Penalized, shrinkage, and preliminary test strategies in nonlinear and proportional hazard regression models for low and high-dimensional data

In regression analysis with many regressors, it is expected that some may not significantly contribute to predicting the response variable. This is called uncertain prior information (UPI) and may be obtained by variable selection. The use of UPI to produce a submodel or restricted model has received increasing attention in statistical models. In practice, the full, or unrestricted, model may be overfitted, with too many predictors included, when prior information uncertainty proves correct. A submodel including only important regressors may be more practical and feasible, but concerns remain that the submodel may be inappropriate when UPI is incorrect.The objective of this study was to propose novel estimators that are more efficient in estimation than the classical estimator. In addition, the study also attempted to optimally incorporate the full model and submodel for parameter estimation using preliminary test and shrinkage strategies. This idea will improve regression parameter estimation efficiency, even with uncertain prior information accuracy. Proposed estimators were applied with the Cobb-Douglas, exponential, and monomolecular multiple nonlinear regression models and the Cox proportional hazards regression model (special chapter) under UPI in low-dimensional and high-dimensional data regimes.The proposed estimator's performance was compared theoretically by deriving asymptotic distributional quadratic bias and risk under the sequence of local alternatives. In addition, Monte Carlo simulations were conducted to evaluate the numerical proposed estimator performance. Numerical comparisons were also made with penalty estimation strategies: least absolute shrinkage and selection operator (LASSO) and adaptive LASSO. Finally, the proposed estimators were applied to real data examples to verify their usefulness. Regardless of prior information correctness, the proposed estimators were shown to perform significantly better than classical estimators which are severely affected by information uncertainty.