Question

Due to right censoring, the survival ( T ) may not obtained exactly. The observable right-censored survival time is ( Y = min(T,C) ), where ( C ) is the censoring time and is assumed independent of ( T ). Now suppose we have ( n ) observations from ( Y ) with event indicators: ( (y_1, delta_1), dots, (y_n, delta_n) ), where ( delta_i = 1 ) if ( y_i ) is the survival time and ( delta_i = 0 ) if ( y_i ) is the right censoring time. Show that the log-likelihood function from these observations is [ l = sum_{i=1}^{n} { delta_i log f_T(y_i) + (1 - delta_i) log S_T(y_i) }. ]

          Due to right censoring, the survival ( T ) may not obtained exactly. The observable right-censored survival time is ( Y = min(T,C) ), where ( C ) is the censoring time and is assumed independent of ( T ). Now suppose we have ( n ) observations from ( Y ) with event indicators: ( (y_1, delta_1), dots, (y_n, delta_n) ), where ( delta_i = 1 ) if ( y_i ) is the survival time and ( delta_i = 0 ) if ( y_i ) is the right censoring time. Show that the log-likelihood function from these observations is

[ l = sum_{i=1}^{n} { delta_i log f_T(y_i) + (1 - delta_i) log S_T(y_i) }. ]

Due to right censoring, the survival ( T ) may not obtained exactly. The observable right-censored survival time is ( Y = min(T,C) ), where ( C ) is the censoring time and is assumed independent of ( T ). Now suppose we have ( n ) observations from ( Y ) with event indicators: ( (y1, delta1), dots, (yn, deltan) ), where ( deltai = 1 ) if ( yi ) is the survival time and ( deltai = 0 ) if ( yi ) is the right censoring time. Show that the log-likelihood function from these observations is

[ l = sumi=1^n deltai log fT(yi) + (1 - deltai) log ST(yi) . ]

Added by Joshua R.

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