数理论证结果表明,空间回归模型参数最小二乘估计量是最佳线性无偏估计量。空间回归模型的回归参数可以在估计量为正态性的条件下而实施显著性检验,而空间效应参数则不可以用此方法进行检验。     

Least squares variogram fitting by spatial subsampling

Summary. Least squares methods are popular for fitting valid variogram models to spatial data. The paper proposes a new least squares method based on spatial subsampling for variogram model fitting. We show that the method proposed is statistically efficient among a class of least squares methods, including the generalized least squares method. Further, it is computationally much simpler than the generalized least squares method. The method produces valid variogram estimators under very mild regularity conditions on the underlying random field and may be applied with different choices of the generic variogram estimator without analytical calculation. An extension of the method proposed to a class of spatial regression models is illustrated with a real data example. Results from a simulation study on finite sample properties of the method are also reported.     

A Simple Approach to Inference in Random Coefficient Models

Random coefficient regression models have been used to analyze cross-sectional and longitudinal data in economics and growth-curve data from biological and agricultural experiments. In the literature several estimators, including the ordinary least squares and the estimated generalized least squares (EGLS), have been considered for estimating the parameters of the mean model. Based on the asymptotic properties of the EGLS estimators, test statistics have been proposed for testing linear hypotheses involving the parameters of the mean model. An alternative estimator, the simple mean of the individual regression coefficients, provides estimation and hypothesis-testing procedures that are simple to compute and teach. The large sample properties of this simple estimator are shown to be similar to that of the EGLS estimator. The performance of the proposed estimator is compared with that of the existing estimators by Monte Carlo simulation.     

A useful tool for statistical estimation: genetic algorithms

In this article, we introduce genetic algorithms (GAs) as a viable tool in estimating parameters in a wide array of statistical models. We performed simulation studies that compared the bias and variance of GAs with classical tools, namely, the steepest descent, Gauss–Newton, Levenberg–Marquardt and don't use derivative methods. In our simulation studies, we used the least squares criterion as the optimizing function. The performance of the GAs and classical methods were compared under the logistic regression model; non-linear Gaussian model and non-linear non-Gaussian model. We report that the GAs' performance is competitive to the classical methods under these three models.     

Adaptive varying-coefficient linear models

Summary. Varying-coefficient linear models arise from multivariate nonparametric regression, non-linear time series modelling and forecasting, functional data analysis, longitudinal data analysis and others. It has been a common practice to assume that the varying coefficients are functions of a given variable, which is often called an index . To enlarge the modelling capacity substantially, this paper explores a class of varying-coefficient linear models in which the index is unknown and is estimated as a linear combination of regressors and/or other variables. We search for the index such that the derived varying-coefficient model provides the least squares approximation to the underlying unknown multidimensional regression function. The search is implemented through a newly proposed hybrid backfitting algorithm. The core of the algorithm is the alternating iteration between estimating the index through a one-step scheme and estimating coefficient functions through one-dimensional local linear smoothing. The locally significant variables are selected in terms of a combined use of the t -statistic and the Akaike information criterion. We further extend the algorithm for models with two indices. Simulation shows that the methodology proposed has appreciable flexibility to model complex multivariate non-linear structure and is practically feasible with average modern computers. The methods are further illustrated through the Canadian mink–muskrat data in 1925–1994 and the pound–dollar exchange rates in 1974–1983.     

Heteroskedastic linear regression model with compositional response and covariates

Compositional data are known as a sort of complex multidimensional data with the feature that reflect the relative information rather than absolute information. There are a variety of models for regression analysis with compositional variables. Similar to the traditional regression analysis, the heteroskedasticity still exists in these models. However, the existing heteroskedastic regression analysis methods cannot apply in these models with compositional error term. In this paper, we mainly study the heteroskedastic linear regression model with compositional response and covariates. The parameter estimator is obtained through weighted least squares method. For the hypothesis test of parameter, the test statistic is based on the original least squares estimator and corresponding heteroskedasticity-consistent covariance matrix estimator. When the proposed method is applied to both simulation and real example, we use the original least squares method as a comparison during the whole process. The results implicate the model's practicality and effectiveness in regression analysis with heteroskedasticity.     

Regression analysis of autocorrelated poisson-distribution) data

A regression model assuming Poisson-dia distributed data. with autocorrelated errors falls into the class of regression models that; have the error structure which is both heteroscedastic and autocorrelated. In general, this class of regression models are not estimable. However, due to the properties of the Poisson distribution that the variance is equal to the mean, this regression model on Poisson-distributed data with autocorrelated. errors is estimable. In this note the special structure of the covarlance matrix of the model with the first order auto-correlated error Is derived utilizing this property, A method based on the least squares method of Frome, Kutner, and Beauchamp (1973), supplemented by steps for handling autocorrelation in studies of time series analysis, nonlinear regression, and econometrics is presented for obtaining generalized least squares estimates for the parameters of the model.     

Combining non-linear regressions that have unequal error variances and some parameters in common

Methods of estimation and inference are presented for the situation where two non-linear regression models with unequal error variances contain some parameters in common. Such a situation arises in structural chemistry, when bond lengths are available for three nearly collinear atoms in crystals and a model is required to quantify the extent and form of the relationship between the longer and the shorter bond. Some atomic triples are symmetric and require a different model and error variance from those required by the asymmetric triples. The profile likelihood for the regression parameters is a weighted sum of the logarithms of the sums-of-squares functions from each model, and the estimates can be obtained by using a simple modification to a standard non-linear least squares program. A likelihood ratio test for assessing whether the parameters in common are equal is described. When these techniques are applied to two data sets consisting of bond lengths for bromine–tellurium–bromine and sulphur–tellurium–sulphur triples, there is no evidence against the equality hypothesis. An extension to the model to allow for a non-constant variance is required for proper analysis of the sulphur–tellurium–sulphur data.     

Least squares parameter estimation in multiplicative noise models

A simple multiplicative noise model with a constant signal has become a basic mathematical model in processing synthetic aperture radar images. The purpose of this paper is to examine a general multiplicative noise model with linear signals represented by a number of unknown parameters. The ordinary least squares (LS) and weighted LS methods are used to estimate the model parameters. The biases of the weighted LS estimates of the parameters are derived. The biases are then corrected to obtain a second-order unbiased estimator, which is shown to be exactly equivalent to the maximum log quasi-likelihood estimation, though the quasi-likelihood function is founded on a completely different theoretical consideration and is known, at the present time, to be a uniquely acceptable theory for multiplicative noise models. Synthetic simulations are carried out to confirm theoretical results and to illustrate problems in processing data contaminated by multiplicative noises. The sensitivity of the LS and weighted LS methods to extremely noisy data is analysed through the simulated examples.     

Efficient Computation of Reduced Regression Models

We consider settings where it is of interest to fit and assess regression submodels that arise as various explanatory variables are excluded from a larger regression model. The larger model is referred to as the full model; the submodels are the reduced models. We show that a computationally efficient approximation to the regression estimates under any reduced model can be obtained from a simple weighted least squares (WLS) approach based on the estimated regression parameters and covariance matrix from the full model. This WLS approach can be considered an extension to unbiased estimating equations of a first-order Taylor series approach proposed by Lawless and Singhal. Using data from the 2010 Nationwide Inpatient Sample (NIS), a 20% weighted, stratified, cluster sample of approximately 8 million hospital stays from approximately 1000 hospitals, we illustrate the WLS approach when fitting interval censored regression models to estimate the effect of type of surgery (robotic versus nonrobotic surgery) on hospital length-of-stay while adjusting for three sets of covariates: patient-level characteristics, hospital characteristics, and zip-code level characteristics. Ordinarily, standard fitting of the reduced models to the NIS data takes approximately 10 hours; using the proposed WLS approach, the reduced models take seconds to fit.     

A Bayesian hierarchical approach to dual response surface modelling

In modern quality engineering, dual response surface methodology is a powerful tool to model an industrial process by using both the mean and the standard deviation of the measurements as the responses. The least squares method in regression is often used to estimate the coefficients in the mean and standard deviation models, and various decision criteria are proposed by researchers to find the optimal conditions. Based on the inherent hierarchical structure of the dual response problems, we propose a Bayesian hierarchical approach to model dual response surfaces. Such an approach is compared with two frequentist least squares methods by using two real data sets and simulated data.     

Partial least squares Cox regression for genome-wide data

Most methods for survival prediction from high-dimensional genomic data combine the Cox proportional hazards model with some technique of dimension reduction, such as partial least squares regression (PLS). Applying PLS to the Cox model is not entirely straightforward, and multiple approaches have been proposed. The method of Park et al. (Bioinformatics 18(Suppl. 1):S120–S127, 2002) uses a reformulation of the Cox likelihood to a Poisson type likelihood, thereby enabling estimation by iteratively reweighted partial least squares for generalized linear models. We propose a modification of the method of park et al. (2002) such that estimates of the baseline hazard and the gene effects are obtained in separate steps. The resulting method has several advantages over the method of park et al. (2002) and other existing Cox PLS approaches, as it allows for estimation of survival probabilities for new patients, enables a less memory-demanding estimation procedure, and allows for incorporation of lower-dimensional non-genomic variables like disease grade and tumor thickness. We also propose to combine our Cox PLS method with an initial gene selection step in which genes are ordered by their Cox score and only the highest-ranking k% of the genes are retained, obtaining a so-called supervised partial least squares regression method. In simulations, both the unsupervised and the supervised version outperform other Cox PLS methods.     

Formulas for the Exact LMS and LQS Estimators

The author presents the derivation of formulas for the calculation of critical values of the median function or the general version of it, namely, the quantile functions. In statistics, these functions are used to detect outliers in the data set and to make predictions that are resistant to outliers. Therefore, these formulas can also be used as estimators for these regressions. The fact that these formulas are able to calculate the global optimum gives the exact least median squares or the exact least quantile of squares estimators. The author provides the theoretical background for deriving these estimator formulas and derives the estimator formulas for regression models up to three parameters. In addition, the author provides guides for the derivation of formulas for other models, illustrates the use of these formulas, and emphasizes their properties that are useful for future works. One important conclusion is that each regression model has its own set of formulas.     

Local estimation for varying-coefficient models with longitudinal data

Varying-coefficient models are very useful for longitudinal data analysis. In this paper, we focus on varying-coefficient models for longitudinal data. We develop a new estimation procedure using Cholesky decomposition and profile least squares techniques. Asymptotic normality for the proposed estimators of varying-coefficient functions has been established. Monte Carlo simulation studies show excellent finite-sample performance. We illustrate our methods with a real data example.     

Estimation for the parameters of the generalized half-normal distribution

As an applicable and flexible lifetime model, the two-parameter generalized half-normal (GHN) distribution has been received wide attention in the field of reliability analysis and lifetime study. In this paper maximum likelihood estimates of the model parameters are discussed and we also proposed corresponding bias-corrected estimates. Unweighted and weighted least squares estimates for the parameters of the GHN distribution are also presented for comparison purpose. Moreover, the likelihood ratio test is provided as complementary. Simulation study and illustrative examples are provided to compare the performance of the proposed methods.     

Fitting the generalized Pareto distribution to data based on transformations of order statistics

Generalized Pareto distribution (GPD) has been widely used to model exceedances over thresholds. In this article we propose a new method called weighted nonlinear least squares (WNLS) to estimate the parameters of the GPD. The WNLS estimators always exist and are simple to compute. Some asymptotic results of the proposed method are provided. The simulation results indicate that the proposed method performs well compared to existing methods in terms of mean squared error and bias. Its advantages are further illustrated through the analysis of two real data sets.     

Improving survey-weighted least squares regression

The weighted least squares (WLS) estimator is often employed in linear regression using complex survey data to deal with the bias in ordinary least squares (OLS) arising from informative sampling. In this paper a 'quasi-Aitken WLS' (QWLS) estimator is proposed. QWLS modifies WLS in the same way that Cragg's quasi-Aitken estimator modifies OLS. It weights by the usual inverse sample inclusion probability weights multiplied by a parameterized function of covariates, where the parameters are chosen to minimize a variance criterion. The resulting estimator is consistent for the superpopulation regression coefficient under fairly mild conditions and has a smaller asymptotic variance than WLS.     

Application of Quasi-Least Squares to Analyse Replicated Autoregressive Time Series Regression Models

Time series regression models have been widely studied in the literature by several authors. However, statistical analysis of replicated time series regression models has received little attention. In this paper, we study the application of the quasi-least squares method to estimate the parameters in a replicated time series model with errors that follow an autoregressive process of order p. We also discuss two other established methods for estimating the parameters: maximum likelihood assuming normality and the Yule-Walker method. When the number of repeated measurements is bounded and the number of replications n goes to infinity, the regression and the autocorrelation parameters are consistent and asymptotically normal for all three methods of estimation. Basically, the three methods estimate the regression parameter efficiently and differ in how they estimate the autocorrelation. When p=2, for normal data we use simulations to show that the quasi-least squares estimate of the autocorrelation is undoubtedly better than the Yule-Walker estimate. And the former estimate is as good as the maximum likelihood estimate almost over the entire parameter space.