Exponential Distribution

• Let T = time until next (y=1) occur
• Fixes the “event” and says “how much time goes by until an event occurs”

\[F(t) = P(T \leq t) = 1 - e^{-\lambda t}\]

Remember this about the survival function for exponential distribution:

\(S(t) = P(T>t)\) \(P(T>t) = 1-P(T \leq t)\) \(1-P(T \leq t) = 1-F(t)\) \(1-F(t) = 1-[1 - e^{-\lambda t}]\) \(S(t) = e^{-\lambda t}\)

The lambda sign in the survival function is the hazard. So it can be re-written as

\[S(t) = e^{HAZ * t}\]

In order to estimate the survival function for exponential model, we think of the survival model as a negative exponential curve. To estimate the rate at which survival is decreasing, we can model the hazard in two ways:

\(HAZ = e^{b_0 + b_1 X_1 ... b_k X_k}\) \(ln(HAZ) = b_0 + b_1 X_1 ... b_k X_k\)

• b-naught is the the log-hazard for reference at t=0