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In this study, we suggest policy implications for managing the risk of bankruptcy in housing presale projects in Korea. In conclusion, we find that the competing risk Cox proportional hazard model (PHM) is the best model for explaining the bankruptcy factors of projects and the banks that lend construction funds and policy authorities need to manage presale projects by considering bankruptcy factors (the initial presale rate, regional factor, the credit rating of the developers, the housing common factor). This study differs from previous studies in that it uses the competing risk Cox proportional hazard model and single-index model methodologies for empirical analysis, distinguishes the causes of business failure as default and the default of repurchase agreement options that are similar to actual defaults, and suggests policy implications based on the analysis. [ABSTRACT FROM AUTHOR]

This article considers the estimation of quantile treatment effects under the assumption of unconfoundedness given quasi-experimental data. We propose a semiparametric single-index method to estimate the propensity score. Our approach overcomes the curse of dimensionality issue of a nonparametric propensity score and can handle a moderately large dimension of covariates. It is more flexible than the parametric propensity score and thereby alleviates the possible model misspecification problem. We derive the asymptotic distribution of the quantile treatment effect estimator that is based on the semiparametric propensity score. We also propose a consistent variance estimator and construct the confidence intervals for the QTE estimator. Monte Carlo simulation results show that the proposed estimator performs well in finite samples and the confidence intervals have adequate coverage rates. We demonstrate the usefulness of our method by applying it to a study of the quantile treatment effects of college education on income. [ABSTRACT FROM AUTHOR]

We propose a flexible functional varying-coefficient single-index quantile regression model where the functional covariates of the linear part have time-varying coefficients and the single-index component offers great model flexibility in data analysis while circumventing the curse of dimensionality. The proposed model includes many existing quantile regression models for functional/longitudinal data as special cases. We use B-splines to estimate the link and coefficient functions. Under some mild conditions, we establish the asymptotic normality of the estimated index parameter vector, and obtain the convergence rates of the estimated link and coefficient functions. Moreover, we propose a score test to examine whether the effects of some covariates on the functional response are time-varying. Finally, we provide some numerical studies including Monte Carlo simulations and an empirical application to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

Single-index model is one of the most popular semiparametric models in applied econometrics. Estimation of the derivative function is often of crucial importance, as studying "marginal effects" serves as a cornerstone of microeconomics. A prerequisite for the successful application of nonparametric/semiparametric kernel estimation methods is to select smoothing parameters properly to balance the estimation squared bias and variance. Henderson et al. (2015) propose a novel method for selecting the smoothing parameters optimally for derivative function estimation. However, their method suffers from the "curse of diemnsionality" problem in a multivariate nonparametric regression model. In this article, we extend the work of Henderson et al. (2015) to estimation of the derivative function of a single-index model. Specifically, we propose a data-driven method to select smoothing parameters optimally for single-index model derivative function estimation. We also derive the asymptotic distribution of the resulting local linear estimator of the derivative function. Both simulations and empirical applications show that the proposed method works well in practice. [ABSTRACT FROM AUTHOR]

This paper explores a methodology for dimension reduction in regression models for a treatment outcome, specifically to capture covariates' moderating impact on the treatment-outcome association. The motivation behind this stems from the field of precision medicine, where a comprehensive understanding of the interactions between a treatment variable and pretreatment covariates is essential for developing individualized treatment regimes (ITRs). We provide a review of sufficient dimension reduction methods suitable for capturing treatment-covariate interactions and establish connections with linear model-based approaches for the proposed model. Within the framework of single-index regression models, we introduce a sparse estimation method for a dimension reduction vector to tackle the challenges posed by high-dimensional covariate data. Our methods offer insights into dimension reduction techniques specifically for interaction analysis, by providing a semiparametric framework for approximating the minimally sufficient subspace for interactions. [ABSTRACT FROM AUTHOR]

In health and clinical research, medical indices (eg, BMI) are commonly used for monitoring and/or predicting health outcomes of interest. While single‐index modeling can be used to construct such indices, methods to use single‐index models for analyzing longitudinal data with multiple correlated binary responses are underdeveloped, although there are abundant applications with such data (eg, prediction of multiple medical conditions based on longitudinally observed disease risk factors). This article aims to fill the gap by proposing a generalized single‐index model that can incorporate multiple single indices and mixed effects for describing observed longitudinal data of multiple binary responses. Compared to the existing methods focusing on constructing marginal models for each response, the proposed method can make use of the correlation information in the observed data about different responses when estimating different single indices for predicting response variables. Estimation of the proposed model is achieved by using a local linear kernel smoothing procedure, together with methods designed specifically for estimating single‐index models and traditional methods for estimating generalized linear mixed models. Numerical studies show that the proposed method is effective in various cases considered. It is also demonstrated using a dataset from the English Longitudinal Study of Aging project. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we consider the problem of hypothesis testing in high-dimensional single-index models. First, we study the feasibility of applying the classical F-test to a single-index model when the dimension of covariate vector and sample size are of the same order, and derive its asymptotic null distribution and asymptotic local power function. For the ultrahigh-dimensional single-index model, we construct F-statistics based on lower-dimensional random projections of the data, and establish the asymptotic null distribution and the asymptotic local power function of the proposed test statistics for the hypothesis testing of global and partial parameters. The new proposed test possesses the advantages of having a simple structure as well as being easy to compute. We compare the proposed test with other high-dimensional tests and provide sufficient conditions under which the proposed tests are more efficient. We conduct simulation studies to evaluate the finite-sample performances of the proposed tests and demonstrate that it has higher power than some existing methods in the models we consider. The application of real high-dimensional gene expression data is also provided to illustrate the effectiveness of the method. for this article are available online. [ABSTRACT FROM AUTHOR]

Beta distributions are commonly used to model proportion valued response variables, often encountered in longitudinal studies. In this article, we develop semi-parametric Beta regression models for proportion valued responses, where the aggregate covariate effect is summarized and flexibly modeled, using a interpretable monotone time-varying single index transform of a linear combination of the potential covariates. We utilize the potential of single index models, which are effective dimension reduction tools and accommodate link function misspecification in generalized linear mixed models. Our Bayesian methodology incorporates the missing-at-random feature of the proportion response and utilize Hamiltonian Monte Carlo sampling to conduct inference. We explore finite-sample frequentist properties of our estimates and assess the robustness via detailed simulation studies. Finally, we illustrate our methodology via application to a motivating longitudinal dataset on obesity research recording proportion body fat. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two semiparametric single-index models for spatially and temporally correlated data. Our first model has spatially and temporally correlated random effects that are additive to the nonparametric function, which we refer to as the "semiparametric spatio-temporal single-index model (ST-SIM)". The second model integrates the spatially correlated effects into the nonparametric function, and the time random effects are additive to the single-index function. We refer to our second model as the "semiparametric integrated spatio-temporal single-index model (IST-SIM)". Two algorithms based on a Markov chain expectation maximization are introduced to simultaneously estimate the model parameters, spatial effects, and time effects of the two models. We compare the performance of our models using several simulation studies. The proposed models are then applied to mortality data from six major cities in South Korea. Our results suggest that IST-SIM (1) is more flexible than ST-SIM because the former can estimate various nonparametric functions for different locations, while ST-SIM enforces the mortality functions having the same shape over locations; (2) provides better estimation and prediction, and (3) does not need restrictions for the single-index coefficients to fix the identifiability problem. [ABSTRACT FROM AUTHOR]

This paper studies the inference problem of index coefficient in single-index models under massive dataset. Analysis of massive dataset is challenging owing to formidable computational costs or memory requirements. A natural method is the averaging divide-and-conquer approach, which splits data into several blocks, obtains the estimators for each block and then aggregates the estimators via averaging. However, there is a restriction on the number of blocks. To overcome this limitation, this paper proposed a computationally efficient method, which only requires an initial estimator and then successively refines the estimator via multiple rounds of aggregations. The proposed estimator achieves the optimal convergence rate without any restriction on the number of blocks. We present both theoretical analysis and experiments to explore the property of the proposed method.

In this paper, we conduct a theoretical examination of a low-rank matrix single-index model. This model has recently been introduced in the field of biostatistics, but its theoretical properties for jointly estimating the link function and the coefficient matrix have not yet been fully explored. In this paper, we make use of the PAC-Bayesian bounds technique to provide a thorough theoretical understanding of the joint estimation of the link function and the coefficient matrix. This allows us to gain a deeper insight into the properties of this model and its potential applications in different fields. [ABSTRACT FROM AUTHOR]

Model checking methods based on non-parametric estimation are widely used because of their tractable limiting null distributions and being sensitive to high-frequency oscillation alternative models. However, this kind of test suffers from the curse of dimensionality, resulting in slow convergence, especially for functional data with infinite dimensional features. In this paper, we propose an adaptive-to-model test for a parametric functional single-index model by using the orthogonality of residual and its conditional expectation. The test achieves model adaptation by sufficient dimension reduction which utilizes functional sliced inverse regression. This test procedure can be easily extended to other non-parametric test methods. Under certain conditions, we prove the asymptotic properties of the test statistic under the null hypothesis, fixed alternative hypothesis and local alternative hypothesis. Simulations show that our test has better performance than the method that does not use functional sufficient dimension reduction. An analysis of COVID-19 data verifies our conclusion. [ABSTRACT FROM AUTHOR]

Single-index quantile regression (QR) models can avoid the curse of dimensionality in nonparametric problems by assuming that the response is only related to a single linear combination of the covariates. Like the standard parametric or nonparametric QR whose estimated curves may cross, the single-index QR can also suffer quantile crossing, leading to an invalid distribution for the response. This issue has attracted considerable attention in the literature in the recent year. In this article, we consider single-index models, develop methods for QR that guarantee noncrossing quantile curves, and extend the methods and results to composite quantile regression. The asymptotic properties of the proposed estimators are derived and their advantages over existing methods are explained. Simulation studies and a real data application are conducted to illustrate the finite sample performance of the proposed methods. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A Distributional (Single) Index Model (DIM) is a semiparametric model for distributional regression, that is, estimation of conditional distributions given covariates. The method is a combination of classical single-index models for the estimation of the conditional mean of a response given covariates, and isotonic distributional regression. The model for the index is parametric, whereas the conditional distributions are estimated nonparametrically under a stochastic ordering constraint. We show consistency of our estimators and apply them to a highly challenging dataset on the length of stay (LoS) of patients in intensive care units. We use the model to provide skillful and calibrated probabilistic predictions for the LoS of individual patients, which outperform the available methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two semiparametric single-index models for spatially and temporally correlated data. Our first model has spatially and temporally correlated random effects that are additive to the nonparametric function, which we refer to as the “semiparametric spatio-temporal single-index model (ST-SIM)”. The second model integrates the spatially correlated effects into the nonparametric function, and the time random effects are additive to the single-index function. We refer to our second model as the “semiparametric integrated spatio-temporal single-index model (IST-SIM)”. Two algorithms based on a Markov chain expectation maximization are introduced to simultaneously estimate the model parameters, spatial effects, and time effects of the two models. We compare the performance of our models using several simulation studies. The proposed models are then applied to mortality data from six major cities in South Korea. Our results suggest that IST-SIM (1) is more flexible than ST-SIM because the former can estimate various nonparametric functions for different locations, while ST-SIM enforces the mortality functions having the same shape over locations; (2) provides better estimation and prediction, and (3) does not need restrictions for the single-index coefficients to fix the identifiability problem.

In many stochastic problems, the output of interest depends on an input random vector mainly through a single random variable (or index) via an appropriate univariate transformation of the input. We exploit this feature by proposing an importance sampling method that makes rare events more likely by changing the distribution of the chosen index. Further variance reduction is guaranteed by combining this single-index importance sampling approach with stratified sampling. The dimension-reduction effect of single-index importance sampling also enhances the effectiveness of quasi-Monte Carlo methods. The proposed method applies to a wide range of financial or risk management problems. We demonstrate its efficiency for estimating large loss probabilities of a credit portfolio under a normal and t-copula model and show that our method outperforms the current standard for these problems. [ABSTRACT FROM AUTHOR]

The partially linear single-index model (PLSIM) is a flexible and powerful model for analyzing the relationship between the response and the multivariate covariates. This paper considers the PLSIM with measurement error possibly in all the variables. The authors propose a new efficient estimation procedure based on the local linear smoothing and the simulation-extrapolation method, and further establish the asymptotic normality of the proposed estimators for both the index parameter and nonparametric link function. The authors also carry out extensive Monte Carlo simulation studies to evaluate the finite sample performance of the new method, and apply it to analyze the osteoporosis prevention data. [ABSTRACT FROM AUTHOR]

The purpose of this paper is two fold. First, the authors investigate quantile regression (QR) estimation for single-index QR models when the response is subject to random left truncation. The random weights are introduced to deal with left truncated data and the associated iteration estimation method is proposed. The asymptotic properties for the proposed QR estimates of the index parameter and unknown link function are both obtained. Further, by combining the QR loss function and the adaptive LASSO penalization, a variable selection procedure for the index parameter is introduced and its oracle property is established. Second, a weighted empirical log-likelihood ratio of the index parameter based on the QR method is introduced and is proved to be asymptotic standard chi-square distribution. Furthermore, confidence regions of the index parameter can be constructed. The finite sample performance of the proposed methods are demonstrated. A real data analysis is also conducted to show the usefulness of the proposed approaches. [ABSTRACT FROM AUTHOR]

This article considers the issue of assessing non-linearity in the relationship between calorie consumption and income using non-parametric and semi-parametric approaches. These methodologies are implemented on the cross-sectional household survey data conducted in Pakistan in 2010–2011. This framework takes account of the heterogeneity among families and potential non-linearity in the relationship. The findings show that the calorie–income elasticity is considerable and statistically significant across estimating methodologies. The results also demonstrate that the elasticity is larger for the substantially poorer households of the sample. By incorporating the explanatory variables in a manageable way in the parametric section of regression procedures, the semi-parametric analysis also reveals a slight increase in calorie response to increases in income at various income levels.

We study the non-parametric estimation of partially linear generalized single-index functional models, where the systematic component of the model has a flexible functional semi-parametric form with a general link function. We suggest an efficient and practical approach to estimate (I) the single-index link function, (II) the single-index coefficients as well as (III) the non-parametric functional component of the model. The estimation procedure is developed by applying quasi-likelihood, polynomial splines and kernel smoothings. We then derive the asymptotic properties, with rates, of the estimators of each component of the model. Their asymptotic normality is also established. By making use of the splines approximation and the Fisher scoring algorithm, we show that our approach has numerical advantages in terms of the practical efficiency and the computational stability. A computational study on data is provided to illustrate the good practical behavior of our methodology. [ABSTRACT FROM AUTHOR]

Research aims: This study aims to compare the performance of mean-variance and single-index models in creating the optimal portfolio. Design/Methodology/Approach: This study creates optimal portfolios using the mean-variance and single-index models with daily stock return data of 38 companies listed on the LQ45 index, IDX Composite index and Bank Indonesia's 7-Day (Reverse) Repo Rate from January 1, 2012 to December 31, 2019. The two models are compared using the Sharpe ratio. Research findings: The result shows that the single-index model dominates the Indonesian Stock Exchange (IDX), more so than the mean-variance model. BBCA has the highest proportion for both mean-variance and single-index portfolios. Theoretical contribution/Originality: This study compares two popular portfolio models in the Indonesian stock market. Practitioner/Policy implication: This study helps investors to create optimal portfolios using a model that is more suited to the IDX. Research limitation/Implication: This study creates the optimal portfolio without differentiating risk preferences (i.e., risk averse, risk moderate and risk taker). In addition, this research only uses daily return data and does not compare it with weekly and monthly data. [ABSTRACT FROM AUTHOR]

Single-index models are potentially important tools for multivariate non-parametric regression analysis. They generalize linear regression models by replacing the linear combination α0⊤X with a non-parametric component η0α0⊤X, where η0(·) is an unknown univariate link function. In this article, we generalize these models to have a functional component, replacing the generalized partially linear single index models η0α0⊤X+β0⊤Z, where α is a vector in IRd, η0(·) and β0(·) are unknown functions that are to be estimated. We propose estimates of the unknown parameter α0, the unknown functions β0(·) and η0(·) and establish their asymptotic distributions, and furthermore, a simulation study is carried out to evaluate the models and the effectiveness of the proposed estimation methodology.

In this paper, we conduct a theoretical examination of a low-rank matrix single-index model. This model has recently been introduced in the field of biostatistics, but its theoretical properties for jointly estimating the link function and the coefficient matrix have not yet been fully explored. In this paper, we make use of the PAC-Bayesian bounds technique to provide a thorough theoretical understanding of the joint estimation of the link function and the coefficient matrix. This allows us to gain a deeper insight into the properties of this model and its potential applications in different fields.

Model checking methods based on non-parametric estimation are widely used because of their tractable limiting null distributions and being sensitive to high-frequency oscillation alternative models. However, this kind of test suffers from the curse of dimensionality, resulting in slow convergence, especially for functional data with infinite dimensional features. In this paper, we propose an adaptive-to-model test for a parametric functional single-index model by using the orthogonality of residual and its conditional expectation. The test achieves model adaptation by sufficient dimension reduction which utilizes functional sliced inverse regression. This test procedure can be easily extended to other non-parametric test methods. Under certain conditions, we prove the asymptotic properties of the test statistic under the null hypothesis, fixed alternative hypothesis and local alternative hypothesis. Simulations show that our test has better performance than the method that does not use functional sufficient dimension reduction. An analysis of COVID-19 data verifies our conclusion.

Composite quantile regression (CQR) is a robust and efficient estimation method. This paper studies CQR method for single-index models with ultra-high-dimensional data. We propose a penalized CQR estimator for single-index models and combine the debiasing technique with the CQR method to construct an estimator that is asymptotically normal, which enables the construction of valid confidence intervals and hypothesis testing. Both simulations and data analysis are conducted to illustrate the finite sample performance of the proposed methods. [ABSTRACT FROM AUTHOR]

This article develops a single-index approach for modeling the expectile-based value at risk (EVaR). EVaR has an advantage over the conventional quantile-based VaR (QVaR) of being more sensitive to the magnitude of extreme losses. EVaR can also be used for calculating QVaR and expected shortfall (ES) by exploiting the one-to-one mapping from expectiles to quantiles and the relationship between VaR and ES. We develop an asymmetric least squares technique for estimating the unknown regression parameter and link function in a single-index model, and establish the asymptotic normality of the resultant estimators. Simulation studies and real data applications are conducted to illustrate the finite sample performance of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel estimation for heteroscedastic single-index models with covariate measurement errors, in which both the conditional mean and conditional variance functions of the response given the covariates have a single-index structure. When covariates are directly observable, we show that the index parameter vector can be estimated consistently by the estimation obtained by fitting a misspecified linear quantile regression model under some mild regularity conditions. It is well known that naively treating mismeasured covariates as error-free usually leads to inconsistent estimators. To account for measurement errors in covariates, we establish a new estimation procedure based on corrected quantile loss function, and obtain the asymptotic consistency and normality of the resulting estimators. Finally, the finite sample performance of the proposed estimation method is illustrated by simulation studies and an empirical analysis of a real dataset. [ABSTRACT FROM AUTHOR]

In this paper, we study partially varying-coefficient single-index model where both the response and predictors are observed with multiplicative distortions which depend on a commonly observable confounding variable. Due to the existence of measurement errors, the existing methods cannot be directly applied, so we recommend using the nonparametric regression to estimate the distortion functions and obtain the calibrated variables accordingly. With these corrected variables, the initial estimators of unknown coefficient and link functions are estimated by assuming that the parameter vector β is known. Furthermore, we can obtain the least square estimators of unknown parameters. Moreover, we establish the asymptotic properties of the proposed estimators. Simulation studies and real data analysis are given to illustrate the advantage of our proposed method. [ABSTRACT FROM AUTHOR]

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Summary: This article develops a penalised asymmetric least squares estimator for single‐index expectile model. The oracle property of the proposed estimator is established. Moreover, the debiasing technique is used to construct an estimator that is asymptotically normal, which enables the construction of valid confidence intervals and hypothesis testing. Simulation studies and one real data application are conducted to illustrate the finite sample performance of the proposed methods. [ABSTRACT FROM AUTHOR]

Single-index models are potentially important tools for multivariate non-parametric regression analysis. They generalize linear regression models by replacing the linear combination ... with a non-parametric component ..., where h0( ) is an unknown univariate link function. In this article, we generalize these models to have a functional component, replacing the generalized partially linear single index models ..., where a is a vector in Rd, η0(⋅) and β 0(⋅) are unknown functions that are to be estimated. We propose estimates of the unknown parameter α0, the unknown functions η0(⋅) and β0(⋅) and establish their asymptotic distributions, and furthermore, a simulation study is carried out to evaluate the models and the effectiveness of the proposed estimation methodology. [ABSTRACT FROM AUTHOR]

As ultra high‐dimensional longitudinal data are becoming ever more apparent in fields such as public health and bioinformatics, developing flexible methods with a sparse model is of high interest. In this setting, the dimension of the covariates can potentially grow exponentially as exp(n1/2) with respect to the number of clusters n. We consider a flexible semiparametric approach, namely, partially linear single‐index models, for ultra high‐dimensional longitudinal data. Most importantly, we allow not only the partially linear covariates but also the single‐index covariates within the unknown flexible function estimated nonparametrically to be ultra high dimensional. Using penalized generalized estimating equations, this approach can capture correlation within subjects, can perform simultaneous variable selection and estimation with a smoothly clipped absolute deviation penalty, and can capture nonlinearity and potentially some interactions among predictors. We establish asymptotic theory for the estimators including the oracle property in ultra high dimension for both the partially linear and nonparametric components, and we present an efficient algorithm to handle the computational challenges. We show the effectiveness of our method and algorithm via a simulation study and a yeast cell cycle gene expression data. [ABSTRACT FROM AUTHOR]

This study aims to find whether Sharpe's single-index model of portfolio construction offers better investment alternatives to the investors of the Dhaka Stock Exchange (DSE). For this purpose, month-ended closing price data of 178 companies listed on the DSE, the prime bourse of Bangladesh, and the month-ended index value of DSEX have been used for the period starting from January 2013 to February 2018. The stocks selected for this study belong to 16 industrial sectors, and purposive sampling technique has been used to select these sectors. Sharpe's model formulates a unique cut-off rate and selects the stocks having an excess return-to-beta ratio above that rate. In this study, 54 stocks qualified to be a part of the optimal portfolio. Hence, the proportion of investment to be made on each of the stock is calculated according to the model. The study reveals that three industries occupy a hefty chunk (65.78%) of the proposed investment portfolio. The constructed portfolio offers a monthly return of 2.1489% and carries 1.9516% risk as measured by standard deviation. The beta of the optimal portfolio is only 0.124003. The constructed portfolio outperforms every individual stock as well as the market index in terms of offering the optimal risk-return combinations. Therefore, this five-and-a-half-decade-old model offers a great opportunity for Bangladeshi investors to optimize return and diversify risk in an efficient manner.

Multivariate nonparametric regression faces great challenges when overburdened with large amounts of covariates. For this reason, single-index models (SIMs) have been frequently used for reducing the parameters to be estimated in nonparametric and semiparametric models. In this paper, we develop a scale-space statistical tool, known as significant zero crossings of derivatives (SiZer), for SIM. It offers the comme il faut methodology for finding important structure within data in scale-space. The average derivative is used as the coefficient vector of the SIM, which is estimated by a local-likelihood approach weighted by kernel functions. By demonstration, the proposed SiZer for the SIM successfully captures trends in simulation and in real applications. Furthermore, we provide an extension from our proposed SiZer methodology of regression to generalized linear models. [ABSTRACT FROM AUTHOR]

We consider a single‐index regression model, uniquely constrained to estimate interactions between a set of pretreatment covariates and a treatment variable on their effects on a response variable, in the context of analyzing data from randomized clinical trials. We represent interaction effect terms of the model through a set of treatment‐specific flexible link functions on a linear combination of the covariates (a single index), subject to the constraint that the expected value given the covariates equals 0, while leaving the main effects of the covariates unspecified. We show that the proposed semiparametric estimator is consistent for the interaction term of the model, and that the efficiency of the estimator can be improved with an augmentation procedure. The proposed single‐index regression provides a flexible and interpretable modeling approach to optimizing individualized treatment rules based on patients' data measured at baseline, as illustrated by simulation examples and an application to data from a depression clinical trial. [ABSTRACT FROM AUTHOR]

Panel current status data arise frequently in biomedical studies when the occurrence of a particular clinical condition is only examined at several prescheduled visit times. Existing methods for analyzing current status data have largely focused on regression modeling based on commonly used survival models such as the proportional hazards model and the accelerated failure time model. However, these procedures have the limitations of being difficult to implement and performing sub‐optimally in relatively small sample sizes. The performance of these procedures is also unclear under model misspecification. In addition, no methods currently exist to evaluate the prediction performance of estimated risk models with panel current status data. In this paper, we propose a simple estimator under a general class of nonparametric transformation (NPT) models by fitting a logistic regression working model and demonstrate that our proposed estimator is consistent for the NPT model parameter up to a scale multiplier. Furthermore, we propose nonparametric estimators for evaluating the prediction performance of the risk score derived from model fitting, which is valid regardless of the adequacy of the fitted model. Extensive simulation results suggest that our proposed estimators perform well in finite samples and the regression parameter estimators outperform existing estimators under various scenarios. We illustrate the proposed procedures using data from the Framingham Offspring Study. [ABSTRACT FROM AUTHOR]

A single-index model is given by y = g*(< x, θ* >) + ε : The scalar response y depends on the covariate vector x both through an unknown (vector) parameter θ* as well as an unknown, non-parametric, univariate link-function g* ∈ G. We study the problem of recovering the parameter θ* from i.i.d. samples of the model when the covariates are drawn from a normal distribution. Our focus is on leveraging information about the (known) function class G in order to design a procedure that adapts to the noise level in the problem, thereby reducing the bias of parameter estimation. We show that when given access to a natural “labeling oracle”, our procedure recovers the underlying parameter at a rate that depends explicitly on how well we are able to estimate a suitably defined “inverse” link function. Both the procedure and its analysis framework are flexible, admitting any black-box estimator for the inverse link function. The resulting rate of parameter estimation significantly improves upon the risk of classical semi-parametric procedures whenever consistent estimates of the inverse link function can be obtained. When the function class G is appropriately structured and empirical risk minimization is used to estimate the inverse function, we provide quantitative upper bounds on the risk that depend on natural complexity measures of the class of inverse functions. We specialize our framework to the case where G is a sub-class of monotone single-index models, showing a computationally efficient, end-to-end algorithm that achieves very fast rates of parameter estimation in the regime in which the signal-to-noise ratio in the problem is large. We also pay particular attention to parameter identifiability in the noiseless model, deriving sharper upper bounds as well as information-theoretic lower bounds. Consequences for some unimodal SIMs and the (real) phase retrieval problem are also discussed. [ABSTRACT FROM AUTHOR]

An important factor in house prices is its location. However, measurement errors arise frequently in the process of observing variables such as the latitude and longitude of the house. The single-index models with measurement errors are used to study the relationship between house location and house price. We obtain the estimators by a SIMEX method based on the local linear method and the estimating equation. To test the significance of the index coefficient and the linearity of the link function, we establish the generalized likelihood ratio (GLR) tests for the models. We demonstrate that the asymptotic null distributions of the established GLR tests follow χ 2 -distributions which are independent of nuisance parameters or functions. Finally, two simulated examples and a real estate valuation data set are given to illustrate the effect of GLR tests. [ABSTRACT FROM AUTHOR]

This study proposes a partial functional partially linear single-index model that consists of a functional linear component and a linear single-index component. This model generalizes many well-known existing models, and is suitable for more complicated data structures. We develop a new estimation procedure that combines a functional principal component analysis of the functional predictors, B-spline model for the parameters, and profile estimation of the unknown parameters and functions in the model. We establish the consistency and asymptotic normality of the parametric estimators. Furthermore, we derive the global convergence rate of the proposed estimator of the linear slope function, and establish that it is optimal in the minimax sense. We implement a two-stage procedure to estimate the nonparametric link function of the single-index component of the model; here, we find that the resulting estimator possesses the optimal global rate of convergence. Then, we obtain the convergence rate of the mean squared prediction error for a predictor. We study the empirical properties of the proposed procedures using Monte Carlo simulations. The proposed method is illustrated by analyzing a diffusion tensor imaging data set from the Alzheimer's Disease Neuroimaging Initiative database. [ABSTRACT FROM AUTHOR]

There has been a lot of interest in sufficient dimension reduction (SDR) methodologies, as well as nonlinear extensions in the statistics literature. The SDR methodology has previously been motivated by several considerations: (a) finding data-driven subspaces that capture the essential facets of regression relationships; (b) analyzing data in a ‘model-free’ manner. In this article, we develop an approach to interpreting SDR techniques using information theory. Such a framework leads to a more assumption-lean understanding of what SDR methods do and also allows for some connections to results in the information theory literature.