On the Strong Consistency of Ridge Estimates

In this article, we establish strong consistency of the ridge estimates using extended results for the strong consistency of the least squares estimates in multiple regression models which discard the usual assumption of null mean value for the errors and only requires them to be i.i.d. with absolute moment of order r (0 < r ? 1).     

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.     

Strong consistency of least squares estimates with i.i.d. errors with mean values not necessarily defined

We establish strong consistency of the least squares estimates in multiple regression models discarding the usual assumption of the errors having null mean value. Thus, we required them to be i.i.d. with absolute moment of order r, 0<r<2, and null mean value when r>1. Only moderately restrictive conditions are imposed on the model matrix. In our treatment, we use an extension of the Marcinkiewicz–Zygmund strong law to overcome the errors mean value not being defined. In this way, we get a unified treatment for the case of i.i.d. errors extending the results of some previous papers.     

Pricing of American options in discrete time using least squares estimates with complexity penalties

Pricing of American options in discrete time is considered, where the option is allowed to be based on several underlying stocks. It is assumed that the price processes of the underlying stocks are given by Markov processes. We use the Monte Carlo approach to generate artificial sample paths of these price processes, and then we use nonparametric regression estimates to estimate from this data so-called continuation values, which are defined as mean values of the American option for given values of the underlying stocks at time t subject to the constraint that the option is not exercised at time t. As nonparametric regression estimates we use least squares estimates with complexity penalties, which include as special cases least squares spline estimates, least squares neural networks, smoothing splines and orthogonal series estimates. General results concerning rate of convergence are presented and applied to derive results for the special cases mentioned above. Furthermore the pricing of American options is illustrated by simulated data.     

Temporal prediction of future state occupation in a multistate model from high-dimensional baseline covariates via pseudo-value regression

In many complex diseases such as cancer, a patient undergoes various disease stages before reaching a terminal state (say disease free or death). This fits a multistate model framework where a prognosis may be equivalent to predicting the state occupation at a future time t. With the advent of high-throughput genomic and proteomic assays, a clinician may intent to use such high-dimensional covariates in making better prediction of state occupation. In this article, we offer a practical solution to this problem by combining a useful technique, called pseudo-value (PV) regression, with a latent factor or a penalized regression method such as the partial least squares (PLS) or the least absolute shrinkage and selection operator (LASSO), or their variants. We explore the predictive performances of these combinations in various high-dimensional settings via extensive simulation studies. Overall, this strategy works fairly well provided the models are tuned properly. Overall, the PLS turns out to be slightly better than LASSO in most settings investigated by us, for the purpose of temporal prediction of future state occupation. We illustrate the utility of these PV-based high-dimensional regression methods using a lung cancer data set where we use the patients’ baseline gene expression values.     

Comparison of computer programs for simple linear L 1 regression

A number of efficient computer codes are available for the simple linear L ₁ regression problem. However, a number of these codes can be made more efficient by utilizing the least squares solution. In fact, a couple of available computer programs already do so.

We report the results of a computational study comparing several openly available computer programs for solving the simple linear L ₁ regression problem with and without computing and utilizing a least squares solution.     

Model selection for infinite variance time series

In this we consider the problem of model selection for infinite variance time series. We introduce a group of model selection critera based on a general loss function Ψ. This family includes various generalizations of predictive least square and AIC Parameter estimation is carried out using Ψ. We use two loss functions commonly used in robust estimation and show that certain criteria out perform the conventional approach based on least squares or Yule-Walker estima­tion for heavy tailed innovations. Our conclusions are based on a comprehensive study of the performance of competing criteria for a wide selection of AR(2) models. We also consider the performance of these techniques when the ‘true’ model is not contained in the family of candidate models.     

Estimation of the linear-plateau segmented regression model in the presence of measurement error

It is well known that when the true values of the independent variable are unobservable due to measurement error, the least squares estimator for a regression model is biased and inconsistent. When repeated observations on each x_i are taken, consistent estimators for the linear-plateau model can be formed. The repeated observations are required to classify each observation to the appropriate line segment. Two cases of repeated observations are treated in detail. First, when a single value of y_i is observed with the repeated observations of x_i the least squares estimator using the mean of the repeated x_i observations is consistent and asymptotically normal. Second, when repeated observations on the pair (x_i, y_i ) are taken the least squares estimator is inconsistent, but two consistent estimators are proposed: one that consistently estimates the bias of the least squares estimator and adjusts accordingly; the second is the least squares estimator using the mean of the repeated observations on each pair.     

Estimating the parameter in the Pauling equation

Four procedures are suggested for estimating the parameter ‘a’ in the Pauling equation:

e-X/a+e ? Y/a = 1.

The procedures are: using the mean of individual solutions, least squares with Y the subject of the equation, least squares with X the subject of the equation and maximum likelihood using a statistical model. In order to compare these estimates, we use Efron's bootstrap technique (1979), since distributional results are not available. This example also illustrates the role of the bootstrap in statistical inference.     

Superiority of the r–k Class Estimator Over Some Estimators In A Linear Model

In regression analysis, to overcome the problem of multicollinearity, the r ? k class estimator is proposed as an alternative to the ordinary least squares estimator which is a general estimator including the ordinary ridge regression estimator, the principal components regression estimator and the ordinary least squares estimator. In this article, we derive the necessary and sufficient conditions for the superiority of the r ? k class estimator over each of these estimators under the Mahalanobis loss function by the average loss criterion. Then, we compare these estimators with each other using the same criterion. Also, we suggest to test to verify if these conditions are indeed satisfied. Finally, a numerical example and a Monte Carlo simulation are done to illustrate the theoretical results.     

Minimax estimators of regression functions under normalized quadratic loss functions and inequality restrictions

This paper deals with the estimation of a regression function f of unknown form by shrinked least squares estimates calculated on the basis of possibly replicated observa-tions, The estimation loss is chosen in a somewhat more realistic manner then the usual quadratic losses and is given by an adequately weighted sum of squared errors in estimat-ing the values of f at the design points, normalized by the squared norm of the regression function, Shrinked least squares (as special ridge estimators) have been proved by the suthors in special cases to be minimax under all estimatiors.

We investigate the shrinking of least squares estimators with the objective of minimiz-ing the least favourable risk. Here we assume a known lower bound for the magnitude of f and a known upper bound for the difference between f and some simple function approxi-mating f, e.g. we know that f is the sum of a quadratic polynomial and of some     

Least squares and shrinkage estimation under bimonotonicity constraints

In this paper we describe active set type algorithms for minimization of a smooth function under general order constraints, an important case being functions on the set of bimonotone r×s matrices. These algorithms can be used, for instance, to estimate a bimonotone regression function via least squares or (a smooth approximation of) least absolute deviations. Another application is shrinkage estimation in image denoising or, more generally, regression problems with two ordinal factors after representing the data in a suitable basis which is indexed by pairs (i,j)∈{1,…,r}×{1,…,s}. Various numerical examples illustrate our methods.     

The use of a reference variety for comparisons in incomplete series of crop variety trials

In a series of crop variety trials, ‘test varieties’ are compared with one another and with a ‘reference’ variety that is included in all trials. The series is typically analyzed with a linear mixed model and the method of generalized least squares. Usually, the estimates of the expected differences between the test varieties and the reference variety are presented. When the series is incomplete, i.e. when all test varieties were not included in all trials, the method of generalized least squares may give estimates of expected differences to the reference variety that do not appear to accord with observed differences. The present paper draws attention to this phenomenon and explores the recurrent idea of comparing test varieties indirectly through the use of the reference. A new ‘reference treatment method’ was specified and compared with the method of generalized least squares when applied to a five-year series of 85 spring wheat trials. The reference treatment method provided estimates of differences to the reference variety that agreed with observed differences, but was considerably less efficient than the method of generalized least squares.     

A Note on Two-Way ECM Estimation of SUR Systems on Unbalanced Panel Data

This article considers the two-way error components model (ECM) estimation of seemingly unrelated regressions (SUR) on unbalanced panel by generalized least squares (GLS). As suggested by Biørn (2004 Biørn , E. ( 2004 ). Regression systems for unbalanced panel data: a stepwise maximum likelihood procedure . Journal of Econometrics 122 : 281 – 291 . [Google Scholar]) for the one-way case, in order to use the standard results for the balanced case the individuals are arranged in groups according to the number of times they are observed. Thus, the GLS estimator can be interpreted as a matrix weighted average of the group specific GLS estimators with weights equal to the inverse of their respective covariance matrices.     

Generalized Least Squares Model Averaging

In this article, we propose a method of averaging generalized least squares estimators for linear regression models with heteroskedastic errors. The averaging weights are chosen to minimize Mallows’ C_p-like criterion. We show that the weight vector selected by our method is optimal. It is also shown that this optimality holds even when the variances of the error terms are estimated and the feasible generalized least squares estimators are averaged. The variances can be estimated parametrically or nonparametrically. Monte Carlo simulation results are encouraging. An empirical example illustrates that the proposed method is useful for predicting a measure of firms’ performance.     

Performance of Kibria's Method for the Heteroscedastic Ridge Regression Model: Some Monte Carlo Evidence

It is common for a linear regression model that the error terms display some form of heteroscedasticity and at the same time, the regressors are also linearly correlated. Both of these problems have serious impact on the ordinary least squares (OLS) estimates. In the presence of heteroscedasticity, the OLS estimator becomes inefficient and the similar adverse impact can also be found on the ridge regression estimator that is alternatively used to cope with the problem of multicollinearity. In the available literature, the adaptive estimator has been established to be more efficient than the OLS estimator when there is heteroscedasticity of unknown form. The present article proposes the similar adaptation for the ridge regression setting with an attempt to have more efficient estimator. Our numerical results, based on the Monte Carlo simulations, provide very attractive performance of the proposed estimator in terms of efficiency. Three different existing methods have been used for the selection of biasing parameter. Moreover, three different distributions of the error term have been studied to evaluate the proposed estimator and these are normal, Student's t and F distribution.     

The regression dilemma

A substantial fraction of the statistical analyses and in particular statistical computing is done under the heading of multiple linear regression. That is the fitting of equations to multivariate data using the least squares technique for estimating parameters The optimality properties of these estimates are described in an ideal setting which is not often realized in practice.

Frequently, we do not have "good" data in the sense that the errors are non-normal or the variance is non-homogeneous. The data may contain outliers or extremes which are not easily detectable but variables in the proper functional, and we. may have the linearity

Prior to the mid-sixties regression programs provided just the basic least squares computations plus possibly a step-wise algorithm for variable selection. The increased interest in regression prompted by dramatic improvements in computers has led to a vast amount of literatur describing alternatives to least squares improved variable selection methods and extensive diagnostic procedures

The purpose of this paper is to summarize and illustrate some of these recent developments. In particular we shall review some of the potential problems with regression data discuss the statistics and techniques used to detect these problems and consider some of the proposed solutions. An example is presented to illustrate the effectiveness of these diagnostic methods in revealing such problems and the potential consequences of employing the proposed methods.     

SHRINKAGE, PRETEST AND ABSOLUTE PENALTY ESTIMATORS IN PARTIALLY LINEAR MODELS

We consider a partially linear model in which the vector of coefficients β in the linear part can be partitioned as ( β ₁, β ₂) , where β ₁ is the coefficient vector for main effects (e.g. treatment effect, genetic effects) and β ₂ is a vector for ‘nuisance’ effects (e.g. age, laboratory). In this situation, inference about β ₁ may benefit from moving the least squares estimate for the full model in the direction of the least squares estimate without the nuisance variables (Steinian shrinkage), or from dropping the nuisance variables if there is evidence that they do not provide useful information (pretesting). We investigate the asymptotic properties of Stein‐type and pretest semiparametric estimators under quadratic loss and show that, under general conditions, a Stein‐type semiparametric estimator improves on the full model conventional semiparametric least squares estimator. The relative performance of the estimators is examined using asymptotic analysis of quadratic risk functions and it is found that the Stein‐type estimator outperforms the full model estimator uniformly. By contrast, the pretest estimator dominates the least squares estimator only in a small part of the parameter space, which is consistent with the theory. We also consider an absolute penalty‐type estimator for partially linear models and give a Monte Carlo simulation comparison of shrinkage, pretest and the absolute penalty‐type estimators. The comparison shows that the shrinkage method performs better than the absolute penalty‐type estimation method when the dimension of the β ₂ parameter space is large.     

Using Liu-Type Estimator to Combat Collinearity

Abstract

Linear regression model and least squares method are widely used in many fields of natural and social sciences. In the presence of collinearity, the least squares estimator is unstable and often gives misleading information. Ridge regression is the most common method to overcome this problem. We find that when there exists severe collinearity, the shrinkage parameter selected by existing methods for ridge regression may not fully address the ill conditioning problem. To solve this problem, we propose a new two-parameter estimator. We show using both theoretic results and simulation that our new estimator has two advantages over ridge regression. First, our estimator has less mean squared error (MSE). Second, our estimator can fully address the ill conditioning problem. A numerical example from literature is used to illustrate the results.     

Variable Selection via Regression Trees in the Presence of Irrelevant Variables

Many tree algorithms have been developed for regression problems. Although they are regarded as good algorithms, most of them suffer from loss of prediction accuracy when there are many irrelevant variables and the number of predictors exceeds the number of observations. We propose the multistep regression tree with adaptive variable selection to handle this problem. The variable selection step and the fitting step comprise the multistep method.

The multistep generalized unbiased interaction detection and estimation (GUIDE) with adaptive forward selection (fg) algorithm, as a variable selection tool, performs better than some of the well-known variable selection algorithms such as efficacy adaptive regression tube hunting (EARTH), FSR (false selection rate), LSCV (least squares cross-validation), and LASSO (least absolute shrinkage and selection operator) for the regression problem. The results based on simulation study show that fg outperforms other algorithms in terms of selection result and computation time. It generally selects the important variables correctly with relatively few irrelevant variables, which gives good prediction accuracy with less computation time.