Folder:
113 Math
File:
113.020.010.10.20 Regression models - Exponential Distribution
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
Source:
- Me