  In this paper we continue our recent study of a manifold endowed with a
singular or regular distribution, determined as the image of the tangent bundle
under a smooth endomorphism, and generalize Bochner's technique to the case of
a distribution with a statistical type structure. Following the theory of
statistical structures on Riemannian manifolds and construction of an almost
Lie algebroid on a vector bundle, we define the modified statistical connection
and exterior derivative on tensors. Then we introduce the Weitzenbock type
curvature operator on tensors and derive the Bochner-Weitzenbock type formula.
These allow us to obtain vanishing theorems about the null space of the Hodge
type Laplacian on a distribution.
