SOA Exam ALTAM Fall 2023, Question 4
Study guide: Exam ALTAM
Twenty individuals with independent future lifetimes, all exact age 70, are subject to the following multi-state nursing home model:
You are using maximum likelihood estimation to estimate model transition intensities, assuming constant transition intensities.
You are given:
(i) Life A is in State 0 at age 70 and remains in State 0 until she dies at age 70.9.
(ii) Life B is in State 1 at age 70. At age 70.2, she moves into State 0 where she remains until age 70.5. At age 70.5, she moves into State 1 and remains there until she dies at age 70.9.
(iii) Life C is in State 0 at age 70. At age 70.3, she moves into State 1 where she remains until age 70.6. At age 70.6, she moves into State 2 and remains there until age 71.
(iv) Life D is in State 0 at age 70. At age 70.5, she moves into State 1 where she remains until age 70.7 at which point she dies.
(v) Six additional lives are in State 0 at age 70 and remain there until age 71.
(vi) Nine additional lives are in State 1 at age 70 and remain there until age 71.
(vii) One additional life is in State 2 at age 70 and remains there until age 71.
Part a (1 point)
Explain in words why the contribution to the likelihood of a life who remains in State 0 from age 70 to age 71 is e^{-(1)\left(\mu_{70}^{01} + \mu_{70}^{03}\right)}.
Part b (2 points)
Write an expression for the contribution to the likelihood of Life B expressed entirely in terms of transition intensities.
Part c (2 points)
Calculate the maximum likelihood estimate of \mu_{70}^{01}.
Part d (2 points)
Calculate the maximum likelihood estimate of {}_1p_{70}^{\overline{00}}.
Part e (1 point)
Calculate the estimated standard deviation of the maximum likelihood estimate of \mu_{70}^{01}.
Part f (1 point)
The maximum likelihood estimate of \mu_{70}^{23} is 0.
Explain why this transition intensity is estimated to be 0 despite the transition from State 2 to State 3 being a possible transition in the model.
Answer this question and see the worked solution