IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Mortality projection
In a Lee-Carter model fitted by singular value decomposition, the matrix of centred log death rates Z(x,t) = ln m(x,t) - a(x) has first singular value s1 and the first left and right singular vectors u1 and v1. Which statement about the fit is correct?
In the SVD fit, the age vector b(x) is proportional to the first left singular vector u1, while k(t) is proportional to s1 times the first right singular vector v1, then both are rescaled to meet the identifiability constraints. The fit is only a rank-one approximation.
- Ab(x) is proportional to u1 and k(t) is proportional to s1 times v1, before normalising the constraintsCorrect
- Bb(x) is proportional to v1 and k(t) is proportional to u1, with no role for s1
- Ca(x) is obtained from the second singular vectors
- Dk(t) is the average of Z(x,t) over x, regardless of the singular vectors
- The fit is exact whenever s1 is positive
Explanation
The rank-one approximation of Z is s1 u1 v1', with ages as rows. So b(x) is proportional to u1 and k(t) to s1 v1, then rescaled so b sums to one and k to zero. The fit is exact only if the matrix has rank one, so the final option is wrong.
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