Background: I have a basic to moderate knowledge on probability and statistics. I have familiarity with R programming language and optimization routines.

I'm reading an excellent article "Jobs, Strikes, and Wars: Probability Models for Duration" by Morrison and Schmittlein.. The authors of this article go on to develop probability models to fit duration data like strikes and jobs with an emphasis on mixture models to capture heterogeneity. They provide multiple detailed examples. One of them is listed below.

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The data is aggregated, and captures the Job duration of Firm 1 in months. The sample size is 1206. For convenience, I have extracted the data.

Time (months) Observed Duration  
    0-3      242 
    3+-6     152 
    6+-9     104 
    9+-12    73 
    12+-15   52 
    15+-18   47 
    18+-21   49 
    >21      487

In the same article and else where we know for weibull distribution, the CDF, PDF, Hazard function and cumulative hazard function are as follows:

enter image description here

I'm not sure on how to write the log likelihood function for a data such as above and also how to handle the censoring at the end.

Below are my questions:

  1. How to fit a weibull distrubtion to aggregate data like the one above ?. How to write a maximum likelihood function?.
  2. How to handle the censoring in the last data point i.e., 487 (>21).

The author also provided the weibull shape and scale parameters. Please ignroe the other distribution parameters, the weibull parameters are (0.700, 0.108):

enter image description here

I'm familiar with R,Excel and SAS. If someone could demonstrate the parameter estimation process in anyone of the software that would be greatly appreciated.


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