Using (6.11), show that the Legendre transformation K*(x; ) of the exponentially tilted density satisfies the partial

Using (6.11), show that the Legendre transformation K*(x; θ) of the exponentially tilted density satisfies the partial differential equations Hence show that in the univariate case, K∗ (x; θ) = ∫

x

μ

x − t v (t)

dt, where μ = Κ′(θ) and ν(μ) = Κ″(θ), (Wedderbum, 1974; Nelder & Pregibon, 1986).