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Cumulative Distribution Function


Techniques

Cumulative Hazard Function

Weibull Distribution

Normal Distribution

Lognormal Distribution

Exponential Distribution

Exam

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The cumulative distribution function, F(x), is usually estimated from the median rank, but other estimates such as the mean rank and the Kaplan-Meier product limit estimator are also used. The median rank estimate for F(x) is

where Oi is the modified order of failure of the ith data point.

A modified order of failure is only needed if censored data is involved; if not the original order of failure, i, is equivalent to the modified order of failure. The logic for a modified order of failure is as follows. Consider three items; the first was tested for 3 hours and the test was stopped without failure, the second item was tested and failed after 4 hours, and the third item was tested and failed after 4.5 hours. For this data set the failure order is unclear. The first item could have been either the first failure, the second failure or the third failure; thus it is not certain that the first item to fail, the second item, is the first ordered failure. The modified order of failure is computed from the expression

where IiIi is the increment for the ith failure, and is computed from the expression

where: n is the total number of points in the data set, both censored and uncensored,
c is the number of points remaining in the data set including the current point, and
Op is the order of the previous failure.

Example
Twelve items were tested with failures occurring at times of 43 hours, 67 hours, 92 hours 94 hours, and 149 hours. At a time of 149 hours, the testing was stopped for the remaining seven items.  Estimate the cumulative distribution function.

Solution
The cumulative distribution function estimates are shown in the table below.

Time to Fail Ii Oi F(x)
43 1 1 0.0565
67 2 2 0.1371
92 3 3 0.2177
94 4 4 0.2984
149 5 5 0.3790
149 c      
149 c      
149 c      
149 c      
149 c      
149 c      
149 c      

Example
Use probability plotting to determine the parameters of the normal distribution given the following data set. A "c" following an entry indicates the censoring.

150 c 183 c 235
157 c 209 235
167 c 216 c 248 c
179 217 c 257

Solution
The cumulative distribution function estimates are shown in the table below.

Time to Fail Ii Oi F(x)
150 c      
157 c      
167 c      
179 1.3000 1.3000 0.0806
183 c      
209 1.4625 2.7625 0.1986
216 c      
217 c      
235 2.0475 4.8100 0.3637
235.1 2.0475 6.8575 0.5288
248 c      
257 3.0713 9.9288 0.7765
 

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