In this paper, it is aimed to broad Xiong's scope of analysis to the threshold models and develop an adjusted Mallows criterion to select the garrote parameter vector. Compared with other penalized least-squares methods, the nonnegative garrote (NG) method has a natural penalty and tuning parameter. Furthermore, we show the asymptotic optimality of the NG estimator by referring to the idea of the asymptotic optimality of the model averaging estimator under some regular conditions. Our investigation of finite-sample performance demonstrates that the proposed method exhibits very favorable properties with respect to the ratio of the correct number of zero estimated components (RCNZ) and the root mean square error (RMSE) compared to the least absolute shrinkage and selection operator (LASSO), the smoothly clipped absolute deviation (SCAD) and the minimax concave penalty (MCP) penalized regression methods. The proposed method is applied to the analysis of body fat data, soil respiration data and the US unemployment rate data and performs well.

Chatterjee's new coefficient proposed by Chatterjee [2021, 'A New Coefficient of Correlation', Journal of the American Statistical Association, 116(536), 2009-2022.] is used to measure the degree of dependence between two scalars by rank correlation. However, the independence test based on Chatterjee's rank correlation may lose power since it only considers the distance between the nearest neighbours and ignores the other neighbours. In this paper, we propose an improvement to Chatterjee's new coefficient by incorporating the inverse distance-weighting, and further obtain the asymptotic normality of the improved coefficient and the Berry-Esseen bound under the null hypothesis. The proposed method is evaluated on the simulated as well as the real data on Yeast Gene Expression. The results show that the proposed method is superior to Chatterjee's new coefficient under various alternative hypotheses.

In this paper, we investigate model selection and model averaging based on rank regression. Under mild conditions, we propose a focused information criterion and a frequentist model averaging estimator for the focused parameters in rank regression model. Compared to the least squares method, the new method is not only highly efficient but also robust. The large sample properties of the proposed procedure are established. The finite sample properties are investigated via extensive Monte Claro simulation study. Finally, we use the Boston Housing Price Dataset to illustrate the use of the proposed rank methods.