Let {P : N } be a full, 1-parameter exponential family of distributions indexed by the natural

parameter, so that each P has density with respect to some -finite measure () of the form p(x) = es(x)K() .

Define a new parametric family {Q : H} with density q(x) = e()s(x)K(())

that is, through a change of parameters governed by the smooth function () which is three times continuously differentiable and maps the (new) parameter space H, assumed to be an open set, into the interior of the natural parameter space N.

Show that the LAN property holds at each interior point of H, being sure to clearly describe the score and information matrix.