Computations and analysis in robust regression model selection using stochastic complexity

Summary

A stochastic complexity approach for model selection in robust linear regression is studied in this paper. Computational aspects and applications of this approach are the focuses of the study. Particularly, we provide both procedures and a package of S language programs for computing the stochastic complexity and for proceeding with the associated model selection. On the other hand, we discuss how a probability distribution on the set of candidate models may be induced by stochastic complexity and how this distribution may be used in diagnosis to measure the likelihood that a candidate model is selected. We also discuss some strategies for model selection when large number of potential explanatory variables are available. Finally, examples and a simulation study are presented for assessing the finite sample performance of our methods.

Appendix. Proof of Inequality (9)

Define = F(h) = ρ_c(t+h) − ρ_c(t) -− hψ_c(t) − min{1/2, c(c+∣t∣)⁻¹}h² for any given t. Straightforward calculations give the following expressions: When t≥ c

$$F(h)=\left\{\begin{array}{ll}{-2 c(t+h)-c(c+t)^{-1} h^{2},} & {h \leq-c-t} \\ {\frac{1}{2}(t+h-c)^{2}-c(c+t)^{-1} h^{2},} & {-c-t<h<c-t} \\ {-c(c+t)^{-1} h^{2},} & {h \geq c-t}\end{array}\right.$$

When t ≤ −c

$$F(h)=\left\{\begin{array}{ll}{-c(c-t)^{-1} h^{2},} & {h \leq-c-t} \\ {\frac{1}{2}(t+h+c)^{2}-c(c-t)^{-1} h^{2},} & {-c-t<h<c-t} \\ {2 c(t+h)-c(c-t)^{-1} h^{2},} & {h \geq c-t}\end{array}\right.$$

When ∣t∣| < c

$$F(h)=\left\{\begin{array}{ll}{-\frac{1}{2}(t+h+c)^{2},} & {h \leq-c-t} \\ {0,} & {-c-t<h<c-t} \\ {-\frac{1}{2}(t+h-c)^{2},} & {h \geq c-t}\end{array}\right.$$

It is easy to show that each of the above expressions is not great than 0. For example, given that t ≥ c and h ≤ − c − t, we have F(−c − t) = c² − ct ≤ 0 and F′(h) = −2c(c+t+ h)(c +t)⁻¹ ≥ 0, thus F(h) ≥ 0. Therefore, the assertion of (9) follows.