Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 34 / 5 / b

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- Solution b

Solution

Average the covariate over its possible event subjects using the conditional probabilities from part (a). Define the risk-weighted sums

S_{r} (t, β) = \sum_{i \in R (t)} z_{i}^{r} e^{β z_{i}}, r = 0, 1, 2,

(1)

with

z_{i}^{0} = 1

. Then

\overline{z} (t^{*}, β) = \frac{S _{1} ( t ^{*} , β )}{S _{0} ( t ^{*} , β )} = \frac{\sum _{i \in R (t^{*})} z _{i} e ^{β z_{i}}}{\sum _{i \in R (t^{*})} e ^{β z_{i}}} .

(2)

This is the mean covariate of the next event subject, conditional on the current risk set and an event. It is generally not the unweighted average over the original sample. For

β = 0

it reduces to the unweighted current risk-set mean.

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Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 34 / 5 / b

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