Xn from a p.f. Let x, ..., n denote the observed values of n p-dimensional observations...

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Xn from a p.f. Let x₁, ..., în denote the observed values of n p-dimensional observations X₁, or p.d.f. f(x; 0), where is a d-dimensional parameter. If f(x; 0) belongs to the d-dimensional exponential family, then the likelihood function L(0) for has the form L(x₁,...,xn; 0) = b(x₁, ..., xn) exp{c(0)TT(x₁,...,xn)}/a(0), where c(0) is a q × 1 vector function of the parameter vector and where a(0) and b(x₁,...,xn) are non-negative scalar functions. (i) What is the condition on q and on the Jacobian of c(0) for the likelihood function (or equivalently, f(x; 0)) to belong to the regular exponential family? [3 marks] (ii) Discuss the uniqueness of the so-called natural or canonical parameter c(0) and the corresponding choice of the sufficient statistic T. [3 marks] (iii) Assuming henceforth that the likelihood function belongs to the regular exponential family, show that the likelihood equation can be expressed as alog L(0)/20 = 0 T(x1, E{T(X₁,..., Xn)}; that is, the maximum likelihood (ML) estimate of 0 satisfies the equation, T(x1, ..., xn) = Eĝ{T(X1, · Xn)}; Xn) = where E implies expectation using for 0. [7 marks] (iv) In the current case where L(0) belongs to the regular exponential family, show that the observed information matrix I() is equal to the estimated Fisher (expected) information matrix I (8), that is, I (Ô) = I (Ô). Xn from a p.f. Let x₁, ..., în denote the observed values of n p-dimensional observations X₁, or p.d.f. f(x; 0), where is a d-dimensional parameter. If f(x; 0) belongs to the d-dimensional exponential family, then the likelihood function L(0) for has the form L(x₁,...,xn; 0) = b(x₁, ..., xn) exp{c(0)TT(x₁,...,xn)}/a(0), where c(0) is a q × 1 vector function of the parameter vector and where a(0) and b(x₁,...,xn) are non-negative scalar functions. (i) What is the condition on q and on the Jacobian of c(0) for the likelihood function (or equivalently, f(x; 0)) to belong to the regular exponential family? [3 marks] (ii) Discuss the uniqueness of the so-called natural or canonical parameter c(0) and the corresponding choice of the sufficient statistic T. [3 marks] (iii) Assuming henceforth that the likelihood function belongs to the regular exponential family, show that the likelihood equation can be expressed as alog L(0)/20 = 0 T(x1, E{T(X₁,..., Xn)}; that is, the maximum likelihood (ML) estimate of 0 satisfies the equation, T(x1, ..., xn) = Eĝ{T(X1, · Xn)}; Xn) = where E implies expectation using for 0. [7 marks] (iv) In the current case where L(0) belongs to the regular exponential family, show that the observed information matrix I() is equal to the estimated Fisher (expected) information matrix I (8), that is, I (Ô) = I (Ô).