Study on adaptive model selection through generalized degrees of freedom in nested linear regression models

Various model selection criteria have been proposed to fit models to data, such as
AIC (Akaike 1974), BIC (Schwarz 1978), and Mallows’ Cp (1973). For linear regression
with suitable regularity conditions, we can combine those criterion into general
final prediction error criterion with different lambda. If we consider all possible general
final prediction error criterion over an interval including λ = 2 and λ = log n.
Shen and Ye (2002) proposed the adaptive model selection by determining proper
lambda through general final prediction error.
In this thsis, we will introduce the adaptive models selection criterion through
generalized degrees of freedom which proposed by Shen and Ye (2002) and evaluate
the performance of this criterion in a most widely used linear regression model with
normal error and some further bias and sample size assumption. We will demonstrate
that the adaptive model selection criterion is not fully adaptive. As a remedy, we
suggest that the interval should be restricted. We will provide some simulation results
to show the performance of adaptive model selection through generalized degrees of
freedom in nested linear regression models and the conclusions. We will provide
some simulation results to motivate the procedure of solving problems and support
our conclusions.