In this paper we introduce the generalized oscillator model (GOM) as a family of exactly solvable models useful to investigate theoretical aspects related to the statistical description of the aging state. GOMs are defined by a potential function V(x) and characterized by a zero-temperature relaxation determined by entropy barriers and partial equilibration. Analytic expressions for the effective temperature can be derived using a fluctuation theorem valid in the aging regime without the need to solve the dynamical equations for correlations and responses. Two classes of models are investigated in detail: the homogeneous potential model with V(x) = (k/2p)x(2p) (p being a positive integer) and the wedge potential model (V(x) = k\x\ where V(x) has a singularity at the ground-state coordinate x = 0). For the latter, we present some numerical simulations that reinforce the validity of the main analytical results. GOMs offer a conceptual framework to develop a statistical description of the spontaneous relaxation process that has been recently proposed(4) to be at the root of the intermittency phenomenon observed in glasses and colloids.