Inference in High Dimensional Generalized Linear Models based on Soft-Thresholding

We propose a new method for estimation of a high number of coefficients within the generalized linear model framework. The estimator leads to an adaptive selection of model terms without substantial variance inflation. Our proposal extends the soft-thresholding strategy from Donoho and Johnstone (1994) to generalized linear models and multiple predictor variables. Furthermore, we develop an estimator for the covariance matrix of the estimated coefficients, which can even be used for terms dropped from the model. Used in connection with basis functions, the proposed methodology provides an alternative to other generalized function estimators. It leads to an adaptive economical description of the results in terms of basis functions. Specifically, it is shown how adaptive regression splines and qualitative restrictions can be incorporated. Our approach is demonstrated by applications to solvency prognosis and rental guides

A nonparametric multiplicative hazard model for event history analysis

A major issue in exploring and analyzing life history data with multiple states and events is the development and availability of flexible methods that allow simultaneous incorporation and estimation of baseline hazards, detection and modelling of nonlinear functional forms of covariates and time-varying effects, and the possibility to include time-dependent covariates. In this paper we consider a nonparametric multiplicative hazard model that takes into account these aspects. Embedded in the counting process approach, estimation is based on penalized likelihoods and splines. The methods are illustrated by two real data applications, one to a more conventional survival data set with two absorbing states, and one to more complex sleep-electroencephalography data with multiple recurrent states of sleep

Additive, Dynamic and Multiplicative Regression

We survey and compare model-based approaches to regression for cross-sectional and longitudinal data which extend the classical parametric linear model for Gaussian responses in several aspects and for a variety of settings. Additive models replace the sum of linear functions of regressors by a sum of smooth functions. In dynamic or state space models, still linear in the regressors, coefficients are allowed to vary smoothly with time according to a Bayesian smoothness prior. We show that this is equivalent to imposing a roughness penalty on time-varying coefficients. Admitting the coefficients to vary with the values of other covariates, one obtains a class of varying-coefficient models (Hastie and Tibshirani, 1993), or in another interpretation, multiplicative models. The roughness penalty approach to non- and semiparametric modelling, together with Bayesian justifications, is used as a unifying and general framework for estimation. The methodological discussion is illustrated by some real data applications

Inference in High Dimensional Generalized Linear Models based on Soft-Thresholding

We propose a new method for estimation of a high number of coefficients within the generalized linear model framework. The estimator leads to an adaptive selection of model terms without substantial variance inflation. Our proposal extends the soft-thresholding strategy from Donoho and Johnstone (1994) to generalized linear models and multiple predictor variables. Furthermore, we develop an estimator for the covariance matrix of the estimated coefficients, which can even be used for terms dropped from the model. Used in connection with basis functions, the proposed methodology provides an alternative to other generalized function estimators. It leads to an adaptive economical description of the results in terms of basis functions. Specifically, it is shown how adaptive regression splines and qualitative restrictions can be incorporated. Our approach is demonstrated by applications to solvency prognosis and rental guides

Generalized Soft-Thresholding and Varying-coefficient Models

We propose a new method for estimation of unknown functions within the generalized linear model framework. The estimator leads to an adaptive economical description of the results in terms of basis functions. Our proposal extends the soft--thresholding strategy from ordinary wavelet regression to generalized linear models and multiple predictor variables. Several sets of basis functions, tailored to specific purposes, can be incorporated into our methodology. We discuss semiparametric statistical inference based on generalized soft--thresholding. An algorithm which produces a sequence of estimates corresponding to increasing model complexity is developed. Advantages of our approach are demonstrated by an application to German labour market data