SiZer Inference for Varying Coefficient Models

Abstract

Varying coefficient models are a useful statistical tool to explore dynamic patterns of a regression relationship, in which the variation features of the regression coefficients are taken as the main evidence to reflect the dynamic relationship between the response and the explanatory variables. In this study, we propose a SiZer approach as a visually diagnostic device to uncover the statistically significant features of the coefficients. This method can highlight the significant structures of the coefficients under different scales and can therefore extract relatively full information in the data. The simulation studies and real-world data analysis show that the SiZer approach performs satisfactorily in mining the significant features of the coefficients.

Robust SiZer Approach for Varying Coefficient Models

“…The above findings are very similar to those obtained by the local least-squares-based SiZer method in Zhang and Mei [35] except the SiZer map of the coefficient 3 ( ). For intermediate bandwidths, the local least-squares-based SiZer map of 3 ( ) displays a strong increase trend on the time range (250, 450), but this significant characteristic does not appear in the robust SiZer map in Figure 7(c), which may indicate that there are outliers in the data.…”

“…Additionally, in order to make a comparison of the robustness between the local LAD-based SiZer approach and the local least-squares-based SiZer method suggested by Zhang and Mei [35], Figure 5 displays the local leastsquares-based SiZer maps and the families of the smooths of the coefficients in Group 2 with the model errors drawn, respectively, from the Cauchy distribution and the contaminated normal distribution. It can be seen that the SiZer maps cannot correctly reflect the features of either the cosine curve or the parabola.…”

“…A comparison with the local least-square-based SiZer method in Zhang and Mei [35] is also made in this case to demonstrate the robustness of the proposed SiZer approach. Like the situation of the second group coefficient functions, the local least-squares-based SiZer maps of the coefficient functions are seriously distorted by outliers.…”

“…The coefficients 1 ( ) and 2 ( ) in each of the above groups have different degrees of smoothness and they have been used in Zhang and Mei [35] to demonstrate the local least-squares-based SiZer analysis for the varying coefficient model. Figure 1 depicts the true curves of the coefficient functions 1 ( ) and 2 ( ) in Groups 1-3.…”

“…The other specific SiZer tools include SiZer for additive model [19], comparison of curves [30], time series analysis [31], SiZer for regression quantiles [32], analysis of random signals [33], and spatially dependent images [34], among others. Recently, Zhang and Mei [35] developed a SiZer approach for the varying coefficient models to explore the statistically significant features of the coefficient functions under different bandwidths. Although the study shows that the SiZer method is efficient at uncovering the genuine features such as monotonicity, peaks, valleys, and even the degree of smoothness of the coefficients, it did not consider the situation where outliers are present in the data.…”