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Stochastic Models Waiting for Disaster Walt Pohl Universität Zürich Department of Business Administration May 16, 2013 Survival Analysis We will use a few concepts from survival analysis. Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 2/7 Survival Analysis We will use a few concepts from survival analysis. Survival analysis models waiting for an event to happen. It was first developed in medicine, where it was used to model how long terminally ill patients had to live. The associated terminology is grim. Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 2/7 Survival Analysis We will use a few concepts from survival analysis. Survival analysis models waiting for an event to happen. It was first developed in medicine, where it was used to model how long terminally ill patients had to live. The associated terminology is grim. We will use it to model how long companies have to live, so for us the terminology fits. Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 2/7 Survival Let T be a random variable that represents the amount of time until an event occurs (death, for example), and let Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 3/7 Survival Let T be a random variable that represents the amount of time until an event occurs (death, for example), and let F (t) = P(T < t) be its CDF. Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 3/7 Survival Let T be a random variable that represents the amount of time until an event occurs (death, for example), and let F (t) = P(T < t) be its CDF. Then the probability that the event doesn’t happen before time t, F̂ , is Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 3/7 Survival Let T be a random variable that represents the amount of time until an event occurs (death, for example), and let F (t) = P(T < t) be its CDF. Then the probability that the event doesn’t happen before time t, F̂ , is F̂ (t) = P(T > t) = 1 − F (t). Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 3/7 Survival Let T be a random variable that represents the amount of time until an event occurs (death, for example), and let F (t) = P(T < t) be its CDF. Then the probability that the event doesn’t happen before time t, F̂ , is F̂ (t) = P(T > t) = 1 − F (t). F̂ is known as the survival function for T . Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 3/7 Hazard Rate Before the event occurs, the only information we have is how long we’ve already waited. This makes the conditional probability distribution, Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 4/7 Hazard Rate Before the event occurs, the only information we have is how long we’ve already waited. This makes the conditional probability distribution, P(T ≤ t 0 |T > t), interesting. Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 4/7 Hazard Rate Before the event occurs, the only information we have is how long we’ve already waited. This makes the conditional probability distribution, P(T ≤ t 0 |T > t), interesting. In discrete time, all we need to know is that probability of the event happening at time t + 1, given that it hasn’t happened at time t. Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 4/7 Hazard Rate, cont’d The analoguous concept in continuous time is the hazard rate, which is the rate of change in the probability of the event happening in the next instant, given that it has not yet happened. Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 5/7 Hazard Rate, cont’d The analoguous concept in continuous time is the hazard rate, which is the rate of change in the probability of the event happening in the next instant, given that it has not yet happened. The hazard rate is defined as P(t < T < t + ∆t|T = t) f (t) lim = , ∆t→0 ∆t 1 − F (t) where f is the pdf of T . Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 5/7 Hazard Rate, cont’d The analoguous concept in continuous time is the hazard rate, which is the rate of change in the probability of the event happening in the next instant, given that it has not yet happened. The hazard rate is defined as P(t < T < t + ∆t|T = t) f (t) lim = , ∆t→0 ∆t 1 − F (t) where f is the pdf of T . Given the hazard rate h(t), we can reconstruct the original distribution, F (t) = 1 − e − Walt Pohl (UZH QBA) Rt Stochastic Models 0 h(s)ds . May 16, 2013 5/7 Exponential Distribution Of particular interest is the case of a constant hazard rate, h(t) = λ, which describes the exponential distribution, P(t ≤ T ) = 1 − e −λt . Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 6/7 Exponential Distribution Of particular interest is the case of a constant hazard rate, h(t) = λ, which describes the exponential distribution, P(t ≤ T ) = 1 − e −λt . Lots of things follow an exponential distribution. For example, the waiting time for the first jump in a Poisson (or compound Poisson) process is exponentially distributed. Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 6/7 Estimation and Prediction The main purpose of survival analysis is prediction: to predict the probability of the event happening within a prescribed period, given that it has not yet happened. Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 7/7 Estimation and Prediction The main purpose of survival analysis is prediction: to predict the probability of the event happening within a prescribed period, given that it has not yet happened. If we have a large prior history, we can use that history to fit a model for the event. Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 7/7 Estimation and Prediction The main purpose of survival analysis is prediction: to predict the probability of the event happening within a prescribed period, given that it has not yet happened. If we have a large prior history, we can use that history to fit a model for the event. This is how survival analysis is used to calculate actuarial rates for life insurance, for example. But what if we don’t have a long history of data? Walt Pohl (UZH QBA) Stochastic Models May 16, 2013 7/7