The use of studentized diagnostics in regression

Recent work by LaMotte (1999) uncovered redundancies and inconsistencies in the current practice of selected deletion diagnostics in regression. The present study extends earlier work to include further diagnostics on using different methods. Benchmarks adjusted to the scale of each diagnostic are given to assure consistency across diagnostics. Case studies illustrate anomalies in the use of these diagnostics as currently practiced. Alternative diagnostics are given to gauge effects of single-case deletions on variances and biases in prediction and estimation. Received: November 1999     

Logarithmic calibration for multiplicative distortion measurement errors regression models

In this article, we propose a new identifiability condition by using the logarithmic calibration for the distortion measurement error models, where neither the response variable nor the covariates can be directly observed but are measured with multiplicative measurement errors. Under the logarithmic calibration, the direct-plug-in estimators of parameters and empirical likelihood based confidence intervals are proposed, and we studied the asymptotic properties of the proposed estimators. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the restricted estimator and test statistic are established. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its practical usage.     

The smoothing dichotomy in nonparametric regression under long-memory errors

The phenomenon of smoothing dichotomy in random-design nonparametric regression is exposed in nontechnical terms from two recent papers published jointly with Jan Mielniczuk. This concerns the asymptotic distribution of kernel estimators when the errors exhibit long-range dependence, being instantaneous functions either of Gaussian sequences or of infinite-order moving averages, depending on the amount of smoothing.     

Statistical inference on restricted partial linear regression models with partial distortion measurement errors

We consider the estimation and hypothesis testing problems for the partial linear regression models when some variables are distorted with errors by some unknown functions of commonly observable confounding variable. The proposed estimation procedure is designed to accommodate undistorted as well as distorted variables. To test a hypothesis on the parametric components, a restricted least squares estimator is proposed under the null hypothesis. Asymptotic properties for the estimators are established. A test statistic based on the difference between the residual sums of squares under the null and alternative hypotheses is proposed, and we also obtain the asymptotic properties of the test statistic. A wild bootstrap procedure is proposed to calculate critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure, and a real example is analyzed for an illustration.     

Linear and nonlinear regression with stable errors

In this paper we describe methods and evaluate programs for linear regression by maximum likelihood when the errors have a heavy tailed stable distribution. The asymptotic Fisher information matrix for both the regression coefficients and the error distribution parameters are derived, giving large sample confidence intervals for all parameters. Simulated examples are shown where the errors are stably distributed and also where the errors are heavy tailed but are not stable, as well as a real example using financial data. The results are then extended to nonlinear models and to non-homogeneous error terms.     

A note on the maximum deviation of the scale-contaminated normal to the best normal distribution

In this paper we consider the case of the scale-contaminated normal (mixture of two normals with equal mean components but different component variances: (1−p)N(μ,σ²)+pN(μ,τ²) with σ and τ being non-negative and 0≤p≤1). Here is the scale error and p denotes the amount with which this error occurs. It's maximum deviation to the best normal distribution is studied and shown to be montone increasing with increasing scale error. A closed-form expression is derived for the proportion which maximizes the maximum deviation of the mixture of normals to the best normal distribution. Implications to power studies of tests for normality are pointed out. Received May 2001     

Rank tests for testing randomness of a regression coefficient in a linear regression model

We propose a class of nonparametric tests for testing non-stochasticity of the regression parameterβ in the regression modely _i =βx _i +ɛ _i ,i=1, ...,n. We prove that the test statistics are asymptotically normally distributed both underH ₀ and under contiguous alternatives. The asymptotic relative efficiencies (in the Pitman sense) with respect to the best parametric test have also been computed and they are quite high. Some simulation studies are carried out to illustrate the results. Research was supported by the University Grants Commission, India.     

Distribution of correlation coefficient under mixture normal model

The present paper obtains the nonnull distribution of the product moment correlation coefficient r when sample is drawn from a mixture of two bivariate Gaussian distributions. The moments of 1−r ² have been used to derive the nonnull density of r. Received September 2000     

Robustness issues in multilevel regression analysis

A multilevel problem concerns a population with a hierarchical structure. A sample from such a population can be described as a multistage sample. First, a sample of higher level units is drawn (e.g. schools or organizations), and next a sample of the sub‐units from the available units (e.g. pupils in schools or employees in organizations). In such samples, the individual observations are in general not completely independent. Multilevel analysis software accounts for this dependence and in recent years these programs have been widely accepted. Two problems that occur in the practice of multilevel modeling will be discussed. The first problem is the choice of the sample sizes at the different levels. What are sufficient sample sizes for accurate estimation? The second problem is the normality assumption of the level‐2 error distribution. When one wants to conduct tests of significance, the errors need to be normally distributed. What happens when this is not the case? In this paper, simulation studies are used to answer both questions. With respect to the first question, the results show that a small sample size at level two (meaning a sample of 50 or less) leads to biased estimates of the second‐level standard errors. The answer to the second question is that only the standard errors for the random effects at the second level are highly inaccurate if the distributional assumptions concerning the level‐2 errors are not fulfilled. Robust standard errors turn out to be more reliable than the asymptotic standard errors based on maximum likelihood.     

Errors of misclassification in discrimination of dimensional coherent elliptic random field observations

The problem of classification of dimensional coherent elliptic random field observations into one of two populations specified by different regression mean models and common stationary scale matrix is considered, under the further assumption that the observations to be classified are dependent on the training samples. In this statistical frame, the behaviour of linear discriminant function is studied and an asymptotic expression for the distribution function of the probabilities of misclassification is derived.     

Collapsibility hypotheses and diagnostic bounds in regression analysis

Deleted-case diagnostic statistics in regression analysis are based on changes in estimates due to deleting one or more cases. Bounds on these statistics, suggested in the literature for identifying influential cases, are widely used.  In a linear regression model for Y in terms of X and Z, the model is “collapsible” with respect to Z if the Y−X relation is unchanged by deleting Z from the model. Deleted-case diagnostic statistics can be viewed as test statistics for collapsibility hypotheses in the mean shift outlier model. It follows that, for any given case, all deleted-case statistics test the same hypothesis, hence all have the same p-value, while the bounds correspond to different levels of significance among the several statistics. Furthermore, the bound for any particular deleted-case statistic gives widely varying levels of significance over the cases in the data set. Received: April 1999     

Robustness of normal theory statistics in structural equation models*

A condition is given by which optimal normal theory methods, such as the maximum likelihood methods, are robust against violation of the normality assumption in a general linear structural equation model. Specifically, the estimators and the goodness of fit test are robust. The estimator is efficient within some defined class, and its standard errors can be obtained by a correction formula applied to the inverse of the information matrix. Some special models, like the factor analysis model and path models, are discussed in more detail. A method for evaluating the robustness condition is given.     

Combining information for prediction in linear regression

This article deals with the prediction problem in linear regression where the measurements are obtained using k different devices or collected from k different independent sources. For the case of k=2, a Graybill-Deal type combined estimtor for the regression parameters is shown to dominate the individual least squares estimators under the covariance criterion. Two predictors ŷ _c and ŷ _p are proposed. ŷ _c is based on a combined estimator of the regression coefficient vector, and ŷ _p is obtained by combining the individual predictors from different models. Prediction mean square errors of both predictors are derived. It is shown that the predictor ŷ _p is better than the individual predictors for k≥2 and the predictor ŷ _c is better than the individual predictors for k=2. Numerical comparison between ŷ _c and ŷ _p shows that the former is superior to the latter for the case k=2.     

The effect of a Durbin–Watson pretest on confidence intervals in regression

Consider a linear regression model and suppose that our aim is to find a confidence interval for a specified linear combination of the regression parameters. In practice, it is common to perform a Durbin–Watson pretest of the null hypothesis of zero first‐order autocorrelation of the random errors against the alternative hypothesis of positive first‐order autocorrelation. If this null hypothesis is accepted then the confidence interval centered on the ordinary least squares estimator is used; otherwise the confidence interval centered on the feasible generalized least squares estimator is used. For any given design matrix and parameter of interest, we compare the confidence interval resulting from this two‐stage procedure and the confidence interval that is always centered on the feasible generalized least squares estimator, as follows. First, we compare the coverage probability functions of these confidence intervals. Second, we compute the scaled expected length of the confidence interval resulting from the two‐stage procedure, where the scaling is with respect to the expected length of the confidence interval centered on the feasible generalized least squares estimator, with the same minimum coverage probability. These comparisons are used to choose the better confidence interval, prior to any examination of the observed response vector.     

Variance estimation in the change analysis of a linear regression model

We study a “direct test” of Chu and White (1992) proposed for detecting changes in the trend of a linear regression model. The power of this test strongly depends on a suitable estimation of the variance of the error variables involved. We discuss various types of variance estimators and derive their asymptotic properties under the null-hypothesis of “no change” as well as under the alternative of “a change in linear trend”. A small simulation study illustrates the estimators' finite sample behaviour.     

Algorithms to compute CM- and S-estimates for regression

Constrained M-estimators for regression were introduced by Mendes and Tyler in 1995 as an alternative class of robust regression estimators with high breakdown point and high asymptotic efficiency. To compute the CM-estimate, the global minimum of an objective function with an inequality constraint has to be localized. To find the S-estimate for the same problem, we instead restrict ourselves to the boundary of the feasible region. The algorithm presented for computing CM-estimates can easily be modified to compute S-estimates as well. Testing is carried out with a comparison to the algorithm SURREAL by Ruppert.     

Model selection in regression analysis,regression diagnostics and prior knowledge: A book review article with comments from Anthony C. Atkinson,D.R. Cox And John McDonald

Model selection of a regression model typically includes both selecting which variables to include in a model and the functional form of the relationship between the variables. In recent years regression diagnostics (a catch-all phrase that includes such topics as identifying influential observations) have been increasingly used as an aid in selecting a regression model. In evaluating the recent book Residuals and influence in regression, by Cook and Weisberg, Belsley argues that such analyses, while extremely useful, should not be used in a theoretical vacuum. Effective model building requires the analyst to make substantial use of any prior information relevant to the problem in hand. The commentators, while accepting the general thrust of Belsley's comments, are on the whole more sceptical — perhaps the prior knowledge is poor and in conflict with the data; perhaps the use of a wrong theory is more dangerous than ‘letting the data speak for themselves’.     

The mixture index of fit and minimax regression

A measure of the fit of a statistical model can be obtained by estimating the relative size of the largest fraction of the population where a distribution belonging to the model may be valid. This is the mixture index of fit that was suggested for models for contingency tables by Rudas, Clogg, Lindsay (1994) and it is extended here for models involving continuous observations. In particular, the approach is applied to regression models with normal and uniform error structures. Best fit, as measured by the mixture index of fit, is obtained with minimax estimation of the regression parameters. Therefore, whenever minimax estimation is used for these problems, the mixture index of fit provides a natural approach for measuring model fit and for variable selection. Received: September 1997     

北京新常态经济增长的金融支持问题研究

从金融资源空间布局、金融与其他产业的关联性以及金融对经济增长的DEA效率等方面,该文分析了北京金融对经济的整体支持情况与各区县支持情况,发现北京金融业发展的波动性、金融业与其他产业的较低关联性及各区县金融资源空间布局的不平衡性不利于新常态经济增长。建议顶层做好总体规划,同时,各区县应根据整体发展需要与自身优势,提高金融对经济增长的支持效率与质量。     

On bootstrapping the mode in the nonparametric regression model with random design

In the nonparametric regression model with random design and based on i.i.d. pairs of observations (X _i, Y _i), where the regression function m is given by m(x)=?(Y _i|X _i=x), estimation of the location θ (mode) of a unique maximum of m by the location of a maximum of the Nadaraya-Watson kernel estimator for the curve m is considered. In order to obtain asymptotic confidence intervals for θ, the suitably normalized distribution of is bootstrapped in two ways: we present a paired bootstrap (PB) where resampling is done from the empirical distribution of the pairs of observations and a smoothed paired bootstrap (SPB) where the bootstrap variables are generated from a smooth bivariate density based on the pairs of observations. While the PB requires only relatively small computational effort when carried out in practice, it is shown to work only in the case of vanishing asymptotic bias, i.e. of “undersmoothing” when compared to optimal smoothing for mode estimation. On the other hand, the SPB, although causing more intricate computations, is able to capture the correct amount of bias if the pilot estimator for m oversmoothes. Received: May 2000