5.6 In the density (5.1) (a) For s = 1 show that E [T (X)] = B...

5.6 In the density (5.1)

(a) For s = 1 show that Eθ [T (X)] = B

(θ)/η

(θ) and varθ [T (X)] = B(θ)

[η(θ)]2 − η(θ)B

(θ)

[η(θ)]3 .

(b) For s > 1, show that Eθ [T (X)] = J −1∇B where J is the Jacobian matrix defined by J = {

∂ηj

∂θi

} and ∇B is the gradient vector ∇B = { ∂

∂θi B(θ)}.

(See Johnson, Ladalla, and Liu (1979) for a general treatment of these identities.)