113 Math
File: 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

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