Estimation of Weibull Parameters for Censored Data

[James J]

Could you please help me with the estimation of Weibull parameters for Censored data? Can I estimate the parameters for data with zero failures? When should I use 3-parameter Weibull instead of 2-parameter Weibull? Thanks.

[RS Tech Support]

Using ReliaSoft's Weibull++ 6 it is possible to analyze data with zero failures using MLE with the 1-parameter Weibull distribution. If you would like more information on how to do this or if you would like more information regarding Weibull++ 6 please contact us at ReliaSoft via phone at (520) 886-0366 or via email at support@reliasoft.com.

The 3-parameter Weibull distribution incorporates the parameter gamma (location parameter). Gamma can be positive or negative. A positive gamma corresponds to a failure free period in the life of the product. A negative gamma indicates that failures can occur before time zero. You can use the 3-parameter Weibull distribution if it makes sense given what you know about your product.

[DCraggs]

The Likelihood function (L) is defined as the product of the probability density function (PDF) for each failure time, if there is no censoring. However, if there is right censoring, L is the product of the PDF of the failures and the Reliability (1-CDF) of the suspensions. Why?

Could one have used CDF for the failures and (1-CDF for the suspensions? I am primarily concerned with the logic of the switch from PDF to the CDF.

[RS Support]

No! One must use it just the way it is presented. Keep in mind that in MLE we try to find the values of the parameters that would have most likely produced the data we observed (i.e. x1, x2 … ).

What you may be confused on is the switch from the CDF[F(x)] to PDF[f(x)] for failures. Keep in mind that for the failures, and lets say a failure at t1, one wants the probability of obtaining exactly t1, not the probability that t is less than t1, as the CDF would give you.

Having said that, do note that for a continuous pdf the probability of obtaining exactly t1 is zero – but one could look at the probability of t1 occurring in an interval of length dx which is f(x)dx. Since dx does not depend on the parameters we are estimating it may be dropped out of the likelihood equation formulation (that we are going to maximize) and thus just use f(x). (For more details see Wetherill, G. B. (1981). Intermediate Statistical Methods. Chapman & Hall. ISBN 0-412-16450-7.)