65 b.Find the maximum likelihood estimator of 0,0(If there exists a closed form solution,then present it.If not,indicate how a solution can be found). To find the MLE,solve the score equations.The log-likelihood for an individual is 1e:x,r）x-l1oga)-2ox-m2+r(a+m）-log(1+exp(a+x》 1 The score vector for an individual is al(0;x,r） r exp(a+Bx) 8a 1+exp(a+Bx) al(0;x,r) x(r- exp(a+Bx) b(x,T;0)= 83 1+exp(a+Bx)) ∂l(0;c,r x一业 ∂μ 8l(0;x,r） 8o2 动 2G465 b. Find the maximum likelihood estimator of θ, ˆ θn (If there exists a closed form solution, then present it. If not, indicate how a solution can be found). To find the MLE, solve the score equations. The log-likelihood for an individual is l(θ; x, r) ∝ − log(σ)− 12σ2 (x−µ)2 +r(α+βx)−log(1+ exp(α+βx)) The score vector for an individual is ψ(x, r; θ) = ⎡⎢ ⎢⎢⎢ ⎢⎣ ∂l(θ;x,r) ∂α ∂l(θ;x,r) ∂β ∂l(θ;x,r) ∂µ ∂l(θ;x,r) ∂σ2 ⎤⎥ ⎥⎥⎥ ⎥⎦ = ⎡⎢ ⎢⎢⎢ ⎢⎣ r − exp(α+βx) 1+exp(α+βx) x(r − exp(α+βx) 1+exp(α+βx) ) x−µ σ2 − 1 2σ2 + (x−µ)2 2σ4 ⎤⎥ ⎥⎥⎥ ⎥⎦