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## Failure Time Analysis Outline Lecture Twelve

Coker College, US has reference to this Academic Journal, Lecture Twelve Outline Failure Time Analysis Linear Probability Model Poisson Distribution Failure Time Analysis Example: Duration of Expansions Issue: does the probability of an expansion ending depend on how long it has lasted? Exponential distribution: assumes the answer since the hazard rate is constant Weibull distribution allows a test so that be performed

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Part II: Failure Time Analysis Exponential survival function hazard rate Weibull Exploratory Data Analysis, Lab Seven Duration of Post-War Economic Expansions in Months

Estimated Survivor Function in consideration of Ten Post-War Expansions

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Exponential Distribution Hazard rate: ratio of density function so that the survivor function: h(t) = f(t)/S(t) measure of probability of failure at time t given that you have survived that long in consideration of the exponential it is a constant: h(t) =

Interval hazard rate=#ending/#at risk Cumulative Hazard Function In general: For the exponential,

Weibull Distribution F(t) = 1 – exp[ S(t) = ln S(t) = – (t/a)b h(t) = f(t)/S(t) f(t) = dF(t)/dt = – exp[-(t/a)b](-b/a)(t/a)b-1 h(t) = (b/a)(t/a)b-1 if b = 1, h(t) = constant if b>1, h(t) is increasing function if b

Weibull Distribution Cumulative Hazard Function

Dependent Variable: LNCUMHAZ Method: Least Squares Sample: 2 11 Included observations: 10 Variable Coefficient Std. Error t-Statistic Prob. LNDUR 1.436662 0.103558 13.87303 0.0000 C -5.920740 0.403303 -14.68061 0.0000 R-squared 0.960092 Mean dependent var -0.409591 Adjusted R-squared 0.955103 S.D. dependent var 1.038386 S.E. of regression 0.220022 Akaike info criterion -0.013326 Sum squared resid 0.387276 Schwarz criterion 0.047191 Log likelihood 2.066628 F-statistic 192.4609 Durbin-Watson stat 1.210695 Prob(F-statistic) 0.000001 Is Beta More Than One? H0: beta=1 HA: beta>1, in addition to hazard rate is increasing alongside time, i.e. expansions are more likely so that end the longer they last t = ( 1.437 – 1)/0.104 = 4.20 Conclude Economic expansions are at increasing risk the longer they last the business cycle is not dead so much in consideration of the new economics maybe Karl Marx was right, capitalism is an inherently unstable system, subject so that cycles

Lab Seven

Cumulative Hazard Rate in consideration of Fan Failure y = 4E-05x + 0.0089 R 2 = 0.9816 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Duration in Hours Cumulative Hazard Lambda= 3.89 x 10-5, mean=25,707 hrs Regress Cumulative Hazard on Duration

Part IV. Poisson Approximation so that Binomial Conditions: f(x) = {exp[-m] mx }/x! Assumptions: the number of events occurring in non-overlapping intervals are independent the probability of a single event occurring in a small interval is approximately proportional so that the interval the probability of more than one event in an interval is negligible Example Ten % of tools produced in a manufacturing process are defective. What is the probability of finding exactly two defectives in a random sample of 10? Binomial: p(k=2) = 10!/(8!2!)(0.1)2(0.9)8 = 0.194 Poisson , where the mean of the Poisson, m, equals n*p = 0.1 p(k=2) = {exp[-1] 12 }/2! = 0.184

## Kostroff, Michael Managing Editor

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