The name is absent

43

₌ Λ f ⅛)exp(⅜ )^θ'^{faj
j W} ⅛l∑Lι^(¾)expW)/

and,

L∞(∕5) = ∏<⅝

J=I

_ππ( ⅜-)exp(⅜ }^d^

⅛MlΣLι^(¾)θχp(^)∫

where ʧɔ(/?) is the likelihood function for the time point x_i.

To make the above equation well-defined, we need to make several adjustments.
First, we set P⅛(¾∙) = 1 for x_i ≤ k. Because by definition, P⅛(¾) represents the
probability of the subject к survived at time Xj for Xj > k, given that it survives until
time k. That is for Xj ≤ k, the subject k survives, i.e. Pk(xj) = 1 ■

Second, please note that observations have been sorted in ascending order already.
That is, for i = 1,..., n, the subject i failed at time x_i. Therefore, for a failed subject
j, we set P_i(xj) = 0 for j > i.

Third, regarding D_i(xj) for a failed subject i, instead of using Equation 3.6, we
set D_i(x_i) = 1 and D_i(xj) = 0 for j > i. That is, the subject i losses all of its unit
mass when it fails at time x_i. As a result, a failed subject contributes only one time
in the extended Cox partial likelihood function.

The counterpart of the extended Cox partial likelihood function is shown below,

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