A smoothed maximum rank correlation estimator for deep ordinal choice models

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

Preface. 1. Introduction. 1.1. Ordinal Categorical Scales. 1.2. Advantages of Using Ordinal Methods. 1.3. Ordinal Modeling Versus Ordinary Regession Analysis. 1.4. Organization of This Book. 2. Ordinal Probabilities, Scores, and Odds Ratios. 2.1. Probabilities and Scores for an Ordered Categorical Scale. 2.2. Ordinal Odds Ratios for Contingency Tables. 2.3. Confidence Intervals for Ordinal Association Measures. 2.4. Conditional Association in Three-Way Tables. 2.5. Category Choice for Ordinal Variables. Chapter Notes. Exercises. 3. Logistic Regression Models Using Cumulative Logits. 3.1. Types of Logits for An Ordinal Response. 3.2. Cumulative Logit Models. 3.3. Proportional Odds Models: Properties and Interpretations. 3.4. Fitting and Inference for Cumulative Logit Models. 3.5. Checking Cumulative Logit Models. 3.6. Cumulative Logit Models Without Proportional Odds. 3.7. Connections with Nonparametric Rank Methods. Chapter Notes. Exercises. 4. Other Ordinal Logistic Regression Models. 4.1. Adjacent-Categories Logit Models. 4.2. Continuation-Ratio Logit Models. 4.3. Stereotype Model: Multiplicative Paired-Category Logits. Chapter Notes. Exercises. 5. Other Ordinal Multinomial Response Models. 5.1. Cumulative Link Models. 5.2. Cumulative Probit Models. 5.3. Cumulative Log-Log Links: Proportional Hazards Modeling. 5.4. Modeling Location and Dispersion Effects. 5.5. Ordinal ROC Curve Estimation. 5.6. Mean Response Models. Chapter Notes. Exercises. 6. Modeling Ordinal Association Structure. 6.1. Ordinary Loglinear Modeling. 6.2. Loglinear Model of Linear-by-Linear Association. 6.3. Row or Column Effects Association Models. 6.4. Association Models for Multiway Tables. 6.5. Multiplicative Association and Correlation Models. 6.6. Modeling Global Odds Ratios and Other Associations. Chapter Notes. Exercises. 7. Non-Model-Based Analysis of Ordinal Association. 7.1. Concordance and Discordance Measures of Association. 7.2. Correlation Measures for Contingency Tables. 7.3. Non-Model-Based Inference for Ordinal Association Measures. 7.4. Comparing Singly Ordered Multinomials. 7.5. Order-Restricted Inference with Inequality Constraints. 7.6. Small-Sample Ordinal Tests of Independence. 7.7. Other Rank-Based Statistical Methods for Ordered Categories. Appendix: Standard Errors for Ordinal Measures. Chapter Notes. Exercises. 8. Matched-Pairs Data with Ordered Categories. 8.1. Comparing Marginal Distributions for Matched Pairs. 8.2. Models Comparing Matched Marginal Distributions. 8.3. Models for The Joint Distribution in A Square Table. 8.4. Comparing Marginal Distributions for Matched Sets. 8.5. Analyzing Rater Agreement on an Ordinal Scale. 8.6. Modeling Ordinal Paired Preferences. Chapter Notes. Exercises. 9. Clustered Ordinal Responses: Marginal Models. 9.1. Marginal Ordinal Modeling with Explanatory Variables. 9.2. Marginal Ordinal Modeling: GEE Methods. 9.3. Transitional Ordinal Modeling, Given the Past. Chapter Notes. Exercises. 10. Clustered Ordinal Responses: Random Effects Models. 10.1. Ordinal Generalized Linear Mixed Models. 10.2. Examples of Ordinal Random Intercept Models. 10.3. Models with Multiple Random Effects. 10.4. Multilevel (Hierarchical) Ordinal Models. 10.5. Comparing Random Effects Models and Marginal Models. Chapter Notes. Exercises. 11. Bayesian Inference for Ordinal Response Data. 11.1. Bayesian Approach to Statistical Inference. 11.2. Estimating Multinomial Parameters. 11.3. Bayesian Ordinal Regression Modeling. 11.4. Bayesian Ordinal Association Modeling. 11.5. Bayesian Ordinal Multivariate Regression Modeling. 11.6. Bayesian Versus Frequentist Approaches to Analyzing Ordinal Data. Chapter Notes. Exercises. Appendix Software for Analyzing Ordinal Categorical Data. Bibliography. Example Index. Subject Index.

Semiparametric maximum likelihood estimation of polychotomous and sequential choice models

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.

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.

Analyzing Ordinal Data: Support for a Bayesian Approach

Several semiparametric estimators recently developed in the econometrics literature are based on the rank correlation between the dependent and explanatory variables. Examples include the maximum rank correlation estimator (MRC) of Han [1987], the monotone rank estimator (MR) of Cavanagh and Sherman [1998], the pairwise-difference rank estimators (PDR) of Abrevaya [2003], and others. These estimators apply to various monotone semiparametric single-index models, such as the binary choice models, the censored regression models, the nonlinear regression models, and the transformation and duration models, among others, without imposing functional form restrictions on the unknown functions and distributions. This work provides several new results on the theory of rank-based estimators. In Chapter 2 we prove that the quantiles and the variances of their asymptotic distributions can be consistently estimated by the nonparametric bootstrap. In Chapter 3 we investigate the accuracy of inference based on the asymptotic normal and bootstrap approximations, 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, however, belong to a special subclass of rank estimators for which the bound is vanishing with the rate close to n^(-1/2). In Chapter 4 we study the efficiency properties of rank estimators and propose weighted rank estimators that improve efficiency. We show that the optimally weighted MR attains the semiparametric efficiency bound in the nonlinear regression model and the binary choice model. Optimally weighted MRC 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 practically relevant 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 MRC in the transformation model and for MRC and MR in the binary choice model, and can be large for MRC and MR in the monotone regression model. Throughout, the theoretical results are illustrated with Monte-Carlo experiments and real data examples

Most mobile robots for indoor use rely on 2D laser scanners for localization, mapping and navigation. These sensors, however, cannot detect transparent surfaces or measure the full occupancy of complex objects such as tables. Deep Neural Networks have recently been proposed to overcome this limitation by learning to estimate object occupancy. These estimates are nevertheless subject to uncertainty, making the evaluation of their confidence an important issue for these measures to be useful for autonomous navigation and mapping. In this work we approach the problem from two sides. First we discuss uncertainty estimation in deep models, proposing a solution based on a fully convolutional neural network. The proposed architecture is not restricted by the assumption that the uncertainty follows a Gaussian model, as in the case of many popular solutions for deep model uncertainty estimation, such as Monte-Carlo Dropout. We present results showing that uncertainty over obstacle distances is actually better modeled with a Laplace distribution. Then, we propose a novel approach to build maps based on Deep Neural Network uncertainty models. In particular, we present an algorithm to build a map that includes information over obstacle distance estimates while taking into account the level of uncertainty in each estimate. We show how the constructed map can be used to increase global navigation safety by planning trajectories which avoid areas of high uncertainty, enabling higher autonomy for mobile robots in indoor settings.

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.

Quasi-random maximum simulated likelihood estimation of the mixed multinomial logit model

Introduction This chapter provides a review of the methods in practice, and advances in recent years, of ways of combining revealed preference (RP) and stated preference (SP) data in the estimation and application of choice models. The focus is on both the theory underlying the pooling of data sources as a guide to relevant practice, as well as a step by step outline of how choice models are structured and estimated. We use a mode choice example involving existing and new modes to illustrate the practicalities of application. Choice model specification, estimation, and application has a very long history, centered originally on the use of RP data. Behavior observed in an actual market through the collection of RP data contains information about a current market equilibrium process. Figure 19.1(a) shows a simple transport example of a market with five modes (walk, bicycle, bus, train, and car) and certain cost and speed characteristics. The technology frontier reflected in choice data collected from an existing market can be characterized by the following (Louviere et al . 2000): Technological relationships : By definition, RP data describes only those alternatives that exist, which implies that existing attribute levels and correlations between attributes will be in any model estimated from such data.

One of the most popular approaches to model choices in the airline industry is the multinomial logit (MNL) model and its variations because it has key properties for businesses: acceptable accuracy and high interpretability. On the other hand, recent research has proven the interest of considering choice models based on deep neural networks as these provide better out-of-sample predictive power. However, these models typically lack direct business interpretability. One useful way to get insights for consumer behavior is by estimating and studying the price elasticity in different choice situations. In this research, we present a new methodology to estimate price elasticity from Deep Learning-based choice models. The approach leverages the automatic differentiation capabilities of deep learning libraries. We test our approach on data extracted from a global distribution system (GDS) on European market data. The results show clear differences in price elasticity between leisure and business trips. Overall, the demand for trips is price elastic for leisure and inelastic for the business segment. Moreover, the approach is flexible enough to study elasticity on different dimensions, showing that the demand for business trips could become highly elastic in some contexts like departures during weekends, international destinations, or when the reservation is done with enough anticipation. All these insights are of a particular interest for travel providers (e.g., airlines) to better adapt their offer, not only to the segment but also to the context.

One of the most popular approaches to model choices in the airline industry is the multinomial logit (MNL) model and its variations because it has key properties for businesses: acceptable accuracy and high interpretability. On the other hand, recent research has proven the interest of considering choice models based on deep neural networks as these provide better out-of-sample predictive power. However, these models typically lack direct business interpretability. One useful way to get insights for consumer behavior is by estimating and studying the price elasticity in different choice situations. In this research, we present a new methodology to estimate price elasticity from Deep Learning-based choice models. The approach leverages the automatic differentiation capabilities of deep learning libraries. We test our approach on data extracted from a global distribution system (GDS) on European market data. The results show clear differences in price elasticity between leisure and business trips. Overall, the demand for trips is price elastic for leisure and inelastic for the business segment. Moreover, the approach is flexible enough to study elasticity on different dimensions, showing that the demand for business trips could become highly elastic in some contexts like departures during weekends, international destinations, or when the reservation is done with enough anticipation. All these insights are of a particular interest for travel providers (e.g., airlines) to better adapt their offer, not only to the segment but also to the context.

Whole-Slide Histopathology Image (WSI) is generally considered the gold standard for cancer diagnosis and prognosis. Given the large inter-operator variation among pathologists, there is an imperative need to develop machine learning models based on WSIs for consistently predicting patient prognosis. The existing WSI-based prediction methods do not utilize the ordinal ranking loss to train the prognosis model, and thus cannot model the strong ordinal information among different patients in an efficient way. Another challenge is that a WSI is of large size (e.g., 100,000-by-100,000 pixels) with heterogeneous patterns but often only annotated with a single WSI-level label, which further complicates the training process. To address these challenges, we consider the ordinal characteristic of the survival process by adding a ranking-based regularization term on the Cox model and propose a weakly supervised deep ordinal Cox model (BDOCOX) for survival prediction from WSIs. Here, we generate amounts of bags from WSIs, and each bag is comprised of the image patches representing the heterogeneous patterns of WSIs, which is assumed to match the WSI-level labels for training the proposed model. The effectiveness of the proposed method is well validated by theoretical analysis as well as the prognosis and patient stratification results on three cancer datasets from The Cancer Genome Atlas (TCGA).

Dynastic models have long provided a framework for the study of equilibria with intergenerational transfers, social mobility, and inequality. However with a few exceptions, there has been very little work on the estimations of these models. With the advent of data sets like the PSID that tracks households over more than one generation estimation of these models are now feasible. This paper explores the estimation of a class of life-cycle discrete choice intergenerational models. It proposes a new semiparametric estimation technique that circumvents the need for full solution of the dynamic programming problem. As is standard in this class of estimators, we show that it is p N consistent and asymptotically normally distributed. We compare our estimator to a modified version of the full solution maximum likelihood estimator in a Monte Carlo study. Our estimator performs comparable to ML in finite sample but greatly reduces the computational cost. To demonstrate the applicability of the estimator, a dynastic model of intergenerational transmission of human capital with unitary households is estimated. (Preliminary and Incomplete)