Oct 27 2005 LEC 11 ECON 240A 1 L Phillips Weibull Distribution Transformations Poisson Distribution I Introduction In Lecture Ten we introduced the exponential distribution as a parametric approach to estimating the distribution of time until failure This distribution has one parameter lambda and the reciprocal of lambda is the mean time until failure So the exponential is parsimonious in parameters to estimate but this simplicity came at a price of two assumptions First the hazard rate is constant for the exponential which is restrictive Second the exponential has the no memory feature which means that the survival time to date does not affect the expected time remaining before failure The Weibull is a distribution that permits a little more flexibility but at a price of two parameters The survivor function is also nonlinear in these parameters which raises a question about whether we can linearize the function through transformation We can not That leaves the question of how to estimate the equation Lastly we turn to another distribution the Poisson which can be used as an approximation to the binomial for rare events It is useful for modeling problems such as the number of defects on a foot of magnetic recording tape and other applications to quality control II Failure Time Models and the Weibull Distribution The Weibull Distribution has the cumulative distribution function F t 1 exp t 1 And so the survivor function is S t 1 F t exp t 2 Taking the derivative of the cumulative distribution function yields the density function dF t dt f t t 1 exp t 3 Oct 27 2005 LEC 11 ECON 240A 2 L Phillips Weibull Distribution Transformations Poisson Distribution and so the hazard rate is h t f t S t t 1 4 Thus the hazard rate is a power function of the duration t If beta equals one then the hazard rate is a constant 1 If beta is greater than one then the hazard rate increases with survival time t If beta is less than one then the hazard rate is a decreasing function of t So depending on the value of this parameter three different patterns of behavior for the hazard rate can be explained This does not cover all possibilities such as a situation where the hazard rate may first increase and then decrease with survival time but it is more flexible than the exponential III Transformations We saw in the previous lecture that taking the logarithms of both sides of the equation for the Weibull s survivor function did not result in an equation linear in the parameters We could try this transformation on the hazard rate ln h t ln ln 1 ln t 5 This is not linear in alpha and beta but it is close if we do not try and separately identify those two parameters and let the intercept equal ln ln and the slope equal beta minus one i e ln h t a b ln t 6 We are interested in the slope If it is not significantly different from zero then we can accept the null hypothesis that beta equals one The logarithm of the hazard rate is plotted against the logarithm of survival time duration in Figure1 Oct 27 2005 LEC 11 ECON 240A 3 L Phillips Weibull Distribution Transformations Poisson Distribution Figure 1 Log Hazard Rate Vs Log Duration 0 0 0 5 1 1 5 2 2 5 3 3 5 4 4 5 5 1 Log Hazard Rate 2 y 0 366x 5 1181 R2 0 0549 3 4 Log Hazard Rate Linear Regression 5 6 Log Duration The coefficient of determination is only 5 5 The F distribution statistic calculated from this R2 is not significant and Student s t statistic for the null hypothesis that the slope is zero is only 0 64 so we can not reject the hypothesis that beta equals one and the hazard rate is constant Of course there are only a few observations but the exponential seems appropriate II Cumulative Hazard Function There is another test for whether the exponential distribution is appropriate for the duration of post war expansions It is the cumulative hazard function H t which is the sum of the hazard function h t t H t h u du 0 7 Oct 27 2005 LEC 11 ECON 240A 4 L Phillips Weibull Distribution Transformations Poisson Distribution And applying this to the exponential where from Eq 20 of chapter ten the hazard rate is the constant lambda t H t t h u du du t 0 8 0 So the cumulative hazard function for the exponential is a linear function of the time until failure We can calculate the cumulative hazard rate using the values we calculated for the interval hazard rate in Table 5 of the previous chapter reproduced in part in Table 1 below The cumulative hazard rate is just the running sum of the hazard rate so at Table 1 Estimated Hazard Rate and Cumulative Hazard Rate Post War Expansions Duration 0 12 24 36 37 39 45 58 92 106 125 Ending 0 1 1 1 1 1 1 1 1 1 at Risk Interval Hazard Rate 10 10 0 1000 9 0 1111 8 0 1250 7 0 1429 6 0 1667 5 0 2000 4 0 2500 3 0 3333 2 0 5000 1 Cumulative Hazard Rate 0 1000 0 2111 0 3361 0 4790 0 6456 0 8456 1 0956 1 4290 1 9290 the expansion of duration 12 months the cumulative hazard rate equals the hazard rate at the expansion of duration 24 months the cumulative hazard rate is equal to the hazard rate at that duration plus the previous hazard rate i e 0 1111 0 10000 and so on This cumulative hazard rate is plotted against duration in Figure 2 Oct 27 2005 LEC 11 ECON 240A 5 L Phillips Weibull Distribution Transformations Poisson Distribution Figure 2 Cumulative Hazard Plot for Expansions 2 5 2 Cumulative Hazard y 0 0192x 0 1736 R2 0 9562 1 5 1 Cumulative Hazard Function Linear Regression 0 5 0 0 20 40 60 80 100 120 Duration in Nonths Note that the linear regression is a good fit with an R2 of 0 96 supporting evidence that the expansion durations are distributed exponentially Note also that the slope which is an estimate of the hazard rate or lambda is 0 0192 Since the reciprocal of lambda is the mean time to failure we get an estimate of 52 months So the cumulative hazard function is quite informative and we can learn a lot using exploratory graphical analysis III Poisson Distribution The Poisson is a useful approximation to the binomial for events with a low probability i e p small one minus p or q approaching one and a sample size n of fifty or more or the product of n and p equal to five or less The density for the Poisson is f x exp x x 9 Oct 27 2005 LEC 11 ECON 240A 6 L Phillips Weibull Distribution Transformations Poisson Distribution The assumptions underlying the use of the …
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