Maximum Rank Correlation Estimator Research Articles

Connections between two classes of estimators for single‐index models

Single‐index model is a very popular and powerful semiparametric model. As an improvement of the maximum rank correlation estimator, Shen et al. proposed the linearized maximum rank correlation estimator. We show that this estimator has some interesting connections with the distribution‐transformed least‐squares estimator for single‐index models. We also propose a rescaled distribution‐transformed least‐squares estimator, which is mathematically equivalent to the linearized maximum rank correlation estimator when the distribution of the response is absolutely continuous. Despite some nontrivial connections, the two estimation procedures are different in terms of motivations, interpretations, and applications. We discuss some of the differences between the two estimation procedures.

Linearized maximum rank correlation estimation

Summary We propose a linearized maximum rank correlation estimator for the single-index model. Unlike the existing maximum rank correlation and other rank-based methods, the proposed estimator has a closed-form expression, making it appealing in theory and computation. The proposed estimator is robust to outliers in the response and its construction does not need knowledge of the unknown link function or the error distribution. Under mild conditions, it is shown to be consistent and asymptotically normal when the predictors satisfy the linearity of the expectation assumption. A more general class of estimators is also studied. Inference procedures based on the plug-in rule or random weighting resampling are employed for variance estimation. The proposed method can be easily modified to accommodate censored data. It can also be extended to deal with high-dimensional data combined with a penalty function. Extensive simulation studies provide strong evidence that the proposed method works well in various practical situations. Its application is illustrated with the Beijing PM 2.5 dataset.

Exact computation of maximum rank correlation estimator

SummaryIn this paper we provide a computation algorithm to get a global solution for the maximum rank correlation estimator using the mixed integer programming (MIP) approach. We construct a new constrained optimization problem by transforming all indicator functions into binary parameters to be estimated and show that it is equivalent to the original problem. We also consider an application of the best subset rank prediction and show that the original optimization problem can be reformulated as MIP. We derive the nonasymptotic bound for the tail probability of the predictive performance measure. We investigate the performance of the MIP algorithm by an empirical example and Monte Carlo simulations.

Exact Computation of Maximum Rank Correlation Estimator

In this paper we provide a computation algorithm to get a global solution for the maximum rank correlation estimator using the mixed integer programming (MIP) approach. We construct a new constrained optimization problem by transforming all indicator functions into binary parameters to be estimated and show that it is equivalent to the original problem. We also consider an application of the best subset rank prediction and show that the original optimization problem can be reformulated as MIP. We derive the non-asymptotic bound for the tail probability of the predictive performance measure. We investigate the performance of the MIP algorithm by an empirical example and Monte Carlo simulations.

A robust conditional maximum likelihood estimator for generalized linear models with a dispersion parameter

Highly robust and efficient estimators for generalized linear models with a dispersion parameter are proposed. The estimators are based on three steps. In the first step, the maximum rank correlation estimator is used to consistently estimate the slopes up to a scale factor. The scale factor, the intercept, and the dispersion parameter are robustly estimated using a simple regression model. Then, randomized quantile residuals based on the initial estimators are used to define a region S such that observations out of S are considered as outliers. Finally, a conditional maximum likelihood (CML) estimator given the observations in S is computed. We show that, under the model, S tends to the whole space for increasing sample size. Therefore, the CML estimator tends to the unconditional maximum likelihood estimator and this implies that this estimator is asymptotically fully efficient. Moreover, the CML estimator maintains the high degree of robustness of the initial one. The negative binomial regression case is studied in detail.

Statistical inference on transformation models: a self-induced smoothing approach

ABSTRACTThis paper deals with a general class of transformation models that contains many important semiparametric regression models as special cases. It develops a self-induced smoothing for the maximum rank correlation estimator, resulting in simultaneous point and variance estimation. The self-induced smoothing does not require bandwidth selection, yet provides the right amount of smoothness so that the estimator is asymptotically normal with mean zero (unbiased) and variance–covariance matrix consistently estimated by the usual sandwich-type estimator. An iterative algorithm is given for the variance estimation and shown to numerically converge to a consistent limiting variance estimator. The approach is applied to a data set involving survival times of primary biliary cirrhosis patients. Simulation results are reported, showing that the new method performs well under a variety of scenarios.

Order‐constrained linear optimization

Despite the fact that data and theories in the social, behavioural, and health sciences are often represented on an ordinal scale, there has been relatively little emphasis on modelling ordinal properties. The most common analytic framework used in psychological science is the general linear model, whose variants include ANOVA, MANOVA, and ordinary linear regression. While these methods are designed to provide the best fit to the metric properties of the data, they are not designed to maximally model ordinal properties. In this paper, we develop an order-constrained linear least-squares (OCLO) optimization algorithm that maximizes the linear least-squares fit to the data conditional on maximizing the ordinal fit based on Kendall's τ. The algorithm builds on the maximum rank correlation estimator (Han, 1987, Journal of Econometrics, 35, 303) and the general monotone model (Dougherty & Thomas, 2012, Psychological Review, 119, 321). Analyses of simulated data indicate that when modelling data that adhere to the assumptions of ordinary least squares, OCLO shows minimal bias, little increase in variance, and almost no loss in out-of-sample predictive accuracy. In contrast, under conditions in which data include a small number of extreme scores (fat-tailed distributions), OCLO shows less bias and variance, and substantially better out-of-sample predictive accuracy, even when the outliers are removed. We show that the advantages of OCLO over ordinary least squares in predicting new observations hold across a variety of scenarios in which researchers must decide to retain or eliminate extreme scores when fitting data.

Sparse concordance-assisted learning for optimal treatment decision

To find optimal decision rule, Fan et al. (2016) proposed an innovative concordance-assisted learning algorithm which is based on maximum rank correlation estimator. It makes better use of the available information through pairwise comparison. However the objective function is discontinuous and computationally hard to optimize. In this paper, we consider a convex surrogate loss function to solve this problem. In addition, our algorithm ensures sparsity of decision rule and renders easy interpretation. We derive the L 2 error bound of the estimated coefficients under ultra-high dimension. Simulation results of various settings and application to STAR*D both illustrate that the proposed method can still estimate optimal treatment regime successfully when the number of covariates is large.

Model Selection Based on FDR-Thresholding Optimizing the Area under the ROC-Curve

We evaluate variable selection by multiple tests controlling the false discovery rate (FDR) to build a linear score for prediction of clinical outcome in high-dimensional data. Quality of prediction is assessed by the receiver operating characteristic curve (ROC) for prediction in independent patients. Thus we try to combine both goals: prediction and controlled structure estimation. We show that the FDR-threshold which provides the ROC-curve with the largest area under the curve (AUC) varies largely over the different parameter constellations not known in advance. Hence, we investigated a new cross validation procedure based on the maximum rank correlation estimator to determine the optimal selection threshold. This procedure (i) allows choosing an appropriate selection criterion, (ii) provides an estimate of the FDR close to the true FDR and (iii) is simple and computationally feasible for rather moderate to small sample sizes. Low estimates of the cross validated AUC (the estimates generally being positively biased) and large estimates of the cross validated FDR may indicate a lack of sufficiently prognostic variables and/or too small sample sizes. The method is applied to an oncology dataset.

Weighted Rank Estimators

Rank-based estimators are important tools of robust estimation in popular semiparametric models under monotonicity constraints. Here we study weighted versions of such estimators. Optimally weighted monotone rank estimator (MR) of Cavanagh and Sherman (1998) attains the semiparametric efficiency bound in the nonlinear regression model and the binary choice model. Optimally weighted maximum rank correlation estimator (MRC) of Han (1987) has the asymptotic variance close to the semiparametric efficiency bound in single-index models under independence when the distribution of the errors is close to normal, and is consistent under deviations from the single index assumption. Under moderate nonlinearities and nonsmoothness in the data, the efficiency gains from weighting are likely to be small for MR and MRC in the binary choice model and for MRC in the transformation model, and can be large for MR and MRC in the monotone regression model.

Asymptotic and Bootstrap Properties of Rank Regressions

The paper develops the bootstrap theory and extends the asymptotic theory of rank estimators, such as the Maximum Rank Correlation Estimator (MRC) of Han (1987), Monotone Rank Estimator (MR) of Cavanagh and Sherman (1998) or Pairwise-Difference Rank Estimators (PDR) of Abrevaya (2003). It is known that under general conditions these estimators have asymptotic normal distributions, but the asymptotic variances are difficult to find. Here we prove that the quantiles and the variances of the asymptotic distributions can be consistently estimated by the nonparametric bootstrap. We investigate the accuracy of inference based on the asymptotic approximation and the bootstrap, and provide bounds on the associated error. In the case of MRC and MR, the bound is a function of the sample size of order close to n^(-1/6). The PDR estimators belong to a special subclass of rank estimators for which the bound is vanishing with the rate close to n^(-1/2). The theoretical findings are illustrated with Monte-Carlo experiments and a real data example.

A note on iterative marginal optimization: a simple algorithm for maximum rank correlation estimation

The maximum rank correlation (MRC) estimator was originally studied by Han [1987. Nonparametric analysis of a generalized regression model. J. Econometrics 35, 303–316] and Sherman [1993. The limiting distribution of the maximum rank correlation estimator. Econometrica 61, 123–137] from the econometrics point of view, and most recently attracted much attention from the classification literature due to its close relationship with the receiver operating characteristics (ROC) curve [Baker, 2003. The central role of receiver operating characteristics (ROC) curves in evaluating tests for the early detection of cancer. J. Nat. Cancer Inst. 95, 511–515; Pepe, 2003. The Statistical Evaluation of Medical Tests for Classification and Prediction. Oxford University Press, Oxford; Pepe et al., 2004. Combining predictors for classification using the area under the ROC curve. University of Washington Biostatistics Working Paper Series]. Compared with its nice theoretical properties and successful applications, the MRC estimator's computational aspects are not trivial. This is because the MRC objective function is neither smooth nor continuous. Therefore, the traditional Newton–Raphson type algorithm cannot be used to find the MRC estimator. As an easy solution, we propose in this article a very simple fitting algorithm named iterative marginal optimization (IMO), which guarantees a monotone increasing of the MRC objective function at each iteration step in a very efficient manner. We show via extensive simulation that the proposed IMO algorithm is not only computationally stable but also reasonably fast. Moreover, real data about the China stock market are analyzed to further illustrate the usefulness of the proposed.

Nonparametric regression with current status data

We apply nonparametric regression to current status data, which often arises in survival analysis and reliability analysis. While no parametric assumption on the distributions has been imposed, most authors have employed parametric models like linear models to measure the covariate effects on failure times in regression analysis with current status data. We construct a nonparametric estimator of the regression function by modifying the maximum rank correlation (MRC) estimator. Our estimator can deal with the cases where the other estimators do not work. We present the asymptotic bias and the asymptotic distribution of the estimator by adapting a result on equicontinuity of degenerate U-processes to the setup of this paper.

Computation of the maximum rank correlation estimator

This paper shows that the objective function for the maximum rank correlation estimator can be evaluated in O(n log n) calculations (where n is the sample size). Previously, O( n 2) calculations were thought necessary for computation of the objective function.

Misclassification of the dependent variable in a discrete-response setting

Misclassification of dependent variables in a discrete-response model causes inconsistent coefficient estimates when traditional estimation techniques (e.g., probit or logit) are used. A modified maximum likelihood estimator that corrects for misclassification is proposed. A semiparametric approach, which combines the maximum rank correlation estimator of Han (1987) (Journal of Econometrics 35, 303–316) with isotonic regression, allows for more general forms of misclassification than the maximum likelihood approach. The parametric and semiparametric estimation techniques are applied to a model of job change with two commonly used data sets, the Current Population Survey (CPS) and the Panel Study of Income Dynamics (PSID).

The Limiting Distribution of the Maximum Rank Correlation Estimator

Han's maximum rank correlation (MRC) estimator is shown to be 4/_-consistent and asymptotically normal. The proof rests on a general method for determining the asymptotic distribution of a maximization estimator, a simple U-statistic decomposition, and a uniform bound for degenerate U-processes. A consistent estimator of the asymptotic covariance matrix is provided, along with a result giving the explicit form of this matrix for any model within the scope of the MRC estimator. The latter result is applied to the binary choice model, and it is found that the MRC estimator does not achieve the semiparametric efficiency bound.

Non-parametric analysis of a generalized regression model: The maximum rank correlation estimator

The paper considers estimation of a model y i = D · F( x′ i β 0, u i ), where the composite transformation D · F is only specified that D: R → R is non-degenerate monotonic and F: R 2 → R is strictly monotonic in each of its variables. The paper thus generalizes standard data analysis which assumes that the functional form of D · F is known and additive. The estimator which it proposes is the maximum rank correlation estimator which is non-parametric in the functional form of D · F and non-parametric in the distribution of the error terms, u i . The estimator is shown to be strongly consistent for the parameters β 0 up to a scale coefficient.