A Common Model Selection Criterion

We consider the linear model situation (1.1) $$ {\rm{}} = {{\rm{}}_{\rm{t}}}{{\rm{\beta }}_{\rm{t}}} + {{\rm{}}_{\rm{t}}} $$ where y = (y1,y2,…,yn)′ is an n × 1 vector of response observations, Xt is an n × pt matrix of predictor variable values, $$ {\rm{n}} > {{\rm{p}}_{\rm{t}}}{\rm{,}}{{\rm{}}_{\rm{t}}} $$ is a pt × 1 vector of regression parameters to be estimated and $$ {{\rm{}}_{\rm{t}}} $$ is distribt uted $$ {\rm{N}}({{\rm{}}_{\rm{t}}}{\rm{.}}{{\rm{\sigma }}^{\rm{2}}}{{\rm{I}}_{\rm{n}}}) $$ . We shall distinguish between problems in which (a) $$ {{\rm{}}_{\rm{t}}} = $$ for all t, and (b) $$\mathop{\delta }\limits_{{ \sim t}} = {{\left( {\mathop{{0'}}\limits_{ \sim } ,\mathop{{a'}}\limits_{{ \sim t}} } \right)}^{\prime }} $$ , for all t, where the elements of the $$ {{\rm{}}_{\rm{t}}} $$ are non-zero and each $$ {{\rm{}}_{\rm{t}}} $$ vector is size k × 1 where, typically, k ≪ n/2. The generic notation t denotes a general indexing which will be made specific for particular problems to be discussed below. Each choice of t will provide a model Mt, say, defined by (1.1). The general problem, given a specific indexing system for t, is to decide, from data made available on $$\mathop{y}\limits_{ \sim } $$ and the $$ {{\rm{}}_{\rm{t}}}^\prime {\rm{s}} $$ , which Mt “best represents” the data.